.. 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 Click :ref:`here ` 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 ` 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:: default import numpy as np from openturns.usecases import stressed_beam 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 28-29 We load the model from the usecases module : .. GENERATED FROM PYTHON SOURCE LINES 29-31 .. code-block:: default 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:: default 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:: default 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:: default 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:: default 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 59-64 .. code-block:: default inputRandomVector = ot.RandomVector(myDistribution) outputRandomVector = ot.CompositeRandomVector( limitStateFunction, inputRandomVector) myEvent = ot.ThresholdEvent(outputRandomVector, ot.Less(), 0.0) .. GENERATED FROM PYTHON SOURCE LINES 65-67 Using Monte Carlo simulations ----------------------------- .. GENERATED FROM PYTHON SOURCE LINES 69-78 .. code-block:: default 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 79-80 For statistics about the algorithm .. GENERATED FROM PYTHON SOURCE LINES 80-82 .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() .. GENERATED FROM PYTHON SOURCE LINES 83-84 Perform the analysis. .. GENERATED FROM PYTHON SOURCE LINES 86-88 .. code-block:: default algoMC.run() .. GENERATED FROM PYTHON SOURCE LINES 89-98 .. code-block:: default 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 Out: .. code-block:: none Number of calls to the limit state = 13583 Pf = 0.028638739600971738 CV = 0.049970717914001414 .. GENERATED FROM PYTHON SOURCE LINES 99-103 .. code-block:: default 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 104-106 Using FORM analysis ------------------- .. GENERATED FROM PYTHON SOURCE LINES 108-109 We create a NearestPoint algorithm .. GENERATED FROM PYTHON SOURCE LINES 109-118 .. code-block:: default 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 119-120 For statistics about the algorithm .. GENERATED FROM PYTHON SOURCE LINES 120-122 .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() .. GENERATED FROM PYTHON SOURCE LINES 123-124 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 126-128 .. code-block:: default algoFORM = ot.FORM(myCobyla, myEvent, myDistribution.getMean()) .. GENERATED FROM PYTHON SOURCE LINES 129-130 Perform the analysis. .. GENERATED FROM PYTHON SOURCE LINES 132-134 .. code-block:: default algoFORM.run() .. GENERATED FROM PYTHON SOURCE LINES 135-142 .. code-block:: default 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 Out: .. code-block:: none Number of calls to the limit state = 98 Pf = 0.02998278558231473 .. GENERATED FROM PYTHON SOURCE LINES 143-146 .. code-block:: default 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:: default 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:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() .. GENERATED FROM PYTHON SOURCE LINES 165-166 Perform the analysis. .. GENERATED FROM PYTHON SOURCE LINES 168-170 .. code-block:: default algoDS.run() .. GENERATED FROM PYTHON SOURCE LINES 171-180 .. code-block:: default 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 Out: .. code-block:: none Number of calls to the limit state = 8773 Pf = 0.029529907622332107 CV = 0.049879032319933896 .. GENERATED FROM PYTHON SOURCE LINES 181-185 .. code-block:: default 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 186-188 Using importance sampling with FORM design point: FORM-IS --------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 190-191 The `getStandardSpaceDesignPoint` method returns the design point in the U-space. .. GENERATED FROM PYTHON SOURCE LINES 193-196 .. code-block:: default standardSpaceDesignPoint = resultFORM.getStandardSpaceDesignPoint() standardSpaceDesignPoint .. raw:: html

[-1.59355,0.999463]



.. GENERATED FROM PYTHON SOURCE LINES 197-198 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 200-203 .. code-block:: default dimension = myDistribution.getDimension() dimension .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 2 .. GENERATED FROM PYTHON SOURCE LINES 204-208 .. code-block:: default myImportance = ot.Normal(dimension) myImportance.setMean(standardSpaceDesignPoint) myImportance .. raw:: html

Normal(mu = [-1.59355,0.999463], sigma = [1,1], R = [[ 1 0 ]
[ 0 1 ]])



.. GENERATED FROM PYTHON SOURCE LINES 209-210 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 212-214 .. code-block:: default experiment = ot.ImportanceSamplingExperiment(myImportance) .. GENERATED FROM PYTHON SOURCE LINES 215-216 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 218-220 .. code-block:: default standardEvent = ot.StandardEvent(myEvent) .. GENERATED FROM PYTHON SOURCE LINES 221-222 We then create the simulation algorithm. .. GENERATED FROM PYTHON SOURCE LINES 224-228 .. code-block:: default algo = ot.ProbabilitySimulationAlgorithm(standardEvent, experiment) algo.setMaximumCoefficientOfVariation(cv) algo.setMaximumOuterSampling(40000) .. GENERATED FROM PYTHON SOURCE LINES 229-230 For statistics about the algorithm .. GENERATED FROM PYTHON SOURCE LINES 230-232 .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() .. GENERATED FROM PYTHON SOURCE LINES 233-235 .. code-block:: default algo.run() .. GENERATED FROM PYTHON SOURCE LINES 236-237 retrieve results .. GENERATED FROM PYTHON SOURCE LINES 237-246 .. code-block:: default 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 Out: .. code-block:: none Number of calls to the limit state = 940 Pf = 0.028337050932472036 CV = 0.04997559964567614 .. GENERATED FROM PYTHON SOURCE LINES 247-249 Conclusion ---------- .. GENERATED FROM PYTHON SOURCE LINES 251-252 We now compare the different methods in terms of accuracy and speed. .. GENERATED FROM PYTHON SOURCE LINES 257-258 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 260-265 .. code-block:: default def computeLogRelativeError(exact, computed): logRelativeError = -np.log10(abs(exact - computed) / abs(exact)) return logRelativeError .. GENERATED FROM PYTHON SOURCE LINES 266-267 The following function prints the results. .. GENERATED FROM PYTHON SOURCE LINES 269-283 .. code-block:: default 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 284-292 .. code-block:: default printMethodSummary("Monte-Carlo", probabilityMonteCarlo, numberOfFunctionEvaluationsMonteCarlo) printMethodSummary("FORM", probabilityFORM, numberOfFunctionEvaluationsFORM) printMethodSummary("DirectionalSampling", probabilityDirectionalSampling, numberOfFunctionEvaluationsDirectionalSampling) printMethodSummary("FORM-IS", probabilityFORMIS, numberOfFunctionEvaluationsFORMIS) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none --- Monte-Carlo : Number of calls to the limit state = 13583 Pf = 0.028638739600971738 Number of correct digits=1.718 Performance=1.26e-04 (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 = 8773 Pf = 0.029529907622332107 Number of correct digits=1.945 Performance=2.22e-04 (correct digits/evaluation) --- FORM-IS : Number of calls to the limit state = 940 Pf = 0.028337050932472036 Number of correct digits=1.530 Performance=1.63e-03 (correct digits/evaluation) .. GENERATED FROM PYTHON SOURCE LINES 293-299 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. 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. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.541 seconds) .. _sphx_glr_download_auto_reliability_sensitivity_reliability_plot_axial_stressed_beam.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_axial_stressed_beam.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_axial_stressed_beam.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_