.. 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 =========================================================================== 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. Define the model ---------------- .. code-block:: default from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) We load the model from the usecases module : .. code-block:: default from openturns.usecases import stressed_beam as stressed_beam sm = stressed_beam.AxialStressedBeam() The limit state function is defined in the model field of the data class : .. code-block:: default limitStateFunction = sm.model The probabilistic model of the axial stressed beam is defined in the data class. We get the first marginal and draw it : .. code-block:: default R_dist = sm.distribution_R graph = R_dist.drawPDF() view = viewer.View(graph) .. image:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_001.png :alt: plot axial stressed beam :class: sphx-glr-single-img We get the second marginal and draw it : .. code-block:: default F_dist = sm.distribution_F graph = F_dist.drawPDF() view = viewer.View(graph) .. image:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_002.png :alt: plot axial stressed beam :class: sphx-glr-single-img These independent marginals define the joint distribution of the input parameters : .. code-block:: default myDistribution = sm.distribution We create a RandomVector from the Distribution, then a composite random vector. Finally, we create a ThresholdEvent from this RandomVector. .. code-block:: default inputRandomVector = ot.RandomVector(myDistribution) outputRandomVector = ot.CompositeRandomVector(limitStateFunction, inputRandomVector) myEvent = ot.ThresholdEvent(outputRandomVector, ot.Less(), 0.0) Using Monte Carlo simulations ----------------------------- .. 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) For statistics about the algorithm .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() Perform the analysis. .. code-block:: default algoMC.run() .. 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 = 12837 Pf = 0.030225130482199877 CV = 0.04999419733127496 .. code-block:: default graph = algoMC.drawProbabilityConvergence() graph.setLogScale(ot.GraphImplementation.LOGX) view = viewer.View(graph) .. image:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_003.png :alt: ProbabilitySimulationAlgorithm convergence graph at level 0.95 :class: sphx-glr-single-img Using FORM analysis ------------------- We create a NearestPoint algorithm .. 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) For statistics about the algorithm .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() 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. .. code-block:: default algoFORM = ot.FORM(myCobyla, myEvent, myDistribution.getMean()) Perform the analysis. .. code-block:: default algoFORM.run() .. 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 = 0.0299827855823147 .. code-block:: default graph = resultFORM.drawImportanceFactors() view = viewer.View(graph) .. image:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_004.png :alt: Importance Factors from Design Point - Unnamed :class: sphx-glr-single-img Using Directional sampling -------------------------- Resolution options: .. code-block:: default cv = 0.05 NbSim = 10000 algoDS = ot.DirectionalSampling(myEvent) algoDS.setMaximumOuterSampling(NbSim) algoDS.setBlockSize(1) algoDS.setMaximumCoefficientOfVariation(cv) For statistics about the algorithm .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() Perform the analysis. .. code-block:: default algoDS.run() .. 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 = 8832 Pf = 0.02995686422511512 CV = 0.04985883421541657 .. code-block:: default graph = algoDS.drawProbabilityConvergence() graph.setLogScale(ot.GraphImplementation.LOGX) view = viewer.View(graph) .. image:: /auto_reliability_sensitivity/reliability/images/sphx_glr_plot_axial_stressed_beam_005.png :alt: DirectionalSampling convergence graph at level 0.95 :class: sphx-glr-single-img Using importance sampling with FORM design point: FORM-IS --------------------------------------------------------- The getStandardSpaceDesignPoint method returns the design point in the U-space. .. code-block:: default standardSpaceDesignPoint = resultFORM.getStandardSpaceDesignPoint() standardSpaceDesignPoint .. raw:: html

[-1.59355,0.999463]

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. .. code-block:: default dimension = myDistribution.getDimension() dimension .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 2 .. 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 ]])

Create the design of experiment corresponding to importance sampling. This generates a WeightedExperiment with weights corresponding to the importance distribution. .. code-block:: default experiment = ot.ImportanceSamplingExperiment(myImportance) Create the standard event corresponding to the event. This transforms the original problem into the U-space, with Gaussian independent marginals. .. code-block:: default standardEvent = ot.StandardEvent(myEvent) We then create the simulation algorithm. .. code-block:: default algo = ot.ProbabilitySimulationAlgorithm(standardEvent, experiment) algo.setMaximumCoefficientOfVariation(cv) algo.setMaximumOuterSampling(40000) For statistics about the algorithm .. code-block:: default initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber() .. code-block:: default algo.run() retrieve results .. 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 = 872 Pf = 0.02957937670652859 CV = 0.049923000188140665 Conclusion ---------- We now compare the different methods in terms of accuracy and speed. .. code-block:: default import numpy as np The following function computes the number of correct base-10 digits in the computed result compared to the exact result. .. code-block:: default def computeLogRelativeError(exact, computed): logRelativeError = -np.log10(abs(exact - computed) / abs(exact)) return logRelativeError The following function prints the results. .. 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 .. 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 = 12837 Pf = 0.030225130482199877 Number of correct digits=1.454 Performance=1.13e-04 (correct digits/evaluation) --- FORM : Number of calls to the limit state = 98 Pf = 0.0299827855823147 Number of correct digits=1.571 Performance=1.60e-02 (correct digits/evaluation) --- DirectionalSampling : Number of calls to the limit state = 8832 Pf = 0.02995686422511512 Number of correct digits=1.585 Performance=1.79e-04 (correct digits/evaluation) --- FORM-IS : Number of calls to the limit state = 872 Pf = 0.02957937670652859 Number of correct digits=1.884 Performance=2.16e-03 (correct digits/evaluation) 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 0.593 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 _