.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_reliability_sensitivity/sensitivity_analysis/plot_sensitivity_ancova.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_reliability_sensitivity_sensitivity_analysis_plot_sensitivity_ancova.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_reliability_sensitivity_sensitivity_analysis_plot_sensitivity_ancova.py:


Use the ANCOVA indices
======================

.. GENERATED FROM PYTHON SOURCE LINES 6-33

In this example we are going to use the ANCOVA decomposition to estimate
sensitivity indices from a model with correlated inputs.

ANCOVA allows one to estimate the Sobol' indices, and thanks to a functional
decomposition of the model it allows one to separate the part of variance
explained by a variable itself from the part of variance explained by a
correlation which is due to its correlation with the other input parameters.


In theory, ANCOVA indices range is :math:`\left[0; 1\right]` ; the closer to 1 the index is,
the greater the model response sensitivity to the variable is.
These indices are a sum of a physical part :math:`S_i^U` and correlated part :math:`S_i^C`.
The correlation has a weak influence on the contribution of :math:`X_i`, if :math:`|S_i^C|`
is low and :math:`S_i` is close to :math:`S_i^U`.
On the contrary, the correlation has a strong influence on the contribution of
the input :math:`X_i`, if :math:`|S_i^C|` is high and :math:`S_i` is close to :math:`S_i^C`.

The ANCOVA indices :math:`S_i` evaluate the importance of one variable at a time
(:math:`d` indices stored, with :math:`d` the input dimension of the model).
The :math:`d` uncorrelated parts of variance of the output due to each input :math:`S_i^U`
and the effects of the correlation are represented by the indices resulting
from the subtraction of these two previous lists.

To evaluate the indices, the library needs of a functional chaos result
approximating the model response with uncorrelated inputs and a sample with
correlated inputs used to compute the real values of the output.


.. GENERATED FROM PYTHON SOURCE LINES 35-41

.. code-block:: Python

    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt

    ot.Log.Show(ot.Log.NONE)








.. GENERATED FROM PYTHON SOURCE LINES 42-43

Create the model (x1,x2) --> (y) = (4.*x1+5.*x2)

.. GENERATED FROM PYTHON SOURCE LINES 43-45

.. code-block:: Python

    model = ot.SymbolicFunction(["x1", "x2"], ["4.*x1+5.*x2"])








.. GENERATED FROM PYTHON SOURCE LINES 46-47

Create the input independent joint distribution

.. GENERATED FROM PYTHON SOURCE LINES 47-50

.. code-block:: Python

    distribution = ot.Normal(2)
    distribution.setDescription(["X1", "X2"])








.. GENERATED FROM PYTHON SOURCE LINES 51-52

Create the correlated input distribution

.. GENERATED FROM PYTHON SOURCE LINES 52-58

.. code-block:: Python

    S = ot.CorrelationMatrix(2)
    S[1, 0] = 0.3
    R = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(S)
    copula = ot.NormalCopula(R)
    distribution_corr = ot.ComposedDistribution([ot.Normal()] * 2, copula)








.. GENERATED FROM PYTHON SOURCE LINES 59-60

ANCOVA needs a functional decomposition of the model

.. GENERATED FROM PYTHON SOURCE LINES 60-75

.. code-block:: Python

    enumerateFunction = ot.LinearEnumerateFunction(2)
    productBasis = ot.OrthogonalProductPolynomialFactory(
        [ot.HermiteFactory()] * 2, enumerateFunction
    )
    adaptiveStrategy = ot.FixedStrategy(
        productBasis, enumerateFunction.getStrataCumulatedCardinal(4)
    )
    samplingSize = 250
    experiment = ot.MonteCarloExperiment(distribution, samplingSize)
    X = experiment.generate()
    Y = model(X)
    algo = ot.FunctionalChaosAlgorithm(X, Y, distribution, adaptiveStrategy)
    algo.run()
    result = ot.FunctionalChaosResult(algo.getResult())








.. GENERATED FROM PYTHON SOURCE LINES 76-77

Create the input sample taking account the correlation

.. GENERATED FROM PYTHON SOURCE LINES 77-80

.. code-block:: Python

    size = 2000
    sample = distribution_corr.getSample(size)








.. GENERATED FROM PYTHON SOURCE LINES 81-82

Perform the decomposition

.. GENERATED FROM PYTHON SOURCE LINES 82-90

.. code-block:: Python

    ancova = ot.ANCOVA(result, sample)
    # Compute the ANCOVA indices (first order and uncorrelated indices are computed together)
    indices = ancova.getIndices()
    # Retrieve uncorrelated indices
    uncorrelatedIndices = ancova.getUncorrelatedIndices()
    # Retrieve correlated indices:
    correlatedIndices = indices - uncorrelatedIndices








.. GENERATED FROM PYTHON SOURCE LINES 91-92

Print Sobol' indices, the physical part, and the correlation part

.. GENERATED FROM PYTHON SOURCE LINES 92-96

.. code-block:: Python

    print("ANCOVA indices ", indices)
    print("ANCOVA uncorrelated indices ", uncorrelatedIndices)
    print("ANCOVA correlated indices ", correlatedIndices)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ANCOVA indices  [0.422633,0.577367]
    ANCOVA uncorrelated indices  [0.296946,0.451679]
    ANCOVA correlated indices  [0.125687,0.125687]




.. GENERATED FROM PYTHON SOURCE LINES 97-102

.. code-block:: Python

    graph = ot.SobolIndicesAlgorithm.DrawImportanceFactors(
        indices, distribution.getDescription(), "ANCOVA indices (Sobol')"
    )
    view = viewer.View(graph)




.. image-sg:: /auto_reliability_sensitivity/sensitivity_analysis/images/sphx_glr_plot_sensitivity_ancova_001.png
   :alt: ANCOVA indices (Sobol')
   :srcset: /auto_reliability_sensitivity/sensitivity_analysis/images/sphx_glr_plot_sensitivity_ancova_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 103-110

.. code-block:: Python

    graph = ot.SobolIndicesAlgorithm.DrawImportanceFactors(
        uncorrelatedIndices,
        distribution.getDescription(),
        "ANCOVA uncorrelated indices\n(part of physical variance in the model)",
    )
    view = viewer.View(graph)




.. image-sg:: /auto_reliability_sensitivity/sensitivity_analysis/images/sphx_glr_plot_sensitivity_ancova_002.png
   :alt: ANCOVA uncorrelated indices (part of physical variance in the model)
   :srcset: /auto_reliability_sensitivity/sensitivity_analysis/images/sphx_glr_plot_sensitivity_ancova_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 111-118

.. code-block:: Python

    graph = ot.SobolIndicesAlgorithm.DrawImportanceFactors(
        correlatedIndices,
        distribution.getDescription(),
        "ANCOVA correlated indices\n(part of variance due to the correlation)",
    )
    view = viewer.View(graph)
    plt.show()



.. image-sg:: /auto_reliability_sensitivity/sensitivity_analysis/images/sphx_glr_plot_sensitivity_ancova_003.png
   :alt: ANCOVA correlated indices (part of variance due to the correlation)
   :srcset: /auto_reliability_sensitivity/sensitivity_analysis/images/sphx_glr_plot_sensitivity_ancova_003.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_reliability_sensitivity_sensitivity_analysis_plot_sensitivity_ancova.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_sensitivity_ancova.ipynb <plot_sensitivity_ancova.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_sensitivity_ancova.py <plot_sensitivity_ancova.py>`