ExpectationSimulationAlgorithm¶

class ExpectationSimulationAlgorithm(*args)¶

Expectation computation using sampling.

Incremental Monte Carlo sampling algorithm to estimate the mean $\Expect{\vect{X}}$ of a random vector $\vect{X}$ .

Parameters

XRandomVector: The random vector to study.

See also

ExpectationSimulationResult

Notes

The algorithm can operate on a multivariate random vector $\vect{X}$ .

There are 3 mathematical stopping criteria available:

through an operator on the coefficient of variation
through an operator on the standard deviation
on the maximum standard deviation per component

The criterion on the coefficient of variation is defined using either:

The maximum: $\max_i \frac{\sigma_i}{|\mu_i|} \leq \max_{cov}$

The norm-1: $\frac{1}{d} \sum_1^{d} \frac{\sigma_i}{|\mu_i|} \leq \max_{cov}$

The norm-2: $\sqrt{\frac{1}{d} \sum_1^{d} \left(\frac{\sigma_i}{|\mu_i|}\right)^2} \leq \max_{cov}$

The type of operator on the coefficient of variation is set using setCoefficientOfVariationCriterionType().

The default type is set by the ExpectationSimulationAlgorithm-DefaultCoefficientOfVariationCriterionType ResourceMap key.

The threshold $\max_{cov}$ can be set using setMaximumCoefficientOfVariation().

The criterion on the standard deviation is defined using either:

The maximum: $\max_i \sigma_i \leq \max_{\sigma}$

The norm-1: $\frac{1}{d} \sum_1^{d} |\sigma_i| \leq \max_{\sigma}$

The norm-2: $\sqrt{\frac{1}{d} \sum_1^{d} \sigma_i^2} \leq \max_{\sigma}$

The type of operator on the coefficient of variation can be set using setStandardDeviationCriterionType().

The default type is set by the ExpectationSimulationAlgorithm-DefaultStandardDeviationCriterionType ResourceMap key.

The threshold $\max_{\sigma}$ can be set using setMaximumStandardDeviation().

The criterion on the maximum deviation per component is defined by $\sigma_i \leq \max_{\sigma_i}$

The thresholds $\max_{\sigma_i}$ can be set using setMaximumStandardDeviationPerComponent().

By default this criterion is disabled.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> # Create a composite random vector
>>> model = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3*E*I)'])
>>> distribution = ot.Normal([50.0, 1.0, 10.0, 5.0], [1.0]*4, ot.IdentityMatrix(4))
>>> vect = ot.RandomVector(distribution)
>>> X = ot.CompositeRandomVector(model, vect)
>>> algo = ot.ExpectationSimulationAlgorithm(X)
>>> algo.setMaximumOuterSampling(1000)
>>> algo.setBlockSize(1)
>>> algo.setCoefficientOfVariationCriterionType('NONE')
>>> algo.run()
>>> result = algo.getResult()
>>> expectation = result.getExpectationEstimate()
>>> print(expectation)
[-1.39543]
>>> expectationDistribution = result.getExpectationDistribution()

Methods

`drawExpectationConvergence`(*args)	Draw the expectation convergence at a given level.
`getBlockSize`()	Accessor to the block size.
`getClassName`()	Accessor to the object’s name.
`getCoefficientOfVariationCriterionType`()	Accessor to the criterion operator.
`getConvergenceStrategy`()	Accessor to the convergence strategy.
`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.
`getMaximumStandardDeviationPerComponent`()	Accessor to the maximum standard deviation.
`getName`()	Accessor to the object’s name.
`getRandomVector`()	Accessor to the random vector.
`getResult`()	Accessor to the result.
`getShadowedId`()	Accessor to the object’s shadowed id.
`getStandardDeviationCriterionType`()	Accessor to the criterion operator.
`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.
`setCoefficientOfVariationCriterionType`(…)	Accessor to the criterion operator.
`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.
`setMaximumStandardDeviationPerComponent`(…)	Accessor to the maximum standard deviation.
`setName`(name)	Accessor to the object’s name.
`setProgressCallback`(*args)	Set up a progress callback.
`setShadowedId`(id)	Accessor to the object’s shadowed id.
`setStandardDeviationCriterionType`(criterionType)	Accessor to the criterion operator.
`setStopCallback`(*args)	Set up a stop callback.
`setVerbose`(verbose)	Accessor to verbosity.
`setVisibility`(visible)	Accessor to the object’s visibility state.

__init__(*args)¶: Initialize self. See help(type(self)) for accurate signature.

drawExpectationConvergence(*args)¶

Draw the expectation convergence at a given level.

Parameters

marginalIndexint: Index of the random vector component to consider
levelfloat, optional: The expectation convergence is drawn at this given confidence length level. By default level is 0.95.

Returns

grapha Graph: expectation 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__).

getCoefficientOfVariationCriterionType()¶

Accessor to the criterion operator.

Returns

resultstr: The criterion operator.

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.

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.

getMaximumStandardDeviationPerComponent()¶

Accessor to the maximum standard deviation.

Returns

sigmaMaxsequence of float: The maximum standard deviation on each component.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getRandomVector()¶

Accessor to the random vector.

Returns

XRandomVector: Random vector we want to study.

getResult()¶

Accessor to the result.

Returns

resultExpectationSimulationResult: The simulation result.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getStandardDeviationCriterionType()¶

Accessor to the criterion operator.

Returns

resultstr: The criterion operator.

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 on 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.

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 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.

setCoefficientOfVariationCriterionType(criterionType)¶

Accessor to the criterion operator.

Parameters

resultstr: The criterion operator, either NONE, MAX, NORM1 or NORM2.

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.

setMaximumStandardDeviationPerComponent(maximumStandardDeviation)¶

Accessor to the maximum standard deviation.

Parameters

sigmaMaxsequence of float

The maximum standard deviation on each component.

If empty, the stopping criterion is not applied.

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()

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setStandardDeviationCriterionType(criterionType)¶

Accessor to the criterion operator.

Parameters

resultstr: The criterion operator, either NONE, MAX, NORM1 or NORM2

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.

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