# SobolSimulationAlgorithm¶

class SobolSimulationAlgorithm(*args)

Sobol indices computation using iterative sampling.

The algorithm uses sampling of the distribution of the random vector through the model to iteratively estimate the Sobol indices.

At each iteration a fixed number of replications inputs is generated. These inputs are evaluated by blocks of size through the model . 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 , 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 $j=1,…,n_X$, let us denote by $Phi_j^F$ (resp. $Phi_j^T$) the cumulated distribution function of the gaussian asymptotic distribution of the estimator of the Sobol’ first (resp. total) order indice. Let $alphain[0,1]$ be the level of the confidence interval and $epsilonin(0,1]$ the length of this confidence interval. The algorithms stops when, on all components, at least one of the following conditions is satisfied: - first and total order indices haved been estimated with enough precision or - the first order indices can be separated from the total order indices.

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

(\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 .

The first order indices can be separated from the total order indices if: .. math:

\Phi^{T}_j(\alpha) \ge \Phi^{F}_j(1-\alpha)


for all .

The second criteria allows to stop when the algorithm has detected an interaction between input variables with sufficient precision.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.ComposedDistribution([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(self, \*args) Draw the first order Sobol index convergence at a given level. drawTotalOrderIndexConvergence(self, \*args) Draw the total order Sobol index convergence at a given level. getBatchSize(self) Accessor to the batch size. getBlockSize(self) Accessor to the block size. getClassName(self) Accessor to the object’s name. Accessor to the convergence strategy. Accessor to the batch size. getEstimator(self) Sobol estimator accessor. getId(self) Accessor to the object’s id. Accessor to the criterion operator. Accessor to the quantile level. Accessor to the maximum coefficient of variation. Accessor to the maximum sample size. Accessor to the maximum standard deviation. getName(self) Accessor to the object’s name. getResult(self) Accessor to the result. getShadowedId(self) Accessor to the object’s shadowed id. 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. setBatchSize(self, replicationSize) Accessor to the batch size. setBlockSize(self, blockSize) Accessor to the block size. setConvergenceStrategy(self, convergenceStrategy) Accessor to the convergence strategy. setEstimator(self, estimator) Sobol estimator accessor. setIndexQuantileEpsilon(self, …) Accessor to the quantile tolerance. setIndexQuantileLevel(self, indexQuantileLevel) Accessor to the quantile level. Accessor to the maximum coefficient of variation. setMaximumOuterSampling(self, …) Accessor to the maximum sample size. 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. 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.

drawFirstOrderIndexConvergence(self, *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
graph

expectation convergence graph

drawTotalOrderIndexConvergence(self, *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
graph

expectation convergence graph

getBatchSize(self)

Accessor to the batch size.

Returns
batchSizeint

Number of points evaluated simultaneously.

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__).

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.

getDistribution(self)

Accessor to the batch size.

Returns
distibutionDistribution

Distribution of the random variable.

getEstimator(self)

Sobol estimator accessor.

Returns
estimatorSobolIndicesAlgorithm

The estimator of the indices.

getId(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getIndexQuantileEpsilon(self)

Accessor to the criterion operator.

Returns
epsilonfloat

The quantile tolerance

getIndexQuantileLevel(self)

Accessor to the quantile level.

Returns
alphafloat

The quantile level.

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,

Maximum standard deviation of the estimator.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getResult(self)

Accessor to the result.

Returns
resultSobolSimulationResult

The simulation result.

getShadowedId(self)

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

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 by a combination of blockSize and outerSampling.

setBatchSize(self, replicationSize)

Accessor to the batch size.

Parameters
batchSizeint

Number of points evaluated simultaneously.

setBlockSize(self, blockSize)

Accessor to the block size.

Parameters
blockSizeint,

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.

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.

setEstimator(self, estimator)

Sobol estimator accessor.

Parameters
estimatorSobolIndicesAlgorithm

The estimator of the indices.

setIndexQuantileEpsilon(self, indexQuantileEpsilon)

Accessor to the quantile tolerance.

Parameters
epsilonfloat

The quantile tolerance

setIndexQuantileLevel(self, indexQuantileLevel)

Accessor to the quantile level.

Parameters
alphafloat

The quantile level.

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,

Maximum standard deviation of the estimator.

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.

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.