SubsetInverseSampling¶

class otrobopt.SubsetInverseSampling(*args)¶

Subset inverse simulation.

The goal is to estimate the threshold of the following target probability :

$P_f = \int_{\mathcal D_f} f_{\uX}(\ux)\di{\ux}\\ = \int_{\mathbb R^{n_X}} \mathbf{1}_{\{g(\ux,\underline{d}) \:\leq 0\: \}}f_{\uX}(\ux)\di{\ux}\\ = \Prob {\{g(\uX,\underline{d}) \leq q\}}$

The idea of the subset simulation method is to replace simulating a rare failure event in the original probability space by a sequence of simulations of more frequent conditional events $F_i$ :

$F_1 \supset F_2 \supset \dots \supset F_m = F$

The original probability estimate rewrites :

$P_f = P(F_m) = P(\bigcap \limits_{i=1}^m F_i) = P(F_1) \prod_{i=2}^m P(F_i|F_{i-1})$

And each conditional subset failure region is chosen by setting the threshold $g_i$ so that $P(F_i|F_{i-1})$ leads to a conditional failure probability of order $0.1$ :

$F_i =\Prob {\{g(\uX,\underline{d}) \leq g_i\}}$

The conditional samples are generated by the means of Markov Chains, using the Metropolis Hastings algorithm.

$N$ being the number of simulations per subset, and $p_{0i}$ the conditional probability of each subset event, and $\gamma_i$ the autocorrelation between Markov chain samples.

$\delta^2 = \sum_{i=1}^m \delta^2_i = \sum_{i=1}^m (1+\gamma_i) \frac{1-p_{0i}}{p_{0i}N}$

The first event $F_1$ not being conditional, $\delta^2_1$ expresses as the classic Monte Carlo c.o.v.

Parameters:

eventRandomVector: Event we are computing the probability of. The threshold of the event is not used.
targetProbabilityfloat: The wanted final probability.
proposalRangefloat, optional: Proposal range length
conditionalProbabilityfloat, optional: Value of $P(F_i|F_{i-1})$ between successive steps

Methods

`drawProbabilityConvergence`(*args)	Draw the probability convergence at a given level.
`getBlockSize`()	Accessor to the block size.
`getClassName`()	Accessor to the object's name.
`getCoefficientOfVariationPerStep`()	Coefficient of variation per step accessor.
`getConditionalProbability`()	Conditional probability accessor.
`getConvergenceStrategy`()	Accessor to the convergence strategy.
`getEvent`()	Accessor to the event.
`getEventInputSample`()	Input sample accessor.
`getEventOutputSample`()	Output sample accessor.
`getGammaPerStep`()	Autocorrelation accessor.
`getInputSample`()	All input sample accessor.
`getMaximumCoefficientOfVariation`()	Accessor to the maximum coefficient of variation.
`getMaximumOuterSampling`()	Accessor to the maximum iterations number.
`getMaximumStandardDeviation`()	Accessor to the maximum standard deviation.
`getMaximumTimeDuration`()	Accessor to the maximum duration.
`getName`()	Accessor to the object's name.
`getNumberOfSteps`()	Subset steps number accesor.
`getOutputSample`()	All output sample accessor.
`getProbabilityEstimatePerStep`()	Probability estimate accessor.
`getProposalRange`()	Proposal range length accessor.
`getResult`()	Accessor to the results.
`getTargetProbability`()	Final target probability accessor.
`getThresholdCoefficientOfVariationPerStep`()	Threshold coefficient of variation per step accessor.
`getThresholdConfidenceLength`(*args)	Accessor to the confidence length of the threshold.
`getThresholdPerStep`()	Threshold accessor.
`hasName`()	Test if the object is named.
`run`()	Run the algorithm.
`setBetaMin`(betaMin)	Exclusion hypersphere radius accessor.
`setBlockSize`(blockSize)	Accessor to the block size.
`setConditionalProbability`(conditionalProbability)	Conditional probability accessor.
`setConvergenceStrategy`(convergenceStrategy)	Accessor to the convergence strategy.
`setISubset`(iSubset)	Conditional simulation activation accessor.
`setKeepEventSample`(keepEventSample)	Sample storage accessor.
`setMaximumCoefficientOfVariation`(...)	Accessor to the maximum coefficient of variation.
`setMaximumOuterSampling`(maximumOuterSampling)	Accessor to the maximum iterations number.
`setMaximumStandardDeviation`(...)	Accessor to the maximum standard deviation.
`setMaximumTimeDuration`(maximumTimeDuration)	Accessor to the maximum duration.
`setName`(name)	Accessor to the object's name.
`setProgressCallback`(*args)	Set up a progress callback.
`setProposalRange`(proposalRange)	Proposal range length accessor.
`setStopCallback`(*args)	Set up a stop callback.
`setTargetProbability`(targetProbability)	Final target probability accessor.

