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, SimulationAlgorithm

Notes

Let $\vect{X}\in\Rset^{n_X}$ be a random vector. This algorithm estimates $\Expect{\vect{X}}$ using the smallest possible sample size which satisfies pre-defined stopping rules. It increases the sample size $n$ incrementally until a stopping criterion is met. Hence, both the mean and the variance must be finite. For example, consider the Student distribution with $\nu$ degrees of freedom. If $\nu=1$ (where the expectation is undefined) or $\nu=2$ (where the variance is undefined), then the algorithm cannot succeed.

The algorithm is based on two nested loops:

the outer loop sets the number of iterations of the algorithm; this can be configured using setMaximumOuterSampling(),
the inner loop sets the number of function calls which can be parallelized; this can be configured using setBlockSize().

The algorithm allows one to get the best possible performance on distributed supercomputers and multi-core workstations. For example, if the block size is equal to 100, then the sample size is successively equal to 100, 200, etc… Each block of evaluation of the outputs can be parallelized, which may improve the performance of the algorithm. We suggest to set the block size as a multiple of the number of cores.

The algorithm stops either when the maximum number of outer iterations is reached or when the target precision is met. The latter stopping criterion is based on either the coefficient of variation of the sample mean (relative criterion) or the standard deviation of the sample mean (absolute criterion).

We consider $\vect{n}$ independent realizations $\vect{x}^{(1)}$ , …, $\vect{x}^{(n)}\in\Rset^{n_X}$ of the random vector $\vect{X}$ . We estimate $\Expect{\vect{X}}$ with the sample mean:

$\overline{\vect{x}} = \frac{1}{n} \sum_{j=1}^n \vect{x}^{(j)}.$

We estimate $\Var{X_i}$ for $i = 1, \ldots, n_X$ with the unbiased sample variance:

$\hat{\sigma}^2_i = \frac{1}{n - 1} \sum_{j=1}^n \left(x^{(j)}_i - \overline{x}_i\right)^2.$

for $i = 1, \ldots, n_X$ .

The expected value of the sample mean is:

$\Expect{\overline{X}_i} = \Expect{X_i}$

and, since the observations are independent, the variance of the sample mean is:

$\Var{\overline{X}_i} = \frac{\Var{X_i}}{n}.$

for $i = 1, \ldots, n_X$ , since the observations are independent.

Moreover, we can estimate the standard deviation of the sample mean with:

$\hat{\sigma}\left(\overline{X}_i\right) = \frac{\hat{\sigma}_i}{\sqrt{n}}$

for $i = 1, \ldots, n_X$ .

If the expectation $\Expect{X_i}$ is nonzero, the coefficient of variation of the sample mean is:

$CV\left(\overline{X}_i\right) = \frac{\sqrt{\Var{\overline{X}_i}}}{\left|\Expect{\overline{X}_i}\right|} = \frac{\sqrt{\Var{X_i}}/\sqrt{n}}{\left|\Expect{X_i}\right|}$

for $i = 1, \ldots, n_X$ .

We can estimate it with

$\widehat{CV}\left(\overline{X}_i\right) = \frac{\hat{\sigma}_i/\sqrt{n}}{\left|\overline{x}_i\right|}$

for $i = 1, \ldots, n_X$ .

When the sample size $n$ increases, the sample standard deviation and the sample coefficient of variation decrease to zero at the Monte-Carlo rate of $1/\sqrt{n}$ .

There are 3 mathematical stopping criteria available:

through an operator on the componentwise coefficients of variation (by default),
through an operator on the componentwise standard deviations,
on the maximum standard deviation per component.

A low coefficient of variation guarantees relative accuracy, while a low standard deviation guarantees absolute accuracy.

If the chosen condition is found to be true, the algorithm stops.

Let $n_X$ be the dimension of the random vector $\vect{X}$ . Let $\max_{cov}\in\Rset$ be the maximum coefficient of variation. The criterion on the componentwise coefficients of variation is defined using either:

the maximum (by default):

$\max_{i=1,\ldots,n_X} \frac{\hat{\sigma}_i / \sqrt{n}}{|\overline{x}_i|} \leq \max_{cov},$

the norm-1:

$\sum_{i=1}^{n_X} \frac{\hat{\sigma}_i / \sqrt{n}}{|\overline{x}_i|} \leq \max_{cov},$

the norm-2:

$\sqrt{\sum_{i=1}^{n_X} \left(\frac{\hat{\sigma}_i / \sqrt{n}}{|\overline{x}_i|}\right)^2} \leq \max_{cov},$

or disabled.

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

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

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

Let $\max_{\sigma}\in\Rset$ be the maximum value of the standard deviation. The criterion on the componentwise standard deviations is defined using either:

the maximum (by default):

$\max_{i=1,\ldots,n_X} \hat{\sigma}_i / \sqrt{n} \leq \max_{\sigma},$

the norm-1:

$\sum_{i=1}^{n_X} \left|\hat{\sigma}_i / \sqrt{n}\right| \leq \max_{\sigma},$

the norm-2:

$\sqrt{\sum_{i=1}^{n_X} \left(\hat{\sigma}_i / \sqrt{n} \right)^2} \leq \max_{\sigma},$

or disabled.

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

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

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

Let $\max_{\sigma_1}, \ldots, \max_{\sigma_{n_X}}$ be the componentwise maximum standard deviations. The criterion on the maximum deviation per component is defined by:

$\sigma_i \leq \max_{\sigma_i}$

for $i = 1, \ldots, n_X$ .

