From b827cffa2eb603bb8d849989c44f367138dc7e4b Mon Sep 17 00:00:00 2001
From: Hu Zhao <zhao@mbd.rwth-aachen.de>
Date: Sat, 22 Apr 2023 10:33:24 +0200
Subject: [PATCH] docs: track built auto_examples
---
.gitignore | 2 +-
.../auto_examples/emulator/index.doctree | Bin 0 -> 9089 bytes
.../emulator/plot_ppgasp.doctree | Bin 0 -> 31650 bytes
.../emulator/plot_scalargasp.doctree | Bin 0 -> 29894 bytes
.../emulator/sg_execution_times.doctree | Bin 0 -> 6536 bytes
.../auto_examples/inference/index.doctree | Bin 0 -> 8809 bytes
.../inference/plot_active_learning.doctree | Bin 0 -> 106807 bytes
.../inference/plot_bayes_inference.doctree | Bin 0 -> 24905 bytes
.../inference/sg_execution_times.doctree | Bin 0 -> 6651 bytes
.../auto_examples/sampler/index.doctree | Bin 0 -> 10184 bytes
.../auto_examples/sampler/plot_latin.doctree | Bin 0 -> 18996 bytes
.../sampler/plot_metropolis_hastings.doctree | Bin 0 -> 22160 bytes
.../sampler/plot_saltelli.doctree | Bin 0 -> 19180 bytes
.../sampler/sg_execution_times.doctree | Bin 0 -> 7782 bytes
.../auto_examples/sensitivity/index.doctree | Bin 0 -> 7391 bytes
.../sensitivity/plot_sobol_analyze.doctree | Bin 0 -> 34899 bytes
.../sensitivity/sg_execution_times.doctree | Bin 0 -> 5398 bytes
.../auto_examples/simulator/index.doctree | Bin 0 -> 11909 bytes
.../simulator/plot_mass_point_model.doctree | Bin 0 -> 24737 bytes
.../simulator/plot_ravaflow24.doctree | Bin 0 -> 33188 bytes
.../plot_run_mass_point_model.doctree | Bin 0 -> 40356 bytes
.../simulator/plot_run_ravaflow24.doctree | Bin 0 -> 51659 bytes
.../simulator/sg_execution_times.doctree | Bin 0 -> 9179 bytes
.../auto_examples/emulator/index.rst.txt | 75 ++
.../emulator/plot_ppgasp.rst.txt | 280 ++++
.../emulator/plot_scalargasp.rst.txt | 289 ++++
.../emulator/sg_execution_times.rst.txt | 14 +
.../auto_examples/inference/index.rst.txt | 75 ++
.../inference/plot_active_learning.rst.txt | 1155 ++++++++++++++++
.../inference/plot_bayes_inference.rst.txt | 231 ++++
.../inference/sg_execution_times.rst.txt | 14 +
.../auto_examples/sampler/index.rst.txt | 93 ++
.../auto_examples/sampler/plot_latin.rst.txt | 209 +++
.../sampler/plot_metropolis_hastings.rst.txt | 248 ++++
.../sampler/plot_saltelli.rst.txt | 193 +++
.../sampler/sg_execution_times.rst.txt | 16 +
.../auto_examples/sensitivity/index.rst.txt | 57 +
.../sensitivity/plot_sobol_analyze.rst.txt | 358 +++++
.../sensitivity/sg_execution_times.rst.txt | 12 +
.../auto_examples/simulator/index.rst.txt | 112 ++
.../simulator/plot_mass_point_model.rst.txt | 259 ++++
.../simulator/plot_ravaflow24.rst.txt | 304 +++++
.../plot_run_mass_point_model.rst.txt | 376 ++++++
.../simulator/plot_run_ravaflow24.rst.txt | 444 +++++++
.../simulator/sg_execution_times.rst.txt | 18 +
.../html/auto_examples/emulator/index.html | 147 +++
.../auto_examples/emulator/plot_ppgasp.html | 259 ++++
.../emulator/plot_scalargasp.html | 267 ++++
.../emulator/sg_execution_times.html | 131 ++
.../html/auto_examples/inference/index.html | 147 +++
.../inference/plot_active_learning.html | 1174 +++++++++++++++++
.../inference/plot_bayes_inference.html | 244 ++++
.../inference/sg_execution_times.html | 131 ++
.../html/auto_examples/sampler/index.html | 151 +++
.../auto_examples/sampler/plot_latin.html | 217 +++
.../sampler/plot_metropolis_hastings.html | 239 ++++
.../auto_examples/sampler/plot_saltelli.html | 217 +++
.../sampler/sg_execution_times.html | 135 ++
.../html/auto_examples/sensitivity/index.html | 143 ++
.../sensitivity/plot_sobol_analyze.html | 363 +++++
.../sensitivity/sg_execution_times.html | 127 ++
.../html/auto_examples/simulator/index.html | 155 +++
.../simulator/plot_mass_point_model.html | 261 ++++
.../simulator/plot_ravaflow24.html | 280 ++++
.../simulator/plot_run_mass_point_model.html | 304 +++++
.../simulator/plot_run_ravaflow24.html | 333 +++++
.../simulator/sg_execution_times.html | 139 ++
67 files changed, 10397 insertions(+), 1 deletion(-)
create mode 100644 docs/source/_build/doctrees/auto_examples/emulator/index.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/emulator/plot_ppgasp.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/emulator/plot_scalargasp.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/emulator/sg_execution_times.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/inference/index.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/inference/plot_active_learning.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/inference/plot_bayes_inference.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/inference/sg_execution_times.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sampler/index.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sampler/plot_latin.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sampler/plot_metropolis_hastings.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sampler/plot_saltelli.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sampler/sg_execution_times.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sensitivity/index.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sensitivity/plot_sobol_analyze.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/sensitivity/sg_execution_times.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/simulator/index.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/simulator/plot_mass_point_model.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/simulator/plot_ravaflow24.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/simulator/plot_run_mass_point_model.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/simulator/plot_run_ravaflow24.doctree
create mode 100644 docs/source/_build/doctrees/auto_examples/simulator/sg_execution_times.doctree
create mode 100644 docs/source/_build/html/_sources/auto_examples/emulator/index.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/emulator/plot_ppgasp.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/emulator/plot_scalargasp.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/emulator/sg_execution_times.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/inference/index.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/inference/plot_active_learning.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/inference/plot_bayes_inference.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/inference/sg_execution_times.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sampler/index.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sampler/plot_latin.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sampler/plot_metropolis_hastings.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sampler/plot_saltelli.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sampler/sg_execution_times.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sensitivity/index.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sensitivity/plot_sobol_analyze.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/sensitivity/sg_execution_times.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/simulator/index.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/simulator/plot_mass_point_model.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/simulator/plot_ravaflow24.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/simulator/plot_run_mass_point_model.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/simulator/plot_run_ravaflow24.rst.txt
create mode 100644 docs/source/_build/html/_sources/auto_examples/simulator/sg_execution_times.rst.txt
create mode 100644 docs/source/_build/html/auto_examples/emulator/index.html
create mode 100644 docs/source/_build/html/auto_examples/emulator/plot_ppgasp.html
create mode 100644 docs/source/_build/html/auto_examples/emulator/plot_scalargasp.html
create mode 100644 docs/source/_build/html/auto_examples/emulator/sg_execution_times.html
create mode 100644 docs/source/_build/html/auto_examples/inference/index.html
create mode 100644 docs/source/_build/html/auto_examples/inference/plot_active_learning.html
create mode 100644 docs/source/_build/html/auto_examples/inference/plot_bayes_inference.html
create mode 100644 docs/source/_build/html/auto_examples/inference/sg_execution_times.html
create mode 100644 docs/source/_build/html/auto_examples/sampler/index.html
create mode 100644 docs/source/_build/html/auto_examples/sampler/plot_latin.html
create mode 100644 docs/source/_build/html/auto_examples/sampler/plot_metropolis_hastings.html
create mode 100644 docs/source/_build/html/auto_examples/sampler/plot_saltelli.html
create mode 100644 docs/source/_build/html/auto_examples/sampler/sg_execution_times.html
create mode 100644 docs/source/_build/html/auto_examples/sensitivity/index.html
create mode 100644 docs/source/_build/html/auto_examples/sensitivity/plot_sobol_analyze.html
create mode 100644 docs/source/_build/html/auto_examples/sensitivity/sg_execution_times.html
create mode 100644 docs/source/_build/html/auto_examples/simulator/index.html
create mode 100644 docs/source/_build/html/auto_examples/simulator/plot_mass_point_model.html
create mode 100644 docs/source/_build/html/auto_examples/simulator/plot_ravaflow24.html
create mode 100644 docs/source/_build/html/auto_examples/simulator/plot_run_mass_point_model.html
create mode 100644 docs/source/_build/html/auto_examples/simulator/plot_run_ravaflow24.html
create mode 100644 docs/source/_build/html/auto_examples/simulator/sg_execution_times.html
diff --git a/.gitignore b/.gitignore
index b3693e1..bb6a509 100644
--- a/.gitignore
+++ b/.gitignore
@@ -125,4 +125,4 @@ cython_debug/
.vscode/
# auto_examples
-auto_examples/
\ No newline at end of file
+docs/source/auto_examples/
\ No newline at end of file
diff --git a/docs/source/_build/doctrees/auto_examples/emulator/index.doctree b/docs/source/_build/doctrees/auto_examples/emulator/index.doctree
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diff --git a/docs/source/_build/doctrees/auto_examples/simulator/plot_run_mass_point_model.doctree b/docs/source/_build/doctrees/auto_examples/simulator/plot_run_mass_point_model.doctree
new file mode 100644
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diff --git a/docs/source/_build/doctrees/auto_examples/simulator/plot_run_ravaflow24.doctree b/docs/source/_build/doctrees/auto_examples/simulator/plot_run_ravaflow24.doctree
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literal 0
HcmV?d00001
diff --git a/docs/source/_build/html/_sources/auto_examples/emulator/index.rst.txt b/docs/source/_build/html/_sources/auto_examples/emulator/index.rst.txt
new file mode 100644
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@@ -0,0 +1,75 @@
+:orphan:
+
+Emulator
+========
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnails">
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="PPGaSP: GP emulation for multi-output functions">
+
+.. only:: html
+
+ .. image:: /auto_examples/emulator/images/thumb/sphx_glr_plot_ppgasp_thumb.png
+ :alt: PPGaSP: GP emulation for multi-output functions
+
+ :ref:`sphx_glr_auto_examples_emulator_plot_ppgasp.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">PPGaSP: GP emulation for multi-output functions</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="ScalarGaSP: GP emulation for a single-output function">
+
+.. only:: html
+
+ .. image:: /auto_examples/emulator/images/thumb/sphx_glr_plot_scalargasp_thumb.png
+ :alt: ScalarGaSP: GP emulation for a single-output function
+
+ :ref:`sphx_glr_auto_examples_emulator_plot_scalargasp.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">ScalarGaSP: GP emulation for a single-output function</div>
+ </div>
+
+
+.. raw:: html
+
+ </div>
+
+
+.. toctree::
+ :hidden:
+
+ /auto_examples/emulator/plot_ppgasp
+ /auto_examples/emulator/plot_scalargasp
+
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-gallery
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download all examples in Python source code: emulator_python.zip </auto_examples/emulator/emulator_python.zip>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download all examples in Jupyter notebooks: emulator_jupyter.zip </auto_examples/emulator/emulator_jupyter.zip>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/emulator/plot_ppgasp.rst.txt b/docs/source/_build/html/_sources/auto_examples/emulator/plot_ppgasp.rst.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,280 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/emulator/plot_ppgasp.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_emulator_plot_ppgasp.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_emulator_plot_ppgasp.py:
+
+
+PPGaSP: GP emulation for multi-output functions
+===============================================
+
+.. GENERATED FROM PYTHON SOURCE LINES 9-23
+
+This example shows how to apply Gaussian process emulation to a multi-output
+function using class :class:`.PPGaSP`.
+
+The multi-output function that we are going to look at is the `DIAMOND`
+(diplomatic and military operations in a non-warfighting domain) computer model.
+It is used as a testbed to illustrate the `PP GaSP` emulator in the `R` package
+`RobustGaSP`, see :cite:t:`Gu2019` for more detail.
+
+The simulator has :math:`13` input parameters and :math:`5` outputs. Namely,
+:math:`\mathbf{y}=f(\mathbf{x})` where :math:`\mathbf{x}=(x_1,\ldots,x_{13})^T`
+and :math:`\mathbf{y}=(y_1,\ldots,y_5)^T`.
+
+The training and testing data are provided in the folder '.../tests/data/'.
+We first load the training and testing data.
+
+.. GENERATED FROM PYTHON SOURCE LINES 24-40
+
+.. code-block:: default
+
+
+ import numpy as np
+ import os
+
+ dir_data = os.path.abspath('../../../tests/data/')
+
+ humanityX = np.genfromtxt(os.path.join(dir_data, 'humanityX.csv'), delimiter=',')
+ humanityY = np.genfromtxt(os.path.join(dir_data, 'humanityY.csv'), delimiter=',')
+ print(f"Number of training data points: ", humanityX.shape[0])
+ print(f"Input dimension: ", humanityX.shape[1])
+ print(f"Output dimension: ", humanityY.shape[1])
+
+ humanityXt = np.genfromtxt(os.path.join(dir_data, 'humanityXt.csv'), delimiter=',')
+ humanityYt = np.genfromtxt(os.path.join(dir_data, 'humanityYt.csv'), delimiter=',')
+ print(f"Number of testing data points: ", humanityXt.shape[0])
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Number of training data points: 120
+ Input dimension: 13
+ Output dimension: 5
+ Number of testing data points: 120
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 41-43
+
+.. note:: You may need to modify ``dir_data`` according to where you save them
+ on your local machine.
+
+.. GENERATED FROM PYTHON SOURCE LINES 45-50
+
+``humanityX`` and ``humanitY`` are the training data, corresponding to ``design``
+and ``response`` respectively. ``humanityXt`` are testing input data, at which
+we are going to make predictions once the emulator is trained. ``humanityYt``
+are the true outputs at ``humanityXt``, which is then used to validate the
+performance of the trained emulator.
+
+.. GENERATED FROM PYTHON SOURCE LINES 52-54
+
+To build a `PP GaSP` emulator for the above simulator, first import class
+:class:`.PPGaSP` by
+
+.. GENERATED FROM PYTHON SOURCE LINES 55-58
+
+.. code-block:: default
+
+
+ from psimpy.emulator import PPGaSP
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 59-63
+
+Then, create an instance of :class:`.PPGaSP`. The parameter ``ndim``
+(dimension of function input ``x``) must be specified. Optional parameters, such
+as ``method``, ``kernel_type``, etc., can be set up if desired. Here, we leave
+all the optional parameters to their default values.
+
+.. GENERATED FROM PYTHON SOURCE LINES 64-68
+
+.. code-block:: default
+
+
+ emulator = PPGaSP(ndim=humanityX.shape[1])
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 69-71
+
+Next, we train the `PP GaSP` emulator based on the training data using
+:py:meth:`.PPGaSP.train` method.
+
+.. GENERATED FROM PYTHON SOURCE LINES 72-75
+
+.. code-block:: default
+
+
+ emulator.train(design=humanityX, response=humanityY)
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ The upper bounds of the range parameters are 296,974 297,1814 294,7672 295,9345 296,563 295,935 297,3198 296,4444 296,9323 298,2373 298,0619 298,7249 298,7249
+ The initial values of range parameters are 5,93948 5,943628 5,895344 5,918691 5,93126 5,918701 5,946395 5,928888 5,938645 5,964745 5,961238 5,974498 5,974498
+ Start of the optimization 1 :
+ The number of iterations is 44
+ The value of the marginal posterior function is -5279,534
+ Optimized range parameters are 25,46132 2,891428 5,30812 26,95519 291,1828 48,17274 84,56133 2,874385 39,12121 51,81304 0,5407651 1,728696 1,020605
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 12,37503 12,38367 12,28307 12,33172 12,3579 12,33174 12,38944 12,35296 12,37329 12,42767 12,42037 12,44799 12,44799
+ Start of the optimization 2 :
+ The number of iterations is 45
+ The value of the marginal posterior function is -5279,534
+ Optimized range parameters are 25,46132 2,891428 5,308121 26,95519 291,183 48,17274 84,56133 2,874385 39,12121 51,81304 0,5407651 1,728696 1,020605
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 76-79
+
+With the trained emulator, we can make predictions for any arbitrary set of
+input points using :py:meth:`PPGaSP.predict` method.
+Here, we make predictions at testing input points ``humanityXt``.
+
+.. GENERATED FROM PYTHON SOURCE LINES 80-83
+
+.. code-block:: default
+
+
+ predictions = emulator.predict(humanityXt)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 84-86
+
+We can validate the performance of the trained emulator based on the true outputs
+``humanityYt``.
+
+.. GENERATED FROM PYTHON SOURCE LINES 87-106
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ fig, ax = plt.subplots(5, 1, figsize=(6,15))
+
+ for i in range(humanityY.shape[1]):
+
+ ax[i].set_xlabel(f'Actual $y_{i+1}$')
+ ax[i].set_ylabel(f'Emulator-predicted $y_{i+1}$')
+ ax[i].set_xlim(np.min(humanityYt[:,i]),np.max(humanityYt[:,i]))
+ ax[i].set_ylim(np.min(humanityYt[:,i]),np.max(humanityYt[:,i]))
+
+ _ = ax[i].plot([np.min(humanityYt[:,i]),np.max(humanityYt[:,i])], [np.min(humanityYt[:,i]),np.max(humanityYt[:,i])])
+ _ = ax[i].errorbar(humanityYt[:,i], predictions[:,i,0], predictions[:,i,3], fmt='.', linestyle='', label='prediction and std')
+ _ = ax[i].legend()
+
+ plt.tight_layout()
+
+
+
+
+
+.. image-sg:: /auto_examples/emulator/images/sphx_glr_plot_ppgasp_001.png
+ :alt: plot ppgasp
+ :srcset: /auto_examples/emulator/images/sphx_glr_plot_ppgasp_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 107-109
+
+We can also draw any number of samples at testing input ``humanityXt`` using
+:py:meth:`.PPGaSPsample()` method.
+
+.. GENERATED FROM PYTHON SOURCE LINES 110-112
+
+.. code-block:: default
+
+
+ samples = emulator.sample(humanityXt, nsamples=10)
+ print("Shape of samples: ", samples.shape)
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Shape of samples: (120, 5, 10)
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 5.333 seconds)
+
+
+.. _sphx_glr_download_auto_examples_emulator_plot_ppgasp.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_ppgasp.py <plot_ppgasp.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_ppgasp.ipynb <plot_ppgasp.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/emulator/plot_scalargasp.rst.txt b/docs/source/_build/html/_sources/auto_examples/emulator/plot_scalargasp.rst.txt
new file mode 100644
index 0000000..95b144b
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/emulator/plot_scalargasp.rst.txt
@@ -0,0 +1,289 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/emulator/plot_scalargasp.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_emulator_plot_scalargasp.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_emulator_plot_scalargasp.py:
+
+
+ScalarGaSP: GP emulation for a single-output function
+=====================================================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-13
+
+This example shows how to apply Gaussian process emulation to a single-output
+function using class :class:`.ScalarGaSP`.
+
+The task is to build a GP emulator for the function :math:`y = x * sin(x)`
+based on a few number of training data.
+
+.. GENERATED FROM PYTHON SOURCE LINES 16-17
+
+First, import the class :class:`.ScalarGaSP` by
+
+.. GENERATED FROM PYTHON SOURCE LINES 18-21
+
+.. code-block:: default
+
+
+ from psimpy.emulator import ScalarGaSP
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 22-26
+
+Then, create an instance of :class:`.ScalarGaSP`. The parameter ``ndim``
+(dimension of function input ``x``) must be specified. Optional parameters, such
+as ``method``, ``kernel_type``, etc., can be set up if desired. Here, we leave
+all the optional parameters to their default values.
+
+.. GENERATED FROM PYTHON SOURCE LINES 27-30
+
+.. code-block:: default
+
+
+ emulator = ScalarGaSP(ndim=1)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 31-34
+
+Given training input points ``design`` and corresponding output values ``response``,
+the emulator can be trained using :py:meth:`.ScalarGaSP.train`. Below we train
+an emulator using :math:`8` selected points.
+
+.. GENERATED FROM PYTHON SOURCE LINES 35-47
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ def f(x):
+ #return x + 3*np.sin(x/2)
+ return x*np.sin(x)
+
+ x = np.arange(2,10,1)
+ y = f(x)
+
+ emulator.train(design=x, response=y)
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ The upper bounds of the range parameters are 1881,404
+ The initial values of range parameters are 37,62809
+ Start of the optimization 1 :
+ The number of iterations is 9
+ The value of the marginal posterior function is -13,79116
+ Optimized range parameters are 2,237467
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,21875
+ Start of the optimization 2 :
+ The number of iterations is 9
+ The value of the marginal posterior function is -13,79116
+ Optimized range parameters are 2,237467
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 48-50
+
+We can validate the performance of the trained emulator using the leave-one-out
+cross validation method :py:meth:`loo_validate()`.
+
+.. GENERATED FROM PYTHON SOURCE LINES 51-54
+
+.. code-block:: default
+
+
+ validation = emulator.loo_validate()
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 55-57
+
+Let's plot emulator predictions vs actual outputs. The error bar indicates the
+standard deviation.
+
+.. GENERATED FROM PYTHON SOURCE LINES 58-74
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ fig , ax = plt.subplots(figsize=(4,4))
+
+ ax.set_xlabel('Actual y')
+ ax.set_ylabel('Emulator-predicted y')
+ ax.set_xlim(np.min(y)-1,np.max(y)+1)
+ ax.set_ylim(np.min(y)-1,np.max(y)+1)
+
+ _ = ax.plot([np.min(y)-1,np.max(y)+1], [np.min(y)-1,np.max(y)+1])
+ _ = ax.errorbar(y, validation[:,0], validation[:,1], fmt='.', linestyle='', label='prediction and std')
+ _ = plt.legend()
+ plt.tight_layout()
+
+
+
+
+
+.. image-sg:: /auto_examples/emulator/images/sphx_glr_plot_scalargasp_001.png
+ :alt: plot scalargasp
+ :srcset: /auto_examples/emulator/images/sphx_glr_plot_scalargasp_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 75-80
+
+With the trained emulator at our deposit, we can use the
+:py:meth:`.ScalarGaSP.predict()` method to make predictions at
+any arbitrary set of input points (``testing_input``). It should be noted that,
+``testing_trend`` should be set according to ``trend`` used during emulator
+training.
+
+.. GENERATED FROM PYTHON SOURCE LINES 81-95
+
+.. code-block:: default
+
+
+ testing_input = np.arange(0,10,0.1)
+ predictions = emulator.predict(testing_input)
+
+ plt.plot(testing_input, predictions[:, 0], 'r-', label= "mean")
+ plt.scatter(x, y, s=15, c='k', label="training data", zorder=3)
+ plt.plot(testing_input, f(testing_input), 'k:', zorder=2, label="true function")
+ plt.fill_between(testing_input, predictions[:, 1], predictions[:, 2], alpha=0.3, label="95% CI")
+ plt.xlabel('x')
+ plt.ylabel('emulator-predicted y')
+ plt.xlim(testing_input[0], testing_input[-1])
+ _ = plt.legend()
+ plt.tight_layout()
+
+
+
+
+.. image-sg:: /auto_examples/emulator/images/sphx_glr_plot_scalargasp_002.png
+ :alt: plot scalargasp
+ :srcset: /auto_examples/emulator/images/sphx_glr_plot_scalargasp_002.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 96-98
+
+We can also draw any number of samples at ``testing_input``` using
+:py:meth:`.ScalarGaSP.sample()` method.
+
+.. GENERATED FROM PYTHON SOURCE LINES 99-117
+
+.. code-block:: default
+
+
+ nsamples = 5
+ samples = emulator.sample(testing_input, nsamples=nsamples)
+
+ # sphinx_gallery_thumbnail_number = 3
+ for i in range(nsamples):
+ plt.plot(testing_input, samples[:,i], '--', label=f'sample{i+1}')
+
+ plt.scatter(x, y, s=15, c='k', label="training data", zorder=2)
+ plt.plot(testing_input, f(testing_input), 'k:', zorder=2, label="true function")
+ plt.fill_between(testing_input, predictions[:, 1], predictions[:, 2], alpha=0.3, label="95% CI")
+ plt.xlabel('x')
+ plt.ylabel('emulator-predicted y')
+ plt.xlim(testing_input[0], testing_input[-1])
+ _ = plt.legend()
+ plt.tight_layout()
+
+
+
+
+
+.. image-sg:: /auto_examples/emulator/images/sphx_glr_plot_scalargasp_003.png
+ :alt: plot scalargasp
+ :srcset: /auto_examples/emulator/images/sphx_glr_plot_scalargasp_003.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 118-124
+
+.. tip:: Above example shows how to train a GP emulator based on noise-free training data,
+ which is often the case of emulating a deterministic simulator. If you are dealing
+ with noisy training data, you can
+
+ - set the parameter ``nugget`` to a desired value, or
+ - set ``nugget`` to :math:`0` and ``nugget_est`` to `True`, meaning that ``nugget``
+ is estimated from the noisy training data.
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 0.728 seconds)
+
+
+.. _sphx_glr_download_auto_examples_emulator_plot_scalargasp.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_scalargasp.py <plot_scalargasp.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_scalargasp.ipynb <plot_scalargasp.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/emulator/sg_execution_times.rst.txt b/docs/source/_build/html/_sources/auto_examples/emulator/sg_execution_times.rst.txt
new file mode 100644
index 0000000..92925ce
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/emulator/sg_execution_times.rst.txt
@@ -0,0 +1,14 @@
+
+:orphan:
+
+.. _sphx_glr_auto_examples_emulator_sg_execution_times:
+
+Computation times
+=================
+**00:06.062** total execution time for **auto_examples_emulator** files:
+
++------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_emulator_plot_ppgasp.py` (``plot_ppgasp.py``) | 00:05.333 | 0.0 MB |
++------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_emulator_plot_scalargasp.py` (``plot_scalargasp.py``) | 00:00.728 | 0.0 MB |
++------------------------------------------------------------------------------------+-----------+--------+
diff --git a/docs/source/_build/html/_sources/auto_examples/inference/index.rst.txt b/docs/source/_build/html/_sources/auto_examples/inference/index.rst.txt
new file mode 100644
index 0000000..8df5785
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/inference/index.rst.txt
@@ -0,0 +1,75 @@
+:orphan:
+
+Inference
+=========
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnails">
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Bayesian inference">
+
+.. only:: html
+
+ .. image:: /auto_examples/inference/images/thumb/sphx_glr_plot_bayes_inference_thumb.png
+ :alt: Bayesian inference
+
+ :ref:`sphx_glr_auto_examples_inference_plot_bayes_inference.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Bayesian inference</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Active learning">
+
+.. only:: html
+
+ .. image:: /auto_examples/inference/images/thumb/sphx_glr_plot_active_learning_thumb.png
+ :alt: Active learning
+
+ :ref:`sphx_glr_auto_examples_inference_plot_active_learning.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Active learning</div>
+ </div>
+
+
+.. raw:: html
+
+ </div>
+
+
+.. toctree::
+ :hidden:
+
+ /auto_examples/inference/plot_bayes_inference
+ /auto_examples/inference/plot_active_learning
+
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-gallery
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download all examples in Python source code: inference_python.zip </auto_examples/inference/inference_python.zip>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download all examples in Jupyter notebooks: inference_jupyter.zip </auto_examples/inference/inference_jupyter.zip>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/inference/plot_active_learning.rst.txt b/docs/source/_build/html/_sources/auto_examples/inference/plot_active_learning.rst.txt
new file mode 100644
index 0000000..f98c7f2
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/inference/plot_active_learning.rst.txt
@@ -0,0 +1,1155 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/inference/plot_active_learning.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_inference_plot_active_learning.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_inference_plot_active_learning.py:
+
+
+Active learning
+===============
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-18
+
+This example shows how to use the :class:`.ActiveLearning` class to iteratively
+build a Gaussian process emulator for an unnormalized posterior involving a
+simulator. It should be noted that this example is only for illustration
+purpose, rather than a real case. For simplicity, required arguments for
+:class:`.ActiveLearning` (including `simulator`, `likelihood`, `data`, etc.) are purely
+made up. For a realistic case of active learning, one can refer to :cite:t:`Zhao2022`.
+
+First, we define the simulator, prior distribution of its variable parameters,
+observed data, and likelihood function. They basically define the Bayesian
+inference problem.
+
+.. GENERATED FROM PYTHON SOURCE LINES 19-39
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ ndim = 2 # dimension of variable parameters of the simulator
+ bounds = bounds = np.array([[-5,5],[-5,5]]) # bounds of variable parameters
+ data = np.array([1,0])
+
+ def simulator(x1, x2):
+ """Simulator y=f(x)."""
+ y1, y2 = x1, x2
+ return np.array([y1, y2])
+
+ def prior(x):
+ """Uniform prior."""
+ return 1/(10*10)
+
+ def likelihood(y, data):
+ """Likelihood function L(y,data)."""
+ return np.exp(-(y[0]-data[0])**2/100 - (y[0]**2-y[1]-data[1])**2)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 40-56
+
+Imagine that the simulator is a complex solver. It is not computationally feasible
+to compute the posterior distribution of the variable parameters using grid estimation
+or Metropolis Hastings estimation. This is because they require evaluating the likelihood
+many times which essentially leads to many evaluations of the simulator. Therefore, we
+resort to use active learning to build a Gaussian process emulator for the unnormalized
+posterior (prior times likelihood) based on a small number of evaluations of the simulator.
+The the posterior can be estimated using the emulator.
+
+To do so, we need to pass arguments to following parameters of the :class:`.ActiveLearning` class:
+
+ - run_sim_obj : instance of class :class:`.RunSimulator`. It carries information on how
+ to run the simulator.
+ - lhs_sampler : instance of class :class:`.LHS`. It is used to draw initial samples to run
+ simulations in order to train an inital Gaussian process emulator.
