PostAnalyticalControlledImportanceSampling¶

class PostAnalyticalControlledImportanceSampling(*args)¶

Post analytical controlled importance sampling.

Importance sampling algorithm around the design point, controlled by the tangent hyperplane.

Parameters:

analyticalResultAnalyticalResult: Result structure.

Methods

`drawProbabilityConvergence`(*args)	Draw the probability convergence at a given level.
`getAnalyticalResult`()	Accessor to the analytical result.
`getBlockSize`()	Accessor to the block size.
`getClassName`()	Accessor to the object's name.
`getControlProbability`()	Accessor to the control probability.
`getConvergenceStrategy`()	Accessor to the convergence strategy.
`getEvent`()	Accessor to the event.
`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 results.
`hasName`()	Test if the object is named.
`run`()	Launch simulation.
`setBlockSize`(blockSize)	Accessor to the block size.
`setConvergenceStrategy`(convergenceStrategy)	Accessor to the convergence strategy.
`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.

See also

PostAnalyticalImportanceSampling

Notes

Let $\inputRV$ be a random vector of dimension $\inputDim$ , distributed according to the measure $\inputMeasure$ , and $\model: \Rset^\inputDim \rightarrow \Rset$ be the limit state function (where we only wrote the random input parameters). We define the event $\cD_f$ by:

$\cD_f = \{\vect{x} \in \Rset^{\inputDim} \, | \,\model(\vect{x}) \leq 0\}$

The post analytical controlled importance sampling algorithm estimates the probability of the domain $\cD_f$ :

$P_f = \Prob{\model\left( \inputRV \right) \leq 0} = \int_{\Rset^{\inputDim}} \mathbf{1}_{\{\model(\vect{x}) \leq 0 \}}\inputMeasure(\vect{x})\di{\vect{x}}$

The post analytical controlled importance sampling algorithm is a variance reduction sampling method, which is performed in the standard space, where the random vector follows a spherical distribution (see Isoprobabilistic transformations to get more details). It is an additive correction of the FORM approximation of the probability $P_f$ . See Analytical for the description of the first steps of the FORM analysis.

Let $\ell$ be the function defined by:

$\ell(\vect{u}) = \vect{u}^*.\vect{u} - \beta^2$

where $\vect{u}^*$ is the design point in the standard space, and $\beta$ the distance of the design point from the origin of the standard space: $\beta = \| \vect{u}^*\|$ .

The tangent hyperplane at the design point in the standard space is defined by the equation:

$\{ \vect{u} \in \Rset^{\inputDim} \, | \, \ell(\vect{u}) = 0 \}$

Let $G: \Rset^\inputDim \rightarrow \Rset$ be the model in the standard space: if $T$ is the iso-probabilistic transformation such that $T(\vect{X}) = \vect{U}$ , then:

$G(\vect{u}) = \model \circ T^{-1}(\vect{u})$

Let $\cD_f^{std}$ be the domain $\cD_f$ in the standard space. We assume that the domain $\cD_f^{std}$ does not contain the origin of the standard space. Thus, it is defined by:

$\cD_f^{std} & = \{\vect{u} \in \Rset^{\inputDim} \, | \, G(\vect{u}) \leq 0 \} \\ & = \{\vect{u} \in \Rset^{\inputDim} \, | \, \ell(\vect{u}) \geq 0 \} \cup \{\vect{u} \in \Rset^{\inputDim} \, | \, \ell(\vect{u}) \leq 0 , G(\vect{u}) \leq 0 \} \setminus \{\vect{u} \in \Rset^{\inputDim} \, | \, \ell(\vect{u}) \geq 0 , G(\vect{u}) \geq 0 \} Thus, we have:$

$P_f & = \Prob{\ell(\vect{U}) \geq 0} + \Prob{\ell(\vect{U}) \leq 0, G(\RVU) \leq 0} - \Prob{\ell(\vect{U}) \geq 0, G(\RVU) \geq 0}\\ & = \Prob{\ell(\vect{U}) \geq 0} + \delta_1 - \delta_2$

where $\Prob{\ell(\vect{U} \geq 0} = E(-\beta)$ is known exactly. $E$ is the univariate standard CDF of the spherical distribution in the standard space.

If $\delta_1 \ll \Prob{\ell(\vect{U} \geq 0}$ and $\delta_2 \ll \Prob{\ell(\vect{U} \geq 0}$ , then we use $\ell(\vect{U})$ as a controlled variable and we use an importance sampling around the design point in the standard space.