See also

openturns.EventSimulation

Examples

>>> import openturns as ot
>>> import otrobopt

>>> ot.RandomGenerator.SetSeed(0)

Create a performance function with an associated distribution.

>>> limitState = ot.SymbolicFunction(['u1', 'u2'], ['u1-u2'])
>>> dim = limitState.getInputDimension()
>>> mean = ot.Point([7., 2.])
>>> sigma = ot.Point(dim, 1.0)
>>> R = ot.IdentityMatrix(dim)
>>> distribution = ot.Normal(mean, sigma, R)
>>> vect = ot.RandomVector(distribution)
>>> output = ot.CompositeRandomVector(limitState, vect)

Create an event with a fictional threshold value which will not be used.

>>> event = ot.ThresholdEvent(output, ot.Less(), 0.)

Define the target probability for which the threshold will be computed.

>>> targetProbability = 0.0002
>>> algo = otrobopt.SubsetInverseSampling(event, targetProbability)
>>> algo.setMaximumOuterSampling(10000)
>>> algo.run()

Get some results.

>>> result = algo.getResult()
>>> pf = result.getProbabilityEstimate()
>>> threshold = algo.getThresholdPerStep()[-1]
>>> threshold_cl = algo.getThresholdConfidenceLength(0.90)

__init__(*args)¶

drawProbabilityConvergence(*args)¶

Draw the probability convergence at a given level.

Parameters:

levelfloat, optional: The probability convergence is drawn at this given confidence length level. By default level is 0.95.

Returns:

grapha Graph: probability convergence graph

getBlockSize()¶

Accessor to the block size.

Returns:

blockSizeint: Number of simultaneous evaluations of the limit-state function. It is set by default to 1.

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getCoefficientOfVariationPerStep()¶

Coefficient of variation per step accessor.

Returns:

coefopenturns.Point: Coefficient of variation at each subset step.

getConditionalProbability()¶

Conditional probability accessor.

Value of $P(F_i|F_{i-1})$ between successive steps.

Returns:

probfloat: Conditional probability value.

getConvergenceStrategy()¶

Accessor to the convergence strategy.

Returns:

storage_strategyHistoryStrategy: Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.

getEvent()¶

Accessor to the event.

Returns:

eventRandomVector: Event we want to evaluate the probability.

getEventInputSample()¶

Input sample accessor.

Returns:

inputSampleopenturns.Sample: Input sample.

getEventOutputSample()¶

Output sample accessor.

Returns:

outputSampleopenturns.Sample: Ouput sample.

getGammaPerStep()¶

Autocorrelation accessor.

Returns:

gammaopenturns.Point: Autocorrelation values at each subset step.

getInputSample()¶

All input sample accessor.

Returns:

inputSampleopenturns.Sample: Input sample.

getMaximumCoefficientOfVariation()¶

Accessor to the maximum coefficient of variation.

Returns:

coefficientfloat: Maximum coefficient of variation of the simulated sample.

getMaximumOuterSampling()¶

Accessor to the maximum iterations number.

Returns:

outerSamplingint: Maximum number of iterations, each iteration performing a block of evaluations.

getMaximumStandardDeviation()¶

Accessor to the maximum standard deviation.

Returns:

sigmafloat, $\sigma > 0$: Maximum standard deviation of the estimator.

getMaximumTimeDuration()¶

Accessor to the maximum duration.

Returns:

maximumTimeDurationfloat: Maximum optimization duration in seconds.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getNumberOfSteps()¶

Subset steps number accesor.

Returns:

nint: Number of subset steps.

getOutputSample()¶

All output sample accessor.

Returns:

outputSampleopenturns.Sample: Output sample.

getProbabilityEstimatePerStep()¶

Probability estimate accessor.

Returns:

probopenturns.Point: Probability estimate values.

getProposalRange()¶

Proposal range length accessor.

Returns:

rangefloat: Range length.

getResult()¶

Accessor to the results.

