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

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

.. _sphx_glr_auto_reliability_sensitivity_reliability_plot_axial_stressed_beam.py:


Axial stressed beam : comparing different methods to estimate a probability
===========================================================================

.. GENERATED FROM PYTHON SOURCE LINES 6-13

In this example, we compare four methods to estimate the probability in the :ref:`axial stressed beam <use-case-stressed-beam>` example :

* Monte-Carlo simulation,
* FORM,
* directional sampling,
* importance sampling with FORM design point: FORM-IS.


.. GENERATED FROM PYTHON SOURCE LINES 15-17

Define the model
----------------

.. GENERATED FROM PYTHON SOURCE LINES 19-27

.. code-block:: Python

    import numpy as np
    from openturns.usecases import stressed_beam

    import openturns as ot
    import openturns.viewer as viewer

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








.. GENERATED FROM PYTHON SOURCE LINES 28-29

We load the model from the usecases module :

.. GENERATED FROM PYTHON SOURCE LINES 29-31

.. code-block:: Python

    sm = stressed_beam.AxialStressedBeam()








.. GENERATED FROM PYTHON SOURCE LINES 32-33

The limit state function is defined in the `model` field of the data class :

.. GENERATED FROM PYTHON SOURCE LINES 33-35

.. code-block:: Python

    limitStateFunction = sm.model








.. GENERATED FROM PYTHON SOURCE LINES 36-38

The probabilistic model of the axial stressed beam is defined in the data class.
We get the first marginal and draw it :

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: Python

    R_dist = sm.distribution_R
    graph = R_dist.drawPDF()
    view = viewer.View(graph)




.. image-sg:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_001.png
   :alt: plot axial stressed beam
   :srcset: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 43-44

We get the second marginal and draw it :

.. GENERATED FROM PYTHON SOURCE LINES 46-50

.. code-block:: Python

    F_dist = sm.distribution_F
    graph = F_dist.drawPDF()
    view = viewer.View(graph)




.. image-sg:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_002.png
   :alt: plot axial stressed beam
   :srcset: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 51-52

These independent marginals define the joint distribution of the input parameters :

.. GENERATED FROM PYTHON SOURCE LINES 52-55

.. code-block:: Python

    myDistribution = sm.distribution









.. GENERATED FROM PYTHON SOURCE LINES 56-57

We create a `RandomVector` from the `Distribution`, then a composite random vector. Finally, we create a `ThresholdEvent` from this `RandomVector`.

.. GENERATED FROM PYTHON SOURCE LINES 59-63

.. code-block:: Python

    inputRandomVector = ot.RandomVector(myDistribution)
    outputRandomVector = ot.CompositeRandomVector(limitStateFunction, inputRandomVector)
    myEvent = ot.ThresholdEvent(outputRandomVector, ot.Less(), 0.0)








.. GENERATED FROM PYTHON SOURCE LINES 64-66

Using Monte Carlo simulations
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 68-77

.. code-block:: Python

    cv = 0.05
    NbSim = 100000

    experiment = ot.MonteCarloExperiment()
    algoMC = ot.ProbabilitySimulationAlgorithm(myEvent, experiment)
    algoMC.setMaximumOuterSampling(NbSim)
    algoMC.setBlockSize(1)
    algoMC.setMaximumCoefficientOfVariation(cv)








.. GENERATED FROM PYTHON SOURCE LINES 78-79

For statistics about the algorithm

.. GENERATED FROM PYTHON SOURCE LINES 79-81

.. code-block:: Python

    initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber()








.. GENERATED FROM PYTHON SOURCE LINES 82-83

Perform the analysis.

