SubsetSampling

class SubsetSampling(*args)

Subset simulation.

Parameters
eventEvent

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.

getName(self)

Accessor to the object’s name.

getNumberOfSteps(self)

Subset steps number accesor.

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.

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
eventEvent

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.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getNumberOfSteps(self)

Subset steps number accesor.

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.

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.

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.Event(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.Event(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.