.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_reliability_sensitivity/reliability/plot_estimate_probability_importance_sampling.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_reliability_plot_estimate_probability_importance_sampling.py>`
        to download the full example code

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

.. _sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_importance_sampling.py:


Use the Importance Sampling algorithm
=====================================

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In this example we estimate a failure probability with the importance sampling simulation algorithm provided by the `ImportanceSamplingExperiment` class.

The main steps of this method are:

* run a FORM analysis,
* create an importance distribution based on the results of the FORM results,
* run a sampling-based probability estimate algorithm.

We shall consider the analytical example of a :ref:`cantilever beam <use-case-cantilever-beam>`.


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Define the cantilever beam model
--------------------------------

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.. code-block:: Python

    from openturns.usecases import cantilever_beam
    import openturns as ot
    import openturns.viewer as viewer

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








.. GENERATED FROM PYTHON SOURCE LINES 29-30

The cantilever beam example can be accessed in the usecases module :

.. GENERATED FROM PYTHON SOURCE LINES 30-32

.. code-block:: Python

    cb = cantilever_beam.CantileverBeam()








.. GENERATED FROM PYTHON SOURCE LINES 33-34

The joint probability distribution of the input parameters is stored in the object `cb` :

.. GENERATED FROM PYTHON SOURCE LINES 34-36

.. code-block:: Python

    distribution = cb.distribution








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We create the model.

.. GENERATED FROM PYTHON SOURCE LINES 38-40

.. code-block:: Python

    model = cb.model








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Define the event
----------------

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We create the event whose probability we want to estimate.

.. GENERATED FROM PYTHON SOURCE LINES 48-52

.. code-block:: Python

    vect = ot.RandomVector(distribution)
    G = ot.CompositeRandomVector(model, vect)
    event = ot.ThresholdEvent(G, ot.Greater(), 0.30)








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Run a FORM analysis
-------------------

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Define a solver

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.. code-block:: Python

    optimAlgo = ot.Cobyla()
    optimAlgo.setMaximumEvaluationNumber(1000)
    optimAlgo.setMaximumAbsoluteError(1.0e-10)
    optimAlgo.setMaximumRelativeError(1.0e-10)
    optimAlgo.setMaximumResidualError(1.0e-10)
    optimAlgo.setMaximumConstraintError(1.0e-10)








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Run FORM

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.. code-block:: Python

    algo = ot.FORM(optimAlgo, event, distribution.getMean())
    algo.run()
    result = algo.getResult()








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Configure an importance sampling algorithm
------------------------------------------

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The `getStandardSpaceDesignPoint` method returns the design point in the U-space.

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.. code-block:: Python

    standardSpaceDesignPoint = result.getStandardSpaceDesignPoint()
    standardSpaceDesignPoint






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Standard Space Design Point dimension=4 values=[-0.665643,4.31264,1.23029,-1.3689]
    </div>
    <br />
    <br />

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The key point is to define the importance distribution in the U-space. To define it, we use a multivariate standard Gaussian centered around the design point in the U-space.

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.. code-block:: Python

    dimension = distribution.getDimension()
    dimension





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

 .. code-block:: none


    4



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.. code-block:: Python

    myImportance = ot.Normal(dimension)
    myImportance.setMu(standardSpaceDesignPoint)
    myImportance






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    Normal
    <ul>
      <li>name=Normal</li>
      <li>dimension=4</li>
      <li>weight=1</li>
      <li>range=]-inf (-8.31627), (6.98499) +inf[
    ]-inf (-3.33799), (11.9633) +inf[
    ]-inf (-6.42034), (8.88091) +inf[
    ]-inf (-9.01953), (6.28172) +inf[</li>
      <li>description=[X0,X1,X2,X3]</li>
      <li>isParallel=true</li>
      <li>isCopula=false</li>
    </ul>

    </div>
    <br />
    <br />

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Create the design of experiment corresponding to importance sampling. This generates a `WeightedExperiment` with weights fitting to the importance distribution.

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.. code-block:: Python

    experiment = ot.ImportanceSamplingExperiment(myImportance)
    type(experiment)








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Create the standard event corresponding to the event. This pushes the original problem to the U-space, with Gaussian independent marginals.

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.. code-block:: Python

    standardEvent = ot.StandardEvent(event)








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Run the importance sampling simulation
--------------------------------------

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We then create the simulation algorithm.

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.. code-block:: Python

    algo = ot.ProbabilitySimulationAlgorithm(standardEvent, experiment)
    algo.setMaximumCoefficientOfVariation(0.1)
    algo.setMaximumOuterSampling(40000)
    algo.run()








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We can retrieve results of this estimate :

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.. code-block:: Python

    result = algo.getResult()
    probability = result.getProbabilityEstimate()
    print("Probability = ", probability)





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

 .. code-block:: none

    Probability =  5.108962790908332e-07




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In order to compute the confidence interval, we use the `getConfidenceLength` method, which returns the length of the interval.
In order to compute the bounds of the interval, we divide this length by 2.

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.. code-block:: Python

    alpha = 0.05








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.. code-block:: Python

    pflen = result.getConfidenceLength(1 - alpha)
    print(
        "%.2f%% confidence interval = [%.10f,%.10f]"
        % ((1 - alpha) * 100, probability - pflen / 2, probability + pflen / 2)
    )





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

 .. code-block:: none

    95.00% confidence interval = [0.0000004108,0.0000006110]




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We can observe the convergence history of the estimate with the `drawProbabilityConvergence`
method which displays the estimate and confidence interval evolution.

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.. code-block:: Python

    graph = algo.drawProbabilityConvergence()
    graph.setLogScale(ot.GraphImplementation.LOGX)
    view = viewer.View(graph)



.. image-sg:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_estimate_probability_importance_sampling_001.png
   :alt: ProbabilitySimulationAlgorithm convergence graph at level 0.95
   :srcset: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_estimate_probability_importance_sampling_001.png
   :class: sphx-glr-single-img






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.. only:: html

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

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

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

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

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