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.

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(self, \*args)

Draw the expectation convergence at a given level.

getBlockSize(self)

Accessor to the block size.

getClassName(self)

Accessor to the object’s name.

getCoefficientOfVariationCriterionType(self)

Accessor to the criterion operator.

getConvergenceStrategy(self)

Accessor to the convergence strategy.

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.

getMaximumStandardDeviationPerComponent(self)

Accessor to the maximum standard deviation.

getName(self)

Accessor to the object’s name.

getRandomVector(self)

Accessor to the random vector.

getResult(self)

Accessor to the result.

getShadowedId(self)

Accessor to the object’s shadowed id.

getStandardDeviationCriterionType(self)

Accessor to the criterion operator.

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.

setBlockSize(self, blockSize)

Accessor to the block size.

setCoefficientOfVariationCriterionType(self, …)

Accessor to the criterion operator.

setConvergenceStrategy(self, convergenceStrategy)

Accessor to the convergence strategy.

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.

setMaximumStandardDeviationPerComponent(…)

Accessor to the maximum standard deviation.

setName(self, name)

Accessor to the object’s name.

setProgressCallback(self, \*args)

Set up a progress callback.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

setStandardDeviationCriterionType(self, …)

Accessor to the criterion operator.

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.

drawExpectationConvergence(self, *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(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__).

getCoefficientOfVariationCriterionType(self)

Accessor to the criterion operator.

Returns
resultstr

The criterion operator.

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.

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.

getMaximumStandardDeviationPerComponent(self)

Accessor to the maximum standard deviation.

Returns
sigmaMaxsequence of float

The maximum standard deviation on each component.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getRandomVector(self)

Accessor to the random vector.

Returns
XRandomVector

Random vector we want to study.

getResult(self)

Accessor to the result.

Returns
resultExpectationSimulationResult

The simulation result.

getShadowedId(self)

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getStandardDeviationCriterionType(self)

Accessor to the criterion operator.

Returns
resultstr

The criterion operator.

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 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(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.

setCoefficientOfVariationCriterionType(self, criterionType)

Accessor to the criterion operator.

Parameters
resultstr

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

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.

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.

setMaximumStandardDeviationPerComponent(self, 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(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.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(self, id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setStandardDeviationCriterionType(self, criterionType)

Accessor to the criterion operator.

Parameters
resultstr

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

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.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(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.