+ - scalar_gasp : instance of class :class:`.ScalarGaSP`. It sets up the emulator structure.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 57-66
+
+.. code-block:: default
+
+
+ from psimpy.simulator import RunSimulator
+ from psimpy.sampler import LHS
+ from psimpy.emulator import ScalarGaSP
+
+ run_simulator = RunSimulator(simulator, var_inp_parameter=['x1','x2'])
+ lhs_sampler = LHS(ndim=ndim, bounds=bounds, seed=1)
+ scalar_gasp = ScalarGaSP(ndim=ndim)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 67-68
+
+Next, we create an object of the :class:`.ActiveLearning` class by
+
+.. GENERATED FROM PYTHON SOURCE LINES 69-75
+
+.. code-block:: default
+
+
+ from psimpy.inference import ActiveLearning
+
+ active_learner = ActiveLearning(ndim, bounds, data, run_simulator, prior, likelihood,
+ lhs_sampler, scalar_gasp)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 76-81
+
+Then we can call the :py:meth:`.ActiveLearning.initial_simulation` method to run initial
+simulations and call the :py:meth:`.ActiveLearning.iterative_emulation` method to
+iteratively run new simulation and build emulator. Here we allocate 40 simulations for
+initial emulator training and 60 simulations for adaptive training.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 82-91
+
+.. code-block:: default
+
+ n0 = 40
+ niter = 60
+
+ init_var_samples, init_sim_outputs = active_learner.initial_simulation(
+ n0, mode='parallel', max_workers=4)
+
+ var_samples, _, _ = active_learner.iterative_emulation(
+ n0, init_var_samples, init_sim_outputs, niter=niter)
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ The upper bounds of the range parameters are 401,3482 397,4694
+ The initial values of range parameters are 8,026963 7,949389
+ Start of the optimization 1 :
+ The number of iterations is 22
+ The value of the marginal posterior function is -146,5038
+ Optimized range parameters are 30,59853 130,6077
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,9828751 0,9733764
+ Start of the optimization 2 :
+ The number of iterations is 36
+ The value of the marginal posterior function is -146,5038
+ Optimized range parameters are 30,59827 130,6067
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 423,194 420,3365
+ The initial values of range parameters are 8,463881 8,40673
+ Start of the optimization 1 :
+ The number of iterations is 13
+ The value of the marginal posterior function is -148,6792
+ Optimized range parameters are 30,19534 133,3286
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,9589025 0,9524277
+ Start of the optimization 2 :
+ The number of iterations is 12
+ The value of the marginal posterior function is -148,6792
+ Optimized range parameters are 30,19539 133,329
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 422,8426 419,9874
+ The initial values of range parameters are 8,456851 8,399747
+ Start of the optimization 1 :
+ The number of iterations is 34
+ The value of the marginal posterior function is -149,9595
+ Optimized range parameters are 30,64771 135,0597
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,9360715 0,9297509
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -149,9595
+ Optimized range parameters are 30,64696 135,0536
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 423,2876 420,4294
+ The initial values of range parameters are 8,465751 8,408588
+ Start of the optimization 1 :
+ The number of iterations is 25
+ The value of the marginal posterior function is -151,304
+ Optimized range parameters are 31,21605 136,8571
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,9143024 0,9081287
+ Start of the optimization 2 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -151,304
+ Optimized range parameters are 31,21617 136,8537
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 389,6167 386,9859
+ The initial values of range parameters are 7,792334 7,739717
+ Start of the optimization 1 :
+ The number of iterations is 13
+ The value of the marginal posterior function is -152,6009
+ Optimized range parameters are 32,285 141,3784
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,8935228 0,8874894
+ Start of the optimization 2 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -152,6009
+ Optimized range parameters are 32,28096 141,3737
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 368,8855 366,3947
+ The initial values of range parameters are 7,377711 7,327894
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -153,5419
+ Optimized range parameters are 32,43481 141,7852
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,8736668 0,8677675
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -153,5419
+ Optimized range parameters are 32,43357 141,7844
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 339,4555 337,1634
+ The initial values of range parameters are 6,789111 6,743268
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -154,9929
+ Optimized range parameters are 33,69448 151,2103
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,854674 0,848903
+ Start of the optimization 2 :
+ The number of iterations is 22
+ The value of the marginal posterior function is -154,9929
+ Optimized range parameters are 33,69468 151,2111
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 369,9581 367,46
+ The initial values of range parameters are 7,399163 7,349201
+ Start of the optimization 1 :
+ The number of iterations is 26
+ The value of the marginal posterior function is -155,4764
+ Optimized range parameters are 33,90556 152,0139
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,8364895 0,8308412
+ Start of the optimization 2 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -155,4764
+ Optimized range parameters are 33,90541 152,0137
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 299,8308 297,8062
+ The initial values of range parameters are 5,996615 5,956124
+ Start of the optimization 1 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -155,8802
+ Optimized range parameters are 34,04389 152,6133
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,8190626 0,813532
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -155,8802
+ Optimized range parameters are 34,04381 152,6131
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 272,589 270,7484
+ The initial values of range parameters are 5,45178 5,414968
+ Start of the optimization 1 :
+ The number of iterations is 22
+ The value of the marginal posterior function is -156,1383
+ Optimized range parameters are 34,46966 151,4246
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,802347 0,7969293
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -156,1383
+ Optimized range parameters are 34,46972 151,4261
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 217,1413 215,6751
+ The initial values of range parameters are 4,342826 4,313501
+ Start of the optimization 1 :
+ The number of iterations is 16
+ The value of the marginal posterior function is -156,184
+ Optimized range parameters are 34,52228 152,8428
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,7863001 0,7809907
+ Start of the optimization 2 :
+ The number of iterations is 29
+ The value of the marginal posterior function is -156,184
+ Optimized range parameters are 34,5049 152,8145
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 206,6122 205,2171
+ The initial values of range parameters are 4,132244 4,104342
+ Start of the optimization 1 :
+ The number of iterations is 25
+ The value of the marginal posterior function is -156,2348
+ Optimized range parameters are 34,79805 155,1646
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,7708824 0,7656772
+ Start of the optimization 2 :
+ The number of iterations is 22
+ The value of the marginal posterior function is -156,2348
+ Optimized range parameters are 34,82503 155,2481
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 264,8501 263,0618
+ The initial values of range parameters are 5,297002 5,261235
+ Start of the optimization 1 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -156,4063
+ Optimized range parameters are 34,74024 156,9499
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,7560578 0,7509526
+ Start of the optimization 2 :
+ The number of iterations is 35
+ The value of the marginal posterior function is -156,4062
+ Optimized range parameters are 34,71806 156,8712
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 255,2317 253,5083
+ The initial values of range parameters are 5,104633 5,070165
+ Start of the optimization 1 :
+ The number of iterations is 20
+ The value of the marginal posterior function is -156,5552
+ Optimized range parameters are 35,37305 157,5615
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,7417925 0,7367837
+ Start of the optimization 2 :
+ The number of iterations is 37
+ The value of the marginal posterior function is -156,5552
+ Optimized range parameters are 35,37704 157,591
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 234,5139 232,9304
+ The initial values of range parameters are 4,690278 4,658607
+ Start of the optimization 1 :
+ The number of iterations is 25
+ The value of the marginal posterior function is -156,8767
+ Optimized range parameters are 35,91747 161,0899
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,7280556 0,7231396
+ Start of the optimization 2 :
+ The number of iterations is 32
+ The value of the marginal posterior function is -156,8767
+ Optimized range parameters are 35,92577 161,0559
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 237,6023 235,9979
+ The initial values of range parameters are 4,752045 4,719958
+ Start of the optimization 1 :
+ The number of iterations is 15
+ The value of the marginal posterior function is -156,6342
+ Optimized range parameters are 36,30341 161,9859
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,7148183 0,7099916
+ Start of the optimization 2 :
+ The number of iterations is 26
+ The value of the marginal posterior function is -156,6343
+ Optimized range parameters are 36,30117 161,8561
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 231,5375 229,9741
+ The initial values of range parameters are 4,63075 4,599482
+ Start of the optimization 1 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -156,3683
+ Optimized range parameters are 36,27212 162,5847
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,7020536 0,6973131
+ Start of the optimization 2 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -156,3684
+ Optimized range parameters are 36,25962 162,5355
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 237,6502 236,0455
+ The initial values of range parameters are 4,753005 4,720911
+ Start of the optimization 1 :
+ The number of iterations is 14
+ The value of the marginal posterior function is -156,014
+ Optimized range parameters are 36,19577 162,4854
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,6897369 0,6850796
+ Start of the optimization 2 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -156,014
+ Optimized range parameters are 36,19621 162,4877
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 221,4001 219,9051
+ The initial values of range parameters are 4,428002 4,398102
+ Start of the optimization 1 :
+ The number of iterations is 36
+ The value of the marginal posterior function is -155,8456
+ Optimized range parameters are 36,93092 166,708
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,6778449 0,6732679
+ Start of the optimization 2 :
+ The number of iterations is 19
+ The value of the marginal posterior function is -155,8455
+ Optimized range parameters are 37,00954 166,8438
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 207,5545 206,153
+ The initial values of range parameters are 4,15109 4,123061
+ Start of the optimization 1 :
+ The number of iterations is 36
+ The value of the marginal posterior function is -155,2611
+ Optimized range parameters are 37,54574 171,0122
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,666356 0,6618565
+ Start of the optimization 2 :
+ The number of iterations is 26
+ The value of the marginal posterior function is -155,2611
+ Optimized range parameters are 37,52707 170,9901
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 211,1368 209,7111
+ The initial values of range parameters are 4,222736 4,194222
+ Start of the optimization 1 :
+ The number of iterations is 38
+ The value of the marginal posterior function is -154,6105
+ Optimized range parameters are 37,27271 173,4402
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,6552501 0,6508256
+ Start of the optimization 2 :
+ The number of iterations is 38
+ The value of the marginal posterior function is -154,6104
+ Optimized range parameters are 37,31152 173,4271
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 229,1987 227,651
+ The initial values of range parameters are 4,583973 4,553021
+ Start of the optimization 1 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -153,8922
+ Optimized range parameters are 37,52317 173,0118
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,6445083 0,6401563
+ Start of the optimization 2 :
+ The number of iterations is 29
+ The value of the marginal posterior function is -153,8923
+ Optimized range parameters are 37,52772 173,0381
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 200,215 198,8631
+ The initial values of range parameters are 4,004301 3,977262
+ Start of the optimization 1 :
+ The number of iterations is 29
+ The value of the marginal posterior function is -153,0737
+ Optimized range parameters are 37,6626 173,7851
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,634113 0,6298312
+ Start of the optimization 2 :
+ The number of iterations is 38
+ The value of the marginal posterior function is -153,0733
+ Optimized range parameters are 37,57977 172,9914
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 208,2931 206,8866
+ The initial values of range parameters are 4,165862 4,137733
+ Start of the optimization 1 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -152,171
+ Optimized range parameters are 37,55899 172,4488
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,6240477 0,6198339
+ Start of the optimization 2 :
+ The number of iterations is 34
+ The value of the marginal posterior function is -152,1711
+ Optimized range parameters are 37,55216 172,3431
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 190,9562 189,6668
+ The initial values of range parameters are 3,819124 3,793336
+ Start of the optimization 1 :
+ The number of iterations is 26
+ The value of the marginal posterior function is -151,4191
+ Optimized range parameters are 37,87024 177,6757
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,6142969 0,610149
+ Start of the optimization 2 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -151,4191
+ Optimized range parameters are 37,8678 177,4358
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 187,3139 186,0491
+ The initial values of range parameters are 3,746279 3,720983
+ Start of the optimization 1 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -150,3521
+ Optimized range parameters are 37,93228 177,0193
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,6048462 0,6007621
+ Start of the optimization 2 :
+ The number of iterations is 15
+ The value of the marginal posterior function is -150,3521
+ Optimized range parameters are 37,92683 176,9819
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 201,0467 199,6892
+ The initial values of range parameters are 4,020934 3,993783
+ Start of the optimization 1 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -149,229
+ Optimized range parameters are 37,73923 179,7933
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5956819 0,5916596
+ Start of the optimization 2 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -149,2295
+ Optimized range parameters are 37,68347 179,3098
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 208,1589 206,7533
+ The initial values of range parameters are 4,163178 4,135067
+ Start of the optimization 1 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -148,0364
+ Optimized range parameters are 37,10463 181,452
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,5867911 0,5828289
+ Start of the optimization 2 :
+ The number of iterations is 28
+ The value of the marginal posterior function is -148,0353
+ Optimized range parameters are 37,10023 181,8802
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 207,5545 206,153
+ The initial values of range parameters are 4,15109 4,123061
+ Start of the optimization 1 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -147,2549
+ Optimized range parameters are 36,89036 184,8898
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,5781618 0,5742579
+ Start of the optimization 2 :
+ The number of iterations is 18
+ The value of the marginal posterior function is -147,2555
+ Optimized range parameters are 36,87001 184,82
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 166,9279 165,8008
+ The initial values of range parameters are 3,338559 3,316015
+ Start of the optimization 1 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -146,0908
+ Optimized range parameters are 34,10173 165,8008
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5697827 0,5659353
+ Start of the optimization 2 :
+ The number of iterations is 12
+ The value of the marginal posterior function is -146,091
+ Optimized range parameters are 34,11888 165,8008
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 164,9351 163,8214
+ The initial values of range parameters are 3,298703 3,276429
+ Start of the optimization 1 :
+ The number of iterations is 29
+ The value of the marginal posterior function is -144,8362
+ Optimized range parameters are 33,02363 163,8214
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5616429 0,5578505
+ Start of the optimization 2 :
+ The number of iterations is 15
+ The value of the marginal posterior function is -144,8366
+ Optimized range parameters are 32,96846 163,8214
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 161,9899 160,8961
+ The initial values of range parameters are 3,239799 3,217922
+ Start of the optimization 1 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -143,1679
+ Optimized range parameters are 32,60853 160,8961
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5537325 0,5499935
+ Start of the optimization 2 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -143,1673
+ Optimized range parameters are 32,60514 160,8961
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 167,4923 166,3613
+ The initial values of range parameters are 3,349846 3,327227
+ Start of the optimization 1 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -141,4524
+ Optimized range parameters are 32,84554 166,3613
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5460417 0,5423547
+ Start of the optimization 2 :
+ The number of iterations is 10
+ The value of the marginal posterior function is -141,4531
+ Optimized range parameters are 32,87053 166,3613
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 158,1908 157,1226
+ The initial values of range parameters are 3,163815 3,142452
+ Start of the optimization 1 :
+ The number of iterations is 21
+ The value of the marginal posterior function is -140,039
+ Optimized range parameters are 31,62214 157,1226
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5385617 0,5349251
+ Start of the optimization 2 :
+ The number of iterations is 16
+ The value of the marginal posterior function is -140,0387
+ Optimized range parameters are 31,62433 157,1226
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 174,0066 172,8316
+ The initial values of range parameters are 3,480131 3,456632
+ Start of the optimization 1 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -138,1222
+ Optimized range parameters are 33,48823 172,8316
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5312838 0,5276964
+ Start of the optimization 2 :
+ The number of iterations is 28
+ The value of the marginal posterior function is -138,1223
+ Optimized range parameters are 33,40525 172,8316
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 150,4312 149,4154
+ The initial values of range parameters are 3,008624 2,988309
+ Start of the optimization 1 :
+ The number of iterations is 21
+ The value of the marginal posterior function is -139,2082
+ Optimized range parameters are 30,85041 149,4154
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,5242001 0,5206605
+ Start of the optimization 2 :
+ The number of iterations is 25
+ The value of the marginal posterior function is -139,2085
+ Optimized range parameters are 30,93585 149,4154
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 163,2096 162,1076
+ The initial values of range parameters are 3,264193 3,242152
+ Start of the optimization 1 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -136,829
+ Optimized range parameters are 31,64889 162,1076
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5173027 0,5138097
+ Start of the optimization 2 :
+ The number of iterations is 29
+ The value of the marginal posterior function is -136,8292
+ Optimized range parameters are 31,73272 162,1076
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 155,3651 154,316
+ The initial values of range parameters are 3,107301 3,08632
+ Start of the optimization 1 :
+ The number of iterations is 38
+ The value of the marginal posterior function is -137,2507
+ Optimized range parameters are 31,11748 154,316
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5105845 0,5071368
+ Start of the optimization 2 :
+ The number of iterations is 11
+ The value of the marginal posterior function is -137,2507
+ Optimized range parameters are 31,11466 154,316
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 159,0334 157,9595
+ The initial values of range parameters are 3,180667 3,15919
+ Start of the optimization 1 :
+ The number of iterations is 19
+ The value of the marginal posterior function is -134,9592
+ Optimized range parameters are 30,6283 157,9595
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,5040385 0,5006351
+ Start of the optimization 2 :
+ The number of iterations is 20
+ The value of the marginal posterior function is -134,9589
+ Optimized range parameters are 30,62948 157,9595
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 150,6829 149,6655
+ The initial values of range parameters are 3,013659 2,993309
+ Start of the optimization 1 :
+ The number of iterations is 9
+ The value of the marginal posterior function is -133,0322
+ Optimized range parameters are 28,62106 149,6655
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4976583 0,4942979
+ Start of the optimization 2 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -133,0311
+ Optimized range parameters are 28,6202 149,6655
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 143,0785 142,1124
+ The initial values of range parameters are 2,86157 2,842247
+ Start of the optimization 1 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -131,5134
+ Optimized range parameters are 27,05986 142,1124
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,4914376 0,4881192
+ Start of the optimization 2 :
+ The number of iterations is 28
+ The value of the marginal posterior function is -131,5142
+ Optimized range parameters are 27,09861 142,1124
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 149,9901 148,9774
+ The initial values of range parameters are 2,999803 2,979547
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -128,3138
+ Optimized range parameters are 26,94379 148,9774
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4853704 0,482093
+ Start of the optimization 2 :
+ The number of iterations is 11
+ The value of the marginal posterior function is -128,3136
+ Optimized range parameters are 26,93153 148,9774
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 141,9733 141,0146
+ The initial values of range parameters are 2,839465 2,820292
+ Start of the optimization 1 :
+ The number of iterations is 15
+ The value of the marginal posterior function is -126,565
+ Optimized range parameters are 25,84325 141,0146
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4794513 0,4762139
+ Start of the optimization 2 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -126,5658
+ Optimized range parameters are 25,84843 141,0146
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 148,4124 147,4103
+ The initial values of range parameters are 2,968249 2,948206
+ Start of the optimization 1 :
+ The number of iterations is 14
+ The value of the marginal posterior function is -123,2249
+ Optimized range parameters are 26,04536 147,4103
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4736748 0,4704763
+ Start of the optimization 2 :
+ The number of iterations is 37
+ The value of the marginal posterior function is -123,2255
+ Optimized range parameters are 26,11794 147,4103
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 141,3136 140,3594
+ The initial values of range parameters are 2,826273 2,807189
+ Start of the optimization 1 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -121,373
+ Optimized range parameters are 25,34355 140,3594
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,4680358 0,4648754
+ Start of the optimization 2 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -121,3719
+ Optimized range parameters are 25,3222 140,3594
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 146,0605 145,0743
+ The initial values of range parameters are 2,921211 2,901486
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -119,1731
+ Optimized range parameters are 25,38398 145,0743
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4625295 0,4594063
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -119,1742
+ Optimized range parameters are 25,27086 145,0743
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 136,1432 135,2239
+ The initial values of range parameters are 2,722864 2,704479
+ Start of the optimization 1 :
+ The number of iterations is 16
+ The value of the marginal posterior function is -117,4765
+ Optimized range parameters are 23,8631 135,2239
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4571512 0,4540644
+ Start of the optimization 2 :
+ The number of iterations is 31
+ The value of the marginal posterior function is -117,4779
+ Optimized range parameters are 23,87798 135,2239
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 121,8399 121,0172
+ The initial values of range parameters are 2,436797 2,420343
+ Start of the optimization 1 :
+ The number of iterations is 21
+ The value of the marginal posterior function is -117,2996
+ Optimized range parameters are 22,38443 121,0172
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,4518966 0,4488452
+ Start of the optimization 2 :
+ The number of iterations is 19
+ The value of the marginal posterior function is -117,2989
+ Optimized range parameters are 22,37563 121,0172
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 123,1404 122,309
+ The initial values of range parameters are 2,462809 2,446179
+ Start of the optimization 1 :
+ The number of iterations is 18
+ The value of the marginal posterior function is -114,6525
+ Optimized range parameters are 22,32619 122,309
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4467614 0,4437447
+ Start of the optimization 2 :
+ The number of iterations is 38
+ The value of the marginal posterior function is -114,6523
+ Optimized range parameters are 22,33099 122,309
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 117,4477 116,6546
+ The initial values of range parameters are 2,348953 2,333092
+ Start of the optimization 1 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -113,3472
+ Optimized range parameters are 21,62668 116,6546
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,4417416 0,4387588
+ Start of the optimization 2 :
+ The number of iterations is 28
+ The value of the marginal posterior function is -113,3475
+ Optimized range parameters are 21,65104 116,6546
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 115,2499 114,4717
+ The initial values of range parameters are 2,304998 2,289433
+ Start of the optimization 1 :
+ The number of iterations is 12
+ The value of the marginal posterior function is -110,7155
+ Optimized range parameters are 20,98512 114,4717
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4368334 0,4338837
+ Start of the optimization 2 :
+ The number of iterations is 30
+ The value of the marginal posterior function is -110,7149
+ Optimized range parameters are 21,00228 114,4717
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 111,8946 111,139
+ The initial values of range parameters are 2,237892 2,222781
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -108,5065
+ Optimized range parameters are 20,44681 111,139
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,432033 0,4291158
+ Start of the optimization 2 :
+ The number of iterations is 19
+ The value of the marginal posterior function is -108,507
+ Optimized range parameters are 20,44983 111,139
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 93,38518 92,75461
+ The initial values of range parameters are 1,867704 1,855092
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -111,367
+ Optimized range parameters are 18,37032 92,75461
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,427337 0,4244515
+ Start of the optimization 2 :
+ The number of iterations is 18
+ The value of the marginal posterior function is -111,3671
+ Optimized range parameters are 18,37078 92,75461
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 99,64494 98,97211
+ The initial values of range parameters are 1,992899 1,979442
+ Start of the optimization 1 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -106,2562
+ Optimized range parameters are 18,89163 98,97211
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,422742 0,4198875
+ Start of the optimization 2 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -106,2561
+ Optimized range parameters are 18,88065 98,97211
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 99,75194 99,07838
+ The initial values of range parameters are 1,995039 1,981568
+ Start of the optimization 1 :
+ The number of iterations is 17
+ The value of the marginal posterior function is -103,2099
+ Optimized range parameters are 18,65492 99,07838
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4182447 0,4154206
+ Start of the optimization 2 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -103,2113
+ Optimized range parameters are 18,65549 99,07838
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 104,3536 103,649
+ The initial values of range parameters are 2,087073 2,07298
+ Start of the optimization 1 :
+ The number of iterations is 20
+ The value of the marginal posterior function is -98,42375
+ Optimized range parameters are 18,87395 103,649
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4138422 0,4110477
+ Start of the optimization 2 :
+ The number of iterations is 27
+ The value of the marginal posterior function is -98,42241
+ Optimized range parameters are 18,87764 103,649
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The upper bounds of the range parameters are 103,1118 102,4155
+ The initial values of range parameters are 2,062236 2,048311
+ Start of the optimization 1 :
+ The number of iterations is 20
+ The value of the marginal posterior function is -95,30729
+ Optimized range parameters are 18,51323 102,4155
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,4095313 0,406766
+ Start of the optimization 2 :
+ The number of iterations is 29
+ The value of the marginal posterior function is -95,30714
+ Optimized range parameters are 18,54824 102,4155
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 95,06543 94,42352
+ The initial values of range parameters are 1,901309 1,88847
+ Start of the optimization 1 :
+ The number of iterations is 14
+ The value of the marginal posterior function is -95,1497
+ Optimized range parameters are 17,52307 94,42352
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,4053093 0,4025725
+ Start of the optimization 2 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -95,14898
+ Optimized range parameters are 17,55379 94,42352
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 91,10625 90,49107
+ The initial values of range parameters are 1,822125 1,809821
+ Start of the optimization 1 :
+ The number of iterations is 21
+ The value of the marginal posterior function is -93,31858
+ Optimized range parameters are 16,88561 90,49107
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,4011735 0,3984647
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -93,31829
+ Optimized range parameters are 16,83654 90,49107
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 95,06911 94,42717
+ The initial values of range parameters are 1,901382 1,888543
+ Start of the optimization 1 :
+ The number of iterations is 11
+ The value of the marginal posterior function is -88,62383
+ Optimized range parameters are 17,19113 94,42717
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 0,3971213 0,3944398
+ Start of the optimization 2 :
+ The number of iterations is 23
+ The value of the marginal posterior function is -88,62286
+ Optimized range parameters are 17,21816 94,42717
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The upper bounds of the range parameters are 84,25512 83,6862
+ The initial values of range parameters are 1,685102 1,673724
+ Start of the optimization 1 :
+ The number of iterations is 20
+ The value of the marginal posterior function is -90,56369
+ Optimized range parameters are 16,01921 83,6862
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+ The initial values of range parameters are 0,39315 0,3904954
+ Start of the optimization 2 :
+ The number of iterations is 24
+ The value of the marginal posterior function is -90,56328
+ Optimized range parameters are 16,0189 83,6862
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 92-100
+
+Once the active learning process is finished, we obtain the final emulator for the
+logarithm of the unnormalized posterior, which is given by the
+:py:meth:`.ActiveLearning.approx_ln_pxl` method.
+
+We can then estimate the posterior using grid estimation or Metropolis Hastings
+estimation based on the emulator. An example is as follows. The contour plot shows
+the estimated posterior.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 101-127
+
+.. code-block:: default
+
+ from psimpy.inference import GridEstimation
+ import matplotlib.pyplot as plt
+
+ grid_estimator = GridEstimation(ndim, bounds, ln_pxl=active_learner.approx_ln_pxl)
+ posterior, x_ndim = grid_estimator.run(nbins=50)
+
+ fig, ax = plt.subplots(1,1,figsize=(5,4))
+
+ # initial training points
+ ax.scatter(init_var_samples[:,0], init_var_samples[:,1], s=10, c='r', marker='o',
+ zorder=1, alpha=0.8, label='initial training points')
+ # actively picked training points
+ ax.scatter(var_samples[n0:,0], var_samples[n0:,1], s=15, c='k', marker='+',
+ zorder=2, alpha=0.8, label='iterative training points')
+
+ # estimated posterior based on the final emulator
+ posterior = np.where(posterior < 1e-10, None, posterior)
+ contour = ax.contour(x_ndim[0], x_ndim[1], np.transpose(posterior), levels=10, zorder=0)
+ plt.colorbar(contour, ax=ax)
+
+ ax.legend()
+ ax.set_title('Active learning')
+ ax.set_xlabel('x1')
+ ax.set_ylabel('x2')
+ ax.set_xlim([-5,5])
+ ax.set_ylim([-5,5])
+ plt.tight_layout()
+
+
+.. image-sg:: /auto_examples/inference/images/sphx_glr_plot_active_learning_001.png
+ :alt: Active learning
+ :srcset: /auto_examples/inference/images/sphx_glr_plot_active_learning_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 2 minutes 12.653 seconds)
+
+
+.. _sphx_glr_download_auto_examples_inference_plot_active_learning.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_active_learning.py <plot_active_learning.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_active_learning.ipynb <plot_active_learning.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/inference/plot_bayes_inference.rst.txt b/docs/source/_build/html/_sources/auto_examples/inference/plot_bayes_inference.rst.txt
new file mode 100644
index 0000000..f8b71d7
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/inference/plot_bayes_inference.rst.txt
@@ -0,0 +1,231 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/inference/plot_bayes_inference.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_inference_plot_bayes_inference.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_inference_plot_bayes_inference.py:
+
+
+Bayesian inference
+==================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-16
+
+This example shows how to perform Bayesian inference given the uniform prior
+
+:math:`p(\mathbf{x})=p(x_1,x_2)=0.01`
+
+where :math:`x_i \in [-5,5], i=1,2`, and likelihood
+
+:math:`L(\mathbf{x}|\mathbf{d})=\exp \left(-\frac{1}{100}\left(x_1-1\right)^2-\left(x_1^2-x_2\right)^2\right)`.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 17-29
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ ndim = 2
+ bounds = np.array([[-5,5],[-5,5]])
+
+ def prior(x):
+ return 0.01
+
+ def likelihood(x):
+ return np.exp(-(x[0]-1)**2/100 - (x[0]**2-x[1])**2)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 30-33
+
+To estimate the posterior using grid estimation, we need to import the
+:class:`.GridEstimation` class, create an instance, and call the
+:py:meth:`.GridEstimation.run` method.
+
+.. GENERATED FROM PYTHON SOURCE LINES 34-40
+
+.. code-block:: default
+
+
+ from psimpy.inference.bayes_inference import GridEstimation
+
+ grid_estimator = GridEstimation(ndim, bounds, prior, likelihood)
+ posterior, x_ndim = grid_estimator.run(nbins=[50,40])
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 41-43
+
+The following figure plots the estimated posterior.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 44-61
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ fig, ax = plt.subplots(1,1,figsize=(6,4))
+
+ # mask insignificant values
+ posterior = np.where(posterior < 1e-10, None, posterior)
+
+ contour = ax.contour(x_ndim[0], x_ndim[1], np.transpose(posterior), levels=10)
+ plt.colorbar(contour, ax=ax)
+ ax.set_xlim(bounds[0,0], bounds[0,1])
+ ax.set_ylim(bounds[1,0], bounds[1,1])
+ ax.set_title('Grid estimation')
+ ax.set_xlabel('x1')
+ ax.set_ylabel('x2')
+ plt.tight_layout()
+
+
+
+
+.. image-sg:: /auto_examples/inference/images/sphx_glr_plot_bayes_inference_001.png
+ :alt: Grid estimation
+ :srcset: /auto_examples/inference/images/sphx_glr_plot_bayes_inference_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 62-67
+
+To estimate the posterior using Metropolis Hastings estimation, we need to import
+the :class:`.MetropolisHastingsEstimation` class, create an instance, and call the
+:py:meth:`.MetropolisHastingsEstimation.run` method. The
+:py:meth:`.MetropolisHastingsEstimation.run` method has a parameter, ``mh_sampler``,
+which takes an instance of :class:`.MetropolisHastings` as argument.
+
+.. GENERATED FROM PYTHON SOURCE LINES 68-92
+
+.. code-block:: default
+
+
+ from psimpy.inference.bayes_inference import MetropolisHastingsEstimation
+
+ mh_estimator = MetropolisHastingsEstimation(ndim, bounds, prior, likelihood)
+
+ # create a mh_sampler
+ from psimpy.sampler.metropolis_hastings import MetropolisHastings
+ from scipy.stats import multivariate_normal
+
+ init_state = np.array([-4,-4])
+ f_sample = multivariate_normal.rvs
+ nburn = 100
+ nthin = 10
+ seed = 1
+ kwgs_f_sample = {'random_state': np.random.default_rng(seed)}
+
+ mh_sampler = MetropolisHastings(ndim=ndim, init_state=init_state,
+ f_sample=f_sample, bounds=bounds, nburn=nburn, nthin=nthin, seed=seed,
+ kwgs_f_sample=kwgs_f_sample)
+
+ nsamples = 5000
+ mh_samples, mh_accept = mh_estimator.run(nsamples, mh_sampler)
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 93-96
+
+The following figure plots the samples drawn from the unnormalized posterior,
+which can be used to estimate the posterior and its poperties.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 97-110
+
+.. code-block:: default
+
+ import matplotlib.pyplot as plt
+
+ fig, ax = plt.subplots(1,1,figsize=(5,4))
+
+ ax.scatter(mh_samples[:,0], mh_samples[:,1], s=10, c='r', marker='o', alpha=0.1)
+ ax.set_xlim(bounds[0,0], bounds[0,1])
+ ax.set_ylim(bounds[1,0], bounds[1,1])
+ ax.set_title('MH estimation')
+ ax.set_xlabel('x1')
+ ax.set_ylabel('x2')
+ plt.tight_layout()
+
+
+
+
+
+.. image-sg:: /auto_examples/inference/images/sphx_glr_plot_bayes_inference_002.png
+ :alt: MH estimation
+ :srcset: /auto_examples/inference/images/sphx_glr_plot_bayes_inference_002.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 111-116
+
+.. note:: Besides ``prior`` and ``likelihood``, one can also instantiate
+ the :class:`.MetropolisHastingsEstimation` class with
+
+ - ``ln_prior`` and ``ln_likelihood``: Natural logarithm of ``prior`` and ``likelihood``.
+ - ``ln_pxl``: Natural logarithm of the product of ``prior`` and ``likelihood``.
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 9.739 seconds)
+
+
+.. _sphx_glr_download_auto_examples_inference_plot_bayes_inference.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_bayes_inference.py <plot_bayes_inference.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_bayes_inference.ipynb <plot_bayes_inference.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/inference/sg_execution_times.rst.txt b/docs/source/_build/html/_sources/auto_examples/inference/sg_execution_times.rst.txt
new file mode 100644
index 0000000..4f6c429
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/inference/sg_execution_times.rst.txt
@@ -0,0 +1,14 @@
+
+:orphan:
+
+.. _sphx_glr_auto_examples_inference_sg_execution_times:
+
+Computation times
+=================
+**02:22.392** total execution time for **auto_examples_inference** files:
+
++-----------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_inference_plot_active_learning.py` (``plot_active_learning.py``) | 02:12.653 | 0.0 MB |
++-----------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_inference_plot_bayes_inference.py` (``plot_bayes_inference.py``) | 00:09.739 | 0.0 MB |
++-----------------------------------------------------------------------------------------------+-----------+--------+
diff --git a/docs/source/_build/html/_sources/auto_examples/sampler/index.rst.txt b/docs/source/_build/html/_sources/auto_examples/sampler/index.rst.txt
new file mode 100644
index 0000000..6192045
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sampler/index.rst.txt
@@ -0,0 +1,93 @@
+:orphan:
+
+Sampler
+=======
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnails">
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Saltelli sampling">
+
+.. only:: html
+
+ .. image:: /auto_examples/sampler/images/thumb/sphx_glr_plot_saltelli_thumb.png
+ :alt: Saltelli sampling
+
+ :ref:`sphx_glr_auto_examples_sampler_plot_saltelli.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Saltelli sampling</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Latin hypercube sampling">
+
+.. only:: html
+
+ .. image:: /auto_examples/sampler/images/thumb/sphx_glr_plot_latin_thumb.png
+ :alt: Latin hypercube sampling
+
+ :ref:`sphx_glr_auto_examples_sampler_plot_latin.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Latin hypercube sampling</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Metropolis Hastings sampling">
+
+.. only:: html
+
+ .. image:: /auto_examples/sampler/images/thumb/sphx_glr_plot_metropolis_hastings_thumb.png
+ :alt: Metropolis Hastings sampling
+
+ :ref:`sphx_glr_auto_examples_sampler_plot_metropolis_hastings.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Metropolis Hastings sampling</div>
+ </div>
+
+
+.. raw:: html
+
+ </div>
+
+
+.. toctree::
+ :hidden:
+
+ /auto_examples/sampler/plot_saltelli
+ /auto_examples/sampler/plot_latin
+ /auto_examples/sampler/plot_metropolis_hastings
+
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-gallery
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download all examples in Python source code: sampler_python.zip </auto_examples/sampler/sampler_python.zip>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download all examples in Jupyter notebooks: sampler_jupyter.zip </auto_examples/sampler/sampler_jupyter.zip>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/sampler/plot_latin.rst.txt b/docs/source/_build/html/_sources/auto_examples/sampler/plot_latin.rst.txt
new file mode 100644
index 0000000..3c1fd90
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sampler/plot_latin.rst.txt
@@ -0,0 +1,209 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/sampler/plot_latin.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_sampler_plot_latin.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_sampler_plot_latin.py:
+
+
+Latin hypercube sampling
+========================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-11
+
+This example shows how to draw samples using Latin hypercube sampling.
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 14-17
+
+For the illustration purpose, let's have a look at a two-dimensional example
+where we have two random variables X and Y. Each is uniformly distributed in its
+range.
+
+.. GENERATED FROM PYTHON SOURCE LINES 18-25
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ ndim = 2
+ # range of X is 10 to 20, range of Y is -10 to 0
+ bounds = np.array([[10,20], [-10,0]])
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 26-28
+
+Given this setting, we can import :class:`.Latin`, create an instance, and
+call the :py:meth:`.Latin.sample` method to draw required number of samples
+
+.. GENERATED FROM PYTHON SOURCE LINES 29-36
+
+.. code-block:: default
+
+
+ from psimpy.sampler import LHS
+
+ # setting seed leads to same samples every time when the codes are run
+ lhs_sampler = LHS(ndim, bounds, seed=10)
+ lhs_samples = lhs_sampler.sample(nsamples=5)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 37-38
+
+The samples are plotted in the following figure.
+
+.. GENERATED FROM PYTHON SOURCE LINES 39-54
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ fig , ax = plt.subplots(figsize=(6,4))
+
+ ax.scatter(lhs_samples[:,0], lhs_samples[:,1], s=10, c='blue', marker='o')
+
+ ax.set_xlabel('X')
+ ax.set_ylabel('Y')
+ ax.set_xlim(bounds[0])
+ ax.set_ylim(bounds[1])
+ ax.set_title("Latin hypercube samples (criterion='random')")
+ _ = ax.grid(visible=True, which='major', axis='both')
+
+
+
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_latin_001.png
+ :alt: Latin hypercube samples (criterion='random')
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_latin_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 55-58
+
+There are different criterions to pick samples in each hypercube. The default
+is `random`, as used above. Other options are `center` and `maximin`. For instance,
+we can use the `center` criterion to draw :math:`5` samples as follows:
+
+.. GENERATED FROM PYTHON SOURCE LINES 59-75
+
+.. code-block:: default
+
+
+ lhs_sampler = LHS(ndim, bounds, criterion='center', seed=10)
+ lhs_samples = lhs_sampler.sample(nsamples=5)
+
+ fig , ax = plt.subplots(figsize=(6,4))
+
+ ax.scatter(lhs_samples[:,0], lhs_samples[:,1], s=10, c='blue', marker='o')
+
+ ax.set_xlabel('X')
+ ax.set_ylabel('Y')
+ ax.set_xlim(bounds[0])
+ ax.set_ylim(bounds[1])
+ ax.set_title("Latin hypercube samples (criterion='center')")
+ _ = ax.grid(visible=True, which='major', axis='both')
+
+
+
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_latin_002.png
+ :alt: Latin hypercube samples (criterion='center')
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_latin_002.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 76-77
+
+And we can use the `maximin` criterion as follows:
+
+.. GENERATED FROM PYTHON SOURCE LINES 78-91
+
+.. code-block:: default
+
+
+ lhs_sampler = LHS(ndim, bounds, criterion='maximin', seed=10, iteration=500)
+ lhs_samples = lhs_sampler.sample(nsamples=5)
+
+ fig , ax = plt.subplots(figsize=(6,4))
+
+ ax.scatter(lhs_samples[:,0], lhs_samples[:,1], s=10, c='blue', marker='o')
+
+ ax.set_xlabel('X')
+ ax.set_ylabel('Y')
+ ax.set_xlim(bounds[0])
+ ax.set_ylim(bounds[1])
+ ax.set_title("Latin hypercube samples (criterion='maximin')")
+ _ = ax.grid(visible=True, which='major', axis='both')
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_latin_003.png
+ :alt: Latin hypercube samples (criterion='maximin')
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_latin_003.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 0.270 seconds)
+
+
+.. _sphx_glr_download_auto_examples_sampler_plot_latin.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_latin.py <plot_latin.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_latin.ipynb <plot_latin.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/sampler/plot_metropolis_hastings.rst.txt b/docs/source/_build/html/_sources/auto_examples/sampler/plot_metropolis_hastings.rst.txt
new file mode 100644
index 0000000..41772d3
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sampler/plot_metropolis_hastings.rst.txt
@@ -0,0 +1,248 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/sampler/plot_metropolis_hastings.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_sampler_plot_metropolis_hastings.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_sampler_plot_metropolis_hastings.py:
+
+
+Metropolis Hastings sampling
+============================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-22
+
+This example shows how to draw samples using Metropolis Hastings sampling.