.. GENERATED FROM PYTHON SOURCE LINES 85-87

.. code-block:: Python

    algoMC.run()








.. GENERATED FROM PYTHON SOURCE LINES 88-97

.. code-block:: Python

    result = algoMC.getResult()
    probabilityMonteCarlo = result.getProbabilityEstimate()
    numberOfFunctionEvaluationsMonteCarlo = (
        limitStateFunction.getEvaluationCallsNumber() - initialNumberOfCall
    )
    print("Number of calls to the limit state =", numberOfFunctionEvaluationsMonteCarlo)
    print("Pf = ", probabilityMonteCarlo)
    print("CV =", result.getCoefficientOfVariation())





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

 .. code-block:: none

    Number of calls to the limit state = 13931
    Pf =  0.027923336443902068
    CV = 0.04998911613048065




.. GENERATED FROM PYTHON SOURCE LINES 98-102

.. code-block:: Python

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




.. image-sg:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_003.png
   :alt: ProbabilitySimulationAlgorithm convergence graph at level 0.95
   :srcset: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 103-105

Using FORM analysis
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 107-108

We create a NearestPoint algorithm

.. GENERATED FROM PYTHON SOURCE LINES 108-117

.. code-block:: Python

    myCobyla = ot.Cobyla()
    # Resolution options:
    eps = 1e-3
    myCobyla.setMaximumEvaluationNumber(100)
    myCobyla.setMaximumAbsoluteError(eps)
    myCobyla.setMaximumRelativeError(eps)
    myCobyla.setMaximumResidualError(eps)
    myCobyla.setMaximumConstraintError(eps)








.. GENERATED FROM PYTHON SOURCE LINES 118-119

For statistics about the algorithm

.. GENERATED FROM PYTHON SOURCE LINES 119-121

.. code-block:: Python

    initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber()








.. GENERATED FROM PYTHON SOURCE LINES 122-123

We create a FORM algorithm. The first parameter is a NearestPointAlgorithm. The second parameter is an event. The third parameter is a starting point for the design point research.

.. GENERATED FROM PYTHON SOURCE LINES 125-127

.. code-block:: Python

    algoFORM = ot.FORM(myCobyla, myEvent, myDistribution.getMean())








.. GENERATED FROM PYTHON SOURCE LINES 128-129

Perform the analysis.

.. GENERATED FROM PYTHON SOURCE LINES 131-133

.. code-block:: Python

    algoFORM.run()








.. GENERATED FROM PYTHON SOURCE LINES 134-142

.. code-block:: Python

    resultFORM = algoFORM.getResult()
    numberOfFunctionEvaluationsFORM = (
        limitStateFunction.getEvaluationCallsNumber() - initialNumberOfCall
    )
    probabilityFORM = resultFORM.getEventProbability()
    print("Number of calls to the limit state =", numberOfFunctionEvaluationsFORM)
    print("Pf =", probabilityFORM)





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

 .. code-block:: none

    Number of calls to the limit state = 98
    Pf = 0.02998278558231473




.. GENERATED FROM PYTHON SOURCE LINES 143-146

.. code-block:: Python

    graph = resultFORM.drawImportanceFactors()
    view = viewer.View(graph)




.. image-sg:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_004.png
   :alt: Importance Factors from Design Point - Unnamed
   :srcset: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 147-149

Using Directional sampling
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 151-152

Resolution options:

.. GENERATED FROM PYTHON SOURCE LINES 152-160

.. code-block:: Python

    cv = 0.05
    NbSim = 10000

    algoDS = ot.DirectionalSampling(myEvent)
    algoDS.setMaximumOuterSampling(NbSim)
    algoDS.setBlockSize(1)
    algoDS.setMaximumCoefficientOfVariation(cv)








.. GENERATED FROM PYTHON SOURCE LINES 161-162

For statistics about the algorithm

.. GENERATED FROM PYTHON SOURCE LINES 162-164

.. code-block:: Python

    initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber()








.. GENERATED FROM PYTHON SOURCE LINES 165-166

Perform the analysis.

