SubsetSampling

class SubsetSampling(*args)

Subset simulation.

Parameters:
eventRandomVector

Event we are computing the probability.

proposalRangefloat, optional

Proposal range length

targetProbabilityfloat, 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.

getGammaPerStep()

Autocorrelation accessor.

getInitialExperiment()

Initial experiment accessor.

getInputSample(*args)

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.

getMinimumProbability()

Minimum probability accessor.

getName()

Accessor to the object's name.

getOutputSample(*args)

Output sample accessor.

getProbabilityEstimatePerStep()

Probability estimate accessor.

getProposalRange()

Proposal range length accessor.

getResult()

Accessor to the results.

getStepsNumber()

Subset steps number accessor.

getThresholdPerStep()

Threshold accessor.

hasName()

Test if the object is named.

run()

Launch simulation.

setBlockSize(blockSize)

Accessor to the block size.

setConditionalProbability(conditionalProbability)

Conditional probability accessor.

setConvergenceStrategy(convergenceStrategy)

Accessor to the convergence strategy.

setInitialExperiment(initialExperiment)

Initial experiment accessor.

setKeepSample(keepSample)

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.

setMinimumProbability(minimumProbability)

Minimum probability accessor.

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.

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.

__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:
coefPoint

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.

getGammaPerStep()

Autocorrelation accessor.

Returns:
probPoint

Autocorrelation values at each step.

getInitialExperiment()

Initial experiment accessor.

Returns:
initialExperimentWeightedExperiment

Experiment for first step.

getInputSample(*args)

Input sample accessor.

Parameters:
stepint

Iteration index

selectint, optional

Selection flag:

  • EVENT0 : points not realizing the event are selected

  • EVENT1 : points realizing the event are selected

  • BOTH : all points are selected (default)

Returns:
inputSampleSample

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.

getMinimumProbability()

Minimum probability accessor.

Returns:
prob_minfloat

Minimum probability.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getOutputSample(*args)

Output sample accessor.

Parameters:
stepint

Iteration index

selectint, optional

Selection flag:

  • EVENT0 : points not realizing the event are selected

  • EVENT1 : points realizing the event are selected

  • BOTH : all points are selected (default)

Returns:
outputSampleSample

Output sample.

getProbabilityEstimatePerStep()

Probability estimate accessor.

Returns:
probPoint

Probability estimate at each step.

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().

getStepsNumber()

Subset steps number accessor.

Returns:
nint

Number of subset steps, including the initial Monte Carlo sampling.

getThresholdPerStep()

Threshold accessor.

Returns:
thresholdPoint

Threshold values at each step.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

run()

Launch simulation.

See also

openturns.EventSimulation.setBlockSize
openturns.EventSimulation.setMaximumOuterSampling
openturns.ResourceMap
openturns.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.

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 fonction 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.

setInitialExperiment(initialExperiment)

Initial experiment accessor.

Parameters:
initialExperimentWeightedExperiment

Experiment for first step.

setKeepSample(keepSample)

Sample storage accessor.

Parameters:
keepsamplebool

Whether to keep the working samples at each iteration.

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.

setMinimumProbability(minimumProbability)

Minimum probability accessor.

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

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

Minimum probability.

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.

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)
>>> algo.setMaximumTimeDuration(0.1)
>>> algo.run()

Examples using the class

Subset Sampling

Subset Sampling