SubsetInverseSampling¶

class otrobopt.SubsetInverseSampling(*args)¶

Subset inverse simulation.

Parameters:

eventEvent: 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

See also

openturns.Simulation

Notes

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.

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)
>>> myDistribution = ot.Normal(mean, sigma, R)
>>> vect = ot.RandomVector(myDistribution)
>>> output = ot.CompositeRandomVector(limitState, vect)

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

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

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

>>> targetProbability = 0.0002
>>> mySS = otrobopt.SubsetInverseSampling(myEvent, targetProbability)
>>> mySS.setMaximumOuterSampling(10000)
>>> mySS.run()

Get some results.

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

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.
`getId`()	Accessor to the object's id.
`getInputSample`()	All input sample accessor.
`getMaximumCoefficientOfVariation`()	Accessor to the maximum coefficient of variation.
`getMaximumOuterSampling`()	Accessor to the maximum sample size.
`getMaximumStandardDeviation`()	Accessor to the maximum standard deviation.
`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.
`getShadowedId`()	Accessor to the object's shadowed id.
`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.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`run`()	Launch simulation.
`setBetaMin`(betaMin)	Radius of the hypershere accessor.
`setBlockSize`(blockSize)	Accessor to the block size.
`setConditionalProbability`(conditionalProbability)	Conditional probability accessor.
`setConvergenceStrategy`(convergenceStrategy)	Accessor to the convergence strategy.
`setISubset`(iSubset)	Conditonal simulation activation accessor.
`setKeepEventSample`(keepEventSample)	Sample storage accessor.
`setMaximumCoefficientOfVariation`(...)	Accessor to the maximum coefficient of variation.
`setMaximumOuterSampling`(maximumOuterSampling)	Accessor to the maximum sample size.
`setMaximumStandardDeviation`(...)	Accessor to the maximum standard deviation.
`setName`(name)	Accessor to the object's name.
`setProgressCallback`(*args)	Set up a progress callback.
`setProposalRange`(proposalRange)	Proposal range length accessor.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setStopCallback`(*args)	Set up a stop callback.
`setTargetProbability`(targetProbability)	Final target probability accessor.
`setVisibility`(visible)	Accessor to the object's visibility state.

getVerbose
setVerbose

__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 terms in the probability simulation estimator grouped together. 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:

probopenturns.Point: Autocorrelation values.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

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 sample size.

Returns:

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

getMaximumStandardDeviation()¶

Accessor to the maximum standard deviation.

Returns:

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

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().

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

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.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

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

run()¶

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 occurrence 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 one to use efficiently the distribution of the computation as well as it allows one to deal with a sample size $> 2^{32}$ by a combination of blockSize and outerSampling.

setBetaMin(betaMin)¶

Radius of the hypershere accessor.

Parameters:

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

setBlockSize(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 one 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(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)¶

Conditonal simulation activation accessor.

Parameters:

isubsetbool: Activate or not the conditional simulation for the first step of the simulation.

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 sample size.

Parameters:

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

setMaximumStandardDeviation(maximumStandardDeviation)¶

Accessor to the maximum standard deviation.

Parameters:

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

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.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

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.

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()

setTargetProbability(targetProbability)¶

Final target probability accessor.

Value of $P(F_m)$ .

Parameters:

probfloat: Final target probability value.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

SubsetInverseSampling¶

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