+The target probability distribution is
+
+:math:`p(\mathbf{x})=p(x_1, x_2) \propto \exp \left(-\frac{1}{100}\left(x_1-1\right)^2-\left(x_1^2-x_2\right)^2\right)`
+
+where :math:`x_1 \in [-5,5]` and :math:`x_2 \in [-5,5]`.
+
+It should be noted that the right hand side of the equation is an unnormalized
+probability density function since its integral is not equal to :math:`1`.
+This can happen, for example, when the normalization constant is unknown or difficult
+to compute.
+
+We can define the target probability distribution in Python as follows:
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 23-32
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ def target_dist(x):
+ if (x[0]>=-5 and x[0]<=5) and (x[1]>=-5 and x[1]<=5):
+ return np.exp(-0.01*(x[0]-1)**2 - (x[0]**2-x[1])**2)
+ else:
+ return 0
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 33-34
+
+The figure below shows how the target distribution looks like.
+
+.. GENERATED FROM PYTHON SOURCE LINES 35-60
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+ import itertools
+
+ x1_values = np.linspace(-5,5.1,100)
+ x2_values = np.linspace(-5,5.1,100)
+
+ target_values = np.zeros((100, 100))
+ for i, j in itertools.product(range(100), range(100)):
+ x1 = x1_values[i]
+ x2 = x2_values[j]
+ target_values[i,j] = target_dist(np.array([x1, x2]))
+
+ fig , ax = plt.subplots(figsize=(4,4))
+
+ ax.contourf(x1_values, x2_values, np.transpose(target_values), levels=10, cmap='Blues')
+
+ ax.set_title('(unnormalized) target distribution')
+ ax.set_xlabel(r'$x_1$')
+ ax.set_ylabel(r'$x_2$')
+ ax.set_xlim(-5,5)
+ ax.set_ylim(-5,5)
+ plt.tight_layout()
+
+
+
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_metropolis_hastings_001.png
+ :alt: (unnormalized) target distribution
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_metropolis_hastings_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 61-67
+
+To perform Metropolis Hastings sampling, we need to choose a proposal distribution
+which can be used to determine a new state ``x'`` given a current state ``x`` at
+each iteration. This is defined by the parameter ``f_sample`` which should be a
+function. A usual choice is to choose the new state from a Gaussian distribution
+centered at the current state.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 68-75
+
+.. code-block:: default
+
+
+ from scipy.stats import multivariate_normal
+
+ f_sample = multivariate_normal.rvs
+ # make the samples reproducible
+ kwgs_f_sample = {'random_state': np.random.default_rng(seed=1)}
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 76-77
+
+Then, we create an instance of :class:`.MetropolisHastings` class.
+
+.. GENERATED FROM PYTHON SOURCE LINES 78-85
+
+.. code-block:: default
+
+
+ from psimpy.sampler.metropolis_hastings import MetropolisHastings
+
+ mh_sampler = MetropolisHastings(ndim=2, init_state=np.array([-4,-4]),
+ f_sample=f_sample, target=target_dist, nburn=1000, nthin=5,
+ seed=1, kwgs_f_sample=kwgs_f_sample)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 86-88
+
+Next, we call the :py:meth:`.MetropolisHastings.sample` method to draw required
+number of samples.
+
+.. GENERATED FROM PYTHON SOURCE LINES 89-95
+
+.. code-block:: default
+
+
+ mh_samples, mh_accept = mh_sampler.sample(nsamples=5000)
+
+ print("Acceptance ratio: ", np.mean(mh_accept))
+
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Acceptance ratio: 0.3712
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 96-97
+
+The following figure shows how the samples look like.
+
+.. GENERATED FROM PYTHON SOURCE LINES 98-117
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+ import itertools
+
+ # sphinx_gallery_thumbnail_number = 2
+ fig , ax = plt.subplots(figsize=(4,4))
+
+ ax.contourf(x1_values, x2_values, np.transpose(target_values), levels=10, cmap='Blues')
+ ax.scatter(mh_samples[:,0], mh_samples[:,1], s=5, c='r', marker='o',alpha=0.05)
+
+ ax.set_title('Metropolis Hastings samples')
+ ax.set_xlabel(r'$x_1$')
+ ax.set_ylabel(r'$x_2$')
+ ax.set_xlim(-5,5)
+ ax.set_ylim(-5,5)
+ plt.tight_layout()
+
+
+
+
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_metropolis_hastings_002.png
+ :alt: Metropolis Hastings samples
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_metropolis_hastings_002.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 4.769 seconds)
+
+
+.. _sphx_glr_download_auto_examples_sampler_plot_metropolis_hastings.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_metropolis_hastings.py <plot_metropolis_hastings.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_metropolis_hastings.ipynb <plot_metropolis_hastings.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/sampler/plot_saltelli.rst.txt b/docs/source/_build/html/_sources/auto_examples/sampler/plot_saltelli.rst.txt
new file mode 100644
index 0000000..8204636
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sampler/plot_saltelli.rst.txt
@@ -0,0 +1,193 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/sampler/plot_saltelli.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_sampler_plot_saltelli.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_sampler_plot_saltelli.py:
+
+
+Saltelli sampling
+=================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-11
+
+This example shows how to draw samples using Saltelli sampling.
+Assume that there is a three-dimensional problem where X, Y, and Z are the
+three random variables.
+
+.. GENERATED FROM PYTHON SOURCE LINES 12-19
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ ndim = 3
+ # range of X is 10 to 20, range of Y is 100 to 200, range of Z is 1000 to 2000
+ bounds = np.array([[10,20], [100,200], [1000,2000]])
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 20-22
+
+Given this setting, we can import :class:`.Saltelli`, create an instance, and
+call the :py:meth:`.Saltelli.sample` method to draw required number of samples.
+
+.. GENERATED FROM PYTHON SOURCE LINES 23-29
+
+.. code-block:: default
+
+
+ from psimpy.sampler import Saltelli
+
+ saltelli_sampler = Saltelli(ndim, bounds, calc_second_order=False)
+ saltelli_samples = saltelli_sampler.sample(nbase=128)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 30-34
+
+In above codes, we set ``calc_second_order`` to `False`. It means that picked
+samples can be used in following Sobol' analysis to compute first-order and
+total-effect Sobol' indices but not second-order Sobol' indices. It leads to
+:math:`nbase*(ndim+2)` samples as shown below
+
+.. GENERATED FROM PYTHON SOURCE LINES 35-51
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ fig = plt.figure()
+ ax = fig.add_subplot(projection='3d')
+
+ ax.scatter(saltelli_samples[:,0], saltelli_samples[:,1], saltelli_samples[:,2], marker='o')
+ ax.set_xlabel('X')
+ ax.set_ylabel('Y')
+ ax.set_zlabel('Z')
+
+ plt.tight_layout()
+
+ print('Number of samples: ', f'{len(saltelli_samples)}')
+
+
+
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_saltelli_001.png
+ :alt: plot saltelli
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_saltelli_001.png
+ :class: sphx-glr-single-img
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Number of samples: 640
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 52-55
+
+If we want to draw samples which can also be used to compute second-order
+Sobol' indices, we need to set ``calc_second_order`` to `True`.
+It leads to :math:`nbase*(2*ndim+2)` samples.
+
+.. GENERATED FROM PYTHON SOURCE LINES 56-73
+
+.. code-block:: default
+
+
+ saltelli_sampler = Saltelli(ndim, bounds, calc_second_order=True)
+ saltelli_samples = saltelli_sampler.sample(nbase=128)
+
+ fig = plt.figure()
+ ax = fig.add_subplot(projection='3d')
+
+ ax.scatter(saltelli_samples[:,0], saltelli_samples[:,1], saltelli_samples[:,2], marker='o')
+ ax.set_xlabel('X')
+ ax.set_ylabel('Y')
+ ax.set_zlabel('Z')
+
+ plt.tight_layout()
+
+ print('Number of samples: ', f'{len(saltelli_samples)}')
+
+
+
+
+
+.. image-sg:: /auto_examples/sampler/images/sphx_glr_plot_saltelli_002.png
+ :alt: plot saltelli
+ :srcset: /auto_examples/sampler/images/sphx_glr_plot_saltelli_002.png
+ :class: sphx-glr-single-img
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Number of samples: 1024
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 74-78
+
+.. note:: If one has a two-dimensional problem, there is no need to set
+ ``calc_second_order`` to `True`. The reason is that the second-order Sobol'
+ index can be directly computed based on the first-order and total-effect
+ Sobol' index in that case.
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 0.309 seconds)
+
+
+.. _sphx_glr_download_auto_examples_sampler_plot_saltelli.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_saltelli.py <plot_saltelli.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_saltelli.ipynb <plot_saltelli.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/sampler/sg_execution_times.rst.txt b/docs/source/_build/html/_sources/auto_examples/sampler/sg_execution_times.rst.txt
new file mode 100644
index 0000000..4f0d609
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sampler/sg_execution_times.rst.txt
@@ -0,0 +1,16 @@
+
+:orphan:
+
+.. _sphx_glr_auto_examples_sampler_sg_execution_times:
+
+Computation times
+=================
+**00:05.348** total execution time for **auto_examples_sampler** files:
+
++-----------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_sampler_plot_metropolis_hastings.py` (``plot_metropolis_hastings.py``) | 00:04.769 | 0.0 MB |
++-----------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_sampler_plot_saltelli.py` (``plot_saltelli.py``) | 00:00.309 | 0.0 MB |
++-----------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_sampler_plot_latin.py` (``plot_latin.py``) | 00:00.270 | 0.0 MB |
++-----------------------------------------------------------------------------------------------------+-----------+--------+
diff --git a/docs/source/_build/html/_sources/auto_examples/sensitivity/index.rst.txt b/docs/source/_build/html/_sources/auto_examples/sensitivity/index.rst.txt
new file mode 100644
index 0000000..7275650
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sensitivity/index.rst.txt
@@ -0,0 +1,57 @@
+:orphan:
+
+Sensitivity
+===========
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnails">
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Sobol' analysis">
+
+.. only:: html
+
+ .. image:: /auto_examples/sensitivity/images/thumb/sphx_glr_plot_sobol_analyze_thumb.png
+ :alt: Sobol' analysis
+
+ :ref:`sphx_glr_auto_examples_sensitivity_plot_sobol_analyze.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Sobol' analysis</div>
+ </div>
+
+
+.. raw:: html
+
+ </div>
+
+
+.. toctree::
+ :hidden:
+
+ /auto_examples/sensitivity/plot_sobol_analyze
+
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-gallery
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download all examples in Python source code: sensitivity_python.zip </auto_examples/sensitivity/sensitivity_python.zip>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download all examples in Jupyter notebooks: sensitivity_jupyter.zip </auto_examples/sensitivity/sensitivity_jupyter.zip>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/sensitivity/plot_sobol_analyze.rst.txt b/docs/source/_build/html/_sources/auto_examples/sensitivity/plot_sobol_analyze.rst.txt
new file mode 100644
index 0000000..4309db9
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sensitivity/plot_sobol_analyze.rst.txt
@@ -0,0 +1,358 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/sensitivity/plot_sobol_analyze.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_sensitivity_plot_sobol_analyze.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_sensitivity_plot_sobol_analyze.py:
+
+
+Sobol' analysis
+===============
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-16
+
+This example shows how to compute Sobol' sensitivity indices using
+the :class:`.SobolAnalyze` class.
+
+Let us consider the Ishigami function :cite:p:`LeGratiet2014`
+
+:math:`y=f\left(x_1, x_2, x_3\right)=\sin \left(x_1\right)+7 \sin \left(\mathrm{x}_2\right)^2+0.1 x_3^4 \sin \left(x_1\right)`
+
+where :math:`x_i` is uniformly distributed on :math:`[-\pi, \pi]` for :math:`i \in [1,2,3]`. We define the function as follows:
+
+.. GENERATED FROM PYTHON SOURCE LINES 17-24
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ def f(x1,x2,x3):
+ return np.sin(x1) + 7*np.sin(x2)**2 + 0.1*x3**4*np.sin(x1)
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 25-32
+
+The variation of each :math:`x_i` leads to the variation of :math:`y`. Sobol' analysis is used to quantify the contribution
+of each :math:`x_i` and their interactions to the variation of :math:`y`. Steps of a Sobol' analysis are:
+
+ #. Draw samples of input.
+ #. Evaluate the model at each input sample to obtain model output.
+ #. Compute Sobol' indices based on model outputs.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 33-57
+
+.. code-block:: default
+
+
+ from psimpy.sampler import Saltelli
+ from psimpy.sensitivity import SobolAnalyze
+
+ ndim = 3
+ bounds = np.array([[-np.pi, np.pi], [-np.pi, np.pi], [-np.pi, np.pi]])
+ calc_second_order = False
+ nbase = 1024
+
+ # draw Saltelli samples
+ saltelli_sampler = Saltelli(ndim, bounds, calc_second_order)
+ saltelli_samples = saltelli_sampler.sample(nbase)
+
+ # Evaluate the model to obtain model outputs
+ Y = f(saltelli_samples[:,0], saltelli_samples[:,1], saltelli_samples[:, 2])
+
+ # Compute Sobol' indices
+ sobol_analyzer = SobolAnalyze(ndim, Y, calc_second_order, seed=2)
+ S_res = sobol_analyzer.run()
+
+ print("Estimated first-order Sobol' index: ", S_res['S1'][:,0])
+ print("Analytical solution: [0.314, 0.442, 0]")
+
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Estimated first-order Sobol' index: [0.31430871 0.44545644 0.01325799]
+ Analytical solution: [0.314, 0.442, 0]
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 58-62
+
+Figure below shows the results. The bar plots represent estimated first-order
+and total-effect Sobol' indices, as well as their 95% confidence interval. The
+red stars represent true values of the first-order Sobol' indices.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 63-103
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ name_vars = [r'$x_1$',r'$x_2$',r'$x_3$']
+ x = {'S1':[1,4,7], 'ST':[2,5,8]}
+ color = {'S1':'#69b3a2', 'ST':'#3399e6'}
+ width=0.8
+ capsize=3
+
+ fig, ax1 = plt.subplots(figsize=(4,3))
+
+ # estimated first-order Sobol' indices and their confidence interval
+ ax1.bar(x['S1'], S_res['S1'][:,0], width=width, yerr=S_res['S1'][:,2], color=color['S1'], capsize=capsize)
+
+ ax1.axis([0,9,0,0.7])
+ ax1.set_xticks([1.5,4.5,7.5])
+ ax1.set_xticklabels(name_vars)
+ ax1.tick_params(axis='both',bottom=False)
+ ax1.tick_params(axis="y", labelcolor=color['S1'])
+ ax1.yaxis.grid(True, linestyle='--', which='major', color='grey', alpha=.25)
+ ax1.set_ylabel('First-order index', color=color['S1'])
+
+ # analytical solution for first-order indices
+ ax1.scatter(x['S1'], [0.314, 0.442, 0], s=30, c='red', marker='*')
+
+
+ # estimated total-effect Sobol' indices and their confidence interval
+ ax2 = ax1.twinx()
+ ax2.bar(x['ST'], S_res['ST'][:,0], width=width, yerr=S_res['ST'][:,2], color=color['ST'], capsize=capsize)
+
+ ax2.axis([0,9,0,0.7])
+ ax2.set_xticks([1.5,4.5,7.5])
+ ax2.tick_params(axis='both',bottom=False)
+ ax2.tick_params(axis="y", labelcolor=color['ST'])
+ ax2.yaxis.grid(True, linestyle='--', which='major', color='grey', alpha=.25)
+ ax2.set_ylabel('Total-effect index', color=color['ST'])
+
+ plt.tight_layout()
+
+
+
+
+
+.. image-sg:: /auto_examples/sensitivity/images/sphx_glr_plot_sobol_analyze_001.png
+ :alt: plot sobol analyze
+ :srcset: /auto_examples/sensitivity/images/sphx_glr_plot_sobol_analyze_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 104-109
+
+.. note:: The interaction between one parameter and all other parameters can be
+ quantified by the difference between its first-order and total-effet Sobol'
+ indices. If detailed information on interaction between any two parameters
+ is needed, one can set ``calc_second_order`` to `True`.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 112-128
+
+**Many models of interest in real-world applications are very computationally
+expensive. In that case, we resort to emulation techniques in order to efficiently
+conduct Sobol' analysis.** See :cite:t:`Zhao2021a` for the detailed theory.
+
+Steps of such an analysis are:
+
+ #. Draw input samples for emulator training.
+ #. Evaluate the model at each training input point to obtain training data.
+ #. Train an emulator for the computationally expensive model of interest.
+ #. Draw samples of input for Sobol' analysis.
+ #. Evaluate the emulator at each input sample at step 3 to obtain approximated model outputs.
+ #. Compute Sobol' indices based on approximated model outputs.
+
+Assume that the above Ishigami function is computationally expensive, an emulator-based
+Sobol' analysis can be conducted as follows:
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 129-178
+
+.. code-block:: default
+
+
+ import numpy as np
+ from psimpy.sampler import LHS
+ from psimpy.emulator import ScalarGaSP
+ from psimpy.sampler import Saltelli
+ from psimpy.sensitivity import SobolAnalyze
+
+ # define the model
+ def f(x1,x2,x3):
+ return np.sin(x1) + 7*np.sin(x2)**2 + 0.1*x3**4*np.sin(x1)
+
+ ndim = 3
+ bounds = np.array([[-np.pi, np.pi], [-np.pi, np.pi], [-np.pi, np.pi]])
+ seed = 2
+
+ # draw 100 input samples for emulator training using Latin hypercube sampling
+ lhs_sampler = LHS(ndim, bounds, seed, criterion='maximin', iteration=200)
+ design = lhs_sampler.sample(nsamples=100)
+
+ # evaluate the model at each training input point
+ response = f(design[:,0], design[:,1], design[:,2])
+
+ # train an emulator
+ scalar_gasp = ScalarGaSP(ndim)
+ scalar_gasp.train(design, response)
+
+ # the trained emulator needs to be validated before moving on
+ # here we leave out this step for simplicity...
+
+ # draw samples of input for Sobol' analysis
+ calc_second_order = False
+ nbase = 1024
+
+ saltelli_sampler = Saltelli(ndim, bounds, calc_second_order)
+ saltelli_samples = saltelli_sampler.sample(nbase)
+
+ # evaluate the emulator to obtain approximated model output.
+ # in order to take emulator-induced uncertainty into account, we draw 50 realizations
+ # from the emulator
+ Y = scalar_gasp.sample(saltelli_samples, 50)
+
+ # Compute Sobol' indices based on approximated model outputs
+ sobol_analyzer = SobolAnalyze(ndim, Y, calc_second_order, seed=seed)
+ S_res = sobol_analyzer.run(mode='parallel', max_workers=5)
+
+ print("\n")
+ print("Emulator-based first-order Sobol' index: ", S_res['S1'][:,0])
+ print("Analytical solution: [0.314, 0.442, 0]")
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ The upper bounds of the range parameters are 173,6849 174,6991 174,3157
+ The initial values of range parameters are 3,473697 3,493981 3,486314
+ Start of the optimization 1 :
+ The number of iterations is 14
+ The value of the marginal posterior function is -284,0714
+ Optimized range parameters are 3,183999 2,786247 4,315419
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+ The initial values of range parameters are 1,721972 1,732027 1,728226
+ Start of the optimization 2 :
+ The number of iterations is 14
+ The value of the marginal posterior function is -284,0714
+ Optimized range parameters are 3,183999 2,786247 4,315419
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+
+
+ Emulator-based first-order Sobol' index: [0.29064086 0.46853437 0.01096656]
+ Analytical solution: [0.314, 0.442, 0]
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 179-182
+
+The following figure shows how the results of emulator-based Sobol' analysis
+look like.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 182-213
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+
+ name_vars = [r'$x_1$',r'$x_2$',r'$x_3$']
+ x = {'S1':[1,4,7], 'ST':[2,5,8]}
+ color = {'S1':'#69b3a2', 'ST':'#3399e6'}
+ width=0.8
+ capsize=3
+
+ fig, ax1 = plt.subplots(figsize=(4,3))
+
+ ax1.bar(x['S1'], S_res['S1'][:,0], width=width, yerr=S_res['S1'][:,2], color=color['S1'], capsize=capsize)
+ ax1.axis([0,9,0,0.7])
+ ax1.set_xticks([1.5,4.5,7.5])
+ ax1.set_xticklabels(name_vars)
+ ax1.tick_params(axis='both',bottom=False)
+ ax1.tick_params(axis="y", labelcolor=color['S1'])
+ ax1.yaxis.grid(True, linestyle='--', which='major', color='grey', alpha=.25)
+ ax1.set_ylabel('First-order index', color=color['S1'])
+
+ ax1.scatter(x['S1'], [0.314, 0.442, 0], s=30, c='red', marker='*')
+
+ ax2 = ax1.twinx()
+ ax2.bar(x['ST'], S_res['ST'][:,0], width=width, yerr=S_res['ST'][:,2], color=color['ST'], capsize=capsize)
+ ax2.axis([0,9,0,0.7])
+ ax2.set_xticks([1.5,4.5,7.5])
+ ax2.tick_params(axis='both',bottom=False)
+ ax2.tick_params(axis="y", labelcolor=color['ST'])
+ ax2.yaxis.grid(True, linestyle='--', which='major', color='grey', alpha=.25)
+ ax2.set_ylabel('Total-effect index', color=color['ST'])
+
+ plt.tight_layout()
+
+
+.. image-sg:: /auto_examples/sensitivity/images/sphx_glr_plot_sobol_analyze_002.png
+ :alt: plot sobol analyze
+ :srcset: /auto_examples/sensitivity/images/sphx_glr_plot_sobol_analyze_002.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 16.499 seconds)
+
+
+.. _sphx_glr_download_auto_examples_sensitivity_plot_sobol_analyze.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_sobol_analyze.py <plot_sobol_analyze.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_sobol_analyze.ipynb <plot_sobol_analyze.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/sensitivity/sg_execution_times.rst.txt b/docs/source/_build/html/_sources/auto_examples/sensitivity/sg_execution_times.rst.txt
new file mode 100644
index 0000000..77cb35d
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/sensitivity/sg_execution_times.rst.txt
@@ -0,0 +1,12 @@
+
+:orphan:
+
+.. _sphx_glr_auto_examples_sensitivity_sg_execution_times:
+
+Computation times
+=================
+**00:16.499** total execution time for **auto_examples_sensitivity** files:
+
++---------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_sensitivity_plot_sobol_analyze.py` (``plot_sobol_analyze.py``) | 00:16.499 | 0.0 MB |
++---------------------------------------------------------------------------------------------+-----------+--------+
diff --git a/docs/source/_build/html/_sources/auto_examples/simulator/index.rst.txt b/docs/source/_build/html/_sources/auto_examples/simulator/index.rst.txt
new file mode 100644
index 0000000..c994f61
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/simulator/index.rst.txt
@@ -0,0 +1,112 @@
+:orphan:
+
+Simulator
+=========
+
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnails">
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Mass Point Model">
+
+.. only:: html
+
+ .. image:: /auto_examples/simulator/images/thumb/sphx_glr_plot_mass_point_model_thumb.png
+ :alt: Mass Point Model
+
+ :ref:`sphx_glr_auto_examples_simulator_plot_mass_point_model.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Mass Point Model</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="RunSimulator: Mass Point Model">
+
+.. only:: html
+
+ .. image:: /auto_examples/simulator/images/thumb/sphx_glr_plot_run_mass_point_model_thumb.png
+ :alt: RunSimulator: Mass Point Model
+
+ :ref:`sphx_glr_auto_examples_simulator_plot_run_mass_point_model.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">RunSimulator: Mass Point Model</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="Ravaflow24 Mixture Model">
+
+.. only:: html
+
+ .. image:: /auto_examples/simulator/images/thumb/sphx_glr_plot_ravaflow24_thumb.png
+ :alt: Ravaflow24 Mixture Model
+
+ :ref:`sphx_glr_auto_examples_simulator_plot_ravaflow24.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">Ravaflow24 Mixture Model</div>
+ </div>
+
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbcontainer" tooltip="RunSimulator: Ravaflow24 Mixture Model">
+
+.. only:: html
+
+ .. image:: /auto_examples/simulator/images/thumb/sphx_glr_plot_run_ravaflow24_thumb.png
+ :alt: RunSimulator: Ravaflow24 Mixture Model
+
+ :ref:`sphx_glr_auto_examples_simulator_plot_run_ravaflow24.py`
+
+.. raw:: html
+
+ <div class="sphx-glr-thumbnail-title">RunSimulator: Ravaflow24 Mixture Model</div>
+ </div>
+
+
+.. raw:: html
+
+ </div>
+
+
+.. toctree::
+ :hidden:
+
+ /auto_examples/simulator/plot_mass_point_model
+ /auto_examples/simulator/plot_run_mass_point_model
+ /auto_examples/simulator/plot_ravaflow24
+ /auto_examples/simulator/plot_run_ravaflow24
+
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-gallery
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download all examples in Python source code: simulator_python.zip </auto_examples/simulator/simulator_python.zip>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download all examples in Jupyter notebooks: simulator_jupyter.zip </auto_examples/simulator/simulator_jupyter.zip>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/simulator/plot_mass_point_model.rst.txt b/docs/source/_build/html/_sources/auto_examples/simulator/plot_mass_point_model.rst.txt
new file mode 100644
index 0000000..ce38db1
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/simulator/plot_mass_point_model.rst.txt
@@ -0,0 +1,259 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/simulator/plot_mass_point_model.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_simulator_plot_mass_point_model.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_simulator_plot_mass_point_model.py:
+
+
+Mass Point Model
+================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-12
+
+This example shows how to simulate the movement of a masspoint on a topography
+using :class:`.MassPointModel`.
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 15-16
+
+First, import the class :class:`.MassPointModel` and create an instance by
+
+.. GENERATED FROM PYTHON SOURCE LINES 17-22
+
+.. code-block:: default
+
+
+ from psimpy.simulator import MassPointModel
+
+ mpm = MassPointModel()
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 23-31
+
+Required inputs for a simulation using :class:`.MassPointModel` include:
+ 1. topographic data: digital elevation model (in `ESRI ascii` format)
+ 2. friction coefficients: coulomb friction and turbulent friction coefficients
+ 3. initial state: initial location and initial velocity of the masspoint
+ 4. computational parameters: such as time step, end time, etc.
+
+The synthetic topography ``synthetic_topo.asc`` is used here for illustration.
+It is located at the `tests/data/` folder.
+
+.. GENERATED FROM PYTHON SOURCE LINES 32-41
+
+.. code-block:: default
+
+
+ import os
+ import linecache
+ import numpy as np
+
+ dir_data = os.path.abspath('../../../tests/data/')
+ elevation = os.path.join(dir_data, 'synthetic_topo.asc')
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 42-44
+
+.. note:: You may need to modify ``dir_data`` according to where you save
+ ``synthetic_topo.asc`` on your local machine.
+
+.. GENERATED FROM PYTHON SOURCE LINES 46-49
+
+We can load the elevation data and visulize it. The figure below shows how
+the topography looks like, as well as the initial location of the masspoint
+(noted by the red dot).
+
+.. GENERATED FROM PYTHON SOURCE LINES 49-90
+
+.. code-block:: default
+
+
+ header = [linecache.getline(elevation, i) for i in range(1,6)]
+ header_values = [float(h.split()[-1].strip()) for h in header]
+ ncols, nrows, xll, yll, cellsize = header_values
+ ncols = int(ncols)
+ nrows = int(nrows)
+
+ x_values = np.arange(xll, xll+(cellsize*ncols), cellsize)
+ y_values = np.arange(yll, yll+(cellsize*nrows), cellsize)
+
+ z_values = np.loadtxt(elevation, skiprows=6)
+ z_values = np.rot90(np.transpose(z_values))
+
+ # initial location
+ x0 = 200
+ y0 = 2000
+ z0 = z_values[0, int(x0/cellsize)]
+
+ import matplotlib.pyplot as plt
+
+ fig, ax = plt.subplots(2, 1, figsize=(10,6), height_ratios=[3,2])
+
+ fig0 = ax[0].contourf(x_values, y_values, z_values, levels=20)
+ ax[0].set_xlabel('x')
+ ax[0].set_ylabel('y')
+ ax[0].set_title('synthetic topography')
+ ax[0].scatter(x0, y0, s=10, c='r', marker='o')
+ cbar = plt.colorbar(fig0, ax=ax[0], format='%d', orientation='horizontal',
+ fraction=0.1, pad=0.2)
+ cbar.ax.set_ylabel('z')
+
+
+ ax[1].plot(x_values, z_values[0, :])
+ ax[1].scatter(x0, z0, s=10, c='r', marker='o')
+ ax[1].set_xlabel('x')
+ ax[1].set_ylabel('z')
+ ax[1].set_xlim(0, 5000)
+ ax[1].set_title('cross section at any y')
+
+ plt.tight_layout()
+
+
+
+
+.. image-sg:: /auto_examples/simulator/images/sphx_glr_plot_mass_point_model_001.png
+ :alt: synthetic topography, cross section at any y
+ :srcset: /auto_examples/simulator/images/sphx_glr_plot_mass_point_model_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 91-92
+
+We set the friction coefficients as
+
+.. GENERATED FROM PYTHON SOURCE LINES 92-96
+
+.. code-block:: default
+
+
+ mu = 0.15
+ xi = 1000
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 97-101
+
+Given above topography, initial location, and friction coefficients, we can
+call the :py:meth:`.MassPointModel.run` method to perform a simulation. Other
+parameters are set to their default values. (we suppress raised warnings )
+(other parameters are set to their default values).
+
+.. GENERATED FROM PYTHON SOURCE LINES 101-107
+
+.. code-block:: default
+
+
+ import warnings
+
+ warnings.filterwarnings("ignore")
+ output = mpm.run(elevation=elevation, coulomb_friction=mu, turbulent_friction=xi, x0=x0, y0=y0)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 108-110
+
+The simulation returns time history of the mass point's location and velocity.
+Following plots show the simulation results.
+
+.. GENERATED FROM PYTHON SOURCE LINES 110-129
+
+.. code-block:: default
+
+
+ fig, ax = plt.subplots(3, 1, figsize=(10,6))
+
+ ax[0].set_xlabel('time (s)')
+ ax[0].set_ylabel('x')
+ ax[0].set_title("x-t plot")
+ ax[0].plot(output[:,0], output[:,1])
+
+ ax[1].set_xlabel('time (s)')
+ ax[1].set_ylabel('velocity')
+ ax[1].set_title("v-t plot")
+ ax[1].plot(output[:,0], output[:,5])
+
+ ax[2].set_xlabel('x')
+ ax[2].set_ylabel('velocity')
+ ax[2].set_title("x-v plot")
+ ax[2].plot(output[:,1], output[:,5])
+
+ plt.tight_layout()
+
+
+
+.. image-sg:: /auto_examples/simulator/images/sphx_glr_plot_mass_point_model_002.png
+ :alt: x-t plot, v-t plot, x-v plot
+ :srcset: /auto_examples/simulator/images/sphx_glr_plot_mass_point_model_002.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 1.550 seconds)
+
+
+.. _sphx_glr_download_auto_examples_simulator_plot_mass_point_model.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_mass_point_model.py <plot_mass_point_model.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_mass_point_model.ipynb <plot_mass_point_model.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/simulator/plot_ravaflow24.rst.txt b/docs/source/_build/html/_sources/auto_examples/simulator/plot_ravaflow24.rst.txt
new file mode 100644
index 0000000..4737357
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/simulator/plot_ravaflow24.rst.txt
@@ -0,0 +1,304 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/simulator/plot_ravaflow24.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_simulator_plot_ravaflow24.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_simulator_plot_ravaflow24.py:
+
+
+Ravaflow24 Mixture Model
+========================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-12
+
+This example shows how to simulate mass flow on a topography using
+:class:`.Ravaflow24Mixture`.
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 15-16
+
+First, import the class :class:`.Ravaflow24Mixture` by
+
+.. GENERATED FROM PYTHON SOURCE LINES 17-20
+
+.. code-block:: default
+
+
+ from psimpy.simulator import Ravaflow24Mixture
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 21-24
+
+To create an instance of this class, we must specify the parameter ``dir_sim``.
+It represents the directory in which output files generated by r.avaflow will be
+saved.
+
+.. GENERATED FROM PYTHON SOURCE LINES 25-35
+
+.. code-block:: default
+
+
+ import os
+
+ # Here we create a folder called `temp_Ravaflow24Mixture_example` to save output
+ # files generated by r.avaflow
+ cwd = os.getcwd()
+ if not os.path.exists('temp_Ravaflow24Mixture_example'):
+ os.mkdir('temp_Ravaflow24Mixture_example')
+ dir_sim = os.path.join(cwd, 'temp_Ravaflow24Mixture_example')
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 36-40
+
+Now, we can create an instance of :class:`.Ravaflow24Mixture` by given
+``dir_sim`` and leaving other parameters to their default values
+(Other parameters include ``time_step``, ``time_end``, ``curvature_control``,
+``entrainment_control``, ``stopping_control``, etc.)
+
+.. GENERATED FROM PYTHON SOURCE LINES 41-44
+
+.. code-block:: default
+
+
+ voellmy_model = Ravaflow24Mixture(dir_sim)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 45-57
+
+To run a simulation using above `voellmy_model`, one needs to specify
+ 1. ``elevation`` – Name of elevation raster file (including its path).
+ 2. ``hrelease`` – Name of release height raster file (including its path).
+ 3. ``prefix`` – Prefix required by r.avaflow to name output files.
+ 4. If ``elevation`` is not a georeferenced file which can be used to create
+ a `GRASS Location`, ``EPSG`` must be provided.
+
+Other optional parameters include ``internal_friction``, ``basal_friction``,
+``turbulent_friction``, ``entrainment_coef`` etc.
+
+The synthetic topography ``synthetic_topo.tif`` and release mass ``synthetic_rel.tif``
+are used here for illustration. They are located at the `/tests/data/` folder.
+
+.. GENERATED FROM PYTHON SOURCE LINES 58-64
+
+.. code-block:: default
+
+
+ dir_data = os.path.abspath('../../../tests/data/')
+ elevation = os.path.join(dir_data, 'synthetic_topo.tif')
+ hrelease = os.path.join(dir_data, 'synthetic_rel.tif')
+ prefix = 'synthetic'
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 65-67
+
+.. note:: You may need to modify ``dir_data`` according to where you save them
+ on your local machine.