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

By default this criterion is disabled.

The default values of the parameters are based on the following keys of the ResourceMap:

SimulationAlgorithm-DefaultMaximumOuterSampling,
SimulationAlgorithm-DefaultMaximumCoefficientOfVariation,
SimulationAlgorithm-DefaultMaximumStandardDeviation.

In general, criteria based on coefficients of variation (C.O.V.) should be used because they guarantee a relative accuracy on the estimate of the mean. However, we may happen to estimate an expected value equal to zero (perhaps for teaching or research purposes). The default value of SimulationAlgorithm-DefaultMaximumStandardDeviation is $0$ , which means that standard deviation-based criteria are disabled by default (in order to let C.O.V.-based criteria be used by default). If the expected value of $\vect{X}$ is zero, the algorithm is likely to reach the maximum number of outer iterations. If this happens, please configure the maximum standard deviation in order to match the scale of the random variable at hand.

Examples

In the following example, we perform at most $8 \times 10000 = 80000$ evaluations of the model. However, the algorithm may stop earlier if the coefficient of variation of the sample mean falls below the threshold.

>>> 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(10000)
>>> algo.setBlockSize(8)  # The number of cores we usually have.
>>> algo.setMaximumCoefficientOfVariation(0.01)  # 1% C.O.V.
>>> algo.run()
>>> result = algo.getResult()
>>> expectation = result.getExpectationEstimate()
>>> print(expectation)
[-1.41067]

The following statement retrieves the asymptotically Gaussian distribution of the sample mean. This may be useful for users who want to compute a confidence interval of the sample mean.

>>> expectationDistribution = result.getExpectationDistribution()

The following code prints the outer sample (i.e. the number of outer iterations of the algorithm) and the sample size. We see that the algorithm stops based on the accuracy criterion.

>>> outer_sampling = result.getOuterSampling()
>>> print('outer_sampling=', outer_sampling)
outer_sampling= 1662
>>> block_size = result.getBlockSize()
>>> sample_size = outer_sampling * block_size
>>> print('sample_size=', sample_size)
sample_size= 13296

In the following example, we disable the coefficient of variation criterion. Therefore, the only remaining criterion is based on the number of iterations.

>>> algo = ot.ExpectationSimulationAlgorithm(X)
>>> algo.setMaximumOuterSampling(1000)
>>> algo.setBlockSize(1)
>>> algo.setCoefficientOfVariationCriterionType('NONE')
>>> algo.run()
>>> result = algo.getResult()
>>> outer_sampling = result.getOuterSampling()
>>> print('outer_sampling=', outer_sampling)
outer_sampling= 1000

In the following example, we compute the mean of a dimension 4 random vector. We use a standard deviation-based stopping criterion with a different threshold for every component. We must disable the coefficient of variation-based stopping criterion, otherwise, in this example, it will trigger first. The comparison with the exact mean is satisfactory, given the relatively small sample size.

>>> from openturns.usecases import cantilever_beam
>>> cb = cantilever_beam.CantileverBeam()
>>> sigma = cb.distribution.getStandardDeviation()
>>> print('sigma=', sigma)
sigma= [1.73582e+09,30,0.0288675,7.10585e-09]
>>> componentwise_max_sigma = sigma / 32.0
>>> print('componentwise_max_sigma=', componentwise_max_sigma)
componentwise_max_sigma= [5.42445e+07,0.9375,0.00090211,2.22058e-10]
>>> X = ot.RandomVector(cb.distribution)
>>> algo = ot.ExpectationSimulationAlgorithm(X)
>>> algo.setMaximumOuterSampling(1000)
>>> algo.setBlockSize(8)
>>> algo.setCoefficientOfVariationCriterionType('NONE')
>>> algo.setMaximumStandardDeviationPerComponent(componentwise_max_sigma)
>>> algo.run()
>>> result = algo.getResult()
>>> expectation = result.getExpectationEstimate()
>>> print(expectation)
[6.7125e+10,298.637,2.55074,1.45427e-07]
>>> print(cb.distribution.getMean())
[6.70455e+10,300,2.55,1.45385e-07]
>>> outer_sampling = result.getOuterSampling()
>>> print('outer_sampling=', outer_sampling)
outer_sampling= 126

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.
`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.
`getMaximumTimeDuration`()	Accessor to the maximum duration.
`getName`()	Accessor to the object's name.
`getRandomVector`()	Accessor to the random vector.
`getResult`()	Accessor to the result.
`getStandardDeviationCriterionType`()	Accessor to the criterion operator.
`hasName`()	Test if the object is named.
`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.
`setMaximumTimeDuration`(maximumTimeDuration)	Accessor to the maximum duration.
`setName`(name)	Accessor to the object's name.
`setProgressCallback`(*args)	Set up a progress callback.
`setStandardDeviationCriterionType`(criterionType)	Accessor to the criterion operator.
`setStopCallback`(*args)	Set up a stop callback.

__init__(*args)¶

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.

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.

getMaximumTimeDuration()¶

Accessor to the maximum duration.

Returns:

maximumTimeDurationfloat: Maximum optimization duration in seconds.

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.

getStandardDeviationCriterionType()¶

Accessor to the criterion operator.

Returns:

resultstr: The criterion operator.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

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 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.

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.

setMaximumTimeDuration(maximumTimeDuration)¶

Accessor to the maximum duration.

Parameters:

maximumTimeDurationfloat: Maximum optimization duration in seconds.

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()

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)
>>> algo.setMaximumTimeDuration(0.1)
>>> algo.run()