SubsetSampling¶

class SubsetSampling(*args)¶

Subset simulation.

Parameters

eventRandomVector: Event we are computing the probability of.
proposalRangefloat, optional: Proposal range length
targetProbabilityfloat, optional: Value of $P(F_i|F_{i-1})$ between successive steps

See also

EventSimulation

Notes

The goal is to estimate the following probability

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

The idea of the subset simulation method [au2001] 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.

Methods

`drawProbabilityConvergence`(self, \*args)	Draw the probability convergence at a given level.
`getBlockSize`(self)	Accessor to the block size.
`getClassName`(self)	Accessor to the object’s name.
`getCoefficientOfVariationPerStep`(self)	Coefficient of variation per step accessor.
`getConditionalProbability`(self)	Conditional probability accessor.
`getConvergenceStrategy`(self)	Accessor to the convergence strategy.
`getEvent`(self)	Accessor to the event.
`getEventInputSample`(self)	Input sample accessor.
`getEventOutputSample`(self)	Output sample accessor.
`getGammaPerStep`(self)	Autocorrelation accessor.
`getId`(self)	Accessor to the object’s id.
`getMaximumCoefficientOfVariation`(self)	Accessor to the maximum coefficient of variation.
`getMaximumOuterSampling`(self)	Accessor to the maximum sample size.
`getMaximumStandardDeviation`(self)	Accessor to the maximum standard deviation.
`getMinimumProbability`(self)	Minimum probability accessor.
`getName`(self)	Accessor to the object’s name.
`getNumberOfSteps`(self)	Subset steps number accessor.
`getProbabilityEstimatePerStep`(self)	Probability estimate accessor.
`getProposalRange`(self)	Proposal range length accessor.
`getResult`(self)	Accessor to the results.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getThresholdPerStep`(self)	Threshold accessor.
`getVerbose`(self)	Accessor to verbosity.
`getVisibility`(self)	Accessor to the object’s visibility state.
`hasName`(self)	Test if the object is named.
`hasVisibleName`(self)	Test if the object has a distinguishable name.
`run`(self)	Launch simulation.
`setBetaMin`(self, betaMin)	Hypersphere radius accessor.
`setBlockSize`(self, blockSize)	Accessor to the block size.
`setConditionalProbability`(self, …)	Conditional probability accessor.
`setConvergenceStrategy`(self, convergenceStrategy)	Accessor to the convergence strategy.
`setISubset`(self, iSubset)	Conditonal simulation flag accessor.
`setKeepEventSample`(self, keepEventSample)	Sample storage accessor.
`setMaximumCoefficientOfVariation`(self, …)	Accessor to the maximum coefficient of variation.
`setMaximumOuterSampling`(self, …)	Accessor to the maximum sample size.
`setMaximumStandardDeviation`(self, …)	Accessor to the maximum standard deviation.
`setMinimumProbability`(self, minimumProbability)	Minimum probability accessor.
`setName`(self, name)	Accessor to the object’s name.
`setProgressCallback`(self, \*args)	Set up a progress callback.
`setProposalRange`(self, proposalRange)	Proposal range length accessor.
`setShadowedId`(self, id)	Accessor to the object’s shadowed id.
`setStopCallback`(self, \*args)	Set up a stop callback.
`setVerbose`(self, verbose)	Accessor to verbosity.
`setVisibility`(self, visible)	Accessor to the object’s visibility state.

__init__(self, \*args)¶: Initialize self. See help(type(self)) for accurate signature.

drawProbabilityConvergence(self, \*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(self)¶

Accessor to the block size.

Returns

blockSizeint: Number of terms in the probability simulation estimator grouped together. It is set by default to 1.

getClassName(self)¶

Accessor to the object’s name.

Returns

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

getCoefficientOfVariationPerStep(self)¶

Coefficient of variation per step accessor.

Returns

coef~openturns.Point: Coefficient of variation at each subset step.

getConditionalProbability(self)¶

Conditional probability accessor.

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

Returns

probfloat: Conditional probability value.

getConvergenceStrategy(self)¶

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(self)¶

Accessor to the event.

Returns

eventRandomVector: Event we want to evaluate the probability.

getEventInputSample(self)¶

Input sample accessor.

Returns

inputSample~openturns.Sample: Input sample.

getEventOutputSample(self)¶

Output sample accessor.

Returns

outputSample~openturns.Sample: Ouput sample.

getGammaPerStep(self)¶

Autocorrelation accessor.

Returns

prob~openturns.Point: Autocorrelation values.