+
+.. GENERATED FROM PYTHON SOURCE LINES 70-73
+
+Given above inputs, we can create a `GRASS Location` and a shell file by calling
+:py:meth:`.Ravaflow24Mixture.preprocess` method, and run the simulation using
+:py:meth:`.Ravaflow24Mixture.run` method
+
+.. GENERATED FROM PYTHON SOURCE LINES 74-79
+
+.. code-block:: default
+
+
+ grass_location, sh_file = voellmy_model.preprocess(
+ prefix=prefix, elevation=elevation, hrelease=hrelease, EPSG='2326')
+ voellmy_model.run(grass_location, sh_file)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 80-86
+
+Once the simulation is finished, the results are located in the folder
+`/dir_sim/your_prefix_results`. In this case, they are located in
+`/temp_Ravaflow24Mixture_example/synthetic_results`. We can extract desired
+outputs using respective method and visualize them.
+For example, we can extract the overall impact area, maximum flow velocity, or
+maximum flow velocity at specific locations:
+
+.. GENERATED FROM PYTHON SOURCE LINES 87-104
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ # overall impact area
+ impact_area = voellmy_model.extract_impact_area(prefix)
+ print(f"Impact_area is {impact_area} m^2")
+
+ # overall maximum flow velocity
+ v_mmax = voellmy_model.extract_qoi_max(prefix, 'v', aggregate=True)
+ print(f"Maximum flow velocity is {v_mmax} m/s^2")
+
+ # maximum flow velocity at specific locations
+ loc = np.array([[500, 2000], [1500, 2000]])
+ v_max_loc = voellmy_model.extract_qoi_max_loc(prefix, loc, 'v')
+ print(f"Maximum flow velocity at location {loc[0]} is {v_max_loc[0]} m/s^2")
+ print(f"Maximum flow velocity at location {loc[1]} is {v_max_loc[1]} m/s^2")
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Impact_area is 1394800.0 m^2
+ Maximum flow velocity is 41.39 m/s^2
+ Maximum flow velocity at location [ 500 2000] is 34.264 m/s^2
+ Maximum flow velocity at location [1500 2000] is 36.689 m/s^2
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 105-106
+
+We can visulize point-wise maximum flow height and velocity using heatmaps:
+
+.. GENERATED FROM PYTHON SOURCE LINES 107-144
+
+.. code-block:: default
+
+
+ import matplotlib.pyplot as plt
+ import linecache
+
+ # extract head information of result ascii files and compute x and y coordinates
+ hmax_asc = os.path.join(dir_sim, f'{prefix}_results', f'{prefix}_ascii', f'{prefix}_hflow_max.asc')
+
+ header = [linecache.getline(hmax_asc, i) for i in range(1,6)]
+ header_values = [float(h.split()[-1].strip()) for h in header]
+ ncols, nrows, xll, yll, cellsize = header_values
+ ncols = int(ncols)
+ nrows = int(nrows)
+
+ x = np.arange(xll, xll+(cellsize*ncols), cellsize)
+ y = np.arange(yll, yll+(cellsize*nrows), cellsize)
+
+ # point-wise maximum flow height
+ h_max = voellmy_model.extract_qoi_max(prefix, 'h', aggregate=False)
+ # point-wise maximum flow velocity
+ v_max = voellmy_model.extract_qoi_max(prefix, 'v', aggregate=False)
+
+ fig, ax = plt.subplots(2, 1, figsize=(10,6))
+
+ fig0 = ax[0].contourf(x, y, h_max, levels=20)
+ ax[0].set_xlabel('x')
+ ax[0].set_ylabel('y')
+ ax[0].set_title('maximum flow height' + r' $(m/s)$')
+ cbar0 = plt.colorbar(fig0, ax=ax[0], format='%.1f', orientation='vertical', fraction=0.1)
+
+ fig1 = ax[1].contourf(x, y, v_max, levels=20)
+ ax[1].set_xlabel('x')
+ ax[1].set_ylabel('y')
+ ax[1].set_title('maximum flow velocity' + r' ($m/s^2$)')
+ cbar1 = plt.colorbar(fig1, ax=ax[1], format='%.1f', orientation='vertical', fraction=0.1)
+
+ plt.tight_layout()
+
+
+
+
+.. image-sg:: /auto_examples/simulator/images/sphx_glr_plot_ravaflow24_001.png
+ :alt: maximum flow height $(m/s)$, maximum flow velocity ($m/s^2$)
+ :srcset: /auto_examples/simulator/images/sphx_glr_plot_ravaflow24_001.png
+ :class: sphx-glr-single-img
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 145-146
+
+Here we delete the folder `temp_Ravaflow24Mixture_example` and all files therein.
+
+.. GENERATED FROM PYTHON SOURCE LINES 147-151
+
+.. code-block:: default
+
+
+ import shutil
+
+ shutil.rmtree(dir_sim)
+
+
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 5 minutes 56.157 seconds)
+
+
+.. _sphx_glr_download_auto_examples_simulator_plot_ravaflow24.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_ravaflow24.py <plot_ravaflow24.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_ravaflow24.ipynb <plot_ravaflow24.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/simulator/plot_run_mass_point_model.rst.txt b/docs/source/_build/html/_sources/auto_examples/simulator/plot_run_mass_point_model.rst.txt
new file mode 100644
index 0000000..c969e75
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/simulator/plot_run_mass_point_model.rst.txt
@@ -0,0 +1,376 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/simulator/plot_run_mass_point_model.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_simulator_plot_run_mass_point_model.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_simulator_plot_run_mass_point_model.py:
+
+
+RunSimulator: Mass Point Model
+==============================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-12
+
+This example shows how to run multiple MassPointModel simulations in serial
+and parallelly using :class:`.RunSimulator`.
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 16-18
+
+First, we import the class :class:`.MassPointModel` and define a `simulator`
+based on :py:meth:`.MassPointModel.run` method.
+
+.. GENERATED FROM PYTHON SOURCE LINES 19-26
+
+.. code-block:: default
+
+
+ from psimpy.simulator import MassPointModel
+
+ mpm = MassPointModel()
+ mpm_simulator = mpm.run
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 27-39
+
+This simulator takes required input data, solves the mass point model, and
+returns time history of the mass point's location and velocity
+(see :py:meth:`.MassPointModel.run`).
+
+Required inputs for `mpm_simulator` include:
+ 1. topographic data: digital elevation model (in `ESRI ascii` format)
+ 2. friction coefficients: coulomb friction and turbulent friction coefficients
+ 3. initial state: initial location and initial velocity of the masspoint
+ 4. computational parameters: such as time step, end time, etc.
+
+The synthetic topography ``synthetic_topo.asc`` is used here for illustration.
+It is located at the `/tests/data/` folder.
+
+.. GENERATED FROM PYTHON SOURCE LINES 40-46
+
+.. code-block:: default
+
+
+ import os
+
+ dir_data = os.path.abspath('../../../tests/data/')
+ elevation = os.path.join(dir_data, 'synthetic_topo.asc')
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 47-49
+
+.. note:: You may need to modify ``dir_data`` according to where you save
+ ``synthetic_topo.asc`` on your local machine.
+
+.. GENERATED FROM PYTHON SOURCE LINES 51-57
+
+For this example, we are going to run multiple simulations at different values
+of coulomb friction coefficient (``coulomb_friction``) and turbulent friction
+coefficient (``turbulent_friction``) while keep other inputs fixed. Namely,
+``coulomb_friction`` and ``turbulent_friction`` are variable input parameters.
+All other inputs are fixed input parameters.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 58-74
+
+.. code-block:: default
+
+
+ import numpy as np
+ import itertools
+
+ # Variable iput parameters are defined as a list of strings. Their values will be
+ # passed as a numpy array (var_samples) when run simulations.
+ var_inp_parameter = ['coulomb_friction', 'turbulent_friction']
+ coulomb_friction = np.arange(0.1, 0.31, 0.1)
+ turbulent_friction = np.arange(500, 2001, 400)
+ var_samples = np.array(
+ [x for x in itertools.product(coulomb_friction, turbulent_friction)])
+ print("Number of variable input samples are: ", len(var_samples))
+
+ # Fixed input parameters are defined as a dictionary. Their values are given.
+ fix_inp = {'elevation': elevation, 'x0': 200, 'y0': 2000, 'tend': 50}
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Number of variable input samples are: 12
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 75-78
+
+.. note:: Parameters of `mpm_simulator` which are not included in
+ ``var_inp_parameter`` and ``fix_inp``, such as ``ux0``, ``uy0`` and
+ ``curvature``, will automatically take their default values.
+
+.. GENERATED FROM PYTHON SOURCE LINES 80-83
+
+We may want to save outputs returned by `mpm_simulator` at each simulation for
+later inspection or processing. In that case, we need to define ``dir_out`` and
+set ``save_out`` as `True`.
+
+.. GENERATED FROM PYTHON SOURCE LINES 84-95
+
+.. code-block:: default
+
+
+ import os
+
+ # Here we create a folder called `temp_run_MassPointModel_example` to save outputs
+ # returned at each simulation.
+ cwd = os.getcwd()
+ if not os.path.exists('temp_run_MassPointModel_example'):
+ os.mkdir('temp_run_MassPointModel_example')
+ dir_out = os.path.join(cwd, 'temp_run_MassPointModel_example')
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 96-97
+
+Now we can define an object of :class:`.RunSimulator` by
+
+.. GENERATED FROM PYTHON SOURCE LINES 98-104
+
+.. code-block:: default
+
+
+ from psimpy.simulator import RunSimulator
+
+ run_mpm_simulator = RunSimulator(mpm_simulator, var_inp_parameter, fix_inp,
+ dir_out=dir_out, save_out=True)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 105-108
+
+.. note:: Since outputs of each simulation are saved in the same folder ``dir_out``,
+ we need to give each file a unique name in order to avoid conflict. This is
+ realized by defining ``prefixes``.
+
+.. GENERATED FROM PYTHON SOURCE LINES 108-113
+
+.. code-block:: default
+
+
+ serial_prefixes = ["serial"+str(i) for i in range(len(var_samples))]
+ parallel_prefixes = ["parallel"+str(i) for i in range(len(var_samples))]
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 114-117
+
+Using :py:meth:`.RunSimulator.serial_run` method or
+:py:meth:`.RunSimulator.parallel_run` method, we can run the simulations
+in serial or parallelly.
+
+.. GENERATED FROM PYTHON SOURCE LINES 118-136
+
+.. code-block:: default
+
+
+ import time
+
+ start = time.time()
+ run_mpm_simulator.serial_run(var_samples=var_samples, prefixes=serial_prefixes)
+ serial_time = time.time() - start
+ serial_output = run_mpm_simulator.outputs
+
+ start = time.time()
+ # max_workers controls maximum number of tasks running in parallel
+ run_mpm_simulator.parallel_run(var_samples, prefixes=parallel_prefixes, max_workers=4)
+ parallel_time = time.time() - start
+ parallel_output = run_mpm_simulator.outputs
+
+ print("Serial run time: ", serial_time)
+ print("Parallel run time: ", parallel_time)
+
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Serial run time: 1.1941184997558594
+ Parallel run time: 0.4859800338745117
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 137-139
+
+Once a simulation is done, its output is saved to ``dir_out``. All output files
+generated by above simulations are as follows:
+
+.. GENERATED FROM PYTHON SOURCE LINES 140-144
+
+.. code-block:: default
+
+
+ os.listdir(dir_out)
+
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+
+ ['serial0_output.npy', 'serial9_output.npy', 'serial11_output.npy', 'parallel10_output.npy', 'parallel0_output.npy', 'parallel5_output.npy', 'parallel11_output.npy', 'serial1_output.npy', 'serial4_output.npy', 'parallel7_output.npy', 'serial5_output.npy', 'serial8_output.npy', 'parallel6_output.npy', 'parallel4_output.npy', 'parallel2_output.npy', 'serial3_output.npy', 'serial6_output.npy', 'parallel3_output.npy', 'serial2_output.npy', 'parallel9_output.npy', 'serial7_output.npy', 'parallel1_output.npy', 'parallel8_output.npy', 'serial10_output.npy']
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 145-147
+
+Once all simulations are done, their outputs can also be accessed by
+:py:attr:`.RunSimulator.outputs` attribute, which is a list.
+
+.. GENERATED FROM PYTHON SOURCE LINES 148-157
+
+.. code-block:: default
+
+
+ # Here we print the maximum velocity of each simulation in the parallel run.
+ max_v = [np.max(output[:,5]) for output in parallel_output]
+ print(
+ f"Maximum velocity of {parallel_prefixes[0]} to {parallel_prefixes[-1]} are: ",
+ '\n',
+ max_v)
+
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ Maximum velocity of parallel0 to parallel11 are:
+ [16.55505557211892, 22.014114896121015, 26.245546886730782, 29.78487303612985, 15.34198274528081, 20.371864007499866, 24.249957243412815, 27.485487855011687, 14.031665649204944, 18.589324449452967, 22.092803891476187, 24.99727320816709]
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 158-164
+
+.. warning:: If one simulation failed due to whatever reason, the error massage
+ will be printed to the screen but other simulations will continue. In that
+ case, the output file of failed simulation will not be writted to ``dir_out``.
+ Also, the element of :py:attr:`.RunSimulator.outputs` corresponding to that
+ simulation will be a string representing the error message, instead of a
+ numpy array.
+
+.. GENERATED FROM PYTHON SOURCE LINES 167-168
+
+Here we delete the folder `temp_run_MassPointModel_example` and all files therein.
+
+.. GENERATED FROM PYTHON SOURCE LINES 169-185
+
+.. code-block:: default
+
+
+ import shutil
+
+ shutil.rmtree(dir_out)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 0 minutes 1.687 seconds)
+
+
+.. _sphx_glr_download_auto_examples_simulator_plot_run_mass_point_model.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_run_mass_point_model.py <plot_run_mass_point_model.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_run_mass_point_model.ipynb <plot_run_mass_point_model.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/simulator/plot_run_ravaflow24.rst.txt b/docs/source/_build/html/_sources/auto_examples/simulator/plot_run_ravaflow24.rst.txt
new file mode 100644
index 0000000..b727903
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/simulator/plot_run_ravaflow24.rst.txt
@@ -0,0 +1,444 @@
+
+.. DO NOT EDIT.
+.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
+.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
+.. "auto_examples/simulator/plot_run_ravaflow24.py"
+.. LINE NUMBERS ARE GIVEN BELOW.
+
+.. only:: html
+
+ .. note::
+ :class: sphx-glr-download-link-note
+
+ Click :ref:`here <sphx_glr_download_auto_examples_simulator_plot_run_ravaflow24.py>`
+ to download the full example code
+
+.. rst-class:: sphx-glr-example-title
+
+.. _sphx_glr_auto_examples_simulator_plot_run_ravaflow24.py:
+
+
+RunSimulator: Ravaflow24 Mixture Model
+======================================
+
+.. GENERATED FROM PYTHON SOURCE LINES 8-11
+
+This example shows how to run multiple Ravaflow24Mixture simulations in serial
+and parallelly using :class:`.RunSimulator`.
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 14-15
+
+First, we import the class :class:`.Ravaflow24Mixture`.
+
+.. GENERATED FROM PYTHON SOURCE LINES 16-19
+
+.. code-block:: default
+
+
+ from psimpy.simulator import Ravaflow24Mixture
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 20-23
+
+To create an instance of this class, we must specify the parameter ``dir_sim``.
+It represents the directory in which output files generated by r.avaflow will be
+saved.
+
+.. GENERATED FROM PYTHON SOURCE LINES 24-34
+
+.. code-block:: default
+
+
+ import os
+
+ # Here we create a folder called `temp1_run_Ravaflow24Mixture_example` to save
+ # output files generated by r.avaflow
+ cwd = os.getcwd()
+ if not os.path.exists('temp1_run_Ravaflow24Mixture_example'):
+ os.mkdir('temp1_run_Ravaflow24Mixture_example')
+ dir_sim = os.path.join(cwd, 'temp1_run_Ravaflow24Mixture_example')
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 35-38
+
+Given ``dir_sim``, we can create an instance of :class:`.Ravaflow24Mixture`.
+To reduce simulation time, we set ``time_end`` to :math:`50`. Other parameters
+are set to their default values.
+
+.. GENERATED FROM PYTHON SOURCE LINES 39-42
+
+.. code-block:: default
+
+
+ voellmy_model = Ravaflow24Mixture(dir_sim, time_end=50)
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 43-45
+
+The `simulator` in this example is defined based on ``voellmy_model`` as follows.
+It takes required inputs and returns the overall impact area.
+
+.. GENERATED FROM PYTHON SOURCE LINES 46-59
+
+.. code-block:: default
+
+
+ import numpy as np
+
+ def simulator(prefix, elevation, hrelease, basal_friction, turbulent_friction, EPSG):
+ """Preprocess required inputs, run simulation, and return output as a numpy array."""
+ grass_location, sh_file = voellmy_model.preprocess(prefix=prefix, elevation=elevation,
+ hrelease=hrelease, basal_friction=basal_friction, turbulent_friction=turbulent_friction, EPSG=EPSG)
+ voellmy_model.run(grass_location, sh_file) # run this line, r.avaflow will write outputs to dir_sim
+ impact_area = voellmy_model.extract_impact_area(prefix)
+
+ return np.array([impact_area]) # define dir_out and set save_out to True, returned numpy array will be saved to dir_out
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 60-63
+
+To demonstrate how to run multiple simulations using :class:`.RunSimulator`,
+we choose ``basal_friction`` and ``turbulent_friction`` as variable input
+parameters. Each has two different values, leading to four simulations.
+
+.. GENERATED FROM PYTHON SOURCE LINES 64-74
+
+.. code-block:: default
+
+
+ import itertools
+
+ var_inp_parameter = ['basal_friction', 'turbulent_friction']
+ basal_friction = [20, 30]
+ turbulent_friction = [3, 4]
+ var_samples = np.array(
+ [x for x in itertools.product(basal_friction, turbulent_friction)])
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 75-78
+
+Other parameters of the `simulator`, including ``elevation``, ``hrelease``, and
+``EPSG`` are treated as fixed input. It means that their values are the same in
+all simulations.
+
+.. GENERATED FROM PYTHON SOURCE LINES 79-85
+
+.. code-block:: default
+
+
+ dir_data = os.path.abspath('../../../tests/data/')
+ elevation = os.path.join(dir_data, 'synthetic_topo.tif')
+ hrelease = os.path.join(dir_data, 'synthetic_rel.tif')
+ fix_inp = {'elevation': elevation, 'hrelease': hrelease, 'EPSG': '2326'}
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 86-91
+
+The parameter ``prefix`` of the `simulator` is special. It is not involved in
+the computational model, but only used to name output files generated by r.avaflow.
+Such parameter is not defined in ``var_inp_parameter`` or ``fix_inp`` of
+:class:`.RunSimulator`. Instead, we use a seperate parameter, called ``o_parameter``
+for this purpose.
+
+.. GENERATED FROM PYTHON SOURCE LINES 92-96
+
+.. code-block:: default
+
+
+ o_parameter = 'prefix'
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 97-100
+
+We may want to save outputs returned by the `simulator` at each simulation for
+later inspection or processing. In that case, we need to define ``dir_out`` and
+set ``save_out`` as `True`.
+
+.. GENERATED FROM PYTHON SOURCE LINES 101-109
+
+.. code-block:: default
+
+
+ import os
+
+ cwd = os.getcwd()
+ if not os.path.exists('temp2_run_Ravaflow24Mixture_example'):
+ os.mkdir('temp2_run_Ravaflow24Mixture_example')
+ dir_out = os.path.join(cwd, 'temp2_run_Ravaflow24Mixture_example')
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 110-132
+
+.. note:: Please note that ``dir_out`` and ``dir_sim`` are two different
+ parameters for different purposes.
+
+ - ``dir_out`` is a parameter of :class:`.RunSimulator`. The `simulator`
+ returns output of interest as a numpy array for each simulation. If
+ we want to save this returned output of interest (here impact area)
+ to our local machine, we need to specify the value of ``dir_out`` which
+ represents the directory in which output of interest will be saved and
+ set ``save_out`` to `True`. Otherwise, we do not need ``dir_out`` and
+ leave ``save_out`` to `False`.
+ - ``dir_sim`` is a parameter of :class:`.Ravaflow24Mixture`. :class:`.Ravaflow24Mixture`
+ relies on the third party software r.avaflow 2.4. When we call
+ :py:meth:`.Ravaflow24Mixture.run` in the function body of above `simulator`,
+ r.avaflow 2.4 will be run and it generates output files. The value of
+ ``dir_sim`` specifies the directory in which output files generated by
+ r.avaflow 2.4 are going to be saved.
+
+ The value of ``dir_out`` and ``dir_sim`` can be the same if file names
+ have no conflict. We recommend to keep them seperate. In addition, if the function
+ body of the `simulator` does not write files to disk, ``dir_sim`` is not required. Our
+ :class:`.MassPointModel` is an example. Similarly, if we do not want to save
+ returned numpy array of the `simulator`, ``dir_out`` is not needed.
+
+.. GENERATED FROM PYTHON SOURCE LINES 135-136
+
+Now we can define an object of :class:`.RunSimulator` by
+
+.. GENERATED FROM PYTHON SOURCE LINES 137-144
+
+.. code-block:: default
+
+
+ from psimpy.simulator import RunSimulator
+
+ run_simulator = RunSimulator(simulator, var_inp_parameter, fix_inp, o_parameter,
+ dir_out, save_out=True)
+
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 145-148
+
+Before running simulations, we need to specify values of ``prefixes`` which
+will be used to name files generated by each r.avaflow simulation and
+each returned numpy array of `simulator`.
+
+.. GENERATED FROM PYTHON SOURCE LINES 149-153
+
+.. code-block:: default
+
+
+ serial_prefixes = ["serial"+str(i) for i in range(len(var_samples))]
+ parallel_prefixes = ["parallel"+str(i) for i in range(len(var_samples))]
+
+
+
+
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 154-157
+
+Now we can use :py:meth:`.RunSimulator.serial_run` method or
+:py:meth:`.RunSimulator.parallel_run` method to run the simulations
+in serial or parallelly.
+
+.. GENERATED FROM PYTHON SOURCE LINES 158-181
+
+.. code-block:: default
+
+
+ import time
+
+ start = time.time()
+ run_simulator.serial_run(var_samples=var_samples, prefixes=serial_prefixes)
+ serial_time = time.time() - start
+ serial_output = run_simulator.outputs
+ print(f"serial_output: {serial_output}")
+
+ start = time.time()
+ # append setting to True means that simulation outputs of parallel run will be
+ # appended to above serial run outputs
+ run_simulator.parallel_run(var_samples, prefixes=parallel_prefixes,
+ max_workers=2, append=True)
+ parallel_time = time.time() - start
+
+ parallel_output = run_simulator.outputs[len(var_samples):]
+ print(f"parallel_output: {parallel_output}")
+
+
+ print("Serial run time: ", serial_time)
+ print("Parallel run time: ", parallel_time)
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+ serial_output: [array([1008000.]), array([1520800.]), array([674000.]), array([869600.])]
+ parallel_output: [array([1008000.]), array([1520800.]), array([674000.]), array([869600.])]
+ Serial run time: 110.85352993011475
+ Parallel run time: 62.39644193649292
+
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 182-183
+
+All output files returned by the `simulator` are
+
+.. GENERATED FROM PYTHON SOURCE LINES 184-187
+
+.. code-block:: default
+
+
+ os.listdir(dir_out)
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+
+ ['serial0_output.npy', 'parallel0_output.npy', 'serial1_output.npy', 'parallel2_output.npy', 'serial3_output.npy', 'parallel3_output.npy', 'serial2_output.npy', 'parallel1_output.npy']
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 188-190
+
+All output files/directories generated by r.avaflow simulations
+(including files/directories generated during preprocessing) are
+
+.. GENERATED FROM PYTHON SOURCE LINES 191-195
+
+.. code-block:: default
+
+
+ os.listdir(dir_sim)
+
+
+
+
+
+
+.. rst-class:: sphx-glr-script-out
+
+ .. code-block:: none
+
+
+ ['serial0_glocation', 'parallel3_results', 'parallel2_glocation', 'serial3_shell.sh', 'parallel1_results', 'serial3_glocation', 'serial1_shell.sh', 'parallel2_results', 'serial0_shell.sh', 'parallel1_glocation', 'parallel3_shell.sh', 'parallel0_shell.sh', 'serial1_glocation', 'parallel1_shell.sh', 'parallel2_shell.sh', 'serial0_results', 'serial1_results', 'serial2_shell.sh', 'serial2_glocation', 'serial2_results', 'parallel3_glocation', 'parallel0_glocation', 'serial3_results', 'parallel0_results']
+
+
+
+.. GENERATED FROM PYTHON SOURCE LINES 196-202
+
+.. warning:: If one simulation failed due to whatever reason, the error massage
+ will be printed to the screen but other simulations will continue. In that
+ case, the output file of failed simulation will not be writted to ``dir_out``.
+ Also, the element of :py:attr:`.RunSimulator.outputs` corresponding to that
+ simulation will be a string representing the error message, instead of a
+ numpy array.
+
+.. GENERATED FROM PYTHON SOURCE LINES 205-207
+
+In the end, we delete the folder `temp1_run_Ravaflow24Mixture_example` and
+`temp2_run_Ravaflow24Mixture_example` and all files therein.
+
+.. GENERATED FROM PYTHON SOURCE LINES 208-212
+
+.. code-block:: default
+
+
+ import shutil
+
+ shutil.rmtree(dir_sim)
+ shutil.rmtree(dir_out)
+
+
+
+
+
+
+
+.. rst-class:: sphx-glr-timing
+
+ **Total running time of the script:** ( 2 minutes 53.318 seconds)
+
+
+.. _sphx_glr_download_auto_examples_simulator_plot_run_ravaflow24.py:
+
+.. only:: html
+
+ .. container:: sphx-glr-footer sphx-glr-footer-example
+
+
+ .. container:: sphx-glr-download sphx-glr-download-python
+
+ :download:`Download Python source code: plot_run_ravaflow24.py <plot_run_ravaflow24.py>`
+
+ .. container:: sphx-glr-download sphx-glr-download-jupyter
+
+ :download:`Download Jupyter notebook: plot_run_ravaflow24.ipynb <plot_run_ravaflow24.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+ `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
diff --git a/docs/source/_build/html/_sources/auto_examples/simulator/sg_execution_times.rst.txt b/docs/source/_build/html/_sources/auto_examples/simulator/sg_execution_times.rst.txt
new file mode 100644
index 0000000..383f358
--- /dev/null
+++ b/docs/source/_build/html/_sources/auto_examples/simulator/sg_execution_times.rst.txt
@@ -0,0 +1,18 @@
+
+:orphan:
+
+.. _sphx_glr_auto_examples_simulator_sg_execution_times:
+
+Computation times
+=================
+**08:52.712** total execution time for **auto_examples_simulator** files:
+
++---------------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_simulator_plot_ravaflow24.py` (``plot_ravaflow24.py``) | 05:56.157 | 0.0 MB |
++---------------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_simulator_plot_run_ravaflow24.py` (``plot_run_ravaflow24.py``) | 02:53.318 | 0.0 MB |
++---------------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_simulator_plot_run_mass_point_model.py` (``plot_run_mass_point_model.py``) | 00:01.687 | 0.0 MB |
++---------------------------------------------------------------------------------------------------------+-----------+--------+
+| :ref:`sphx_glr_auto_examples_simulator_plot_mass_point_model.py` (``plot_mass_point_model.py``) | 00:01.550 | 0.0 MB |
++---------------------------------------------------------------------------------------------------------+-----------+--------+
diff --git a/docs/source/_build/html/auto_examples/emulator/index.html b/docs/source/_build/html/auto_examples/emulator/index.html
new file mode 100644
index 0000000..ad9a429
--- /dev/null
+++ b/docs/source/_build/html/auto_examples/emulator/index.html
@@ -0,0 +1,147 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
+
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+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="ppgasp-gp-emulation-for-multi-output-functions">
+<span id="sphx-glr-auto-examples-emulator-plot-ppgasp-py"></span><h1>PPGaSP: GP emulation for multi-output functions<a class="headerlink" href="#ppgasp-gp-emulation-for-multi-output-functions" title="Permalink to this heading"></a></h1>
+<p>This example shows how to apply Gaussian process emulation to a multi-output
+function using class <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.PPGaSP" title="psimpy.emulator.robustgasp.PPGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">PPGaSP</span></code></a>.</p>
+<p>The multi-output function that we are going to look at is the <cite>DIAMOND</cite>
+(diplomatic and military operations in a non-warfighting domain) computer model.
+It is used as a testbed to illustrate the <cite>PP GaSP</cite> emulator in the <cite>R</cite> package
+<cite>RobustGaSP</cite>, see <span id="id1">Gu <em>et al.</em> [<a class="reference internal" href="../../refs.html#id6" title="Meng Yang Gu, Jesus Palomo, and James O. Berger. Robustgasp: robust Gaussian stochastic process emulation in R. The R Journal, 11(1):112–136, 2019. doi:10.32614/RJ-2019-011.">2019</a>]</span> for more detail.</p>
+<p>The simulator has <span class="math notranslate nohighlight">\(13\)</span> input parameters and <span class="math notranslate nohighlight">\(5\)</span> outputs. Namely,
+<span class="math notranslate nohighlight">\(\mathbf{y}=f(\mathbf{x})\)</span> where <span class="math notranslate nohighlight">\(\mathbf{x}=(x_1,\ldots,x_{13})^T\)</span>
+and <span class="math notranslate nohighlight">\(\mathbf{y}=(y_1,\ldots,y_5)^T\)</span>.</p>
+<p>The training and testing data are provided in the folder ‘…/tests/data/’.
+We first load the training and testing data.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">os</span>
+
+<span class="n">dir_data</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="s1">'../../../tests/data/'</span><span class="p">)</span>
+
+<span class="n">humanityX</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">genfromtxt</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'humanityX.csv'</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span>
+<span class="n">humanityY</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">genfromtxt</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'humanityY.csv'</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Number of training data points: "</span><span class="p">,</span> <span class="n">humanityX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Input dimension: "</span><span class="p">,</span> <span class="n">humanityX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Output dimension: "</span><span class="p">,</span> <span class="n">humanityY</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="n">humanityXt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">genfromtxt</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'humanityXt.csv'</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span>
+<span class="n">humanityYt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">genfromtxt</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'humanityYt.csv'</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Number of testing data points: "</span><span class="p">,</span> <span class="n">humanityXt</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Number of training data points: 120
+Input dimension: 13
+Output dimension: 5
+Number of testing data points: 120
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>You may need to modify <code class="docutils literal notranslate"><span class="pre">dir_data</span></code> according to where you save them
+on your local machine.</p>
+</div>
+<p><code class="docutils literal notranslate"><span class="pre">humanityX</span></code> and <code class="docutils literal notranslate"><span class="pre">humanitY</span></code> are the training data, corresponding to <code class="docutils literal notranslate"><span class="pre">design</span></code>
+and <code class="docutils literal notranslate"><span class="pre">response</span></code> respectively. <code class="docutils literal notranslate"><span class="pre">humanityXt</span></code> are testing input data, at which
+we are going to make predictions once the emulator is trained. <code class="docutils literal notranslate"><span class="pre">humanityYt</span></code>
+are the true outputs at <code class="docutils literal notranslate"><span class="pre">humanityXt</span></code>, which is then used to validate the
+performance of the trained emulator.</p>
+<p>To build a <cite>PP GaSP</cite> emulator for the above simulator, first import class
+<a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.PPGaSP" title="psimpy.emulator.robustgasp.PPGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">PPGaSP</span></code></a> by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.emulator</span> <span class="kn">import</span> <span class="n">PPGaSP</span>
+</pre></div>
+</div>
+<p>Then, create an instance of <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.PPGaSP" title="psimpy.emulator.robustgasp.PPGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">PPGaSP</span></code></a>. The parameter <code class="docutils literal notranslate"><span class="pre">ndim</span></code>
+(dimension of function input <code class="docutils literal notranslate"><span class="pre">x</span></code>) must be specified. Optional parameters, such
+as <code class="docutils literal notranslate"><span class="pre">method</span></code>, <code class="docutils literal notranslate"><span class="pre">kernel_type</span></code>, etc., can be set up if desired. Here, we leave
+all the optional parameters to their default values.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">emulator</span> <span class="o">=</span> <span class="n">PPGaSP</span><span class="p">(</span><span class="n">ndim</span><span class="o">=</span><span class="n">humanityX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
+</div>
+<p>Next, we train the <cite>PP GaSP</cite> emulator based on the training data using
+<a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.PPGaSP.train" title="psimpy.emulator.robustgasp.PPGaSP.train"><code class="xref py py-meth docutils literal notranslate"><span class="pre">PPGaSP.train()</span></code></a> method.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">emulator</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">design</span><span class="o">=</span><span class="n">humanityX</span><span class="p">,</span> <span class="n">response</span><span class="o">=</span><span class="n">humanityY</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>The upper bounds of the range parameters are 296,974 297,1814 294,7672 295,9345 296,563 295,935 297,3198 296,4444 296,9323 298,2373 298,0619 298,7249 298,7249
+The initial values of range parameters are 5,93948 5,943628 5,895344 5,918691 5,93126 5,918701 5,946395 5,928888 5,938645 5,964745 5,961238 5,974498 5,974498
+Start of the optimization 1 :
+The number of iterations is 44
+ The value of the marginal posterior function is -5279,534
+ Optimized range parameters are 25,46132 2,891428 5,30812 26,95519 291,1828 48,17274 84,56133 2,874385 39,12121 51,81304 0,5407651 1,728696 1,020605
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 12,37503 12,38367 12,28307 12,33172 12,3579 12,33174 12,38944 12,35296 12,37329 12,42767 12,42037 12,44799 12,44799
+Start of the optimization 2 :
+The number of iterations is 45
+ The value of the marginal posterior function is -5279,534
+ Optimized range parameters are 25,46132 2,891428 5,308121 26,95519 291,183 48,17274 84,56133 2,874385 39,12121 51,81304 0,5407651 1,728696 1,020605
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+</pre></div>
+</div>
+<p>With the trained emulator, we can make predictions for any arbitrary set of
+input points using <code class="xref py py-meth docutils literal notranslate"><span class="pre">PPGaSP.predict()</span></code> method.
