SobolSimulationAlgorithm¶

(Source code, png)

class SobolSimulationAlgorithm(*args)¶

Sobol indices computation using iterative sampling.

The algorithm uses sampling of the distribution of the random vector $\vect{X}$ through the model $f$ to iteratively estimate the Sobol indices.

At each iteration a Sobol’ experiment is generated and evaluated by blocks through the model $f$ . Then the distribution of the indices (first and total order) is computed on this current replication sample. At the end of each iteration we update the global distribution of the indices.

Parameters:

XDistribution: The random vector to study.
fFunction: The function to study.
estimatorSobolIndicesAlgorithm: The estimator of the indices.

Methods

`drawFirstOrderIndexConvergence`(*args)	Draw the first order Sobol index convergence at a given level.
`drawTotalOrderIndexConvergence`(*args)	Draw the total order Sobol index convergence at a given level.
`getBlockSize`()	Accessor to the block size.
`getClassName`()	Accessor to the object's name.
`getConvergenceStrategy`()	Accessor to the convergence strategy.
`getDistribution`()	Accessor to the batch size.
`getEstimator`()	Sobol estimator accessor.
`getExperimentSize`()	Accessor to the experiment size.
`getIndexQuantileEpsilon`()	Accessor to the criterion operator.
`getIndexQuantileLevel`()	Accessor to the quantile level.
`getMaximumCoefficientOfVariation`()	Accessor to the maximum coefficient of variation.
`getMaximumOuterSampling`()	Accessor to the maximum iterations number.
`getMaximumStandardDeviation`()	Accessor to the maximum standard deviation.
`getMaximumTimeDuration`()	Accessor to the maximum duration.
`getName`()	Accessor to the object's name.
`getResult`()	Accessor to the result.
`hasName`()	Test if the object is named.
`run`()	Launch simulation.
`setBlockSize`(blockSize)	Accessor to the block size.
`setConvergenceStrategy`(convergenceStrategy)	Accessor to the convergence strategy.
`setEstimator`(estimator)	Sobol estimator accessor.
`setExperimentSize`(experimentSize)	Accessor to the experiment size.
`setIndexQuantileEpsilon`(indexQuantileEpsilon)	Accessor to the quantile tolerance.
`setIndexQuantileLevel`(indexQuantileLevel)	Accessor to the quantile level.
`setMaximumCoefficientOfVariation`(...)	Accessor to the maximum coefficient of variation.
`setMaximumOuterSampling`(maximumOuterSampling)	Accessor to the maximum iterations number.
`setMaximumStandardDeviation`(...)	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.
`setStopCallback`(*args)	Set up a stop callback.

getBatchSize
setBatchSize

See also

SobolSimulationResult, SobolIndicesExperiment, SobolIndicesAlgorithm

Notes

The algorithm can operate on a multivariate model $f$ , in this case it operates on aggregated indices.

Several estimators are available (Saltelli, Jansen, …).

Let us denote by $n_X$ the number of input variables. For any $j = 1, \ldots, n_X$ , let us denote by $\Phi_j^F$ (resp. $\Phi_j^T$ ) the cumulative distribution function of the gaussian asymptotic distribution of the estimator of the Sobol’ first (resp. total) order indice. Let $\alpha\in[0,1]$ be the level of the confidence interval and $\epsilon\in(0,1]$ the length of this confidence interval. The algorithms stops when, on all components, first and total order indices haved been estimated with enough precision.

The precision is said to be sufficient if the length of the $\alpha$ -level confidence interval is smaller than $\epsilon$ :

$(\Phi_j^F)^{-1}(1-\alpha/2) - (\Phi_j^F)^{-1}(\alpha/2) \leq \epsilon \textrm{ and } (\Phi_j^T)^{-1}(1-\alpha/2) - (\Phi_j^T)^{-1}(\alpha/2) \leq \epsilon$

for all $j = 1, \ldots, n_X$ .

The total number of samples evaluated by the algorithm is $n_{iter} \times N \times (2 + n_X)$ (no second order indices) where

$n_{iter}$ is the actual number of outer iterations, as caped by setMaximumOuterSampling()
$N$ is the internal size of the Sobol’ experiment, as set by setExperimentSize()
$n_X$ is the input dimension

That is why unlike other simulation algorithms the total of evaluations is not equal to the product of iterations (obtained by getOuterSampling from the result) by the block size (getBlockSize()).