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getMaximumCoefficientOfVariation(self)¶

Accessor to the maximum coefficient of variation.

Returns

coefficientfloat: Maximum coefficient of variation of the simulated sample.

getMaximumOuterSampling(self)¶

Accessor to the maximum sample size.

Returns

outerSamplingint: Maximum number of groups of terms in the probability simulation estimator.

getMaximumStandardDeviation(self)¶

Accessor to the maximum standard deviation.

Returns

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

getMinimumProbability(self)¶

Minimum probability accessor.

Returns

prob_minfloat: Minimum probability.

getName(self)¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getNumberOfSteps(self)¶

Subset steps number accessor.

Returns

nint: Number of subset steps.

getProbabilityEstimatePerStep(self)¶

Probability estimate accessor.

Returns

prob~openturns.Point: Probability estimate values.

getProposalRange(self)¶

Proposal range length accessor.

Returns

rangefloat: Range length.

getResult(self)¶

Accessor to the results.

Returns

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

getShadowedId(self)¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getThresholdPerStep(self)¶

Threshold accessor.

Returns

threshold~openturns.Point: Threshold values.

getVerbose(self)¶

Accessor to verbosity.

Returns

verbosity_enabledbool: If True, the computation is verbose. By default it is verbose.

getVisibility(self)¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName(self)¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName(self)¶

Test if the object has a distinguishable name.

Returns

hasVisibleNamebool: True if the name is not empty and not the default one.

run(self)¶

Launch simulation.

See also

setBlockSize, setMaximumOuterSampling, ResourceMap, SimulationResult

Notes

It launches the simulation and creates a SimulationResult, structure containing all the results obtained after simulation. It computes the probability of occurence of the given event by computing the empirical mean of a sample of size at most outerSampling * blockSize, this sample being built by blocks of size blockSize. It allows to use efficiently the distribution of the computation as well as it allows to deal with a sample size $> 2^{32}$ by a combination of blockSize and outerSampling.

setBetaMin(self, betaMin)¶

Hypersphere radius accessor.

Parameters

betafloat: Radius value of the exclusion hypershere when the conditional simulation is enabled.

setBlockSize(self, blockSize)¶

Accessor to the block size.

Parameters

blockSizeint, $blockSize \geq 1$: Number of terms in the probability simulation estimator grouped together. It is set by default to 1.

Notes

For Monte Carlo, LHS and Importance Sampling methods, this allows to save space while allowing multithreading, when available we recommend to use the number of available CPUs; for the Directional Sampling, we recommend to set it to 1.

setConditionalProbability(self, conditionalProbability)¶

Conditional probability accessor.

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

Parameters

probfloat: Conditional probability value.

setConvergenceStrategy(self, 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(self, iSubset)¶

Conditonal simulation flag accessor.

Parameters

isubsetbool: Whether to enable conditional simulation for the first step of the simulation.

setKeepEventSample(self, keepEventSample)¶

Sample storage accessor.

Parameters

probbool: Whether to keep the event samples.

setMaximumCoefficientOfVariation(self, maximumCoefficientOfVariation)¶

Accessor to the maximum coefficient of variation.

Parameters

coefficientfloat: Maximum coefficient of variation of the simulated sample.

setMaximumOuterSampling(self, maximumOuterSampling)¶

Accessor to the maximum sample size.

Parameters

outerSamplingint: Maximum number of groups of terms in the probability simulation estimator.

setMaximumStandardDeviation(self, maximumStandardDeviation)¶

Accessor to the maximum standard deviation.

Parameters

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

setMinimumProbability(self, minimumProbability)¶

Minimum probability accessor.

Allows to stop the algorithm if the probability becomes too small.

Parameters

prob_minfloat, defaults to the square root of SpecFunc.MinScalar: Minimum probability.

setName(self, name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setProgressCallback(self, \*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(self, proposalRange)¶

Proposal range length accessor.

Parameters

rangefloat: Range length.

setShadowedId(self, id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setStopCallback(self, \*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.

Examples

Stop a Monte Carlo simulation algorithm using a time limit

>>> 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(10000000)
>>> algo.setMaximumCoefficientOfVariation(-1.0)
>>> timer = ot.TimerCallback(0.1)
>>> algo.setStopCallback(timer)
>>> algo.run()

setVerbose(self, verbose)¶

Accessor to verbosity.

Parameters

verbosity_enabledbool: If True, make the computation verbose. By default it is verbose.

setVisibility(self, visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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