+Here, we make predictions at testing input points <code class="docutils literal notranslate"><span class="pre">humanityXt</span></code>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">predictions</span> <span class="o">=</span> <span class="n">emulator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">humanityXt</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>We can validate the performance of the trained emulator based on the true outputs
+<code class="docutils literal notranslate"><span class="pre">humanityYt</span></code>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">15</span><span class="p">))</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">humanityY</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+
+ <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">f</span><span class="s1">'Actual $y_</span><span class="si">{</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s1">$'</span><span class="p">)</span>
+ <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">f</span><span class="s1">'Emulator-predicted $y_</span><span class="si">{</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s1">$'</span><span class="p">)</span>
+ <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">]),</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">]))</span>
+ <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">]),</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">]))</span>
+
+ <span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">]),</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">])],</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">]),</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">])])</span>
+ <span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">humanityYt</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">predictions</span><span class="p">[:,</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="n">predictions</span><span class="p">[:,</span><span class="n">i</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">'.'</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">''</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'prediction and std'</span><span class="p">)</span>
+ <span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_ppgasp_001.png" srcset="../../_images/sphx_glr_plot_ppgasp_001.png" alt="plot ppgasp" class = "sphx-glr-single-img"/><p>We can also draw any number of samples at testing input <code class="docutils literal notranslate"><span class="pre">humanityXt</span></code> using
+<code class="xref py py-meth docutils literal notranslate"><span class="pre">PPGaSPsample()</span></code> method.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">samples</span> <span class="o">=</span> <span class="n">emulator</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">humanityXt</span><span class="p">,</span> <span class="n">nsamples</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Shape of samples: "</span><span class="p">,</span> <span class="n">samples</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Shape of samples: (120, 5, 10)
+</pre></div>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 5.333 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-emulator-plot-ppgasp-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/bfca202fd55dbe4c4a07ec2f8128e4cc/plot_ppgasp.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_ppgasp.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/9ac7682e0f10ab62297eb4ff29689bc8/plot_ppgasp.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_ppgasp.ipynb</span></code></a></p>
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+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="scalargasp-gp-emulation-for-a-single-output-function">
+<span id="sphx-glr-auto-examples-emulator-plot-scalargasp-py"></span><h1>ScalarGaSP: GP emulation for a single-output function<a class="headerlink" href="#scalargasp-gp-emulation-for-a-single-output-function" title="Permalink to this heading"></a></h1>
+<p>This example shows how to apply Gaussian process emulation to a single-output
+function using class <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP" title="psimpy.emulator.robustgasp.ScalarGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">ScalarGaSP</span></code></a>.</p>
+<p>The task is to build a GP emulator for the function <span class="math notranslate nohighlight">\(y = x * sin(x)\)</span>
+based on a few number of training data.</p>
+<p>First, import the class <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP" title="psimpy.emulator.robustgasp.ScalarGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">ScalarGaSP</span></code></a> by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.emulator</span> <span class="kn">import</span> <span class="n">ScalarGaSP</span>
+</pre></div>
+</div>
+<p>Then, create an instance of <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP" title="psimpy.emulator.robustgasp.ScalarGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">ScalarGaSP</span></code></a>. The parameter <code class="docutils literal notranslate"><span class="pre">ndim</span></code>
+(dimension of function input <code class="docutils literal notranslate"><span class="pre">x</span></code>) must be specified. Optional parameters, such
+as <code class="docutils literal notranslate"><span class="pre">method</span></code>, <code class="docutils literal notranslate"><span class="pre">kernel_type</span></code>, etc., can be set up if desired. Here, we leave
+all the optional parameters to their default values.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">emulator</span> <span class="o">=</span> <span class="n">ScalarGaSP</span><span class="p">(</span><span class="n">ndim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Given training input points <code class="docutils literal notranslate"><span class="pre">design</span></code> and corresponding output values <code class="docutils literal notranslate"><span class="pre">response</span></code>,
+the emulator can be trained using <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP.train" title="psimpy.emulator.robustgasp.ScalarGaSP.train"><code class="xref py py-meth docutils literal notranslate"><span class="pre">ScalarGaSP.train()</span></code></a>. Below we train
+an emulator using <span class="math notranslate nohighlight">\(8\)</span> selected points.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="c1">#return x + 3*np.sin(x/2)</span>
+ <span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+
+<span class="n">emulator</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">design</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">response</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>The upper bounds of the range parameters are 1881,404
+The initial values of range parameters are 37,62809
+Start of the optimization 1 :
+The number of iterations is 9
+ The value of the marginal posterior function is -13,79116
+ Optimized range parameters are 2,237467
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,21875
+Start of the optimization 2 :
+The number of iterations is 9
+ The value of the marginal posterior function is -13,79116
+ Optimized range parameters are 2,237467
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+</pre></div>
+</div>
+<p>We can validate the performance of the trained emulator using the leave-one-out
+cross validation method <code class="xref py py-meth docutils literal notranslate"><span class="pre">loo_validate()</span></code>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">validation</span> <span class="o">=</span> <span class="n">emulator</span><span class="o">.</span><span class="n">loo_validate</span><span class="p">()</span>
+</pre></div>
+</div>
+<p>Let’s plot emulator predictions vs actual outputs. The error bar indicates the
+standard deviation.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span> <span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Actual y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Emulator-predicted y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
+
+<span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">validation</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">validation</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">'.'</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">''</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'prediction and std'</span><span class="p">)</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_scalargasp_001.png" srcset="../../_images/sphx_glr_plot_scalargasp_001.png" alt="plot scalargasp" class = "sphx-glr-single-img"/><p>With the trained emulator at our deposit, we can use the
+<a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP.predict" title="psimpy.emulator.robustgasp.ScalarGaSP.predict"><code class="xref py py-meth docutils literal notranslate"><span class="pre">ScalarGaSP.predict()</span></code></a> method to make predictions at
+any arbitrary set of input points (<code class="docutils literal notranslate"><span class="pre">testing_input</span></code>). It should be noted that,
+<code class="docutils literal notranslate"><span class="pre">testing_trend</span></code> should be set according to <code class="docutils literal notranslate"><span class="pre">trend</span></code> used during emulator
+training.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">testing_input</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
+<span class="n">predictions</span> <span class="o">=</span> <span class="n">emulator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">testing_input</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">predictions</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="s1">'r-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span> <span class="s2">"mean"</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"training data"</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">testing_input</span><span class="p">),</span> <span class="s1">'k:'</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"true function"</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">predictions</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">predictions</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"95% CI"</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'emulator-predicted y'</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">testing_input</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">testing_input</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_scalargasp_002.png" srcset="../../_images/sphx_glr_plot_scalargasp_002.png" alt="plot scalargasp" class = "sphx-glr-single-img"/><p>We can also draw any number of samples at <code class="docutils literal notranslate"><span class="pre">testing_input`</span></code> using
+<a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP.sample" title="psimpy.emulator.robustgasp.ScalarGaSP.sample"><code class="xref py py-meth docutils literal notranslate"><span class="pre">ScalarGaSP.sample()</span></code></a> method.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">nsamples</span> <span class="o">=</span> <span class="mi">5</span>
+<span class="n">samples</span> <span class="o">=</span> <span class="n">emulator</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">nsamples</span><span class="o">=</span><span class="n">nsamples</span><span class="p">)</span>
+
+<span class="c1"># sphinx_gallery_thumbnail_number = 3</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nsamples</span><span class="p">):</span>
+ <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">samples</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="s1">'--'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s1">'sample</span><span class="si">{</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"training data"</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">testing_input</span><span class="p">),</span> <span class="s1">'k:'</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"true function"</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">testing_input</span><span class="p">,</span> <span class="n">predictions</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">predictions</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"95% CI"</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'emulator-predicted y'</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">testing_input</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">testing_input</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_scalargasp_003.png" srcset="../../_images/sphx_glr_plot_scalargasp_003.png" alt="plot scalargasp" class = "sphx-glr-single-img"/><div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Above example shows how to train a GP emulator based on noise-free training data,
+which is often the case of emulating a deterministic simulator. If you are dealing
+with noisy training data, you can</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>set the parameter <code class="docutils literal notranslate"><span class="pre">nugget</span></code> to a desired value, or</p></li>
+<li><p>set <code class="docutils literal notranslate"><span class="pre">nugget</span></code> to <span class="math notranslate nohighlight">\(0\)</span> and <code class="docutils literal notranslate"><span class="pre">nugget_est</span></code> to <cite>True</cite>, meaning that <code class="docutils literal notranslate"><span class="pre">nugget</span></code>
+is estimated from the noisy training data.</p></li>
+</ul>
+</div></blockquote>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 0.728 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-emulator-plot-scalargasp-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/ca97a6211ade4816b175cce5bc31a34b/plot_scalargasp.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_scalargasp.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/8c06e92daa6ae4230f607ca4f47a3769/plot_scalargasp.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_scalargasp.ipynb</span></code></a></p>
+</div>
+</div>
+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
+</section>
+
+
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+ </div>
+ <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
+ <a href="plot_ppgasp.html" class="btn btn-neutral float-left" title="PPGaSP: GP emulation for multi-output functions" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a>
+ <a href="../inference/index.html" class="btn btn-neutral float-right" title="Inference" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a>
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new file mode 100644
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@@ -0,0 +1,131 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
+
+ <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+ <title>Computation times — PSimPy's documentation</title>
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+
+ <section id="computation-times">
+<span id="sphx-glr-auto-examples-emulator-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Permalink to this heading"></a></h1>
+<p><strong>00:06.062</strong> total execution time for <strong>auto_examples_emulator</strong> files:</p>
+<table class="docutils align-default">
+<colgroup>
+<col style="width: 82%" />
+<col style="width: 11%" />
+<col style="width: 8%" />
+</colgroup>
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_ppgasp.html#sphx-glr-auto-examples-emulator-plot-ppgasp-py"><span class="std std-ref">PPGaSP: GP emulation for multi-output functions</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_ppgasp.py</span></code>)</p></td>
+<td><p>00:05.333</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="plot_scalargasp.html#sphx-glr-auto-examples-emulator-plot-scalargasp-py"><span class="std std-ref">ScalarGaSP: GP emulation for a single-output function</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_scalargasp.py</span></code>)</p></td>
+<td><p>00:00.728</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+
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+<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="Bayesian inference"><img alt="Bayesian inference" src="../../_images/sphx_glr_plot_bayes_inference_thumb.png" />
+<p><a class="reference internal" href="plot_bayes_inference.html#sphx-glr-auto-examples-inference-plot-bayes-inference-py"><span class="std std-ref">Bayesian inference</span></a></p>
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+<p><a class="reference internal" href="plot_active_learning.html#sphx-glr-auto-examples-inference-plot-active-learning-py"><span class="std std-ref">Active learning</span></a></p>
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+ <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
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+
+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-inference-plot-active-learning-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="active-learning">
+<span id="sphx-glr-auto-examples-inference-plot-active-learning-py"></span><h1>Active learning<a class="headerlink" href="#active-learning" title="Permalink to this heading"></a></h1>
+<p>This example shows how to use the <a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning" title="psimpy.inference.active_learning.ActiveLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">ActiveLearning</span></code></a> class to iteratively
+build a Gaussian process emulator for an unnormalized posterior involving a
+simulator. It should be noted that this example is only for illustration
+purpose, rather than a real case. For simplicity, required arguments for
+<a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning" title="psimpy.inference.active_learning.ActiveLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">ActiveLearning</span></code></a> (including <cite>simulator</cite>, <cite>likelihood</cite>, <cite>data</cite>, etc.) are purely
+made up. For a realistic case of active learning, one can refer to <span id="id1">Zhao and Kowalski [<a class="reference internal" href="../../refs.html#id16" title="Hu Zhao and Julia Kowalski. Bayesian active learning for parameter calibration of landslide run-out models. Landslides, 19():2033–2045, 2022. doi:https://doi.org/10.1007/s10346-022-01857-z.">2022</a>]</span>.</p>
+<p>First, we define the simulator, prior distribution of its variable parameters,
+observed data, and likelihood function. They basically define the Bayesian
+inference problem.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">ndim</span> <span class="o">=</span> <span class="mi">2</span> <span class="c1"># dimension of variable parameters of the simulator</span>
+<span class="n">bounds</span> <span class="o">=</span> <span class="n">bounds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">],[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">]])</span> <span class="c1"># bounds of variable parameters</span>
+<span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
+
+<span class="k">def</span> <span class="nf">simulator</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">):</span>
+ <span class="sd">"""Simulator y=f(x)."""</span>
+ <span class="n">y1</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span>
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">y1</span><span class="p">,</span> <span class="n">y2</span><span class="p">])</span>
+
+<span class="k">def</span> <span class="nf">prior</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="sd">"""Uniform prior."""</span>
+ <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">10</span><span class="o">*</span><span class="mi">10</span><span class="p">)</span>
+
+<span class="k">def</span> <span class="nf">likelihood</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">data</span><span class="p">):</span>
+ <span class="sd">"""Likelihood function L(y,data)."""</span>
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">100</span> <span class="o">-</span> <span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">data</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Imagine that the simulator is a complex solver. It is not computationally feasible
+to compute the posterior distribution of the variable parameters using grid estimation
+or Metropolis Hastings estimation. This is because they require evaluating the likelihood
+many times which essentially leads to many evaluations of the simulator. Therefore, we
+resort to use active learning to build a Gaussian process emulator for the unnormalized
+posterior (prior times likelihood) based on a small number of evaluations of the simulator.
+The the posterior can be estimated using the emulator.</p>
+<p>To do so, we need to pass arguments to following parameters of the <a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning" title="psimpy.inference.active_learning.ActiveLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">ActiveLearning</span></code></a> class:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>run_sim_obj : instance of class <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a>. It carries information on how
+to run the simulator.</p></li>
+<li><p>lhs_sampler : instance of class <a class="reference internal" href="../../sampler/latin.html#psimpy.sampler.latin.LHS" title="psimpy.sampler.latin.LHS"><code class="xref py py-class docutils literal notranslate"><span class="pre">LHS</span></code></a>. It is used to draw initial samples to run
+simulations in order to train an inital Gaussian process emulator.</p></li>
+<li><p>scalar_gasp : instance of class <a class="reference internal" href="../../emulator/robustgasp.html#psimpy.emulator.robustgasp.ScalarGaSP" title="psimpy.emulator.robustgasp.ScalarGaSP"><code class="xref py py-class docutils literal notranslate"><span class="pre">ScalarGaSP</span></code></a>. It sets up the emulator structure.</p></li>
+</ul>
+</div></blockquote>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">RunSimulator</span>
+<span class="kn">from</span> <span class="nn">psimpy.sampler</span> <span class="kn">import</span> <span class="n">LHS</span>
+<span class="kn">from</span> <span class="nn">psimpy.emulator</span> <span class="kn">import</span> <span class="n">ScalarGaSP</span>
+
+<span class="n">run_simulator</span> <span class="o">=</span> <span class="n">RunSimulator</span><span class="p">(</span><span class="n">simulator</span><span class="p">,</span> <span class="n">var_inp_parameter</span><span class="o">=</span><span class="p">[</span><span class="s1">'x1'</span><span class="p">,</span><span class="s1">'x2'</span><span class="p">])</span>
+<span class="n">lhs_sampler</span> <span class="o">=</span> <span class="n">LHS</span><span class="p">(</span><span class="n">ndim</span><span class="o">=</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="o">=</span><span class="n">bounds</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">scalar_gasp</span> <span class="o">=</span> <span class="n">ScalarGaSP</span><span class="p">(</span><span class="n">ndim</span><span class="o">=</span><span class="n">ndim</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Next, we create an object of the <a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning" title="psimpy.inference.active_learning.ActiveLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">ActiveLearning</span></code></a> class by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.inference</span> <span class="kn">import</span> <span class="n">ActiveLearning</span>
+
+<span class="n">active_learner</span> <span class="o">=</span> <span class="n">ActiveLearning</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">run_simulator</span><span class="p">,</span> <span class="n">prior</span><span class="p">,</span> <span class="n">likelihood</span><span class="p">,</span>
+ <span class="n">lhs_sampler</span><span class="p">,</span> <span class="n">scalar_gasp</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Then we can call the <a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning.initial_simulation" title="psimpy.inference.active_learning.ActiveLearning.initial_simulation"><code class="xref py py-meth docutils literal notranslate"><span class="pre">ActiveLearning.initial_simulation()</span></code></a> method to run initial
+simulations and call the <a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning.iterative_emulation" title="psimpy.inference.active_learning.ActiveLearning.iterative_emulation"><code class="xref py py-meth docutils literal notranslate"><span class="pre">ActiveLearning.iterative_emulation()</span></code></a> method to
+iteratively run new simulation and build emulator. Here we allocate 40 simulations for
+initial emulator training and 60 simulations for adaptive training.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n0</span> <span class="o">=</span> <span class="mi">40</span>
+<span class="n">niter</span> <span class="o">=</span> <span class="mi">60</span>
+
+<span class="n">init_var_samples</span><span class="p">,</span> <span class="n">init_sim_outputs</span> <span class="o">=</span> <span class="n">active_learner</span><span class="o">.</span><span class="n">initial_simulation</span><span class="p">(</span>
+ <span class="n">n0</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'parallel'</span><span class="p">,</span> <span class="n">max_workers</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
+
+<span class="n">var_samples</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">active_learner</span><span class="o">.</span><span class="n">iterative_emulation</span><span class="p">(</span>
+ <span class="n">n0</span><span class="p">,</span> <span class="n">init_var_samples</span><span class="p">,</span> <span class="n">init_sim_outputs</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>The upper bounds of the range parameters are 401,3482 397,4694
+The initial values of range parameters are 8,026963 7,949389
+Start of the optimization 1 :
+The number of iterations is 22
+ The value of the marginal posterior function is -146,5038
+ Optimized range parameters are 30,59853 130,6077
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,9828751 0,9733764
+Start of the optimization 2 :
+The number of iterations is 36
+ The value of the marginal posterior function is -146,5038
+ Optimized range parameters are 30,59827 130,6067
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 423,194 420,3365
+The initial values of range parameters are 8,463881 8,40673
+Start of the optimization 1 :
+The number of iterations is 13
+ The value of the marginal posterior function is -148,6792
+ Optimized range parameters are 30,19534 133,3286
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,9589025 0,9524277
+Start of the optimization 2 :
+The number of iterations is 12
+ The value of the marginal posterior function is -148,6792
+ Optimized range parameters are 30,19539 133,329
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 422,8426 419,9874
+The initial values of range parameters are 8,456851 8,399747
+Start of the optimization 1 :
+The number of iterations is 34
+ The value of the marginal posterior function is -149,9595
+ Optimized range parameters are 30,64771 135,0597
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,9360715 0,9297509
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -149,9595
+ Optimized range parameters are 30,64696 135,0536
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 423,2876 420,4294
+The initial values of range parameters are 8,465751 8,408588
+Start of the optimization 1 :
+The number of iterations is 25
+ The value of the marginal posterior function is -151,304
+ Optimized range parameters are 31,21605 136,8571
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,9143024 0,9081287
+Start of the optimization 2 :
+The number of iterations is 23
+ The value of the marginal posterior function is -151,304
+ Optimized range parameters are 31,21617 136,8537
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 389,6167 386,9859
+The initial values of range parameters are 7,792334 7,739717
+Start of the optimization 1 :
+The number of iterations is 13
+ The value of the marginal posterior function is -152,6009
+ Optimized range parameters are 32,285 141,3784
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,8935228 0,8874894
+Start of the optimization 2 :
+The number of iterations is 17
+ The value of the marginal posterior function is -152,6009
+ Optimized range parameters are 32,28096 141,3737
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 368,8855 366,3947
+The initial values of range parameters are 7,377711 7,327894
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -153,5419
+ Optimized range parameters are 32,43481 141,7852
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,8736668 0,8677675
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -153,5419
+ Optimized range parameters are 32,43357 141,7844
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 339,4555 337,1634
+The initial values of range parameters are 6,789111 6,743268
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -154,9929
+ Optimized range parameters are 33,69448 151,2103
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,854674 0,848903
+Start of the optimization 2 :
+The number of iterations is 22
+ The value of the marginal posterior function is -154,9929
+ Optimized range parameters are 33,69468 151,2111
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 369,9581 367,46
+The initial values of range parameters are 7,399163 7,349201
+Start of the optimization 1 :
+The number of iterations is 26
+ The value of the marginal posterior function is -155,4764
+ Optimized range parameters are 33,90556 152,0139
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,8364895 0,8308412
+Start of the optimization 2 :
+The number of iterations is 23
+ The value of the marginal posterior function is -155,4764
+ Optimized range parameters are 33,90541 152,0137
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 299,8308 297,8062
+The initial values of range parameters are 5,996615 5,956124
+Start of the optimization 1 :
+The number of iterations is 27
+ The value of the marginal posterior function is -155,8802
+ Optimized range parameters are 34,04389 152,6133
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,8190626 0,813532
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -155,8802
+ Optimized range parameters are 34,04381 152,6131
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 272,589 270,7484
+The initial values of range parameters are 5,45178 5,414968
+Start of the optimization 1 :
+The number of iterations is 22
+ The value of the marginal posterior function is -156,1383
+ Optimized range parameters are 34,46966 151,4246
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,802347 0,7969293
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -156,1383
+ Optimized range parameters are 34,46972 151,4261
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 217,1413 215,6751
+The initial values of range parameters are 4,342826 4,313501
+Start of the optimization 1 :
+The number of iterations is 16
+ The value of the marginal posterior function is -156,184
+ Optimized range parameters are 34,52228 152,8428
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,7863001 0,7809907
+Start of the optimization 2 :
+The number of iterations is 29
+ The value of the marginal posterior function is -156,184
+ Optimized range parameters are 34,5049 152,8145
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 206,6122 205,2171
+The initial values of range parameters are 4,132244 4,104342
+Start of the optimization 1 :
+The number of iterations is 25
+ The value of the marginal posterior function is -156,2348
+ Optimized range parameters are 34,79805 155,1646
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,7708824 0,7656772
+Start of the optimization 2 :
+The number of iterations is 22
+ The value of the marginal posterior function is -156,2348
+ Optimized range parameters are 34,82503 155,2481
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 264,8501 263,0618
+The initial values of range parameters are 5,297002 5,261235
+Start of the optimization 1 :
+The number of iterations is 30
+ The value of the marginal posterior function is -156,4063
+ Optimized range parameters are 34,74024 156,9499
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,7560578 0,7509526
+Start of the optimization 2 :
+The number of iterations is 35
+ The value of the marginal posterior function is -156,4062
+ Optimized range parameters are 34,71806 156,8712
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 255,2317 253,5083
+The initial values of range parameters are 5,104633 5,070165
+Start of the optimization 1 :
+The number of iterations is 20
+ The value of the marginal posterior function is -156,5552
+ Optimized range parameters are 35,37305 157,5615
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,7417925 0,7367837
+Start of the optimization 2 :
+The number of iterations is 37
+ The value of the marginal posterior function is -156,5552
+ Optimized range parameters are 35,37704 157,591
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 234,5139 232,9304
+The initial values of range parameters are 4,690278 4,658607
+Start of the optimization 1 :
+The number of iterations is 25
+ The value of the marginal posterior function is -156,8767
+ Optimized range parameters are 35,91747 161,0899
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,7280556 0,7231396
+Start of the optimization 2 :
+The number of iterations is 32
+ The value of the marginal posterior function is -156,8767
+ Optimized range parameters are 35,92577 161,0559
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 237,6023 235,9979
+The initial values of range parameters are 4,752045 4,719958
+Start of the optimization 1 :
+The number of iterations is 15
+ The value of the marginal posterior function is -156,6342
+ Optimized range parameters are 36,30341 161,9859
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,7148183 0,7099916
+Start of the optimization 2 :
+The number of iterations is 26
+ The value of the marginal posterior function is -156,6343
+ Optimized range parameters are 36,30117 161,8561
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 231,5375 229,9741
+The initial values of range parameters are 4,63075 4,599482
+Start of the optimization 1 :
+The number of iterations is 27
+ The value of the marginal posterior function is -156,3683
+ Optimized range parameters are 36,27212 162,5847
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,7020536 0,6973131
+Start of the optimization 2 :
+The number of iterations is 30
+ The value of the marginal posterior function is -156,3684
+ Optimized range parameters are 36,25962 162,5355
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 237,6502 236,0455
+The initial values of range parameters are 4,753005 4,720911
+Start of the optimization 1 :
+The number of iterations is 14
+ The value of the marginal posterior function is -156,014
+ Optimized range parameters are 36,19577 162,4854
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,6897369 0,6850796
+Start of the optimization 2 :
+The number of iterations is 27
+ The value of the marginal posterior function is -156,014
+ Optimized range parameters are 36,19621 162,4877
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 221,4001 219,9051
+The initial values of range parameters are 4,428002 4,398102
+Start of the optimization 1 :
+The number of iterations is 36
+ The value of the marginal posterior function is -155,8456
+ Optimized range parameters are 36,93092 166,708
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,6778449 0,6732679
+Start of the optimization 2 :
+The number of iterations is 19
+ The value of the marginal posterior function is -155,8455
+ Optimized range parameters are 37,00954 166,8438
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 207,5545 206,153
+The initial values of range parameters are 4,15109 4,123061
+Start of the optimization 1 :
+The number of iterations is 36
+ The value of the marginal posterior function is -155,2611
+ Optimized range parameters are 37,54574 171,0122
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,666356 0,6618565
+Start of the optimization 2 :
+The number of iterations is 26
+ The value of the marginal posterior function is -155,2611
+ Optimized range parameters are 37,52707 170,9901
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 211,1368 209,7111
+The initial values of range parameters are 4,222736 4,194222
+Start of the optimization 1 :
+The number of iterations is 38
+ The value of the marginal posterior function is -154,6105
+ Optimized range parameters are 37,27271 173,4402
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,6552501 0,6508256
+Start of the optimization 2 :
+The number of iterations is 38
+ The value of the marginal posterior function is -154,6104
+ Optimized range parameters are 37,31152 173,4271
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 229,1987 227,651
+The initial values of range parameters are 4,583973 4,553021
+Start of the optimization 1 :
+The number of iterations is 30
+ The value of the marginal posterior function is -153,8922
+ Optimized range parameters are 37,52317 173,0118
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,6445083 0,6401563
+Start of the optimization 2 :
+The number of iterations is 29
+ The value of the marginal posterior function is -153,8923
+ Optimized range parameters are 37,52772 173,0381
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 200,215 198,8631
+The initial values of range parameters are 4,004301 3,977262
+Start of the optimization 1 :
+The number of iterations is 29
+ The value of the marginal posterior function is -153,0737
+ Optimized range parameters are 37,6626 173,7851
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,634113 0,6298312
+Start of the optimization 2 :
+The number of iterations is 38
+ The value of the marginal posterior function is -153,0733
+ Optimized range parameters are 37,57977 172,9914
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 208,2931 206,8866
+The initial values of range parameters are 4,165862 4,137733
+Start of the optimization 1 :
+The number of iterations is 31
+ The value of the marginal posterior function is -152,171
+ Optimized range parameters are 37,55899 172,4488
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,6240477 0,6198339
+Start of the optimization 2 :
+The number of iterations is 34
+ The value of the marginal posterior function is -152,1711
+ Optimized range parameters are 37,55216 172,3431
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 190,9562 189,6668
+The initial values of range parameters are 3,819124 3,793336
+Start of the optimization 1 :
+The number of iterations is 26
+ The value of the marginal posterior function is -151,4191
+ Optimized range parameters are 37,87024 177,6757
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,6142969 0,610149
+Start of the optimization 2 :
+The number of iterations is 31
+ The value of the marginal posterior function is -151,4191
+ Optimized range parameters are 37,8678 177,4358
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 187,3139 186,0491
+The initial values of range parameters are 3,746279 3,720983
+Start of the optimization 1 :
+The number of iterations is 27
+ The value of the marginal posterior function is -150,3521
+ Optimized range parameters are 37,93228 177,0193
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,6048462 0,6007621
+Start of the optimization 2 :
+The number of iterations is 15
+ The value of the marginal posterior function is -150,3521
+ Optimized range parameters are 37,92683 176,9819
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 201,0467 199,6892
+The initial values of range parameters are 4,020934 3,993783
+Start of the optimization 1 :
+The number of iterations is 31
+ The value of the marginal posterior function is -149,229
+ Optimized range parameters are 37,73923 179,7933
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5956819 0,5916596
+Start of the optimization 2 :
+The number of iterations is 23
+ The value of the marginal posterior function is -149,2295
+ Optimized range parameters are 37,68347 179,3098
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 208,1589 206,7533
+The initial values of range parameters are 4,163178 4,135067
+Start of the optimization 1 :
+The number of iterations is 31
+ The value of the marginal posterior function is -148,0364
+ Optimized range parameters are 37,10463 181,452
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,5867911 0,5828289
+Start of the optimization 2 :
+The number of iterations is 28
+ The value of the marginal posterior function is -148,0353
+ Optimized range parameters are 37,10023 181,8802
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 207,5545 206,153
+The initial values of range parameters are 4,15109 4,123061
+Start of the optimization 1 :
+The number of iterations is 31
+ The value of the marginal posterior function is -147,2549
+ Optimized range parameters are 36,89036 184,8898
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,5781618 0,5742579
+Start of the optimization 2 :
+The number of iterations is 18
+ The value of the marginal posterior function is -147,2555
+ Optimized range parameters are 36,87001 184,82
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 166,9279 165,8008
+The initial values of range parameters are 3,338559 3,316015
+Start of the optimization 1 :
+The number of iterations is 24
+ The value of the marginal posterior function is -146,0908
+ Optimized range parameters are 34,10173 165,8008
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5697827 0,5659353
+Start of the optimization 2 :
+The number of iterations is 12
+ The value of the marginal posterior function is -146,091
+ Optimized range parameters are 34,11888 165,8008
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 164,9351 163,8214
+The initial values of range parameters are 3,298703 3,276429
+Start of the optimization 1 :
+The number of iterations is 29
+ The value of the marginal posterior function is -144,8362
+ Optimized range parameters are 33,02363 163,8214
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5616429 0,5578505
+Start of the optimization 2 :
+The number of iterations is 15
+ The value of the marginal posterior function is -144,8366
+ Optimized range parameters are 32,96846 163,8214
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 161,9899 160,8961
+The initial values of range parameters are 3,239799 3,217922
+Start of the optimization 1 :
+The number of iterations is 30
+ The value of the marginal posterior function is -143,1679
+ Optimized range parameters are 32,60853 160,8961
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5537325 0,5499935
+Start of the optimization 2 :
+The number of iterations is 23
+ The value of the marginal posterior function is -143,1673
+ Optimized range parameters are 32,60514 160,8961
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 167,4923 166,3613
+The initial values of range parameters are 3,349846 3,327227
+Start of the optimization 1 :
+The number of iterations is 31
+ The value of the marginal posterior function is -141,4524
+ Optimized range parameters are 32,84554 166,3613
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5460417 0,5423547
+Start of the optimization 2 :
+The number of iterations is 10
+ The value of the marginal posterior function is -141,4531
+ Optimized range parameters are 32,87053 166,3613
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 158,1908 157,1226
+The initial values of range parameters are 3,163815 3,142452
+Start of the optimization 1 :
+The number of iterations is 21
+ The value of the marginal posterior function is -140,039
+ Optimized range parameters are 31,62214 157,1226
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5385617 0,5349251
+Start of the optimization 2 :
+The number of iterations is 16
+ The value of the marginal posterior function is -140,0387
+ Optimized range parameters are 31,62433 157,1226
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 174,0066 172,8316
+The initial values of range parameters are 3,480131 3,456632
+Start of the optimization 1 :
+The number of iterations is 23
+ The value of the marginal posterior function is -138,1222
+ Optimized range parameters are 33,48823 172,8316
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5312838 0,5276964
+Start of the optimization 2 :
+The number of iterations is 28
+ The value of the marginal posterior function is -138,1223
+ Optimized range parameters are 33,40525 172,8316
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 150,4312 149,4154
+The initial values of range parameters are 3,008624 2,988309
+Start of the optimization 1 :
+The number of iterations is 21
+ The value of the marginal posterior function is -139,2082
+ Optimized range parameters are 30,85041 149,4154
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,5242001 0,5206605
+Start of the optimization 2 :
+The number of iterations is 25
+ The value of the marginal posterior function is -139,2085
+ Optimized range parameters are 30,93585 149,4154
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 163,2096 162,1076
+The initial values of range parameters are 3,264193 3,242152
+Start of the optimization 1 :
+The number of iterations is 30
+ The value of the marginal posterior function is -136,829
+ Optimized range parameters are 31,64889 162,1076
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5173027 0,5138097
+Start of the optimization 2 :
+The number of iterations is 29
+ The value of the marginal posterior function is -136,8292
+ Optimized range parameters are 31,73272 162,1076
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 155,3651 154,316
+The initial values of range parameters are 3,107301 3,08632
+Start of the optimization 1 :
+The number of iterations is 38
+ The value of the marginal posterior function is -137,2507
+ Optimized range parameters are 31,11748 154,316
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5105845 0,5071368
+Start of the optimization 2 :
+The number of iterations is 11
+ The value of the marginal posterior function is -137,2507
+ Optimized range parameters are 31,11466 154,316
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 159,0334 157,9595
+The initial values of range parameters are 3,180667 3,15919
+Start of the optimization 1 :
+The number of iterations is 19
+ The value of the marginal posterior function is -134,9592
+ Optimized range parameters are 30,6283 157,9595
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,5040385 0,5006351
+Start of the optimization 2 :
+The number of iterations is 20
+ The value of the marginal posterior function is -134,9589
+ Optimized range parameters are 30,62948 157,9595
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 150,6829 149,6655
+The initial values of range parameters are 3,013659 2,993309
+Start of the optimization 1 :
+The number of iterations is 9
+ The value of the marginal posterior function is -133,0322
+ Optimized range parameters are 28,62106 149,6655
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4976583 0,4942979
+Start of the optimization 2 :
+The number of iterations is 27
+ The value of the marginal posterior function is -133,0311
+ Optimized range parameters are 28,6202 149,6655
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 143,0785 142,1124
+The initial values of range parameters are 2,86157 2,842247
+Start of the optimization 1 :
+The number of iterations is 27
+ The value of the marginal posterior function is -131,5134
+ Optimized range parameters are 27,05986 142,1124
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,4914376 0,4881192
+Start of the optimization 2 :
+The number of iterations is 28
+ The value of the marginal posterior function is -131,5142
+ Optimized range parameters are 27,09861 142,1124
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 149,9901 148,9774
+The initial values of range parameters are 2,999803 2,979547
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -128,3138
+ Optimized range parameters are 26,94379 148,9774
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4853704 0,482093
+Start of the optimization 2 :
+The number of iterations is 11
+ The value of the marginal posterior function is -128,3136
+ Optimized range parameters are 26,93153 148,9774
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 141,9733 141,0146
+The initial values of range parameters are 2,839465 2,820292
+Start of the optimization 1 :
+The number of iterations is 15
+ The value of the marginal posterior function is -126,565
+ Optimized range parameters are 25,84325 141,0146
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4794513 0,4762139
+Start of the optimization 2 :
+The number of iterations is 30
+ The value of the marginal posterior function is -126,5658
+ Optimized range parameters are 25,84843 141,0146