Returns:

resultsSimulationResult: Structure containing all the results obtained after simulation and created by the method run().

getTargetProbability()¶

Final target probability accessor.

Value of $P(F_m)$ .

Returns:

probfloat: Final target probability value.

getThresholdCoefficientOfVariationPerStep()¶

Threshold coefficient of variation per step accessor.

Returns:

coefopenturns.Point: Coefficient of variation at each subset step.

getThresholdConfidenceLength(*args)¶

Accessor to the confidence length of the threshold.

Parameters:

levelfloat, $level \in ]0, 1[$: Confidence level. By default, it is $0.95$ .

Returns:

confidenceLengthfloat: Length of the confidence interval at the confidence level level.

getThresholdPerStep()¶

Threshold accessor.

Returns:

thresholdopenturns.Point: Threshold values.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

run()¶

Run the algorithm.

Launches the subset inverse simulation. The result can be retrieved with getResult().

setBetaMin(betaMin)¶

Exclusion hypersphere radius accessor.

Parameters:

betafloat: Radius of the exclusion hypersphere when the conditional simulation is activated.

setBlockSize(blockSize)¶

Accessor to the block size.

Parameters:

blockSizeint, $blockSize \geq 1$: Number of simultaneous evaluations of the limit-state function. It is set by default to 1.

Notes

If the function supports batch evaluations this parameter can be set to the number of available CPUs to benefit from parallel execution (multithreading, multiprocessing, …); except for the Directional Sampling, where we recommend to set it to 1. It also decides the frequency of the stopping criteria and progress callbacks update as they are called once the whole block of function evaluations is completed.

setConditionalProbability(conditionalProbability)¶

Conditional probability accessor.

Value of $P(F_i|F_{i-1})$ between successive steps.

Parameters:

probfloat: Conditional probability value.

setConvergenceStrategy(convergenceStrategy)¶

Accessor to the convergence strategy.

Parameters:

storage_strategyHistoryStrategy: Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.

setISubset(iSubset)¶

Conditional simulation activation accessor.

Parameters:

isubsetbool: Whether to activate the conditional simulation for the first step.

setKeepEventSample(keepEventSample)¶

Sample storage accessor.

Parameters:

probbool: Whether to keep the event samples.

setMaximumCoefficientOfVariation(maximumCoefficientOfVariation)¶

Accessor to the maximum coefficient of variation.

Parameters:

coefficientfloat: Maximum coefficient of variation of the simulated sample.

setMaximumOuterSampling(maximumOuterSampling)¶

Accessor to the maximum iterations number.

Parameters:

outerSamplingint: Maximum number of iterations, each iteration performing a block of evaluations.

setMaximumStandardDeviation(maximumStandardDeviation)¶

Accessor to the maximum standard deviation.

Parameters:

sigmafloat, $\sigma > 0$: Maximum standard deviation of the estimator.

setMaximumTimeDuration(maximumTimeDuration)¶

Accessor to the maximum duration.

Parameters:

maximumTimeDurationfloat: Maximum optimization duration in seconds.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setProgressCallback(*args)¶

Set up a progress callback.

Can be used to programmatically report the progress of a simulation.

Parameters:

callbackcallable: Takes a float as argument as percentage of progress.

Examples

>>> import sys
>>> import openturns as ot
>>> experiment = ot.MonteCarloExperiment()
>>> X = ot.RandomVector(ot.Normal())
>>> Y = ot.CompositeRandomVector(ot.SymbolicFunction(['X'], ['1.1*X']), X)
>>> event = ot.ThresholdEvent(Y, ot.Less(), -2.0)
>>> algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
>>> algo.setMaximumOuterSampling(100)
>>> algo.setMaximumCoefficientOfVariation(-1.0)
>>> def report_progress(progress):
...     sys.stderr.write('-- progress=' + str(progress) + '%\n')
>>> algo.setProgressCallback(report_progress)
>>> algo.run()

setProposalRange(proposalRange)¶

Proposal range length accessor.

Parameters:

rangefloat: Range length.

setStopCallback(*args)¶

Set up a stop callback.

Can be used to programmatically stop a simulation.

Parameters:

callbackcallable: Returns an int deciding whether to stop or continue.

setTargetProbability(targetProbability)¶

Final target probability accessor.

Value of $P(F_m)$ .

Parameters:

probfloat: Final target probability value.

otrobopt

Module otrobopt

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