StandardSpaceCrossEntropyImportanceSampling¶

class StandardSpaceCrossEntropyImportanceSampling(*args)¶

Standard Space Cross-Entropy Importance Sampling.

Warning

This class is experimental and likely to be modified in future releases. To use it, import the openturns.experimental submodule.

Parameters:

eventThresholdEvent: Event we are computing the probability of.
quantileLevelfloat $0<quantileLevel<1$: Intermediate quantile level. The default number can be tweaked with the CrossEntropyImportanceSampling-DefaultQuantileLevel key from ResourceMap.

See also

PhysicalSpaceCrossEntropyImportanceSampling, CrossEntropyImportanceSampling, CrossEntropyResult

Notes

The following explanations are given for a failure event defined as $g(\vect{X})<T$ with $\vect{X}$ a random vector following a joint PDF $f_\vect{X}$ , $T$ a threshold and $g$ a limit state function, without loss of generality.

The Importance Sampling (IS) probability estimate $\widehat{P}^\text{IS}$ is given by:

$\widehat{P}^\text{IS}=\frac{1}{N} \sum_{i=1}^{N} {\vect{1}}_{g(\vect{x}_i)<T} \frac{f_\vect{X}(\vect{x}_i)}{h(\vect{x}_i)},$

with $h$ the auxiliary PDF of Importance Sampling, $N$ the number of independent samples generated with $h$ and ${\vect{1}}_{g(\vect{x}_i)<T}$ the indicator function of the failure domain.

The optimal density $h_{opt}$ minimizing the variance of the estimator $h_{opt}$ is defined as:

$h_{opt}=\frac{{\vect{1}}_{g(\vect{x})<T}f_\vect{X}}{P},$

with $P$ the failure probability which is inaccessible in practice since this probability is the quantity of interest and unknown.

The Standard Space Cross-Entropy Importance Sampling [rubinstein2017] uses a parametric auxiliary distribution $h_\vect{\lambda}$ in order to optimize its parameters to compute the probability of interest with accuracy. This process is performed in the standard space using a Normal distribution as the auxiliary distribution. Indeed, in the standard space, the optimal value of the mean and standard deviations are given analytically, see [morio2015] for details.

We note $G(\vect{u})$ the limit state function in standard space and $g(\vect{x})$ its counterpart in the physical space. $f_\vect{U}$ stands for the initial distribution in standard space, i.e. the Normal law. The algorithm is as follows:

$k=1$ , set the quantile level $\rho \in [0,1]$ and $h_0 = f_{\vect{U}}$
Generate the population $\vect{u}_1^{(k)},...,\vect{u}_N^{(k)}$ according to the PDF $h_{k-1}$ , apply the function $G$ in order to have $y_1^{(k)}=G(\vect{u}_1^{(k)}),...,y_N^{(k)} = G(\vect{u}_N^{(k)})$
Compute the $\rho$ -quantile $q_k=\max(T,y^{(k)}_{\left \lfloor \rho N \right\rfloor})$ using the floor of $\rho N$ .
If $q_k>T$ , go to Step 7

Update the mean and standard deviation of the auxiliary distribution:

$\mu_{k}= \frac{\frac{1}{N} \sum_{i=1}^{N}{\vect{1}}_{G(\vect{u}_i^{(k)}) \leq q_k} \frac{f_\vect{U}(\vect{u}_i^{(k)})}{h_{\vect{\lambda}_{k-1}}(\vect{u}_i^{(k)})} \vect{u}_i^{(k)}}{\frac{1}{N} \sum_{i=1}^{N}{\vect{1}}_{G(\vect{u}_i^{(k)}) \leq q_k} \frac{f_\vect{U}(\vect{u}_i^{(k)})}{h_{\vect{\lambda}_{k-1}}(\vect{u}_i^{(k)})} }$

$\sigma_{k}= \frac{\frac{1}{N} \sum_{i=1}^{N}{\vect{1}}_{G(\vect{u}_i^{(k)}) \leq q_k} \frac{f_\vect{U}(\vect{u}_i^{(k)})}{h_{\vect{\lambda}_{k-1}}(\vect{u}_i^{(k)})} (\vect{u}_i^{(k)}-\mu_k)^2}{\frac{1}{N} \sum_{i=1}^{N}{\vect{1}}_{G(\vect{u}_i^{(k)}) \leq q_k} \frac{f_\vect{U}(\vect{u}_i^{(k)})}{h_{\vect{\lambda}_{k-1}}(\vect{u}_i^{(k)})} }$

$k\leftarrow k+1$ , go to Step 2
Estimate the probability $\widehat{P}^{CE}(g(\vect{\vect{X}})<T)=\widehat{P}^{CE}(G(\vect{\vect{U}})<T)=\frac{1}{N}\displaystyle \sum_{i=1}^{N} \vect{1}_{G(\vect{u}_i^{(k)})<T} \frac{f_\vect{U}(\vect{u}_i^{(k)})}{h_{\vect{\lambda}_{k-1}(\vect{u}_i^{(k)})}}$

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> from openturns.testing import assert_almost_equal
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution_R = ot.LogNormalMuSigma(300.0, 30.0, 0.0).getDistribution()
>>> distribution_F = ot.Normal(75e3, 5e3)
>>> marginals = [distribution_R, distribution_F]
>>> distribution = ot.ComposedDistribution(marginals)
>>> # create the model
>>> model = ot.SymbolicFunction(['R', 'F'], ['R - F / (pi_ * 100.0)'])  
>>> # create the event 
>>> vect = ot.RandomVector(distribution)
>>> g = ot.CompositeRandomVector(model, vect)
>>> event = ot.ThresholdEvent(g, ot.Less(), 0.0)
>>> myIS = otexp.StandardSpaceCrossEntropyImportanceSampling(event, 0.3)
>>> # run StandardSpaceCrossEntropyImportanceSampling
>>> myIS.run()
>>> myResult = myIS.getResult()

Methods

`drawProbabilityConvergence`(*args)	Draw the probability convergence at a given level.
`getBlockSize`()	Accessor to the block size.
`getClassName`()	Accessor to the object's name.
`getConvergenceStrategy`()	Accessor to the convergence strategy.
`getEvent`()	Accessor to the event.
`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.
`getQuantileLevel`()	Accessor to the intermediate quantile level.
`getResult`()	Accessor to the intermediate quantile level.
`getShadowedId`()	Accessor to the object's shadowed id.
`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.
`setBlockSize`(blockSize)	Accessor to the block size.
`setConvergenceStrategy`(convergenceStrategy)	Accessor to the convergence strategy.
`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.
`setQuantileLevel`(quantileLevel)	Accessor to the intermediate quantile level.
`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:

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__).

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.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

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.

getQuantileLevel()¶

Accessor to the intermediate quantile level.

Returns:

quantileLevel: float: Result of Cross Entropy algorithm.

getResult()¶

Accessor to the intermediate quantile level.

Returns:

CrossEntropyResult: class:~openturns.CrossEntropyResult: Contains all the results of the Cross Entropy algorithm simulation.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getVerbose()¶

Accessor to verbosity.

Returns:

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

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.

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.

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.

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()

setQuantileLevel(quantileLevel)¶

Accessor to the intermediate quantile level.

Parameters:

quantileLevel: float: Intermediate quantile level, must belong to the interval (0,1).

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()

setVerbose(verbose)¶

Accessor to verbosity.

Parameters:

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

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Cross Entropy Importance Sampling

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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StandardSpaceCrossEntropyImportanceSampling¶

Examples using the class¶