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 148,4124 147,4103
+The initial values of range parameters are 2,968249 2,948206
+Start of the optimization 1 :
+The number of iterations is 14
+ The value of the marginal posterior function is -123,2249
+ Optimized range parameters are 26,04536 147,4103
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4736748 0,4704763
+Start of the optimization 2 :
+The number of iterations is 37
+ The value of the marginal posterior function is -123,2255
+ Optimized range parameters are 26,11794 147,4103
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 141,3136 140,3594
+The initial values of range parameters are 2,826273 2,807189
+Start of the optimization 1 :
+The number of iterations is 24
+ The value of the marginal posterior function is -121,373
+ Optimized range parameters are 25,34355 140,3594
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,4680358 0,4648754
+Start of the optimization 2 :
+The number of iterations is 27
+ The value of the marginal posterior function is -121,3719
+ Optimized range parameters are 25,3222 140,3594
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 146,0605 145,0743
+The initial values of range parameters are 2,921211 2,901486
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -119,1731
+ Optimized range parameters are 25,38398 145,0743
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4625295 0,4594063
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -119,1742
+ Optimized range parameters are 25,27086 145,0743
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 136,1432 135,2239
+The initial values of range parameters are 2,722864 2,704479
+Start of the optimization 1 :
+The number of iterations is 16
+ The value of the marginal posterior function is -117,4765
+ Optimized range parameters are 23,8631 135,2239
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4571512 0,4540644
+Start of the optimization 2 :
+The number of iterations is 31
+ The value of the marginal posterior function is -117,4779
+ Optimized range parameters are 23,87798 135,2239
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 121,8399 121,0172
+The initial values of range parameters are 2,436797 2,420343
+Start of the optimization 1 :
+The number of iterations is 21
+ The value of the marginal posterior function is -117,2996
+ Optimized range parameters are 22,38443 121,0172
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,4518966 0,4488452
+Start of the optimization 2 :
+The number of iterations is 19
+ The value of the marginal posterior function is -117,2989
+ Optimized range parameters are 22,37563 121,0172
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 123,1404 122,309
+The initial values of range parameters are 2,462809 2,446179
+Start of the optimization 1 :
+The number of iterations is 18
+ The value of the marginal posterior function is -114,6525
+ Optimized range parameters are 22,32619 122,309
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4467614 0,4437447
+Start of the optimization 2 :
+The number of iterations is 38
+ The value of the marginal posterior function is -114,6523
+ Optimized range parameters are 22,33099 122,309
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 117,4477 116,6546
+The initial values of range parameters are 2,348953 2,333092
+Start of the optimization 1 :
+The number of iterations is 23
+ The value of the marginal posterior function is -113,3472
+ Optimized range parameters are 21,62668 116,6546
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,4417416 0,4387588
+Start of the optimization 2 :
+The number of iterations is 28
+ The value of the marginal posterior function is -113,3475
+ Optimized range parameters are 21,65104 116,6546
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 115,2499 114,4717
+The initial values of range parameters are 2,304998 2,289433
+Start of the optimization 1 :
+The number of iterations is 12
+ The value of the marginal posterior function is -110,7155
+ Optimized range parameters are 20,98512 114,4717
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4368334 0,4338837
+Start of the optimization 2 :
+The number of iterations is 30
+ The value of the marginal posterior function is -110,7149
+ Optimized range parameters are 21,00228 114,4717
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 111,8946 111,139
+The initial values of range parameters are 2,237892 2,222781
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -108,5065
+ Optimized range parameters are 20,44681 111,139
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,432033 0,4291158
+Start of the optimization 2 :
+The number of iterations is 19
+ The value of the marginal posterior function is -108,507
+ Optimized range parameters are 20,44983 111,139
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 93,38518 92,75461
+The initial values of range parameters are 1,867704 1,855092
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -111,367
+ Optimized range parameters are 18,37032 92,75461
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,427337 0,4244515
+Start of the optimization 2 :
+The number of iterations is 18
+ The value of the marginal posterior function is -111,3671
+ Optimized range parameters are 18,37078 92,75461
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 99,64494 98,97211
+The initial values of range parameters are 1,992899 1,979442
+Start of the optimization 1 :
+The number of iterations is 23
+ The value of the marginal posterior function is -106,2562
+ Optimized range parameters are 18,89163 98,97211
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,422742 0,4198875
+Start of the optimization 2 :
+The number of iterations is 27
+ The value of the marginal posterior function is -106,2561
+ Optimized range parameters are 18,88065 98,97211
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 99,75194 99,07838
+The initial values of range parameters are 1,995039 1,981568
+Start of the optimization 1 :
+The number of iterations is 17
+ The value of the marginal posterior function is -103,2099
+ Optimized range parameters are 18,65492 99,07838
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4182447 0,4154206
+Start of the optimization 2 :
+The number of iterations is 27
+ The value of the marginal posterior function is -103,2113
+ Optimized range parameters are 18,65549 99,07838
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 104,3536 103,649
+The initial values of range parameters are 2,087073 2,07298
+Start of the optimization 1 :
+The number of iterations is 20
+ The value of the marginal posterior function is -98,42375
+ Optimized range parameters are 18,87395 103,649
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4138422 0,4110477
+Start of the optimization 2 :
+The number of iterations is 27
+ The value of the marginal posterior function is -98,42241
+ Optimized range parameters are 18,87764 103,649
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The upper bounds of the range parameters are 103,1118 102,4155
+The initial values of range parameters are 2,062236 2,048311
+Start of the optimization 1 :
+The number of iterations is 20
+ The value of the marginal posterior function is -95,30729
+ Optimized range parameters are 18,51323 102,4155
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,4095313 0,406766
+Start of the optimization 2 :
+The number of iterations is 29
+ The value of the marginal posterior function is -95,30714
+ Optimized range parameters are 18,54824 102,4155
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 95,06543 94,42352
+The initial values of range parameters are 1,901309 1,88847
+Start of the optimization 1 :
+The number of iterations is 14
+ The value of the marginal posterior function is -95,1497
+ Optimized range parameters are 17,52307 94,42352
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,4053093 0,4025725
+Start of the optimization 2 :
+The number of iterations is 23
+ The value of the marginal posterior function is -95,14898
+ Optimized range parameters are 17,55379 94,42352
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 91,10625 90,49107
+The initial values of range parameters are 1,822125 1,809821
+Start of the optimization 1 :
+The number of iterations is 21
+ The value of the marginal posterior function is -93,31858
+ Optimized range parameters are 16,88561 90,49107
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,4011735 0,3984647
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -93,31829
+ Optimized range parameters are 16,83654 90,49107
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 95,06911 94,42717
+The initial values of range parameters are 1,901382 1,888543
+Start of the optimization 1 :
+The number of iterations is 11
+ The value of the marginal posterior function is -88,62383
+ Optimized range parameters are 17,19113 94,42717
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 0,3971213 0,3944398
+Start of the optimization 2 :
+The number of iterations is 23
+ The value of the marginal posterior function is -88,62286
+ Optimized range parameters are 17,21816 94,42717
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The upper bounds of the range parameters are 84,25512 83,6862
+The initial values of range parameters are 1,685102 1,673724
+Start of the optimization 1 :
+The number of iterations is 20
+ The value of the marginal posterior function is -90,56369
+ Optimized range parameters are 16,01921 83,6862
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+The initial values of range parameters are 0,39315 0,3904954
+Start of the optimization 2 :
+The number of iterations is 24
+ The value of the marginal posterior function is -90,56328
+ Optimized range parameters are 16,0189 83,6862
+ Optimized nugget parameter is 0
+ Convergence: FALSE
+</pre></div>
+</div>
+<p>Once the active learning process is finished, we obtain the final emulator for the
+logarithm of the unnormalized posterior, which is given by the
+<a class="reference internal" href="../../inference/active_learning.html#psimpy.inference.active_learning.ActiveLearning.approx_ln_pxl" title="psimpy.inference.active_learning.ActiveLearning.approx_ln_pxl"><code class="xref py py-meth docutils literal notranslate"><span class="pre">ActiveLearning.approx_ln_pxl()</span></code></a> method.</p>
+<p>We can then estimate the posterior using grid estimation or Metropolis Hastings
+estimation based on the emulator. An example is as follows. The contour plot shows
+the estimated posterior.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.inference</span> <span class="kn">import</span> <span class="n">GridEstimation</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">grid_estimator</span> <span class="o">=</span> <span class="n">GridEstimation</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">ln_pxl</span><span class="o">=</span><span class="n">active_learner</span><span class="o">.</span><span class="n">approx_ln_pxl</span><span class="p">)</span>
+<span class="n">posterior</span><span class="p">,</span> <span class="n">x_ndim</span> <span class="o">=</span> <span class="n">grid_estimator</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">nbins</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="c1"># initial training points</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">init_var_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">init_var_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">,</span>
+ <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'initial training points'</span><span class="p">)</span>
+<span class="c1"># actively picked training points</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">var_samples</span><span class="p">[</span><span class="n">n0</span><span class="p">:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">var_samples</span><span class="p">[</span><span class="n">n0</span><span class="p">:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'+'</span><span class="p">,</span>
+ <span class="n">zorder</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'iterative training points'</span><span class="p">)</span>
+
+<span class="c1"># estimated posterior based on the final emulator</span>
+<span class="n">posterior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">posterior</span> <span class="o"><</span> <span class="mf">1e-10</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">posterior</span><span class="p">)</span>
+<span class="n">contour</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">x_ndim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x_ndim</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">posterior</span><span class="p">),</span> <span class="n">levels</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contour</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Active learning'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x1'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'x2'</span><span class="p">)</span>
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+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="bayesian-inference">
+<span id="sphx-glr-auto-examples-inference-plot-bayes-inference-py"></span><h1>Bayesian inference<a class="headerlink" href="#bayesian-inference" title="Permalink to this heading"></a></h1>
+<p>This example shows how to perform Bayesian inference given the uniform prior</p>
+<p><span class="math notranslate nohighlight">\(p(\mathbf{x})=p(x_1,x_2)=0.01\)</span></p>
+<p>where <span class="math notranslate nohighlight">\(x_i \in [-5,5], i=1,2\)</span>, and likelihood</p>
+<p><span class="math notranslate nohighlight">\(L(\mathbf{x}|\mathbf{d})=\exp \left(-\frac{1}{100}\left(x_1-1\right)^2-\left(x_1^2-x_2\right)^2\right)\)</span>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">ndim</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">bounds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">],[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">]])</span>
+
+<span class="k">def</span> <span class="nf">prior</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="k">return</span> <span class="mf">0.01</span>
+
+<span class="k">def</span> <span class="nf">likelihood</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">100</span> <span class="o">-</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>To estimate the posterior using grid estimation, we need to import the
+<a class="reference internal" href="../../inference/bayes_inference.html#psimpy.inference.bayes_inference.GridEstimation" title="psimpy.inference.bayes_inference.GridEstimation"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridEstimation</span></code></a> class, create an instance, and call the
+<a class="reference internal" href="../../inference/bayes_inference.html#psimpy.inference.bayes_inference.GridEstimation.run" title="psimpy.inference.bayes_inference.GridEstimation.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">GridEstimation.run()</span></code></a> method.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.inference.bayes_inference</span> <span class="kn">import</span> <span class="n">GridEstimation</span>
+
+<span class="n">grid_estimator</span> <span class="o">=</span> <span class="n">GridEstimation</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">prior</span><span class="p">,</span> <span class="n">likelihood</span><span class="p">)</span>
+<span class="n">posterior</span><span class="p">,</span> <span class="n">x_ndim</span> <span class="o">=</span> <span class="n">grid_estimator</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">nbins</span><span class="o">=</span><span class="p">[</span><span class="mi">50</span><span class="p">,</span><span class="mi">40</span><span class="p">])</span>
+</pre></div>
+</div>
+<p>The following figure plots the estimated posterior.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="c1"># mask insignificant values</span>
+<span class="n">posterior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">posterior</span> <span class="o"><</span> <span class="mf">1e-10</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">posterior</span><span class="p">)</span>
+
+<span class="n">contour</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">x_ndim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x_ndim</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">posterior</span><span class="p">),</span> <span class="n">levels</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contour</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Grid estimation'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x1'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'x2'</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_bayes_inference_001.png" srcset="../../_images/sphx_glr_plot_bayes_inference_001.png" alt="Grid estimation" class = "sphx-glr-single-img"/><p>To estimate the posterior using Metropolis Hastings estimation, we need to import
+the <a class="reference internal" href="../../inference/bayes_inference.html#psimpy.inference.bayes_inference.MetropolisHastingsEstimation" title="psimpy.inference.bayes_inference.MetropolisHastingsEstimation"><code class="xref py py-class docutils literal notranslate"><span class="pre">MetropolisHastingsEstimation</span></code></a> class, create an instance, and call the
+<a class="reference internal" href="../../inference/bayes_inference.html#psimpy.inference.bayes_inference.MetropolisHastingsEstimation.run" title="psimpy.inference.bayes_inference.MetropolisHastingsEstimation.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">MetropolisHastingsEstimation.run()</span></code></a> method. The
+<a class="reference internal" href="../../inference/bayes_inference.html#psimpy.inference.bayes_inference.MetropolisHastingsEstimation.run" title="psimpy.inference.bayes_inference.MetropolisHastingsEstimation.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">MetropolisHastingsEstimation.run()</span></code></a> method has a parameter, <code class="docutils literal notranslate"><span class="pre">mh_sampler</span></code>,
+which takes an instance of <a class="reference internal" href="../../sampler/metropolis_hastings.html#psimpy.sampler.metropolis_hastings.MetropolisHastings" title="psimpy.sampler.metropolis_hastings.MetropolisHastings"><code class="xref py py-class docutils literal notranslate"><span class="pre">MetropolisHastings</span></code></a> as argument.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.inference.bayes_inference</span> <span class="kn">import</span> <span class="n">MetropolisHastingsEstimation</span>
+
+<span class="n">mh_estimator</span> <span class="o">=</span> <span class="n">MetropolisHastingsEstimation</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">prior</span><span class="p">,</span> <span class="n">likelihood</span><span class="p">)</span>
+
+<span class="c1"># create a mh_sampler</span>
+<span class="kn">from</span> <span class="nn">psimpy.sampler.metropolis_hastings</span> <span class="kn">import</span> <span class="n">MetropolisHastings</span>
+<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">multivariate_normal</span>
+
+<span class="n">init_state</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span>
+<span class="n">f_sample</span> <span class="o">=</span> <span class="n">multivariate_normal</span><span class="o">.</span><span class="n">rvs</span>
+<span class="n">nburn</span> <span class="o">=</span> <span class="mi">100</span>
+<span class="n">nthin</span> <span class="o">=</span> <span class="mi">10</span>
+<span class="n">seed</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="n">kwgs_f_sample</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'random_state'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="p">)}</span>
+
+<span class="n">mh_sampler</span> <span class="o">=</span> <span class="n">MetropolisHastings</span><span class="p">(</span><span class="n">ndim</span><span class="o">=</span><span class="n">ndim</span><span class="p">,</span> <span class="n">init_state</span><span class="o">=</span><span class="n">init_state</span><span class="p">,</span>
+ <span class="n">f_sample</span><span class="o">=</span><span class="n">f_sample</span><span class="p">,</span> <span class="n">bounds</span><span class="o">=</span><span class="n">bounds</span><span class="p">,</span> <span class="n">nburn</span><span class="o">=</span><span class="n">nburn</span><span class="p">,</span> <span class="n">nthin</span><span class="o">=</span><span class="n">nthin</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">,</span>
+ <span class="n">kwgs_f_sample</span><span class="o">=</span><span class="n">kwgs_f_sample</span><span class="p">)</span>
+
+<span class="n">nsamples</span> <span class="o">=</span> <span class="mi">5000</span>
+<span class="n">mh_samples</span><span class="p">,</span> <span class="n">mh_accept</span> <span class="o">=</span> <span class="n">mh_estimator</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">nsamples</span><span class="p">,</span> <span class="n">mh_sampler</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>The following figure plots the samples drawn from the unnormalized posterior,
+which can be used to estimate the posterior and its poperties.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">mh_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">mh_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'MH estimation'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x1'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'x2'</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_bayes_inference_002.png" srcset="../../_images/sphx_glr_plot_bayes_inference_002.png" alt="MH estimation" class = "sphx-glr-single-img"/><div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Besides <code class="docutils literal notranslate"><span class="pre">prior</span></code> and <code class="docutils literal notranslate"><span class="pre">likelihood</span></code>, one can also instantiate
+the <a class="reference internal" href="../../inference/bayes_inference.html#psimpy.inference.bayes_inference.MetropolisHastingsEstimation" title="psimpy.inference.bayes_inference.MetropolisHastingsEstimation"><code class="xref py py-class docutils literal notranslate"><span class="pre">MetropolisHastingsEstimation</span></code></a> class with</p>
+<blockquote>
+<div><ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">ln_prior</span></code> and <code class="docutils literal notranslate"><span class="pre">ln_likelihood</span></code>: Natural logarithm of <code class="docutils literal notranslate"><span class="pre">prior</span></code> and <code class="docutils literal notranslate"><span class="pre">likelihood</span></code>.</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">ln_pxl</span></code>: Natural logarithm of the product of <code class="docutils literal notranslate"><span class="pre">prior</span></code> and <code class="docutils literal notranslate"><span class="pre">likelihood</span></code>.</p></li>
+</ul>
+</div></blockquote>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 9.739 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-inference-plot-bayes-inference-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/44f0f511484d9bfd5f8d212c8f287e9e/plot_bayes_inference.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_bayes_inference.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/2d9fc590bc1edb11501b144674a0a949/plot_bayes_inference.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_bayes_inference.ipynb</span></code></a></p>
+</div>
+</div>
+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
+</section>
+
+
+ </div>
+ </div>
+ <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
+ <a href="index.html" class="btn btn-neutral float-left" title="Inference" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a>
+ <a href="plot_active_learning.html" class="btn btn-neutral float-right" title="Active learning" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a>
+ </div>
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+
+ <div role="contentinfo">
+ <p>© Copyright 2023, Hu Zhao.</p>
+ </div>
+
+ Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a
+ <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a>
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diff --git a/docs/source/_build/html/auto_examples/inference/sg_execution_times.html b/docs/source/_build/html/auto_examples/inference/sg_execution_times.html
new file mode 100644
index 0000000..f396b9a
--- /dev/null
+++ b/docs/source/_build/html/auto_examples/inference/sg_execution_times.html
@@ -0,0 +1,131 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
+
+ <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+ <title>Computation times — PSimPy's documentation</title>
+ <link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/sg_gallery.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/sg_gallery-binder.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/sg_gallery-dataframe.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/sg_gallery-rendered-html.css" type="text/css" />
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+ <!--[if lt IE 9]>
+ <script src="../../_static/js/html5shiv.min.js"></script>
+ <![endif]-->
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+ <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
+ <script src="../../_static/jquery.js"></script>
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+ <script src="../../_static/sphinx_highlight.js"></script>
+ <script src="../../_static/js/theme.js"></script>
+ <link rel="index" title="Index" href="../../genindex.html" />
+ <link rel="search" title="Search" href="../../search.html" />
+</head>
+
+<body class="wy-body-for-nav">
+ <div class="wy-grid-for-nav">
+ <nav data-toggle="wy-nav-shift" class="wy-nav-side">
+ <div class="wy-side-scroll">
+ <div class="wy-side-nav-search" >
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+ <form id="rtd-search-form" class="wy-form" action="../../search.html" method="get">
+ <input type="text" name="q" placeholder="Search docs" />
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+<li class="toctree-l1"><a class="reference internal" href="../../quickstart.html">Getting Started</a></li>
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+</ul>
+
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+
+ <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" >
+ <i data-toggle="wy-nav-top" class="fa fa-bars"></i>
+ <a href="../../index.html">psimpy</a>
+ </nav>
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+ <ul class="wy-breadcrumbs">
+ <li><a href="../../index.html" class="icon icon-home"></a></li>
+ <li class="breadcrumb-item active">Computation times</li>
+ <li class="wy-breadcrumbs-aside">
+ <a href="../../_sources/auto_examples/inference/sg_execution_times.rst.txt" rel="nofollow"> View page source</a>
+ </li>
+ </ul>
+ <hr/>
+</div>
+ <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+ <div itemprop="articleBody">
+
+ <section id="computation-times">
+<span id="sphx-glr-auto-examples-inference-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Permalink to this heading"></a></h1>
+<p><strong>02:22.392</strong> total execution time for <strong>auto_examples_inference</strong> files:</p>
+<table class="docutils align-default">
+<colgroup>
+<col style="width: 83%" />
+<col style="width: 10%" />
+<col style="width: 7%" />
+</colgroup>
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_active_learning.html#sphx-glr-auto-examples-inference-plot-active-learning-py"><span class="std std-ref">Active learning</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_active_learning.py</span></code>)</p></td>
+<td><p>02:12.653</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="plot_bayes_inference.html#sphx-glr-auto-examples-inference-plot-bayes-inference-py"><span class="std std-ref">Bayesian inference</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_bayes_inference.py</span></code>)</p></td>
+<td><p>00:09.739</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+
+ </div>
+ </div>
+ <footer>
+
+ <hr/>
+
+ <div role="contentinfo">
+ <p>© Copyright 2023, Hu Zhao.</p>
+ </div>
+
+ Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a
+ <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a>
+ provided by <a href="https://readthedocs.org">Read the Docs</a>.
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+
+</footer>
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+ jQuery(function () {
+ SphinxRtdTheme.Navigation.enable(true);
+ });
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+
+</body>
+</html>
\ No newline at end of file
diff --git a/docs/source/_build/html/auto_examples/sampler/index.html b/docs/source/_build/html/auto_examples/sampler/index.html
new file mode 100644
index 0000000..221c96f
--- /dev/null
+++ b/docs/source/_build/html/auto_examples/sampler/index.html
@@ -0,0 +1,151 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
+
+ <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+ <title>Sampler — PSimPy's documentation</title>
+ <link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/sg_gallery.css" type="text/css" />
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+ <link rel="stylesheet" href="../../_static/sg_gallery-dataframe.css" type="text/css" />
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+ <link rel="index" title="Index" href="../../genindex.html" />
+ <link rel="search" title="Search" href="../../search.html" />
+ <link rel="next" title="Saltelli sampling" href="plot_saltelli.html" />
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+ <input type="text" name="q" placeholder="Search docs" />
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+<li class="toctree-l1"><a class="reference internal" href="../../quickstart.html">Getting Started</a></li>
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+<li class="toctree-l1 current"><a class="reference internal" href="../../examples.html">Example Gallery</a><ul class="current">
+<li class="toctree-l2"><a class="reference internal" href="../emulator/index.html">Emulator</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../inference/index.html">Inference</a></li>
+<li class="toctree-l2 current"><a class="current reference internal" href="#">Sampler</a><ul>
+<li class="toctree-l3"><a class="reference internal" href="plot_saltelli.html">Saltelli sampling</a></li>
+<li class="toctree-l3"><a class="reference internal" href="plot_latin.html">Latin hypercube sampling</a></li>
+<li class="toctree-l3"><a class="reference internal" href="plot_metropolis_hastings.html">Metropolis Hastings sampling</a></li>
+</ul>
+</li>
+<li class="toctree-l2"><a class="reference internal" href="../sensitivity/index.html">Sensitivity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../simulator/index.html">Simulator</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../changelog.html">Changes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../refs.html">References</a></li>
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+ <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+ <div itemprop="articleBody">
+
+ <section id="sampler">
+<h1>Sampler<a class="headerlink" href="#sampler" title="Permalink to this heading"></a></h1>
+<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="Saltelli sampling"><img alt="Saltelli sampling" src="../../_images/sphx_glr_plot_saltelli_thumb.png" />
+<p><a class="reference internal" href="plot_saltelli.html#sphx-glr-auto-examples-sampler-plot-saltelli-py"><span class="std std-ref">Saltelli sampling</span></a></p>
+ <div class="sphx-glr-thumbnail-title">Saltelli sampling</div>
+</div><div class="sphx-glr-thumbcontainer" tooltip="Latin hypercube sampling"><img alt="Latin hypercube sampling" src="../../_images/sphx_glr_plot_latin_thumb.png" />
+<p><a class="reference internal" href="plot_latin.html#sphx-glr-auto-examples-sampler-plot-latin-py"><span class="std std-ref">Latin hypercube sampling</span></a></p>
+ <div class="sphx-glr-thumbnail-title">Latin hypercube sampling</div>
+</div><div class="sphx-glr-thumbcontainer" tooltip="Metropolis Hastings sampling"><img alt="Metropolis Hastings sampling" src="../../_images/sphx_glr_plot_metropolis_hastings_thumb.png" />
+<p><a class="reference internal" href="plot_metropolis_hastings.html#sphx-glr-auto-examples-sampler-plot-metropolis-hastings-py"><span class="std std-ref">Metropolis Hastings sampling</span></a></p>
+ <div class="sphx-glr-thumbnail-title">Metropolis Hastings sampling</div>
+</div></div><div class="toctree-wrapper compound">
+</div>
+<div class="sphx-glr-footer sphx-glr-footer-gallery docutils container">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/79d68b726b379b22e24e81ddc36bf480/sampler_python.zip"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">all</span> <span class="pre">examples</span> <span class="pre">in</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">sampler_python.zip</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/9be10f28a70b43dfd67c8234c21c2052/sampler_jupyter.zip"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">all</span> <span class="pre">examples</span> <span class="pre">in</span> <span class="pre">Jupyter</span> <span class="pre">notebooks:</span> <span class="pre">sampler_jupyter.zip</span></code></a></p>
+</div>
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+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
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diff --git a/docs/source/_build/html/auto_examples/sampler/plot_latin.html b/docs/source/_build/html/auto_examples/sampler/plot_latin.html
new file mode 100644
index 0000000..119f626
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+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
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+</div>
+<section class="sphx-glr-example-title" id="latin-hypercube-sampling">
+<span id="sphx-glr-auto-examples-sampler-plot-latin-py"></span><h1>Latin hypercube sampling<a class="headerlink" href="#latin-hypercube-sampling" title="Permalink to this heading"></a></h1>
+<p>This example shows how to draw samples using Latin hypercube sampling.</p>
+<p>For the illustration purpose, let’s have a look at a two-dimensional example
+where we have two random variables X and Y. Each is uniformly distributed in its
+range.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">ndim</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="c1"># range of X is 10 to 20, range of Y is -10 to 0</span>
+<span class="n">bounds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">10</span><span class="p">,</span><span class="mi">20</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
+</pre></div>
+</div>
+<p>Given this setting, we can import <code class="xref py py-class docutils literal notranslate"><span class="pre">Latin</span></code>, create an instance, and
+call the <code class="xref py py-meth docutils literal notranslate"><span class="pre">Latin.sample()</span></code> method to draw required number of samples</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.sampler</span> <span class="kn">import</span> <span class="n">LHS</span>
+
+<span class="c1"># setting seed leads to same samples every time when the codes are run</span>
+<span class="n">lhs_sampler</span> <span class="o">=</span> <span class="n">LHS</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
+<span class="n">lhs_samples</span> <span class="o">=</span> <span class="n">lhs_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nsamples</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>The samples are plotted in the following figure.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span> <span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">lhs_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">lhs_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'blue'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'X'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Latin hypercube samples (criterion='random')"</span><span class="p">)</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">visible</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">)</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_latin_001.png" srcset="../../_images/sphx_glr_plot_latin_001.png" alt="Latin hypercube samples (criterion='random')" class = "sphx-glr-single-img"/><p>There are different criterions to pick samples in each hypercube. The default
+is <cite>random</cite>, as used above. Other options are <cite>center</cite> and <cite>maximin</cite>. For instance,
+we can use the <cite>center</cite> criterion to draw <span class="math notranslate nohighlight">\(5\)</span> samples as follows:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">lhs_sampler</span> <span class="o">=</span> <span class="n">LHS</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">criterion</span><span class="o">=</span><span class="s1">'center'</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
+<span class="n">lhs_samples</span> <span class="o">=</span> <span class="n">lhs_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nsamples</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
+
+<span class="n">fig</span> <span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">lhs_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">lhs_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'blue'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'X'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Latin hypercube samples (criterion='center')"</span><span class="p">)</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">visible</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">)</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_latin_002.png" srcset="../../_images/sphx_glr_plot_latin_002.png" alt="Latin hypercube samples (criterion='center')" class = "sphx-glr-single-img"/><p>And we can use the <cite>maximin</cite> criterion as follows:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">lhs_sampler</span> <span class="o">=</span> <span class="n">LHS</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">criterion</span><span class="o">=</span><span class="s1">'maximin'</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">iteration</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
+<span class="n">lhs_samples</span> <span class="o">=</span> <span class="n">lhs_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nsamples</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
+
+<span class="n">fig</span> <span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">lhs_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">lhs_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'blue'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'X'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">bounds</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Latin hypercube samples (criterion='maximin')"</span><span class="p">)</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">visible</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">)</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_latin_003.png" srcset="../../_images/sphx_glr_plot_latin_003.png" alt="Latin hypercube samples (criterion='maximin')" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 0.270 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-sampler-plot-latin-py">
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+<p><a class="reference download internal" download="" href="../../_downloads/213f7cd5e82a62f8d1da8c3af9caab35/plot_latin.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_latin.py</span></code></a></p>
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+
+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-sampler-plot-metropolis-hastings-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="metropolis-hastings-sampling">
+<span id="sphx-glr-auto-examples-sampler-plot-metropolis-hastings-py"></span><h1>Metropolis Hastings sampling<a class="headerlink" href="#metropolis-hastings-sampling" title="Permalink to this heading"></a></h1>
+<p>This example shows how to draw samples using Metropolis Hastings sampling.
+The target probability distribution is</p>
+<p><span class="math notranslate nohighlight">\(p(\mathbf{x})=p(x_1, x_2) \propto \exp \left(-\frac{1}{100}\left(x_1-1\right)^2-\left(x_1^2-x_2\right)^2\right)\)</span></p>
+<p>where <span class="math notranslate nohighlight">\(x_1 \in [-5,5]\)</span> and <span class="math notranslate nohighlight">\(x_2 \in [-5,5]\)</span>.</p>
+<p>It should be noted that the right hand side of the equation is an unnormalized
+probability density function since its integral is not equal to <span class="math notranslate nohighlight">\(1\)</span>.
+This can happen, for example, when the normalization constant is unknown or difficult
+to compute.</p>
+<p>We can define the target probability distribution in Python as follows:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">target_dist</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="k">if</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">>=-</span><span class="mi">5</span> <span class="ow">and</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o"><=</span><span class="mi">5</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">>=-</span><span class="mi">5</span> <span class="ow">and</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o"><=</span><span class="mi">5</span><span class="p">):</span>
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mf">0.01</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
+ <span class="k">else</span><span class="p">:</span>
+ <span class="k">return</span> <span class="mi">0</span>
+</pre></div>
+</div>
+<p>The figure below shows how the target distribution looks like.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+
+<span class="n">x1_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mf">5.1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
+<span class="n">x2_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mf">5.1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
+
+<span class="n">target_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">),</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">)):</span>
+ <span class="n">x1</span> <span class="o">=</span> <span class="n">x1_values</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+ <span class="n">x2</span> <span class="o">=</span> <span class="n">x2_values</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
+ <span class="n">target_values</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">target_dist</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">]))</span>
+
+<span class="n">fig</span> <span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">x1_values</span><span class="p">,</span> <span class="n">x2_values</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">target_values</span><span class="p">),</span> <span class="n">levels</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'Blues'</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'(unnormalized) target distribution'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x_1$'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x_2$'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_metropolis_hastings_001.png" srcset="../../_images/sphx_glr_plot_metropolis_hastings_001.png" alt="(unnormalized) target distribution" class = "sphx-glr-single-img"/><p>To perform Metropolis Hastings sampling, we need to choose a proposal distribution
+which can be used to determine a new state <code class="docutils literal notranslate"><span class="pre">x'</span></code> given a current state <code class="docutils literal notranslate"><span class="pre">x</span></code> at
+each iteration. This is defined by the parameter <code class="docutils literal notranslate"><span class="pre">f_sample</span></code> which should be a
+function. A usual choice is to choose the new state from a Gaussian distribution
+centered at the current state.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">multivariate_normal</span>
+
+<span class="n">f_sample</span> <span class="o">=</span> <span class="n">multivariate_normal</span><span class="o">.</span><span class="n">rvs</span>
+<span class="c1"># make the samples reproducible</span>
+<span class="n">kwgs_f_sample</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'random_state'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="mi">1</span><span class="p">)}</span>
+</pre></div>
+</div>
+<p>Then, we create an instance of <a class="reference internal" href="../../sampler/metropolis_hastings.html#psimpy.sampler.metropolis_hastings.MetropolisHastings" title="psimpy.sampler.metropolis_hastings.MetropolisHastings"><code class="xref py py-class docutils literal notranslate"><span class="pre">MetropolisHastings</span></code></a> class.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.sampler.metropolis_hastings</span> <span class="kn">import</span> <span class="n">MetropolisHastings</span>
+
+<span class="n">mh_sampler</span> <span class="o">=</span> <span class="n">MetropolisHastings</span><span class="p">(</span><span class="n">ndim</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">init_state</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">]),</span>
+ <span class="n">f_sample</span><span class="o">=</span><span class="n">f_sample</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target_dist</span><span class="p">,</span> <span class="n">nburn</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">nthin</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
+ <span class="n">seed</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">kwgs_f_sample</span><span class="o">=</span><span class="n">kwgs_f_sample</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Next, we call the <a class="reference internal" href="../../sampler/metropolis_hastings.html#psimpy.sampler.metropolis_hastings.MetropolisHastings.sample" title="psimpy.sampler.metropolis_hastings.MetropolisHastings.sample"><code class="xref py py-meth docutils literal notranslate"><span class="pre">MetropolisHastings.sample()</span></code></a> method to draw required
+number of samples.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">mh_samples</span><span class="p">,</span> <span class="n">mh_accept</span> <span class="o">=</span> <span class="n">mh_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nsamples</span><span class="o">=</span><span class="mi">5000</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Acceptance ratio: "</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">mh_accept</span><span class="p">))</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Acceptance ratio: 0.3712
+</pre></div>
+</div>
+<p>The following figure shows how the samples look like.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+
+<span class="c1"># sphinx_gallery_thumbnail_number = 2</span>
+<span class="n">fig</span> <span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">x1_values</span><span class="p">,</span> <span class="n">x2_values</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">target_values</span><span class="p">),</span> <span class="n">levels</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'Blues'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">mh_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">mh_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">,</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Metropolis Hastings samples'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x_1$'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x_2$'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_metropolis_hastings_002.png" srcset="../../_images/sphx_glr_plot_metropolis_hastings_002.png" alt="Metropolis Hastings samples" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 4.769 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-sampler-plot-metropolis-hastings-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/50d02379952730d4e961434a77e23a62/plot_metropolis_hastings.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_metropolis_hastings.py</span></code></a></p>
+</div>
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+<p><a class="reference download internal" download="" href="../../_downloads/72c9aa931cd2028d7edb96ceb292a4a2/plot_metropolis_hastings.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_metropolis_hastings.ipynb</span></code></a></p>
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+ <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+ <div itemprop="articleBody">
+
+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-sampler-plot-saltelli-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="saltelli-sampling">
+<span id="sphx-glr-auto-examples-sampler-plot-saltelli-py"></span><h1>Saltelli sampling<a class="headerlink" href="#saltelli-sampling" title="Permalink to this heading"></a></h1>
+<p>This example shows how to draw samples using Saltelli sampling.
