SobolSimulationAlgorithm

(Source code, png)

../../_images/SobolSimulationAlgorithm.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 fixed number N = blockSize * (d + 2) of replications inputs is generated. These inputs are evaluated by blocks of size batchSize 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.

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.

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) # number of iterations
>>> algo.setBlockSize(100) # size of Sobol experiment at each iteration
>>> algo.setBatchSize(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()

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.

getBatchSize()

Accessor to the batch size.

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.

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 sample size.

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.

setBatchSize(replicationSize)

Accessor to the batch size.

setBlockSize(blockSize)

Accessor to the block size.

setConvergenceStrategy(convergenceStrategy)

Accessor to the convergence strategy.

setEstimator(estimator)

Sobol estimator accessor.

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 sample size.

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.

__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

getBatchSize()

Accessor to the batch size.

Returns:
batchSizeint

Number of points evaluated simultaneously.

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__).

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.

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 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.

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.

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.

setBatchSize(replicationSize)

Accessor to the batch size.

Parameters:
batchSizeint

Number of points evaluated simultaneously.

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.

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.

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 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.

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()