.. GENERATED FROM PYTHON SOURCE LINES 168-170

.. code-block:: Python

    algoDS.run()








.. GENERATED FROM PYTHON SOURCE LINES 171-183

.. code-block:: Python

    result = algoDS.getResult()
    probabilityDirectionalSampling = result.getProbabilityEstimate()
    numberOfFunctionEvaluationsDirectionalSampling = (
        limitStateFunction.getEvaluationCallsNumber() - initialNumberOfCall
    )
    print(
        "Number of calls to the limit state =",
        numberOfFunctionEvaluationsDirectionalSampling,
    )
    print("Pf = ", probabilityDirectionalSampling)
    print("CV =", result.getCoefficientOfVariation())





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

 .. code-block:: none

    Number of calls to the limit state = 9050
    Pf =  0.029665915864919325
    CV = 0.04998871289144034




.. GENERATED FROM PYTHON SOURCE LINES 184-188

.. code-block:: Python

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




.. image-sg:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_005.png
   :alt: DirectionalSampling convergence graph at level 0.95
   :srcset: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 189-191

Using importance sampling with FORM design point: FORM-IS
---------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 193-194

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

.. GENERATED FROM PYTHON SOURCE LINES 196-199

.. code-block:: Python

    standardSpaceDesignPoint = resultFORM.getStandardSpaceDesignPoint()
    standardSpaceDesignPoint






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[-1.59355,0.999463]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 200-202

The key point is to define the importance distribution in the U-space.
To define it, we use a multivariate standard Gaussian and configure it so that the center is equal to the design point in the U-space.

.. GENERATED FROM PYTHON SOURCE LINES 204-207

.. code-block:: Python

    dimension = myDistribution.getDimension()
    dimension





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

 .. code-block:: none


    2



.. GENERATED FROM PYTHON SOURCE LINES 208-212

.. code-block:: Python

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






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>Normal(mu = [-1.59355,0.999463], sigma = [1,1], R = [[ 1 0 ]<br>
     [ 0 1 ]])</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 213-214

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

.. GENERATED FROM PYTHON SOURCE LINES 216-218

.. code-block:: Python

    experiment = ot.ImportanceSamplingExperiment(myImportance)








.. GENERATED FROM PYTHON SOURCE LINES 219-220

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

.. GENERATED FROM PYTHON SOURCE LINES 222-224

.. code-block:: Python

    standardEvent = ot.StandardEvent(myEvent)








.. GENERATED FROM PYTHON SOURCE LINES 225-226

We then create the simulation algorithm.

.. GENERATED FROM PYTHON SOURCE LINES 228-232

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 233-234

For statistics about the algorithm

.. GENERATED FROM PYTHON SOURCE LINES 234-236

.. code-block:: Python

    initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber()








.. GENERATED FROM PYTHON SOURCE LINES 237-239

.. code-block:: Python

    algo.run()








.. GENERATED FROM PYTHON SOURCE LINES 240-241

retrieve results

.. GENERATED FROM PYTHON SOURCE LINES 241-250

.. code-block:: Python

    result = algo.getResult()
    probabilityFORMIS = result.getProbabilityEstimate()
    numberOfFunctionEvaluationsFORMIS = (
        limitStateFunction.getEvaluationCallsNumber() - initialNumberOfCall
    )
    print("Number of calls to the limit state =", numberOfFunctionEvaluationsFORMIS)
    print("Pf = ", probabilityFORMIS)
    print("CV =", result.getCoefficientOfVariation())





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

 .. code-block:: none

    Number of calls to the limit state = 824
    Pf =  0.03192438900613578
    CV = 0.0499854933665548




.. GENERATED FROM PYTHON SOURCE LINES 251-253

Conclusion
----------

.. GENERATED FROM PYTHON SOURCE LINES 255-256

We now compare the different methods in terms of accuracy and speed.

.. GENERATED FROM PYTHON SOURCE LINES 261-262

The following function computes the number of correct base-10 digits in the computed result compared to the exact result.