+Assume that there is a three-dimensional problem where X, Y, and Z are the
+three random variables.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">ndim</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="c1"># range of X is 10 to 20, range of Y is 100 to 200, range of Z is 1000 to 2000</span>
+<span class="n">bounds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">10</span><span class="p">,</span><span class="mi">20</span><span class="p">],</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span><span class="mi">200</span><span class="p">],</span> <span class="p">[</span><span class="mi">1000</span><span class="p">,</span><span class="mi">2000</span><span class="p">]])</span>
+</pre></div>
+</div>
+<p>Given this setting, we can import <a class="reference internal" href="../../sampler/saltelli.html#psimpy.sampler.saltelli.Saltelli" title="psimpy.sampler.saltelli.Saltelli"><code class="xref py py-class docutils literal notranslate"><span class="pre">Saltelli</span></code></a>, create an instance, and
+call the <a class="reference internal" href="../../sampler/saltelli.html#psimpy.sampler.saltelli.Saltelli.sample" title="psimpy.sampler.saltelli.Saltelli.sample"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Saltelli.sample()</span></code></a> method to draw required number of samples.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.sampler</span> <span class="kn">import</span> <span class="n">Saltelli</span>
+
+<span class="n">saltelli_sampler</span> <span class="o">=</span> <span class="n">Saltelli</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">saltelli_samples</span> <span class="o">=</span> <span class="n">saltelli_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nbase</span><span class="o">=</span><span class="mi">128</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>In above codes, we set <code class="docutils literal notranslate"><span class="pre">calc_second_order</span></code> to <cite>False</cite>. It means that picked
+samples can be used in following Sobol’ analysis to compute first-order and
+total-effect Sobol’ indices but not second-order Sobol’ indices. It leads to
+<span class="math notranslate nohighlight">\(nbase*(ndim+2)\)</span> samples as shown below</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'X'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">'Z'</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">'Number of samples: '</span><span class="p">,</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="nb">len</span><span class="p">(</span><span class="n">saltelli_samples</span><span class="p">)</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_saltelli_001.png" srcset="../../_images/sphx_glr_plot_saltelli_001.png" alt="plot saltelli" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Number of samples: 640
+</pre></div>
+</div>
+<p>If we want to draw samples which can also be used to compute second-order
+Sobol’ indices, we need to set <code class="docutils literal notranslate"><span class="pre">calc_second_order</span></code> to <cite>True</cite>.
+It leads to <span class="math notranslate nohighlight">\(nbase*(2*ndim+2)\)</span> samples.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">saltelli_sampler</span> <span class="o">=</span> <span class="n">Saltelli</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">saltelli_samples</span> <span class="o">=</span> <span class="n">saltelli_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nbase</span><span class="o">=</span><span class="mi">128</span><span class="p">)</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'X'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">'Z'</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s1">'Number of samples: '</span><span class="p">,</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="nb">len</span><span class="p">(</span><span class="n">saltelli_samples</span><span class="p">)</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_saltelli_002.png" srcset="../../_images/sphx_glr_plot_saltelli_002.png" alt="plot saltelli" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Number of samples: 1024
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>If one has a two-dimensional problem, there is no need to set
+<code class="docutils literal notranslate"><span class="pre">calc_second_order</span></code> to <cite>True</cite>. The reason is that the second-order Sobol’
+index can be directly computed based on the first-order and total-effect
+Sobol’ index in that case.</p>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 0.309 seconds)</p>
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+<p><a class="reference download internal" download="" href="../../_downloads/6dd07bf8f59ed369971c33b38f420079/plot_saltelli.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_saltelli.py</span></code></a></p>
+</div>
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+<p><a class="reference download internal" download="" href="../../_downloads/e0b3048771af654ea83c8bb3f5bf6c0c/plot_saltelli.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_saltelli.ipynb</span></code></a></p>
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+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
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+<span id="sphx-glr-auto-examples-sampler-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Permalink to this heading"></a></h1>
+<p><strong>00:05.348</strong> total execution time for <strong>auto_examples_sampler</strong> files:</p>
+<table class="docutils align-default">
+<colgroup>
+<col style="width: 84%" />
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+</colgroup>
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_metropolis_hastings.html#sphx-glr-auto-examples-sampler-plot-metropolis-hastings-py"><span class="std std-ref">Metropolis Hastings sampling</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_metropolis_hastings.py</span></code>)</p></td>
+<td><p>00:04.769</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="plot_saltelli.html#sphx-glr-auto-examples-sampler-plot-saltelli-py"><span class="std std-ref">Saltelli sampling</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_saltelli.py</span></code>)</p></td>
+<td><p>00:00.309</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_latin.html#sphx-glr-auto-examples-sampler-plot-latin-py"><span class="std std-ref">Latin hypercube sampling</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_latin.py</span></code>)</p></td>
+<td><p>00:00.270</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
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+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-sensitivity-plot-sobol-analyze-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="sobol-analysis">
+<span id="sphx-glr-auto-examples-sensitivity-plot-sobol-analyze-py"></span><h1>Sobol’ analysis<a class="headerlink" href="#sobol-analysis" title="Permalink to this heading"></a></h1>
+<p>This example shows how to compute Sobol’ sensitivity indices using
+the <a class="reference internal" href="../../sensitivity/sobol.html#psimpy.sensitivity.sobol.SobolAnalyze" title="psimpy.sensitivity.sobol.SobolAnalyze"><code class="xref py py-class docutils literal notranslate"><span class="pre">SobolAnalyze</span></code></a> class.</p>
+<p>Let us consider the Ishigami function <span id="id1">[<a class="reference internal" href="../../refs.html#id9" title="Loic Le Gratiet, Claire Cannamela, and Bertrand Iooss. A Bayesian approach for global sensitivity analysis of (multifidelity) computer codes. SIAM/ASA Journal on Uncertainty Quantification, 2(1):336–363, 2014. doi:10.1137/130926869.">Le Gratiet <em>et al.</em>, 2014</a>]</span></p>
+<p><span class="math notranslate nohighlight">\(y=f\left(x_1, x_2, x_3\right)=\sin \left(x_1\right)+7 \sin \left(\mathrm{x}_2\right)^2+0.1 x_3^4 \sin \left(x_1\right)\)</span></p>
+<p>where <span class="math notranslate nohighlight">\(x_i\)</span> is uniformly distributed on <span class="math notranslate nohighlight">\([-\pi, \pi]\)</span> for <span class="math notranslate nohighlight">\(i \in [1,2,3]\)</span>. We define the function as follows:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">,</span><span class="n">x3</span><span class="p">):</span>
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mf">0.1</span><span class="o">*</span><span class="n">x3</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>The variation of each <span class="math notranslate nohighlight">\(x_i\)</span> leads to the variation of <span class="math notranslate nohighlight">\(y\)</span>. Sobol’ analysis is used to quantify the contribution
+of each <span class="math notranslate nohighlight">\(x_i\)</span> and their interactions to the variation of <span class="math notranslate nohighlight">\(y\)</span>. Steps of a Sobol’ analysis are:</p>
+<blockquote>
+<div><ol class="arabic simple">
+<li><p>Draw samples of input.</p></li>
+<li><p>Evaluate the model at each input sample to obtain model output.</p></li>
+<li><p>Compute Sobol’ indices based on model outputs.</p></li>
+</ol>
+</div></blockquote>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.sampler</span> <span class="kn">import</span> <span class="n">Saltelli</span>
+<span class="kn">from</span> <span class="nn">psimpy.sensitivity</span> <span class="kn">import</span> <span class="n">SobolAnalyze</span>
+
+<span class="n">ndim</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">bounds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">]])</span>
+<span class="n">calc_second_order</span> <span class="o">=</span> <span class="kc">False</span>
+<span class="n">nbase</span> <span class="o">=</span> <span class="mi">1024</span>
+
+<span class="c1"># draw Saltelli samples</span>
+<span class="n">saltelli_sampler</span> <span class="o">=</span> <span class="n">Saltelli</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="p">)</span>
+<span class="n">saltelli_samples</span> <span class="o">=</span> <span class="n">saltelli_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nbase</span><span class="p">)</span>
+
+<span class="c1"># Evaluate the model to obtain model outputs</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">saltelli_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">saltelli_samples</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">])</span>
+
+<span class="c1"># Compute Sobol' indices</span>
+<span class="n">sobol_analyzer</span> <span class="o">=</span> <span class="n">SobolAnalyze</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">S_res</span> <span class="o">=</span> <span class="n">sobol_analyzer</span><span class="o">.</span><span class="n">run</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Estimated first-order Sobol' index: "</span><span class="p">,</span> <span class="n">S_res</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">][:,</span><span class="mi">0</span><span class="p">])</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Analytical solution: [0.314, 0.442, 0]"</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Estimated first-order Sobol' index: [0.31430871 0.44545644 0.01325799]
+Analytical solution: [0.314, 0.442, 0]
+</pre></div>
+</div>
+<p>Figure below shows the results. The bar plots represent estimated first-order
+and total-effect Sobol’ indices, as well as their 95% confidence interval. The
+red stars represent true values of the first-order Sobol’ indices.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">name_vars</span> <span class="o">=</span> <span class="p">[</span><span class="sa">r</span><span class="s1">'$x_1$'</span><span class="p">,</span><span class="sa">r</span><span class="s1">'$x_2$'</span><span class="p">,</span><span class="sa">r</span><span class="s1">'$x_3$'</span><span class="p">]</span>
+<span class="n">x</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'S1'</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span> <span class="s1">'ST'</span><span class="p">:[</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">8</span><span class="p">]}</span>
+<span class="n">color</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'S1'</span><span class="p">:</span><span class="s1">'#69b3a2'</span><span class="p">,</span> <span class="s1">'ST'</span><span class="p">:</span><span class="s1">'#3399e6'</span><span class="p">}</span>
+<span class="n">width</span><span class="o">=</span><span class="mf">0.8</span>
+<span class="n">capsize</span><span class="o">=</span><span class="mi">3</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+
+<span class="c1"># estimated first-order Sobol' indices and their confidence interval</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">],</span> <span class="n">S_res</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">][:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="n">width</span><span class="p">,</span> <span class="n">yerr</span><span class="o">=</span><span class="n">S_res</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">][:,</span><span class="mi">2</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">],</span> <span class="n">capsize</span><span class="o">=</span><span class="n">capsize</span><span class="p">)</span>
+
+<span class="n">ax1</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.7</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">4.5</span><span class="p">,</span><span class="mf">7.5</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">name_vars</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">,</span><span class="n">bottom</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">"y"</span><span class="p">,</span> <span class="n">labelcolor</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'grey'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.25</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'First-order index'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">])</span>
+
+<span class="c1"># analytical solution for first-order indices</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.314</span><span class="p">,</span> <span class="mf">0.442</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'red'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'*'</span><span class="p">)</span>
+
+
+<span class="c1"># estimated total-effect Sobol' indices and their confidence interval</span>
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">],</span> <span class="n">S_res</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">][:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="n">width</span><span class="p">,</span> <span class="n">yerr</span><span class="o">=</span><span class="n">S_res</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">][:,</span><span class="mi">2</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">],</span> <span class="n">capsize</span><span class="o">=</span><span class="n">capsize</span><span class="p">)</span>
+
+<span class="n">ax2</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.7</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">4.5</span><span class="p">,</span><span class="mf">7.5</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">,</span><span class="n">bottom</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">"y"</span><span class="p">,</span> <span class="n">labelcolor</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'grey'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.25</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Total-effect index'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">])</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_sobol_analyze_001.png" srcset="../../_images/sphx_glr_plot_sobol_analyze_001.png" alt="plot sobol analyze" class = "sphx-glr-single-img"/><div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The interaction between one parameter and all other parameters can be
+quantified by the difference between its first-order and total-effet Sobol’
+indices. If detailed information on interaction between any two parameters
+is needed, one can set <code class="docutils literal notranslate"><span class="pre">calc_second_order</span></code> to <cite>True</cite>.</p>
+</div>
+<p><strong>Many models of interest in real-world applications are very computationally
+expensive. In that case, we resort to emulation techniques in order to efficiently
+conduct Sobol’ analysis.</strong> See <span id="id2">Zhao <em>et al.</em> [<a class="reference internal" href="../../refs.html#id14" title="H. Zhao, F. Amann, and J. Kowalski. Emulator-based global sensitivity analysis for flow-like landslide run-out models. Landslides, ():, 2021. doi:https://doi.org/10.1007/s10346-021-01690-w.">2021</a>]</span> for the detailed theory.</p>
+<p>Steps of such an analysis are:</p>
+<blockquote>
+<div><ol class="arabic simple">
+<li><p>Draw input samples for emulator training.</p></li>
+<li><p>Evaluate the model at each training input point to obtain training data.</p></li>
+<li><p>Train an emulator for the computationally expensive model of interest.</p></li>
+<li><p>Draw samples of input for Sobol’ analysis.</p></li>
+<li><p>Evaluate the emulator at each input sample at step 3 to obtain approximated model outputs.</p></li>
+<li><p>Compute Sobol’ indices based on approximated model outputs.</p></li>
+</ol>
+</div></blockquote>
+<p>Assume that the above Ishigami function is computationally expensive, an emulator-based
+Sobol’ analysis can be conducted as follows:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">psimpy.sampler</span> <span class="kn">import</span> <span class="n">LHS</span>
+<span class="kn">from</span> <span class="nn">psimpy.emulator</span> <span class="kn">import</span> <span class="n">ScalarGaSP</span>
+<span class="kn">from</span> <span class="nn">psimpy.sampler</span> <span class="kn">import</span> <span class="n">Saltelli</span>
+<span class="kn">from</span> <span class="nn">psimpy.sensitivity</span> <span class="kn">import</span> <span class="n">SobolAnalyze</span>
+
+<span class="c1"># define the model</span>
+<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">,</span><span class="n">x3</span><span class="p">):</span>
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mf">0.1</span><span class="o">*</span><span class="n">x3</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
+
+<span class="n">ndim</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">bounds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">]])</span>
+<span class="n">seed</span> <span class="o">=</span> <span class="mi">2</span>
+
+<span class="c1"># draw 100 input samples for emulator training using Latin hypercube sampling</span>
+<span class="n">lhs_sampler</span> <span class="o">=</span> <span class="n">LHS</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">seed</span><span class="p">,</span> <span class="n">criterion</span><span class="o">=</span><span class="s1">'maximin'</span><span class="p">,</span> <span class="n">iteration</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span>
+<span class="n">design</span> <span class="o">=</span> <span class="n">lhs_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nsamples</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+
+<span class="c1"># evaluate the model at each training input point</span>
+<span class="n">response</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">design</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">design</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">design</span><span class="p">[:,</span><span class="mi">2</span><span class="p">])</span>
+
+<span class="c1"># train an emulator</span>
+<span class="n">scalar_gasp</span> <span class="o">=</span> <span class="n">ScalarGaSP</span><span class="p">(</span><span class="n">ndim</span><span class="p">)</span>
+<span class="n">scalar_gasp</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">design</span><span class="p">,</span> <span class="n">response</span><span class="p">)</span>
+
+<span class="c1"># the trained emulator needs to be validated before moving on</span>
+<span class="c1"># here we leave out this step for simplicity...</span>
+
+<span class="c1"># draw samples of input for Sobol' analysis</span>
+<span class="n">calc_second_order</span> <span class="o">=</span> <span class="kc">False</span>
+<span class="n">nbase</span> <span class="o">=</span> <span class="mi">1024</span>
+
+<span class="n">saltelli_sampler</span> <span class="o">=</span> <span class="n">Saltelli</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="p">)</span>
+<span class="n">saltelli_samples</span> <span class="o">=</span> <span class="n">saltelli_sampler</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">nbase</span><span class="p">)</span>
+
+<span class="c1"># evaluate the emulator to obtain approximated model output.</span>
+<span class="c1"># in order to take emulator-induced uncertainty into account, we draw 50 realizations</span>
+<span class="c1"># from the emulator</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">scalar_gasp</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">saltelli_samples</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
+
+<span class="c1"># Compute Sobol' indices based on approximated model outputs</span>
+<span class="n">sobol_analyzer</span> <span class="o">=</span> <span class="n">SobolAnalyze</span><span class="p">(</span><span class="n">ndim</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">calc_second_order</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
+<span class="n">S_res</span> <span class="o">=</span> <span class="n">sobol_analyzer</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">mode</span><span class="o">=</span><span class="s1">'parallel'</span><span class="p">,</span> <span class="n">max_workers</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Emulator-based first-order Sobol' index: "</span><span class="p">,</span> <span class="n">S_res</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">][:,</span><span class="mi">0</span><span class="p">])</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Analytical solution: [0.314, 0.442, 0]"</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>The upper bounds of the range parameters are 173,6849 174,6991 174,3157
+The initial values of range parameters are 3,473697 3,493981 3,486314
+Start of the optimization 1 :
+The number of iterations is 14
+ The value of the marginal posterior function is -284,0714
+ Optimized range parameters are 3,183999 2,786247 4,315419
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+The initial values of range parameters are 1,721972 1,732027 1,728226
+Start of the optimization 2 :
+The number of iterations is 14
+ The value of the marginal posterior function is -284,0714
+ Optimized range parameters are 3,183999 2,786247 4,315419
+ Optimized nugget parameter is 0
+ Convergence: TRUE
+
+
+Emulator-based first-order Sobol' index: [0.29064086 0.46853437 0.01096656]
+Analytical solution: [0.314, 0.442, 0]
+</pre></div>
+</div>
+<p>The following figure shows how the results of emulator-based Sobol’ analysis
+look like.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">name_vars</span> <span class="o">=</span> <span class="p">[</span><span class="sa">r</span><span class="s1">'$x_1$'</span><span class="p">,</span><span class="sa">r</span><span class="s1">'$x_2$'</span><span class="p">,</span><span class="sa">r</span><span class="s1">'$x_3$'</span><span class="p">]</span>
+<span class="n">x</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'S1'</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span> <span class="s1">'ST'</span><span class="p">:[</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">8</span><span class="p">]}</span>
+<span class="n">color</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'S1'</span><span class="p">:</span><span class="s1">'#69b3a2'</span><span class="p">,</span> <span class="s1">'ST'</span><span class="p">:</span><span class="s1">'#3399e6'</span><span class="p">}</span>
+<span class="n">width</span><span class="o">=</span><span class="mf">0.8</span>
+<span class="n">capsize</span><span class="o">=</span><span class="mi">3</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+
+<span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">],</span> <span class="n">S_res</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">][:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="n">width</span><span class="p">,</span> <span class="n">yerr</span><span class="o">=</span><span class="n">S_res</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">][:,</span><span class="mi">2</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">],</span> <span class="n">capsize</span><span class="o">=</span><span class="n">capsize</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.7</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">4.5</span><span class="p">,</span><span class="mf">7.5</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">name_vars</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">,</span><span class="n">bottom</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">"y"</span><span class="p">,</span> <span class="n">labelcolor</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">])</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'grey'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.25</span><span class="p">)</span>
+<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'First-order index'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">])</span>
+
+<span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="s1">'S1'</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.314</span><span class="p">,</span> <span class="mf">0.442</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'red'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'*'</span><span class="p">)</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">],</span> <span class="n">S_res</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">][:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">width</span><span class="o">=</span><span class="n">width</span><span class="p">,</span> <span class="n">yerr</span><span class="o">=</span><span class="n">S_res</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">][:,</span><span class="mi">2</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">],</span> <span class="n">capsize</span><span class="o">=</span><span class="n">capsize</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.7</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">4.5</span><span class="p">,</span><span class="mf">7.5</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s1">'both'</span><span class="p">,</span><span class="n">bottom</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">"y"</span><span class="p">,</span> <span class="n">labelcolor</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">])</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'major'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'grey'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.25</span><span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Total-effect index'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">[</span><span class="s1">'ST'</span><span class="p">])</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_sobol_analyze_002.png" srcset="../../_images/sphx_glr_plot_sobol_analyze_002.png" alt="plot sobol analyze" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 16.499 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-sensitivity-plot-sobol-analyze-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/d3e17d759872695d3b37be8319e77dbf/plot_sobol_analyze.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_sobol_analyze.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/e6e32deb84b9140224f7120b34bdafda/plot_sobol_analyze.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_sobol_analyze.ipynb</span></code></a></p>
+</div>
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+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
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+ <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
+ <a href="index.html" class="btn btn-neutral float-left" title="Sensitivity" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a>
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new file mode 100644
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+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
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+ <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+ <title>Computation times — PSimPy's documentation</title>
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+ <script src="../../_static/js/html5shiv.min.js"></script>
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+ <link rel="index" title="Index" href="../../genindex.html" />
+ <link rel="search" title="Search" href="../../search.html" />
+</head>
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+<body class="wy-body-for-nav">
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+ <li class="breadcrumb-item active">Computation times</li>
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+ <a href="../../_sources/auto_examples/sensitivity/sg_execution_times.rst.txt" rel="nofollow"> View page source</a>
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+ <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+ <div itemprop="articleBody">
+
+ <section id="computation-times">
+<span id="sphx-glr-auto-examples-sensitivity-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Permalink to this heading"></a></h1>
+<p><strong>00:16.499</strong> total execution time for <strong>auto_examples_sensitivity</strong> files:</p>
+<table class="docutils align-default">
+<colgroup>
+<col style="width: 83%" />
+<col style="width: 10%" />
+<col style="width: 7%" />
+</colgroup>
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_sobol_analyze.html#sphx-glr-auto-examples-sensitivity-plot-sobol-analyze-py"><span class="std std-ref">Sobol’ analysis</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_sobol_analyze.py</span></code>)</p></td>
+<td><p>00:16.499</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+
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+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
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+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
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+
+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-simulator-plot-mass-point-model-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="mass-point-model">
+<span id="sphx-glr-auto-examples-simulator-plot-mass-point-model-py"></span><h1>Mass Point Model<a class="headerlink" href="#mass-point-model" title="Permalink to this heading"></a></h1>
+<p>This example shows how to simulate the movement of a masspoint on a topography
+using <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel" title="psimpy.simulator.mass_point_model.MassPointModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">MassPointModel</span></code></a>.</p>
+<p>First, import the class <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel" title="psimpy.simulator.mass_point_model.MassPointModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">MassPointModel</span></code></a> and create an instance by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">MassPointModel</span>
+
+<span class="n">mpm</span> <span class="o">=</span> <span class="n">MassPointModel</span><span class="p">()</span>
+</pre></div>
+</div>
+<dl class="simple">
+<dt>Required inputs for a simulation using <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel" title="psimpy.simulator.mass_point_model.MassPointModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">MassPointModel</span></code></a> include:</dt><dd><ol class="arabic simple">
+<li><p>topographic data: digital elevation model (in <cite>ESRI ascii</cite> format)</p></li>
+<li><p>friction coefficients: coulomb friction and turbulent friction coefficients</p></li>
+<li><p>initial state: initial location and initial velocity of the masspoint</p></li>
+<li><p>computational parameters: such as time step, end time, etc.</p></li>
+</ol>
+</dd>
+</dl>
+<p>The synthetic topography <code class="docutils literal notranslate"><span class="pre">synthetic_topo.asc</span></code> is used here for illustration.
+It is located at the <cite>tests/data/</cite> folder.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+<span class="kn">import</span> <span class="nn">linecache</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">dir_data</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="s1">'../../../tests/data/'</span><span class="p">)</span>
+<span class="n">elevation</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'synthetic_topo.asc'</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>You may need to modify <code class="docutils literal notranslate"><span class="pre">dir_data</span></code> according to where you save
+<code class="docutils literal notranslate"><span class="pre">synthetic_topo.asc</span></code> on your local machine.</p>
+</div>
+<p>We can load the elevation data and visulize it. The figure below shows how
+the topography looks like, as well as the initial location of the masspoint
+(noted by the red dot).</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">header</span> <span class="o">=</span> <span class="p">[</span><span class="n">linecache</span><span class="o">.</span><span class="n">getline</span><span class="p">(</span><span class="n">elevation</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">)]</span>
+<span class="n">header_values</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">split</span><span class="p">()[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">strip</span><span class="p">())</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">header</span><span class="p">]</span>
+<span class="n">ncols</span><span class="p">,</span> <span class="n">nrows</span><span class="p">,</span> <span class="n">xll</span><span class="p">,</span> <span class="n">yll</span><span class="p">,</span> <span class="n">cellsize</span> <span class="o">=</span> <span class="n">header_values</span>
+<span class="n">ncols</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ncols</span><span class="p">)</span>
+<span class="n">nrows</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">nrows</span><span class="p">)</span>
+
+<span class="n">x_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">xll</span><span class="p">,</span> <span class="n">xll</span><span class="o">+</span><span class="p">(</span><span class="n">cellsize</span><span class="o">*</span><span class="n">ncols</span><span class="p">),</span> <span class="n">cellsize</span><span class="p">)</span>
+<span class="n">y_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">yll</span><span class="p">,</span> <span class="n">yll</span><span class="o">+</span><span class="p">(</span><span class="n">cellsize</span><span class="o">*</span><span class="n">nrows</span><span class="p">),</span> <span class="n">cellsize</span><span class="p">)</span>
+
+<span class="n">z_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="n">elevation</span><span class="p">,</span> <span class="n">skiprows</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>
+<span class="n">z_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">rot90</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">z_values</span><span class="p">))</span>
+
+<span class="c1"># initial location</span>
+<span class="n">x0</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">y0</span> <span class="o">=</span> <span class="mi">2000</span>
+<span class="n">z0</span> <span class="o">=</span> <span class="n">z_values</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">x0</span><span class="o">/</span><span class="n">cellsize</span><span class="p">)]</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span> <span class="n">height_ratios</span><span class="o">=</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
+
+<span class="n">fig0</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">x_values</span><span class="p">,</span> <span class="n">y_values</span><span class="p">,</span> <span class="n">z_values</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'synthetic topography'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">y0</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+<span class="n">cbar</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">fig0</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'</span><span class="si">%d</span><span class="s1">'</span><span class="p">,</span> <span class="n">orientation</span><span class="o">=</span><span class="s1">'horizontal'</span><span class="p">,</span>
+ <span class="n">fraction</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">pad</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
+
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_values</span><span class="p">,</span> <span class="n">z_values</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:])</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5000</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'cross section at any y'</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_mass_point_model_001.png" srcset="../../_images/sphx_glr_plot_mass_point_model_001.png" alt="synthetic topography, cross section at any y" class = "sphx-glr-single-img"/><p>We set the friction coefficients as</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">mu</span> <span class="o">=</span> <span class="mf">0.15</span>
+<span class="n">xi</span> <span class="o">=</span> <span class="mi">1000</span>
+</pre></div>
+</div>
+<p>Given above topography, initial location, and friction coefficients, we can
+call the <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel.run" title="psimpy.simulator.mass_point_model.MassPointModel.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">MassPointModel.run()</span></code></a> method to perform a simulation. Other
+parameters are set to their default values. (we suppress raised warnings )
+(other parameters are set to their default values).</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">warnings</span>
+
+<span class="n">warnings</span><span class="o">.</span><span class="n">filterwarnings</span><span class="p">(</span><span class="s2">"ignore"</span><span class="p">)</span>
+<span class="n">output</span> <span class="o">=</span> <span class="n">mpm</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">elevation</span><span class="o">=</span><span class="n">elevation</span><span class="p">,</span> <span class="n">coulomb_friction</span><span class="o">=</span><span class="n">mu</span><span class="p">,</span> <span class="n">turbulent_friction</span><span class="o">=</span><span class="n">xi</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="n">x0</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">y0</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>The simulation returns time history of the mass point’s location and velocity.
+Following plots show the simulation results.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'time (s)'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"x-t plot"</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">output</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">output</span><span class="p">[:,</span><span class="mi">1</span><span class="p">])</span>
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'time (s)'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'velocity'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"v-t plot"</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">output</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">output</span><span class="p">[:,</span><span class="mi">5</span><span class="p">])</span>
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'velocity'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"x-v plot"</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">output</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">output</span><span class="p">[:,</span><span class="mi">5</span><span class="p">])</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_mass_point_model_002.png" srcset="../../_images/sphx_glr_plot_mass_point_model_002.png" alt="x-t plot, v-t plot, x-v plot" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 1.550 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-simulator-plot-mass-point-model-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/98a9fafc5e8001527da7060b5766f5b2/plot_mass_point_model.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_mass_point_model.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/201f9a51be1f2378acae03891235a84e/plot_mass_point_model.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_mass_point_model.ipynb</span></code></a></p>
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+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
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+<li class="toctree-l2 current"><a class="reference internal" href="index.html">Simulator</a><ul class="current">
+<li class="toctree-l3"><a class="reference internal" href="plot_mass_point_model.html">Mass Point Model</a></li>
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+<li class="toctree-l3 current"><a class="current reference internal" href="#">Ravaflow24 Mixture Model</a></li>
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+
+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-simulator-plot-ravaflow24-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="ravaflow24-mixture-model">
+<span id="sphx-glr-auto-examples-simulator-plot-ravaflow24-py"></span><h1>Ravaflow24 Mixture Model<a class="headerlink" href="#ravaflow24-mixture-model" title="Permalink to this heading"></a></h1>
+<p>This example shows how to simulate mass flow on a topography using
+<a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a>.</p>
+<p>First, import the class <a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a> by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">Ravaflow24Mixture</span>
+</pre></div>
+</div>
+<p>To create an instance of this class, we must specify the parameter <code class="docutils literal notranslate"><span class="pre">dir_sim</span></code>.
+It represents the directory in which output files generated by r.avaflow will be
+saved.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+
+<span class="c1"># Here we create a folder called `temp_Ravaflow24Mixture_example` to save output</span>
+<span class="c1"># files generated by r.avaflow</span>
+<span class="n">cwd</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">()</span>
+<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="s1">'temp_Ravaflow24Mixture_example'</span><span class="p">):</span>
+ <span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="s1">'temp_Ravaflow24Mixture_example'</span><span class="p">)</span>
+<span class="n">dir_sim</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">cwd</span><span class="p">,</span> <span class="s1">'temp_Ravaflow24Mixture_example'</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now, we can create an instance of <a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a> by given
+<code class="docutils literal notranslate"><span class="pre">dir_sim</span></code> and leaving other parameters to their default values
+(Other parameters include <code class="docutils literal notranslate"><span class="pre">time_step</span></code>, <code class="docutils literal notranslate"><span class="pre">time_end</span></code>, <code class="docutils literal notranslate"><span class="pre">curvature_control</span></code>,
+<code class="docutils literal notranslate"><span class="pre">entrainment_control</span></code>, <code class="docutils literal notranslate"><span class="pre">stopping_control</span></code>, etc.)</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">voellmy_model</span> <span class="o">=</span> <span class="n">Ravaflow24Mixture</span><span class="p">(</span><span class="n">dir_sim</span><span class="p">)</span>
+</pre></div>
+</div>
+<dl class="simple">
+<dt>To run a simulation using above <cite>voellmy_model</cite>, one needs to specify</dt><dd><ol class="arabic simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">elevation</span></code> – Name of elevation raster file (including its path).</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">hrelease</span></code> – Name of release height raster file (including its path).</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">prefix</span></code> – Prefix required by r.avaflow to name output files.</p></li>
+<li><p>If <code class="docutils literal notranslate"><span class="pre">elevation</span></code> is not a georeferenced file which can be used to create
+a <cite>GRASS Location</cite>, <code class="docutils literal notranslate"><span class="pre">EPSG</span></code> must be provided.</p></li>
+</ol>
+</dd>
+</dl>
+<p>Other optional parameters include <code class="docutils literal notranslate"><span class="pre">internal_friction</span></code>, <code class="docutils literal notranslate"><span class="pre">basal_friction</span></code>,
+<code class="docutils literal notranslate"><span class="pre">turbulent_friction</span></code>, <code class="docutils literal notranslate"><span class="pre">entrainment_coef</span></code> etc.</p>
+<p>The synthetic topography <code class="docutils literal notranslate"><span class="pre">synthetic_topo.tif</span></code> and release mass <code class="docutils literal notranslate"><span class="pre">synthetic_rel.tif</span></code>
+are used here for illustration. They are located at the <cite>/tests/data/</cite> folder.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">dir_data</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="s1">'../../../tests/data/'</span><span class="p">)</span>
+<span class="n">elevation</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'synthetic_topo.tif'</span><span class="p">)</span>
+<span class="n">hrelease</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'synthetic_rel.tif'</span><span class="p">)</span>
+<span class="n">prefix</span> <span class="o">=</span> <span class="s1">'synthetic'</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>You may need to modify <code class="docutils literal notranslate"><span class="pre">dir_data</span></code> according to where you save them
+on your local machine.</p>
+</div>
+<p>Given above inputs, we can create a <cite>GRASS Location</cite> and a shell file by calling
+<a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture.preprocess" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture.preprocess"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Ravaflow24Mixture.preprocess()</span></code></a> method, and run the simulation using
+<a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture.run" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Ravaflow24Mixture.run()</span></code></a> method</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">grass_location</span><span class="p">,</span> <span class="n">sh_file</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">preprocess</span><span class="p">(</span>
+ <span class="n">prefix</span><span class="o">=</span><span class="n">prefix</span><span class="p">,</span> <span class="n">elevation</span><span class="o">=</span><span class="n">elevation</span><span class="p">,</span> <span class="n">hrelease</span><span class="o">=</span><span class="n">hrelease</span><span class="p">,</span> <span class="n">EPSG</span><span class="o">=</span><span class="s1">'2326'</span><span class="p">)</span>
+<span class="n">voellmy_model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">grass_location</span><span class="p">,</span> <span class="n">sh_file</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Once the simulation is finished, the results are located in the folder
+<cite>/dir_sim/your_prefix_results</cite>. In this case, they are located in
+<cite>/temp_Ravaflow24Mixture_example/synthetic_results</cite>. We can extract desired
+outputs using respective method and visualize them.
