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:
X : RandomVector

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(*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:
marginalIndex : int

Index of the random vector component to consider

level : float, optional

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

Returns:
graph : a Graph

expectation 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__).

getCoefficientOfVariationCriterionType()

Accessor to the criterion operator.

Returns:
result : str

The criterion operator.

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.

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.

getMaximumStandardDeviationPerComponent()

Accessor to the maximum standard deviation.

Returns:
sigmaMax : sequence of float

The maximum standard deviation on each component.

getName()

Accessor to the object’s name.

Returns:
name : str

The name of the object.

getRandomVector()

Accessor to the random vector.

Returns:
X : RandomVector

Random vector we want to study.

getResult()

Accessor to the result.

Returns:
result : ExpectationSimulationResult

The simulation result.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:
id : int

Internal unique identifier.

getStandardDeviationCriterionType()

Accessor to the criterion operator.

Returns:
result : str

The criterion operator.

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 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:
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.

setCoefficientOfVariationCriterionType(criterionType)

Accessor to the criterion operator.

Parameters:
result : str

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

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.

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.

setMaximumStandardDeviationPerComponent(maximumStandardDeviation)

Accessor to the maximum standard deviation.

Parameters:
sigmaMax : sequence 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:
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.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:
id : int

Internal unique identifier.

setStandardDeviationCriterionType(criterionType)

Accessor to the criterion operator.

Parameters:
result : str

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

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.