.. GENERATED FROM PYTHON SOURCE LINES 264-269

.. code-block:: Python

    def computeLogRelativeError(exact, computed):
        logRelativeError = -np.log10(abs(exact - computed) / abs(exact))
        return logRelativeError









.. GENERATED FROM PYTHON SOURCE LINES 270-271

The following function prints the results.

.. GENERATED FROM PYTHON SOURCE LINES 273-286

.. code-block:: Python

    def printMethodSummary(name, computedProbability, numberOfFunctionEvaluations):
        print("---")
        print(name, ":")
        print("Number of calls to the limit state =", numberOfFunctionEvaluations)
        print("Pf = ", computedProbability)
        exactProbability = 0.02919819462483051
        logRelativeError = computeLogRelativeError(exactProbability, computedProbability)
        print("Number of correct digits=%.3f" % (logRelativeError))
        performance = logRelativeError / numberOfFunctionEvaluations
        print("Performance=%.2e (correct digits/evaluation)" % (performance))
        return









.. GENERATED FROM PYTHON SOURCE LINES 287-298

.. code-block:: Python

    printMethodSummary(
        "Monte-Carlo", probabilityMonteCarlo, numberOfFunctionEvaluationsMonteCarlo
    )
    printMethodSummary("FORM", probabilityFORM, numberOfFunctionEvaluationsFORM)
    printMethodSummary(
        "DirectionalSampling",
        probabilityDirectionalSampling,
        numberOfFunctionEvaluationsDirectionalSampling,
    )
    printMethodSummary("FORM-IS", probabilityFORMIS, numberOfFunctionEvaluationsFORMIS)





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

 .. code-block:: none

    ---
    Monte-Carlo :
    Number of calls to the limit state = 13931
    Pf =  0.027923336443902068
    Number of correct digits=1.360
    Performance=9.76e-05 (correct digits/evaluation)
    ---
    FORM :
    Number of calls to the limit state = 98
    Pf =  0.02998278558231473
    Number of correct digits=1.571
    Performance=1.60e-02 (correct digits/evaluation)
    ---
    DirectionalSampling :
    Number of calls to the limit state = 9050
    Pf =  0.029665915864919325
    Number of correct digits=1.795
    Performance=1.98e-04 (correct digits/evaluation)
    ---
    FORM-IS :
    Number of calls to the limit state = 824
    Pf =  0.03192438900613578
    Number of correct digits=1.030
    Performance=1.25e-03 (correct digits/evaluation)




.. GENERATED FROM PYTHON SOURCE LINES 299-313

We see that all three methods produce the correct probability, but not with the same accuracy.
In this case, we have found the correct order of magnitude of the probability, i.e. between one and two correct digits.
There is, however, a significant difference in computational performance (measured here by the number of function evaluations).

* The fastest method is the FORM method, which produces more than 1 correct
  digit with less than 98 function evaluations with a performance equal to :math:`1.60 \times 10^{-2}` (correct digits/evaluation).
  A practical limitation is that the FORM method does not produce a confidence interval: there is no guarantee that the computed probability is correct.
* The slowest method is Monte-Carlo simulation, which produces more than 1 correct digit with 12806 function evaluations.
  This is associated with a very slow performance equal to :math:`1.11 \times 10^{-4}` (correct digits/evaluation).
  The interesting point with the Monte-Carlo simulation is that the method produces a confidence interval.
* The DirectionalSampling method is somewhat in-between the two previous methods.
* The FORM-IS method produces 2 correct digits and has a small number of function evaluations.i
  It has an intermediate performance equal to :math:`2.37\times 10^{-3}` (correct digits/evaluation).
  It combines the best of the both worlds: it has the small number of function evaluation of FORM computation and the confidence interval of Monte-Carlo simulation.


.. _sphx_glr_download_auto_reliability_sensitivity_reliability_plot_axial_stressed_beam.py:

.. only:: html

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

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

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

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

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