We denote by $(\vect{u}_i)_{1 \leq i \leq \sampleSize}$ a sample generated from the spherical distribution centered on the origin of the standard space, whose pdf is denoted by $e$ . Let $e^*$ be the pdf of the spherical distribution centered on the design point $\vect{u}^*$ .

The estimate of $P_f$ is defined by:

$\tilde{P}_f & = E(-\beta) + \tilde{\delta}_1 - \tilde{\delta}_2\\ & = E(-\beta) + \dfrac{1}{\sampleSize} \sum_{i=1}^\sampleSize \left( 1_{ \left \{ \ell(\vect{u})_i \leq 0, G(\vect{u}_i) \leq 0 \right\} } - 1_{ \left \{ \ell(\vect{u})_i \geq 0, G(\vect{u}_i) \geq 0\right \} } \right) \dfrac{e(\vect{u}_i)}{e^*(\vect{u}_i)}$

The hypotheses $\delta_1 \ll \Prob{\ell(\vect{U} \geq 0}$ and $\delta_2 \ll \Prob{\ell(\vect{U} \geq 0}$ are verified if the FORM approximation is valid.

The coefficient of variation of $\tilde{P}_f$ is:

$\Cov{\tilde{P}_f} = \dfrac{ \sqrt{ \Var{ \tilde{P}_f}}}{\Expect{\tilde{P}_f}} = \dfrac{ \sqrt{ \Var{ \tilde{\delta}_1 - \tilde{\delta}_2 } }} { E(-\beta) + \Expect{\tilde{\delta}_1 - \tilde{\delta}_2} } \ll \dfrac{ \sqrt{ \Var{ \tilde{\delta}_1 - \tilde{\delta}_2 } } } {\Expect{\tilde{\delta}_1 - \tilde{\delta}_2}}$

Examples

>>> import openturns as ot

Create the output random vector $Y = \model(\inputRV)$ :

>>> f = 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))
>>> X = ot.RandomVector(distribution)
>>> Y = ot.CompositeRandomVector(f, X)

Create the event $\cD_f = \{\vect{x} \in \Rset^{\inputDim} \, | \,\model(\vect{x}) \leq -3.0\}$ :

>>> event = ot.ThresholdEvent(Y, ot.Less(), -3.0)

Create the FORM algorithm and launch it:

>>> solver = ot.AbdoRackwitz()
>>> analytical = ot.FORM(solver, event, [50.0, 1.0, 10.0, 5.0])
>>> analytical.run()
>>> analyticalResult = analytical.getResult()

Create the post analytical importance sampling algorithm and launch it:

>>> algo = ot.PostAnalyticalControlledImportanceSampling(analyticalResult)
>>> algo.run()
>>> result = algo.getResult()
>>> pf = result.getProbabilityEstimate()

__init__(*args)¶

drawProbabilityConvergence(*args)¶

Draw the probability convergence at a given level.

Parameters:

levelfloat, optional: The probability convergence is drawn at this given confidence length level. By default level is 0.95.

Returns:

grapha Graph: probability convergence graph

getAnalyticalResult()¶

Accessor to the analytical result.

Returns:

resultAnalyticalResult: Result of the analytical study which has been performed just before the simulation study centered around the importance factor.

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__).

getControlProbability()¶

Accessor to the control probability.

Returns:

pfloat,: The probability of the analytical result.

Notes

The control probability $p$ is deduced from the corresponding analytical result:

$p = E(-\beta_{HL})$

where $\beta_{HL}$ denotes the Hasofer reliability index and $E$ the univariate standard CDF of the elliptical distribution in the standard space.

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.

getEvent()¶

Accessor to the event.

Returns:

eventRandomVector: Event we want to evaluate the probability.

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 results.

Returns:

resultsSimulationResult: Structure containing all the results obtained after simulation and created by the method run().

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

run()¶

Launch simulation.

See also

openturns.EventSimulation.setBlockSize
openturns.EventSimulation.setMaximumOuterSampling
openturns.ResourceMap
openturns.SimulationResult

Notes

It launches the simulation and creates a SimulationResult, structure containing all the results obtained after simulation. It computes the probability of occurrence of the given event by computing the empirical mean of 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.

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¶

Use the post-analytical importance sampling algorithm

Estimate a buckling probability

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

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PostAnalyticalControlledImportanceSampling¶

Examples using the class¶