SubsetSampling

class SubsetSampling(*args)

Subset simulation.

Parameters:

event : Event

Event we are computing the probability of.

proposalRange : float, optional

Proposal range length

targetProbability : float, optional

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

See also

Simulation

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(*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.
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.
getProbabilityEstimatePerStep() Probability estimate accessor.
getProposalRange() Proposal range length accessor.
getResult() Accessor to the results.
getShadowedId() Accessor to the object’s shadowed id.
getThresholdPerStep() Threshold accessor.
getVerbose() Accessor to verbosity.
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) Hypersphere radius accessor.
setBlockSize(blockSize) Accessor to the block size.
setConditionalProbability(conditionalProbability) Conditional probability accessor.
setConvergenceStrategy(convergenceStrategy) Accessor to the convergence strategy.
setISubset(iSubset) Conditonal simulation flag 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.
setVerbose(verbose) Accessor to verbosity.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
drawProbabilityConvergence(*args)

Draw the probability convergence at a given level.

Parameters:

level : float, optional

The probability convergence is drawn at this given confidence length level. By default level is 0.95.

Returns:

graph : a Graph

probability convergence graph

getBlockSize()

Accessor to the block size.

Returns:

blockSize : int

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_name : str

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

getCoefficientOfVariationPerStep()

Coefficient of variation per step accessor.

Returns:

coef : ~openturns.NumericalPoint

Coefficient of variation at each subset step.

getConditionalProbability()

Conditional probability accessor.

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

Returns:

prob : float

Conditional probability value.

getConvergenceStrategy()

Accessor to the convergence strategy.

Returns:

storage_strategy : HistoryStrategy

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

getEvent()

Accessor to the event.

Returns:

event : Event

Event we want to evaluate the probability.

getEventInputSample()

Input sample accessor.

Returns:

inputSample : ~openturns.NumericalSample

Input sample.

getEventOutputSample()

Output sample accessor.

Returns:

outputSample : ~openturns.NumericalSample

Ouput sample.

getGammaPerStep()

Autocorrelation accessor.

Returns:

prob : ~openturns.NumericalPoint

Autocorrelation values.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getMaximumCoefficientOfVariation()

Accessor to the maximum coefficient of variation.

Returns:

coefficient : float

Maximum coefficient of variation of the simulated sample.

getMaximumOuterSampling()

Accessor to the maximum sample size.

Returns:

outerSampling : int

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

getMaximumStandardDeviation()

Accessor to the maximum standard deviation.

Returns:

sigma : float, \sigma > 0

Maximum standard deviation of the estimator.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getNumberOfSteps()

Subset steps number accesor.

Returns:

n : int

Number of subset steps.

getProbabilityEstimatePerStep()

Probability estimate accessor.

Returns:

prob : ~openturns.NumericalPoint

Probability estimate values.

getProposalRange()

Proposal range length accessor.

Returns:

range : float

Range length.

getResult()

Accessor to the results.

Returns:

results : SimulationResult

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

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getThresholdPerStep()

Threshold accessor.

Returns:

threshold : ~openturns.NumericalPoint

Threshold values.

getVerbose()

Accessor to verbosity.

Returns:

verbosity_enabled : bool

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

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

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

run()

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(betaMin)

Hypersphere radius accessor.

Parameters:

beta : float

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

setBlockSize(blockSize)

Accessor to the block size.

Parameters:

blockSize : int, 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(conditionalProbability)

Conditional probability accessor.

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

Parameters:

prob : float

Conditional probability value.

setConvergenceStrategy(convergenceStrategy)

Accessor to the convergence strategy.

Parameters:

storage_strategy : HistoryStrategy

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

setISubset(iSubset)

Conditonal simulation flag accessor.

Parameters:

isubset : bool

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

setKeepEventSample(keepEventSample)

Sample storage accessor.

Parameters:

prob : bool

Whether to keep the event samples.

setMaximumCoefficientOfVariation(maximumCoefficientOfVariation)

Accessor to the maximum coefficient of variation.

Parameters:

coefficient : float

Maximum coefficient of variation of the simulated sample.

setMaximumOuterSampling(maximumOuterSampling)

Accessor to the maximum sample size.

Parameters:

outerSampling : int

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

setMaximumStandardDeviation(maximumStandardDeviation)

Accessor to the maximum standard deviation.

Parameters:

sigma : float, \sigma > 0

Maximum standard deviation of the estimator.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setProgressCallback(*args)

Set up a progress callback.

Parameters:

callback : callable

Takes a float as argument as percentage of progress.

setProposalRange(proposalRange)

Proposal range length accessor.

Parameters:

range : float

Range length.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setStopCallback(*args)

Set up a stop callback.

Parameters:

callback : callable

Returns an int deciding whether to stop or continue.

setVerbose(verbose)

Accessor to verbosity.

Parameters:

verbosity_enabled : bool

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

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.