+For example, we can extract the overall impact area, maximum flow velocity, or
+maximum flow velocity at specific locations:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="c1"># overall impact area</span>
+<span class="n">impact_area</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">extract_impact_area</span><span class="p">(</span><span class="n">prefix</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Impact_area is </span><span class="si">{</span><span class="n">impact_area</span><span class="si">}</span><span class="s2"> m^2"</span><span class="p">)</span>
+
+<span class="c1"># overall maximum flow velocity</span>
+<span class="n">v_mmax</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">extract_qoi_max</span><span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="s1">'v'</span><span class="p">,</span> <span class="n">aggregate</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Maximum flow velocity is </span><span class="si">{</span><span class="n">v_mmax</span><span class="si">}</span><span class="s2"> m/s^2"</span><span class="p">)</span>
+
+<span class="c1"># maximum flow velocity at specific locations</span>
+<span class="n">loc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">500</span><span class="p">,</span> <span class="mi">2000</span><span class="p">],</span> <span class="p">[</span><span class="mi">1500</span><span class="p">,</span> <span class="mi">2000</span><span class="p">]])</span>
+<span class="n">v_max_loc</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">extract_qoi_max_loc</span><span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="n">loc</span><span class="p">,</span> <span class="s1">'v'</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Maximum flow velocity at location </span><span class="si">{</span><span class="n">loc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> is </span><span class="si">{</span><span class="n">v_max_loc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> m/s^2"</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Maximum flow velocity at location </span><span class="si">{</span><span class="n">loc</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2"> is </span><span class="si">{</span><span class="n">v_max_loc</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2"> m/s^2"</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Impact_area is 1394800.0 m^2
+Maximum flow velocity is 41.39 m/s^2
+Maximum flow velocity at location [ 500 2000] is 34.264 m/s^2
+Maximum flow velocity at location [1500 2000] is 36.689 m/s^2
+</pre></div>
+</div>
+<p>We can visulize point-wise maximum flow height and velocity using heatmaps:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">linecache</span>
+
+<span class="c1"># extract head information of result ascii files and compute x and y coordinates</span>
+<span class="n">hmax_asc</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_sim</span><span class="p">,</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">prefix</span><span class="si">}</span><span class="s1">_results'</span><span class="p">,</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">prefix</span><span class="si">}</span><span class="s1">_ascii'</span><span class="p">,</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">prefix</span><span class="si">}</span><span class="s1">_hflow_max.asc'</span><span class="p">)</span>
+
+<span class="n">header</span> <span class="o">=</span> <span class="p">[</span><span class="n">linecache</span><span class="o">.</span><span class="n">getline</span><span class="p">(</span><span class="n">hmax_asc</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">)]</span>
+<span class="n">header_values</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">split</span><span class="p">()[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">strip</span><span class="p">())</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">header</span><span class="p">]</span>
+<span class="n">ncols</span><span class="p">,</span> <span class="n">nrows</span><span class="p">,</span> <span class="n">xll</span><span class="p">,</span> <span class="n">yll</span><span class="p">,</span> <span class="n">cellsize</span> <span class="o">=</span> <span class="n">header_values</span>
+<span class="n">ncols</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ncols</span><span class="p">)</span>
+<span class="n">nrows</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">nrows</span><span class="p">)</span>
+
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">xll</span><span class="p">,</span> <span class="n">xll</span><span class="o">+</span><span class="p">(</span><span class="n">cellsize</span><span class="o">*</span><span class="n">ncols</span><span class="p">),</span> <span class="n">cellsize</span><span class="p">)</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">yll</span><span class="p">,</span> <span class="n">yll</span><span class="o">+</span><span class="p">(</span><span class="n">cellsize</span><span class="o">*</span><span class="n">nrows</span><span class="p">),</span> <span class="n">cellsize</span><span class="p">)</span>
+
+<span class="c1"># point-wise maximum flow height</span>
+<span class="n">h_max</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">extract_qoi_max</span><span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="s1">'h'</span><span class="p">,</span> <span class="n">aggregate</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="c1"># point-wise maximum flow velocity</span>
+<span class="n">v_max</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">extract_qoi_max</span><span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="s1">'v'</span><span class="p">,</span> <span class="n">aggregate</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
+
+<span class="n">fig0</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">h_max</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'maximum flow height'</span> <span class="o">+</span> <span class="sa">r</span><span class="s1">' $(m/s)$'</span><span class="p">)</span>
+<span class="n">cbar0</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">fig0</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'</span><span class="si">%.1f</span><span class="s1">'</span><span class="p">,</span> <span class="n">orientation</span><span class="o">=</span><span class="s1">'vertical'</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+
+<span class="n">fig1</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">v_max</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'y'</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'maximum flow velocity'</span> <span class="o">+</span> <span class="sa">r</span><span class="s1">' ($m/s^2$)'</span><span class="p">)</span>
+<span class="n">cbar1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">fig1</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'</span><span class="si">%.1f</span><span class="s1">'</span><span class="p">,</span> <span class="n">orientation</span><span class="o">=</span><span class="s1">'vertical'</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+<img src="../../_images/sphx_glr_plot_ravaflow24_001.png" srcset="../../_images/sphx_glr_plot_ravaflow24_001.png" alt="maximum flow height $(m/s)$, maximum flow velocity ($m/s^2$)" class = "sphx-glr-single-img"/><p>Here we delete the folder <cite>temp_Ravaflow24Mixture_example</cite> and all files therein.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">shutil</span>
+
+<span class="n">shutil</span><span class="o">.</span><span class="n">rmtree</span><span class="p">(</span><span class="n">dir_sim</span><span class="p">)</span>
+</pre></div>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 5 minutes 56.157 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-simulator-plot-ravaflow24-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/70f25099189c12416e6ef5e4aa3ed8b4/plot_ravaflow24.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_ravaflow24.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/9847b46a66214926039e571c42505921/plot_ravaflow24.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_ravaflow24.ipynb</span></code></a></p>
+</div>
+</div>
+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
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+
+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-simulator-plot-run-mass-point-model-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="runsimulator-mass-point-model">
+<span id="sphx-glr-auto-examples-simulator-plot-run-mass-point-model-py"></span><h1>RunSimulator: Mass Point Model<a class="headerlink" href="#runsimulator-mass-point-model" title="Permalink to this heading"></a></h1>
+<p>This example shows how to run multiple MassPointModel simulations in serial
+and parallelly using <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a>.</p>
+<p>First, we import the class <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel" title="psimpy.simulator.mass_point_model.MassPointModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">MassPointModel</span></code></a> and define a <cite>simulator</cite>
+based on <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel.run" title="psimpy.simulator.mass_point_model.MassPointModel.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">MassPointModel.run()</span></code></a> method.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">MassPointModel</span>
+
+<span class="n">mpm</span> <span class="o">=</span> <span class="n">MassPointModel</span><span class="p">()</span>
+<span class="n">mpm_simulator</span> <span class="o">=</span> <span class="n">mpm</span><span class="o">.</span><span class="n">run</span>
+</pre></div>
+</div>
+<p>This simulator takes required input data, solves the mass point model, and
+returns time history of the mass point’s location and velocity
+(see <a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel.run" title="psimpy.simulator.mass_point_model.MassPointModel.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">MassPointModel.run()</span></code></a>).</p>
+<dl class="simple">
+<dt>Required inputs for <cite>mpm_simulator</cite> include:</dt><dd><ol class="arabic simple">
+<li><p>topographic data: digital elevation model (in <cite>ESRI ascii</cite> format)</p></li>
+<li><p>friction coefficients: coulomb friction and turbulent friction coefficients</p></li>
+<li><p>initial state: initial location and initial velocity of the masspoint</p></li>
+<li><p>computational parameters: such as time step, end time, etc.</p></li>
+</ol>
+</dd>
+</dl>
+<p>The synthetic topography <code class="docutils literal notranslate"><span class="pre">synthetic_topo.asc</span></code> is used here for illustration.
+It is located at the <cite>/tests/data/</cite> folder.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+
+<span class="n">dir_data</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="s1">'../../../tests/data/'</span><span class="p">)</span>
+<span class="n">elevation</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'synthetic_topo.asc'</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>You may need to modify <code class="docutils literal notranslate"><span class="pre">dir_data</span></code> according to where you save
+<code class="docutils literal notranslate"><span class="pre">synthetic_topo.asc</span></code> on your local machine.</p>
+</div>
+<p>For this example, we are going to run multiple simulations at different values
+of coulomb friction coefficient (<code class="docutils literal notranslate"><span class="pre">coulomb_friction</span></code>) and turbulent friction
+coefficient (<code class="docutils literal notranslate"><span class="pre">turbulent_friction</span></code>) while keep other inputs fixed. Namely,
+<code class="docutils literal notranslate"><span class="pre">coulomb_friction</span></code> and <code class="docutils literal notranslate"><span class="pre">turbulent_friction</span></code> are variable input parameters.
+All other inputs are fixed input parameters.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">itertools</span>
+
+<span class="c1"># Variable iput parameters are defined as a list of strings. Their values will be</span>
+<span class="c1"># passed as a numpy array (var_samples) when run simulations.</span>
+<span class="n">var_inp_parameter</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'coulomb_friction'</span><span class="p">,</span> <span class="s1">'turbulent_friction'</span><span class="p">]</span>
+<span class="n">coulomb_friction</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.31</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
+<span class="n">turbulent_friction</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">500</span><span class="p">,</span> <span class="mi">2001</span><span class="p">,</span> <span class="mi">400</span><span class="p">)</span>
+<span class="n">var_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+ <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">coulomb_friction</span><span class="p">,</span> <span class="n">turbulent_friction</span><span class="p">)])</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of variable input samples are: "</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">var_samples</span><span class="p">))</span>
+
+<span class="c1"># Fixed input parameters are defined as a dictionary. Their values are given.</span>
+<span class="n">fix_inp</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'elevation'</span><span class="p">:</span> <span class="n">elevation</span><span class="p">,</span> <span class="s1">'x0'</span><span class="p">:</span> <span class="mi">200</span><span class="p">,</span> <span class="s1">'y0'</span><span class="p">:</span> <span class="mi">2000</span><span class="p">,</span> <span class="s1">'tend'</span><span class="p">:</span> <span class="mi">50</span><span class="p">}</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Number of variable input samples are: 12
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Parameters of <cite>mpm_simulator</cite> which are not included in
+<code class="docutils literal notranslate"><span class="pre">var_inp_parameter</span></code> and <code class="docutils literal notranslate"><span class="pre">fix_inp</span></code>, such as <code class="docutils literal notranslate"><span class="pre">ux0</span></code>, <code class="docutils literal notranslate"><span class="pre">uy0</span></code> and
+<code class="docutils literal notranslate"><span class="pre">curvature</span></code>, will automatically take their default values.</p>
+</div>
+<p>We may want to save outputs returned by <cite>mpm_simulator</cite> at each simulation for
+later inspection or processing. In that case, we need to define <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> and
+set <code class="docutils literal notranslate"><span class="pre">save_out</span></code> as <cite>True</cite>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+
+<span class="c1"># Here we create a folder called `temp_run_MassPointModel_example` to save outputs</span>
+<span class="c1"># returned at each simulation.</span>
+<span class="n">cwd</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">()</span>
+<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="s1">'temp_run_MassPointModel_example'</span><span class="p">):</span>
+ <span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="s1">'temp_run_MassPointModel_example'</span><span class="p">)</span>
+<span class="n">dir_out</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">cwd</span><span class="p">,</span> <span class="s1">'temp_run_MassPointModel_example'</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Now we can define an object of <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a> by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">RunSimulator</span>
+
+<span class="n">run_mpm_simulator</span> <span class="o">=</span> <span class="n">RunSimulator</span><span class="p">(</span><span class="n">mpm_simulator</span><span class="p">,</span> <span class="n">var_inp_parameter</span><span class="p">,</span> <span class="n">fix_inp</span><span class="p">,</span>
+ <span class="n">dir_out</span><span class="o">=</span><span class="n">dir_out</span><span class="p">,</span> <span class="n">save_out</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Since outputs of each simulation are saved in the same folder <code class="docutils literal notranslate"><span class="pre">dir_out</span></code>,
+we need to give each file a unique name in order to avoid conflict. This is
+realized by defining <code class="docutils literal notranslate"><span class="pre">prefixes</span></code>.</p>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">serial_prefixes</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"serial"</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">var_samples</span><span class="p">))]</span>
+<span class="n">parallel_prefixes</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"parallel"</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">var_samples</span><span class="p">))]</span>
+</pre></div>
+</div>
+<p>Using <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.serial_run" title="psimpy.simulator.run_simulator.RunSimulator.serial_run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RunSimulator.serial_run()</span></code></a> method or
+<a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.parallel_run" title="psimpy.simulator.run_simulator.RunSimulator.parallel_run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RunSimulator.parallel_run()</span></code></a> method, we can run the simulations
+in serial or parallelly.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
+
+<span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="n">run_mpm_simulator</span><span class="o">.</span><span class="n">serial_run</span><span class="p">(</span><span class="n">var_samples</span><span class="o">=</span><span class="n">var_samples</span><span class="p">,</span> <span class="n">prefixes</span><span class="o">=</span><span class="n">serial_prefixes</span><span class="p">)</span>
+<span class="n">serial_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
+<span class="n">serial_output</span> <span class="o">=</span> <span class="n">run_mpm_simulator</span><span class="o">.</span><span class="n">outputs</span>
+
+<span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="c1"># max_workers controls maximum number of tasks running in parallel</span>
+<span class="n">run_mpm_simulator</span><span class="o">.</span><span class="n">parallel_run</span><span class="p">(</span><span class="n">var_samples</span><span class="p">,</span> <span class="n">prefixes</span><span class="o">=</span><span class="n">parallel_prefixes</span><span class="p">,</span> <span class="n">max_workers</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
+<span class="n">parallel_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
+<span class="n">parallel_output</span> <span class="o">=</span> <span class="n">run_mpm_simulator</span><span class="o">.</span><span class="n">outputs</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Serial run time: "</span><span class="p">,</span> <span class="n">serial_time</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Parallel run time: "</span><span class="p">,</span> <span class="n">parallel_time</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Serial run time: 1.1941184997558594
+Parallel run time: 0.4859800338745117
+</pre></div>
+</div>
+<p>Once a simulation is done, its output is saved to <code class="docutils literal notranslate"><span class="pre">dir_out</span></code>. All output files
+generated by above simulations are as follows:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">os</span><span class="o">.</span><span class="n">listdir</span><span class="p">(</span><span class="n">dir_out</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>['serial0_output.npy', 'serial9_output.npy', 'serial11_output.npy', 'parallel10_output.npy', 'parallel0_output.npy', 'parallel5_output.npy', 'parallel11_output.npy', 'serial1_output.npy', 'serial4_output.npy', 'parallel7_output.npy', 'serial5_output.npy', 'serial8_output.npy', 'parallel6_output.npy', 'parallel4_output.npy', 'parallel2_output.npy', 'serial3_output.npy', 'serial6_output.npy', 'parallel3_output.npy', 'serial2_output.npy', 'parallel9_output.npy', 'serial7_output.npy', 'parallel1_output.npy', 'parallel8_output.npy', 'serial10_output.npy']
+</pre></div>
+</div>
+<p>Once all simulations are done, their outputs can also be accessed by
+<a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.outputs" title="psimpy.simulator.run_simulator.RunSimulator.outputs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">RunSimulator.outputs</span></code></a> attribute, which is a list.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Here we print the maximum velocity of each simulation in the parallel run.</span>
+<span class="n">max_v</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">output</span><span class="p">[:,</span><span class="mi">5</span><span class="p">])</span> <span class="k">for</span> <span class="n">output</span> <span class="ow">in</span> <span class="n">parallel_output</span><span class="p">]</span>
+<span class="nb">print</span><span class="p">(</span>
+ <span class="sa">f</span><span class="s2">"Maximum velocity of </span><span class="si">{</span><span class="n">parallel_prefixes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> to </span><span class="si">{</span><span class="n">parallel_prefixes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2"> are: "</span><span class="p">,</span>
+ <span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">,</span>
+ <span class="n">max_v</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Maximum velocity of parallel0 to parallel11 are:
+ [16.55505557211892, 22.014114896121015, 26.245546886730782, 29.78487303612985, 15.34198274528081, 20.371864007499866, 24.249957243412815, 27.485487855011687, 14.031665649204944, 18.589324449452967, 22.092803891476187, 24.99727320816709]
+</pre></div>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>If one simulation failed due to whatever reason, the error massage
+will be printed to the screen but other simulations will continue. In that
+case, the output file of failed simulation will not be writted to <code class="docutils literal notranslate"><span class="pre">dir_out</span></code>.
+Also, the element of <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.outputs" title="psimpy.simulator.run_simulator.RunSimulator.outputs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">RunSimulator.outputs</span></code></a> corresponding to that
+simulation will be a string representing the error message, instead of a
+numpy array.</p>
+</div>
+<p>Here we delete the folder <cite>temp_run_MassPointModel_example</cite> and all files therein.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">shutil</span>
+
+<span class="n">shutil</span><span class="o">.</span><span class="n">rmtree</span><span class="p">(</span><span class="n">dir_out</span><span class="p">)</span>
+</pre></div>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 1.687 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-simulator-plot-run-mass-point-model-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/4d495ff630306dd98ca342f01931251c/plot_run_mass_point_model.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_run_mass_point_model.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/39558f3ab53d6f797e2208e7c89063a5/plot_run_mass_point_model.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_run_mass_point_model.ipynb</span></code></a></p>
+</div>
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+<li class="toctree-l2 current"><a class="reference internal" href="index.html">Simulator</a><ul class="current">
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+ <div class="sphx-glr-download-link-note admonition note">
+<p class="admonition-title">Note</p>
+<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-simulator-plot-run-ravaflow24-py"><span class="std std-ref">here</span></a>
+to download the full example code</p>
+</div>
+<section class="sphx-glr-example-title" id="runsimulator-ravaflow24-mixture-model">
+<span id="sphx-glr-auto-examples-simulator-plot-run-ravaflow24-py"></span><h1>RunSimulator: Ravaflow24 Mixture Model<a class="headerlink" href="#runsimulator-ravaflow24-mixture-model" title="Permalink to this heading"></a></h1>
+<p>This example shows how to run multiple Ravaflow24Mixture simulations in serial
+and parallelly using <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a>.</p>
+<p>First, we import the class <a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">Ravaflow24Mixture</span>
+</pre></div>
+</div>
+<p>To create an instance of this class, we must specify the parameter <code class="docutils literal notranslate"><span class="pre">dir_sim</span></code>.
+It represents the directory in which output files generated by r.avaflow will be
+saved.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+
+<span class="c1"># Here we create a folder called `temp1_run_Ravaflow24Mixture_example` to save</span>
+<span class="c1"># output files generated by r.avaflow</span>
+<span class="n">cwd</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">()</span>
+<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="s1">'temp1_run_Ravaflow24Mixture_example'</span><span class="p">):</span>
+ <span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="s1">'temp1_run_Ravaflow24Mixture_example'</span><span class="p">)</span>
+<span class="n">dir_sim</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">cwd</span><span class="p">,</span> <span class="s1">'temp1_run_Ravaflow24Mixture_example'</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Given <code class="docutils literal notranslate"><span class="pre">dir_sim</span></code>, we can create an instance of <a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a>.
+To reduce simulation time, we set <code class="docutils literal notranslate"><span class="pre">time_end</span></code> to <span class="math notranslate nohighlight">\(50\)</span>. Other parameters
+are set to their default values.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">voellmy_model</span> <span class="o">=</span> <span class="n">Ravaflow24Mixture</span><span class="p">(</span><span class="n">dir_sim</span><span class="p">,</span> <span class="n">time_end</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>The <cite>simulator</cite> in this example is defined based on <code class="docutils literal notranslate"><span class="pre">voellmy_model</span></code> as follows.
+It takes required inputs and returns the overall impact area.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">simulator</span><span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="n">elevation</span><span class="p">,</span> <span class="n">hrelease</span><span class="p">,</span> <span class="n">basal_friction</span><span class="p">,</span> <span class="n">turbulent_friction</span><span class="p">,</span> <span class="n">EPSG</span><span class="p">):</span>
+ <span class="sd">"""Preprocess required inputs, run simulation, and return output as a numpy array."""</span>
+ <span class="n">grass_location</span><span class="p">,</span> <span class="n">sh_file</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">preprocess</span><span class="p">(</span><span class="n">prefix</span><span class="o">=</span><span class="n">prefix</span><span class="p">,</span> <span class="n">elevation</span><span class="o">=</span><span class="n">elevation</span><span class="p">,</span>
+ <span class="n">hrelease</span><span class="o">=</span><span class="n">hrelease</span><span class="p">,</span> <span class="n">basal_friction</span><span class="o">=</span><span class="n">basal_friction</span><span class="p">,</span> <span class="n">turbulent_friction</span><span class="o">=</span><span class="n">turbulent_friction</span><span class="p">,</span> <span class="n">EPSG</span><span class="o">=</span><span class="n">EPSG</span><span class="p">)</span>
+ <span class="n">voellmy_model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">grass_location</span><span class="p">,</span> <span class="n">sh_file</span><span class="p">)</span> <span class="c1"># run this line, r.avaflow will write outputs to dir_sim</span>
+ <span class="n">impact_area</span> <span class="o">=</span> <span class="n">voellmy_model</span><span class="o">.</span><span class="n">extract_impact_area</span><span class="p">(</span><span class="n">prefix</span><span class="p">)</span>
+
+ <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">impact_area</span><span class="p">])</span> <span class="c1"># define dir_out and set save_out to True, returned numpy array will be saved to dir_out</span>
+</pre></div>
+</div>
+<p>To demonstrate how to run multiple simulations using <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a>,
+we choose <code class="docutils literal notranslate"><span class="pre">basal_friction</span></code> and <code class="docutils literal notranslate"><span class="pre">turbulent_friction</span></code> as variable input
+parameters. Each has two different values, leading to four simulations.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">itertools</span>
+
+<span class="n">var_inp_parameter</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'basal_friction'</span><span class="p">,</span> <span class="s1">'turbulent_friction'</span><span class="p">]</span>
+<span class="n">basal_friction</span> <span class="o">=</span> <span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">]</span>
+<span class="n">turbulent_friction</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
+<span class="n">var_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+ <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">basal_friction</span><span class="p">,</span> <span class="n">turbulent_friction</span><span class="p">)])</span>
+</pre></div>
+</div>
+<p>Other parameters of the <cite>simulator</cite>, including <code class="docutils literal notranslate"><span class="pre">elevation</span></code>, <code class="docutils literal notranslate"><span class="pre">hrelease</span></code>, and
+<code class="docutils literal notranslate"><span class="pre">EPSG</span></code> are treated as fixed input. It means that their values are the same in
+all simulations.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">dir_data</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="s1">'../../../tests/data/'</span><span class="p">)</span>
+<span class="n">elevation</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'synthetic_topo.tif'</span><span class="p">)</span>
+<span class="n">hrelease</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">dir_data</span><span class="p">,</span> <span class="s1">'synthetic_rel.tif'</span><span class="p">)</span>
+<span class="n">fix_inp</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'elevation'</span><span class="p">:</span> <span class="n">elevation</span><span class="p">,</span> <span class="s1">'hrelease'</span><span class="p">:</span> <span class="n">hrelease</span><span class="p">,</span> <span class="s1">'EPSG'</span><span class="p">:</span> <span class="s1">'2326'</span><span class="p">}</span>
+</pre></div>
+</div>
+<p>The parameter <code class="docutils literal notranslate"><span class="pre">prefix</span></code> of the <cite>simulator</cite> is special. It is not involved in
+the computational model, but only used to name output files generated by r.avaflow.
+Such parameter is not defined in <code class="docutils literal notranslate"><span class="pre">var_inp_parameter</span></code> or <code class="docutils literal notranslate"><span class="pre">fix_inp</span></code> of
+<a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a>. Instead, we use a seperate parameter, called <code class="docutils literal notranslate"><span class="pre">o_parameter</span></code>
+for this purpose.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">o_parameter</span> <span class="o">=</span> <span class="s1">'prefix'</span>
+</pre></div>
+</div>
+<p>We may want to save outputs returned by the <cite>simulator</cite> at each simulation for
+later inspection or processing. In that case, we need to define <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> and
+set <code class="docutils literal notranslate"><span class="pre">save_out</span></code> as <cite>True</cite>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+
+<span class="n">cwd</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">()</span>
+<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="s1">'temp2_run_Ravaflow24Mixture_example'</span><span class="p">):</span>
+ <span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="s1">'temp2_run_Ravaflow24Mixture_example'</span><span class="p">)</span>
+<span class="n">dir_out</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">cwd</span><span class="p">,</span> <span class="s1">'temp2_run_Ravaflow24Mixture_example'</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Please note that <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> and <code class="docutils literal notranslate"><span class="pre">dir_sim</span></code> are two different
+parameters for different purposes.</p>
+<blockquote>
+<div><ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">dir_out</span></code> is a parameter of <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a>. The <cite>simulator</cite>
+returns output of interest as a numpy array for each simulation. If
+we want to save this returned output of interest (here impact area)
+to our local machine, we need to specify the value of <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> which
+represents the directory in which output of interest will be saved and
+set <code class="docutils literal notranslate"><span class="pre">save_out</span></code> to <cite>True</cite>. Otherwise, we do not need <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> and
+leave <code class="docutils literal notranslate"><span class="pre">save_out</span></code> to <cite>False</cite>.</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">dir_sim</span></code> is a parameter of <a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a>. <a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ravaflow24Mixture</span></code></a>
+relies on the third party software r.avaflow 2.4. When we call
+<a class="reference internal" href="../../simulator/ravaflow24.html#psimpy.simulator.ravaflow24.Ravaflow24Mixture.run" title="psimpy.simulator.ravaflow24.Ravaflow24Mixture.run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Ravaflow24Mixture.run()</span></code></a> in the function body of above <cite>simulator</cite>,
+r.avaflow 2.4 will be run and it generates output files. The value of
+<code class="docutils literal notranslate"><span class="pre">dir_sim</span></code> specifies the directory in which output files generated by
+r.avaflow 2.4 are going to be saved.</p></li>
+</ul>
+</div></blockquote>
+<p>The value of <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> and <code class="docutils literal notranslate"><span class="pre">dir_sim</span></code> can be the same if file names
+have no conflict. We recommend to keep them seperate. In addition, if the function
+body of the <cite>simulator</cite> does not write files to disk, <code class="docutils literal notranslate"><span class="pre">dir_sim</span></code> is not required. Our
+<a class="reference internal" href="../../simulator/mass_point_model.html#psimpy.simulator.mass_point_model.MassPointModel" title="psimpy.simulator.mass_point_model.MassPointModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">MassPointModel</span></code></a> is an example. Similarly, if we do not want to save
+returned numpy array of the <cite>simulator</cite>, <code class="docutils literal notranslate"><span class="pre">dir_out</span></code> is not needed.</p>
+</div>
+<p>Now we can define an object of <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator" title="psimpy.simulator.run_simulator.RunSimulator"><code class="xref py py-class docutils literal notranslate"><span class="pre">RunSimulator</span></code></a> by</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psimpy.simulator</span> <span class="kn">import</span> <span class="n">RunSimulator</span>
+
+<span class="n">run_simulator</span> <span class="o">=</span> <span class="n">RunSimulator</span><span class="p">(</span><span class="n">simulator</span><span class="p">,</span> <span class="n">var_inp_parameter</span><span class="p">,</span> <span class="n">fix_inp</span><span class="p">,</span> <span class="n">o_parameter</span><span class="p">,</span>
+ <span class="n">dir_out</span><span class="p">,</span> <span class="n">save_out</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>Before running simulations, we need to specify values of <code class="docutils literal notranslate"><span class="pre">prefixes</span></code> which
+will be used to name files generated by each r.avaflow simulation and
+each returned numpy array of <cite>simulator</cite>.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">serial_prefixes</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"serial"</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">var_samples</span><span class="p">))]</span>
+<span class="n">parallel_prefixes</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"parallel"</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">var_samples</span><span class="p">))]</span>
+</pre></div>
+</div>
+<p>Now we can use <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.serial_run" title="psimpy.simulator.run_simulator.RunSimulator.serial_run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RunSimulator.serial_run()</span></code></a> method or
+<a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.parallel_run" title="psimpy.simulator.run_simulator.RunSimulator.parallel_run"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RunSimulator.parallel_run()</span></code></a> method to run the simulations
+in serial or parallelly.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
+
+<span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="n">run_simulator</span><span class="o">.</span><span class="n">serial_run</span><span class="p">(</span><span class="n">var_samples</span><span class="o">=</span><span class="n">var_samples</span><span class="p">,</span> <span class="n">prefixes</span><span class="o">=</span><span class="n">serial_prefixes</span><span class="p">)</span>
+<span class="n">serial_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
+<span class="n">serial_output</span> <span class="o">=</span> <span class="n">run_simulator</span><span class="o">.</span><span class="n">outputs</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"serial_output: </span><span class="si">{</span><span class="n">serial_output</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
+
+<span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="c1"># append setting to True means that simulation outputs of parallel run will be</span>
+<span class="c1"># appended to above serial run outputs</span>
+<span class="n">run_simulator</span><span class="o">.</span><span class="n">parallel_run</span><span class="p">(</span><span class="n">var_samples</span><span class="p">,</span> <span class="n">prefixes</span><span class="o">=</span><span class="n">parallel_prefixes</span><span class="p">,</span>
+ <span class="n">max_workers</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">append</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">parallel_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
+
+<span class="n">parallel_output</span> <span class="o">=</span> <span class="n">run_simulator</span><span class="o">.</span><span class="n">outputs</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">var_samples</span><span class="p">):]</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"parallel_output: </span><span class="si">{</span><span class="n">parallel_output</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
+
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Serial run time: "</span><span class="p">,</span> <span class="n">serial_time</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Parallel run time: "</span><span class="p">,</span> <span class="n">parallel_time</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>serial_output: [array([1008000.]), array([1520800.]), array([674000.]), array([869600.])]
+parallel_output: [array([1008000.]), array([1520800.]), array([674000.]), array([869600.])]
+Serial run time: 110.85352993011475
+Parallel run time: 62.39644193649292
+</pre></div>
+</div>
+<p>All output files returned by the <cite>simulator</cite> are</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">os</span><span class="o">.</span><span class="n">listdir</span><span class="p">(</span><span class="n">dir_out</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>['serial0_output.npy', 'parallel0_output.npy', 'serial1_output.npy', 'parallel2_output.npy', 'serial3_output.npy', 'parallel3_output.npy', 'serial2_output.npy', 'parallel1_output.npy']
+</pre></div>
+</div>
+<p>All output files/directories generated by r.avaflow simulations
+(including files/directories generated during preprocessing) are</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">os</span><span class="o">.</span><span class="n">listdir</span><span class="p">(</span><span class="n">dir_sim</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>['serial0_glocation', 'parallel3_results', 'parallel2_glocation', 'serial3_shell.sh', 'parallel1_results', 'serial3_glocation', 'serial1_shell.sh', 'parallel2_results', 'serial0_shell.sh', 'parallel1_glocation', 'parallel3_shell.sh', 'parallel0_shell.sh', 'serial1_glocation', 'parallel1_shell.sh', 'parallel2_shell.sh', 'serial0_results', 'serial1_results', 'serial2_shell.sh', 'serial2_glocation', 'serial2_results', 'parallel3_glocation', 'parallel0_glocation', 'serial3_results', 'parallel0_results']
+</pre></div>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>If one simulation failed due to whatever reason, the error massage
+will be printed to the screen but other simulations will continue. In that
+case, the output file of failed simulation will not be writted to <code class="docutils literal notranslate"><span class="pre">dir_out</span></code>.
+Also, the element of <a class="reference internal" href="../../simulator/run_simulator.html#psimpy.simulator.run_simulator.RunSimulator.outputs" title="psimpy.simulator.run_simulator.RunSimulator.outputs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">RunSimulator.outputs</span></code></a> corresponding to that
+simulation will be a string representing the error message, instead of a
+numpy array.</p>
+</div>
+<p>In the end, we delete the folder <cite>temp1_run_Ravaflow24Mixture_example</cite> and
+<cite>temp2_run_Ravaflow24Mixture_example</cite> and all files therein.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">shutil</span>
+
+<span class="n">shutil</span><span class="o">.</span><span class="n">rmtree</span><span class="p">(</span><span class="n">dir_sim</span><span class="p">)</span>
+<span class="n">shutil</span><span class="o">.</span><span class="n">rmtree</span><span class="p">(</span><span class="n">dir_out</span><span class="p">)</span>
+</pre></div>
+</div>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 2 minutes 53.318 seconds)</p>
+<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-simulator-plot-run-ravaflow24-py">
+<div class="sphx-glr-download sphx-glr-download-python docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/06e1d1c67d64bdd5612e33cffc1295c5/plot_run_ravaflow24.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_run_ravaflow24.py</span></code></a></p>
+</div>
+<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
+<p><a class="reference download internal" download="" href="../../_downloads/ad745ba736a627e1e1bb6fcec00ac417/plot_run_ravaflow24.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_run_ravaflow24.ipynb</span></code></a></p>
+</div>
+</div>
+<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
+</section>
+
+
+ </div>
+ </div>
+ <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
+ <a href="plot_ravaflow24.html" class="btn btn-neutral float-left" title="Ravaflow24 Mixture Model" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a>
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diff --git a/docs/source/_build/html/auto_examples/simulator/sg_execution_times.html b/docs/source/_build/html/auto_examples/simulator/sg_execution_times.html
new file mode 100644
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--- /dev/null
+++ b/docs/source/_build/html/auto_examples/simulator/sg_execution_times.html
@@ -0,0 +1,139 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" >
+<head>
+ <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
+
+ <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+ <title>Computation times — PSimPy's documentation</title>
+ <link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
+ <link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
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+ <link rel="stylesheet" href="../../_static/sg_gallery-dataframe.css" type="text/css" />
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+</head>
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+<body class="wy-body-for-nav">
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+ <li class="breadcrumb-item active">Computation times</li>
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+ <a href="../../_sources/auto_examples/simulator/sg_execution_times.rst.txt" rel="nofollow"> View page source</a>
+ </li>
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+ <hr/>
+</div>
+ <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+ <div itemprop="articleBody">
+
+ <section id="computation-times">
+<span id="sphx-glr-auto-examples-simulator-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Permalink to this heading"></a></h1>
+<p><strong>08:52.712</strong> total execution time for <strong>auto_examples_simulator</strong> files:</p>
+<table class="docutils align-default">
+<colgroup>
+<col style="width: 85%" />
+<col style="width: 9%" />
+<col style="width: 6%" />
+</colgroup>
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_ravaflow24.html#sphx-glr-auto-examples-simulator-plot-ravaflow24-py"><span class="std std-ref">Ravaflow24 Mixture Model</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_ravaflow24.py</span></code>)</p></td>
+<td><p>05:56.157</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="plot_run_ravaflow24.html#sphx-glr-auto-examples-simulator-plot-run-ravaflow24-py"><span class="std std-ref">RunSimulator: Ravaflow24 Mixture Model</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_run_ravaflow24.py</span></code>)</p></td>
+<td><p>02:53.318</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="plot_run_mass_point_model.html#sphx-glr-auto-examples-simulator-plot-run-mass-point-model-py"><span class="std std-ref">RunSimulator: Mass Point Model</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_run_mass_point_model.py</span></code>)</p></td>
+<td><p>00:01.687</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="plot_mass_point_model.html#sphx-glr-auto-examples-simulator-plot-mass-point-model-py"><span class="std std-ref">Mass Point Model</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_mass_point_model.py</span></code>)</p></td>
+<td><p>00:01.550</p></td>
+<td><p>0.0 MB</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+
+ </div>
+ </div>
+ <footer>
+
+ <hr/>
+
+ <div role="contentinfo">
+ <p>© Copyright 2023, Hu Zhao.</p>
+ </div>
+
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+ <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a>
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