This class makes use of the following ResourceMap entries:

SobolSimulationAlgorithm-DefaultIndexQuantileLevel: default float value for the confidence level $\alpha$
SobolSimulationAlgorithm-DefaultIndexQuantileEpsilon: default float value for the confidence length $\epsilon$
SobolSimulationAlgorithm-DefaultExperimentSize: default integer value for $N$

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.JointDistribution([ot.Uniform(-1.0, 1.0)] * 3)
>>> model = ot.SymbolicFunction(['x1', 'x2', 'x3'], ['x1*x2+x3'])
>>> estimator = ot.SaltelliSensitivityAlgorithm()
>>> estimator.setUseAsymptoticDistribution(True)
>>> algo = ot.SobolSimulationAlgorithm(distribution, model, estimator)
>>> algo.setMaximumOuterSampling(25)  # maximum number of iterations
>>> algo.setExperimentSize(100)  # size of Sobol experiment at each iteration
>>> algo.setBlockSize(4) # number of points evaluated simultaneously
>>> algo.setIndexQuantileLevel(0.05)  # alpha
>>> algo.setIndexQuantileEpsilon(1e-2)  # epsilon
>>> algo.run()
>>> result = algo.getResult()
>>> fo = result.getFirstOrderIndicesEstimate()
>>> foDist = result.getFirstOrderIndicesDistribution()

__init__(*args)¶

drawFirstOrderIndexConvergence(*args)¶

Draw the first order Sobol index 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

drawTotalOrderIndexConvergence(*args)¶

Draw the total order Sobol index 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 simultaneous evaluations of the limit-state function. 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.

getDistribution()¶

Accessor to the batch size.

Returns:

distributionDistribution: Distribution of the random variable.

getEstimator()¶

Sobol estimator accessor.

Returns:

estimatorSobolIndicesAlgorithm: The estimator of the indices.

getExperimentSize()¶

Accessor to the experiment size.

Returns:

sizeint: Internal size $N$ of the Sobol design of experiment. See SobolIndicesExperiment

getIndexQuantileEpsilon()¶

Accessor to the criterion operator.

Returns:

epsilonfloat: The quantile tolerance

getIndexQuantileLevel()¶

Accessor to the quantile level.

Returns:

alphafloat: The quantile level.

getMaximumCoefficientOfVariation()¶

Accessor to the maximum coefficient of variation.

Returns:

coefficientfloat: Maximum coefficient of variation of the simulated sample.

getMaximumOuterSampling()¶

Accessor to the maximum iterations number.

Returns:

outerSamplingint: Maximum number of iterations, each iteration performing a block of evaluations.

getMaximumStandardDeviation()¶

Accessor to the maximum standard deviation.

Returns:

sigmafloat, $\sigma > 0$: Maximum standard deviation of the estimator.

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.

getResult()¶

Accessor to the result.

Returns:

resultSobolSimulationResult: The simulation result.

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 simultaneous evaluations of the limit-state function. It is set by default to 1.

Notes

If the function supports batch evaluations this parameter can be set to the number of available CPUs to benefit from parallel execution (multithreading, multiprocessing, …); except for the Directional Sampling, where we recommend to set it to 1. It also decides the frequency of the stopping criteria and progress callbacks update as they are called once the whole block of fonction evaluations is completed.

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.

setEstimator(estimator)¶

Sobol estimator accessor.

Parameters:

estimatorSobolIndicesAlgorithm: The estimator of the indices.

setExperimentSize(experimentSize)¶

Accessor to the experiment size.

Parameters:

sizeint: Internal size $N$ of the Sobol design of experiment. See SobolIndicesExperiment

setIndexQuantileEpsilon(indexQuantileEpsilon)¶

Accessor to the quantile tolerance.

Parameters:

epsilonfloat: The quantile tolerance

setIndexQuantileLevel(indexQuantileLevel)¶

Accessor to the quantile level.

Parameters:

alphafloat: The quantile level.

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 iterations number.

Parameters:

outerSamplingint: Maximum number of iterations, each iteration performing a block of evaluations.

setMaximumStandardDeviation(maximumStandardDeviation)¶

Accessor to the maximum standard deviation.

Parameters:

sigmafloat, $\sigma > 0$: Maximum standard deviation of the estimator.

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()

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()

Examples using the class¶

Estimate Sobol’ indices for the beam by simulation algorithm

Estimate Sobol' indices for the beam by simulation algorithm

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

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SobolSimulationAlgorithm¶

Examples using the class¶