Note
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RP89 analysis and 2D graphics¶
The objective of this example is to present problem 89 of the BBRC. We also present graphic elements for the visualization of the limit state surface in 2 dimensions.
import openturns as ot
import openturns.viewer as otv
import otbenchmark as otb
problem = otb.ReliabilityProblem89()
print(problem)
name = RP89
event = class=ThresholdEventImplementation antecedent=class=CompositeRandomVector function=class=Function name=Unnamed implementation=class=FunctionImplementation name=Unnamed description=[x1,x2,gsys] evaluationImplementation=class=SymbolicEvaluation name=Unnamed inputVariablesNames=[x1,x2] outputVariablesNames=[gsys] formulas=[var g1 := -x1^2 - x2 + 8;var g2 := -x1 / 5 - x2 + 6;gsys := min(g1, g2)] gradientImplementation=class=SymbolicGradient name=Unnamed evaluation=class=SymbolicEvaluation name=Unnamed inputVariablesNames=[x1,x2] outputVariablesNames=[gsys] formulas=[var g1 := -x1^2 - x2 + 8;var g2 := -x1 / 5 - x2 + 6;gsys := min(g1, g2)] hessianImplementation=class=SymbolicHessian name=Unnamed evaluation=class=SymbolicEvaluation name=Unnamed inputVariablesNames=[x1,x2] outputVariablesNames=[gsys] formulas=[var g1 := -x1^2 - x2 + 8;var g2 := -x1 / 5 - x2 + 6;gsys := min(g1, g2)] antecedent=class=UsualRandomVector distribution=class=JointDistribution name=JointDistribution dimension=2 copula=class=IndependentCopula name=IndependentCopula dimension=2 marginal[0]=class=Normal name=Normal dimension=1 mean=class=Point name=Unnamed dimension=1 values=[0] sigma=class=Point name=Unnamed dimension=1 values=[1] correlationMatrix=class=CorrelationMatrix dimension=1 implementation=class=MatrixImplementation name=Unnamed rows=1 columns=1 values=[1] marginal[1]=class=Normal name=Normal dimension=1 mean=class=Point name=Unnamed dimension=1 values=[0] sigma=class=Point name=Unnamed dimension=1 values=[1] correlationMatrix=class=CorrelationMatrix dimension=1 implementation=class=MatrixImplementation name=Unnamed rows=1 columns=1 values=[1] operator=class=Less name=Unnamed threshold=0
probability = 0.00543
event = problem.getEvent()
g = event.getFunction()
problem.getProbability()
0.00543
Create the Monte-Carlo algorithm
algoProb = ot.ProbabilitySimulationAlgorithm(event)
algoProb.setMaximumOuterSampling(1000)
algoProb.setMaximumCoefficientOfVariation(0.01)
algoProb.run()
Get the results
resultAlgo = algoProb.getResult()
neval = g.getEvaluationCallsNumber()
print("Number of function calls = %d" % (neval))
pf = resultAlgo.getProbabilityEstimate()
print("Failure Probability = %.4f" % (pf))
level = 0.95
c95 = resultAlgo.getConfidenceLength(level)
pmin = pf - 0.5 * c95
pmax = pf + 0.5 * c95
print("%.1f %% confidence interval :[%.4f,%.4f] " % (level * 100, pmin, pmax))
Number of function calls = 1000
Failure Probability = 0.0040
95.0 % confidence interval :[0.0001,0.0079]
Compute the bounds of the domain¶
inputVector = event.getAntecedent()
distribution = inputVector.getDistribution()
X1 = distribution.getMarginal(0)
X2 = distribution.getMarginal(1)
alphaMin = 0.00001
alphaMax = 1 - alphaMin
lowerBound = ot.Point(
[X1.computeQuantile(alphaMin)[0], X2.computeQuantile(alphaMin)[0]]
)
upperBound = ot.Point(
[X1.computeQuantile(alphaMax)[0], X2.computeQuantile(alphaMax)[0]]
)
nbPoints = [100, 100]
graph = g.draw(lowerBound, upperBound, nbPoints)
graph.setTitle("Iso-values of limit state function")
_ = otv.View(graph)
Plot the iso-values of the distribution¶
_ = otv.View(distribution.drawPDF())
sampleSize = 5000
drawEvent = otb.DrawEvent(event)
cloud = drawEvent.drawSampleCrossCut(sampleSize)
_ = otv.View(cloud)
Draw the limit state surface¶
bounds = ot.Interval(lowerBound, upperBound)
bounds
graph = drawEvent.drawLimitStateCrossCut(bounds)
graph.add(cloud)
_ = otv.View(graph)
domain = drawEvent.fillEventCrossCut(bounds)
_ = otv.View(domain)
domain.add(cloud)
_ = otv.View(domain)
otv.View.ShowAll()
Total running time of the script: (0 minutes 1.922 seconds)