DirectionalSampling¶

class DirectionalSampling(*args)¶

Directional sampling algorithm.

Refer to Directional Simulation.

Available constructors:

DirectionalSampling(event)

DirectionalSampling(event, rootStrategy, samplingStrategy)

Parameters:

eventRandomVector

Event we are computing the probability of.

rootStrategyRootStrategy

Strategy used to evaluate the frontiers of the event along each direction in the standard space.

By default SafeAndSlow.

samplingStrategySamplingStrategy

Strategy used to sample directions in the standard space.

By default RandomDirection.

Methods

`drawProbabilityConvergence`(*args)	Draw the probability convergence at a given level.
`getBlockSize`()	Accessor to the block size.
`getClassName`()	Accessor to the object's name.
`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.
`getRootStrategy`()	Get the root strategy.
`getSamplingStrategy`()	Get the direction sampling strategy.
`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.
`setRootStrategy`(rootStrategy)	Set the root strategy.
`setSamplingStrategy`(samplingStrategy)	Set the direction sampling strategy.
`setStopCallback`(*args)	Set up a stop callback.

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 directional sampling algorithm estimates the probability of the event $\cD_f$ :

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

The directional sampling algorithm is a variance reduction sampling method. As it is performed in the standard space, it uses the iso-probabilistic transformation associated to $\inputMeasure$ (see Isoprobabilistic transformations to get more details).

Let $T$ be the iso-probabilistic transformation: $T(\vect{X}) = \vect{Z}$ . The random vector $\vect{Z}$ follows a spherical distribution, which means that it is written as:

$\vect{Z} = R \vect{U}$

where $R$ is a scalar random variable following the radial distribution, $\vect{U}$ is a random vector that follows the uniform distribution on the unit sphere of dimension $\inputDim$ , denoted by $\cS_\inputDim$ , and $(R, \vect{U})$ are independent.

Then the probability $P_f$ can be written as:

$P_f & = \dfrac{1}{|\cS_\inputDim|} \int_{\cS_\inputDim} \left \{ \int_{\Rset^+} 1_{\left\{\model \circ T^{-1}(r\vect{u}) \leq 0 \right\}} p_R(r) dr \right\} \, d\vect{u} \\$

For a given direction $\vect{u}$ , let $I_{\vect{u}}$ be:

$I_{\vect{u}} = \{r \in \Rset^+ \, | \, \model \circ T^{-1}(r\vect{u})\leq 0 \}$

Then, we have:

$\int_{\Rset^+} 1_{\left\{\model \circ T^{-1}(r\vect{u}) \leq 0\right\}} p_R(r) dr = \Prob{R \in I_{\vect{u}}}$

As the radial distribution of $R$ is known, we can evaluate exactly the probability $\Prob{R \in I_{\vect{u}}}$ as soon as we know the direction $\vect{u}$ .

The integral with respect to the directions is evaluated using a sampling method, defined by the class:~openturns.SamplingStrategy.

The directional sampling algorithm follows the steps:

A sample is generated from the uniform distribution on $\cS_\inputDim$ : from this sample, some directions are defined, according to the specified SamplingStrategy. We denote by $(\vect{u}_i)$ the sample of the directions;
Along each direction $i$ , we find the frontiers of $\cD_f$ , i.e. the solutions of $\model \circ T^{-1}(r\vect{u}_i) = 0$ according to the RootStrategy. From these solutions, we deduce the subset $I_i \subset \Rset^+$ that belongs to $\cD_f$ ;
For each subset $I_i$ , we compute the probability $q_i = \Prob{R \in I_{\vect{u}_i}}$ .

The probability $P_f$ is computed from all the $q_i$ probabilities. If $N$ directions have been used, then:

$\widehat{P}_{f,DS} = \frac{1}{N} \sum_{i=1}^N q_i$

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)

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

>>> myFunction = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3*E*I)'])
>>> myDistribution = ot.Normal([50.0, 1.0, 10.0, 5.0], [1.0]*4, ot.IdentityMatrix(4))
>>> input_RV = ot.RandomVector(myDistribution)
>>> output_RV = ot.CompositeRandomVector(myFunction, input_RV)

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

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

Create the directional sampling algorithm and the convergence criteria:

>>> myAlgo = ot.DirectionalSampling(event, ot.MediumSafe(), ot.OrthogonalDirection())
>>> myAlgo.setMaximumOuterSampling(150)
>>> myAlgo.setBlockSize(4)
>>> myAlgo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation

Run the algorithm and extract the estimated probability:

>>> myAlgo.run()
>>> print('Probability estimate=%.6f' % myAlgo.getResult().getProbabilityEstimate())
Probability estimate=0.169716

__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

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.

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().

getRootStrategy()¶

Get the root strategy.

Returns:

strategyRootStrategy: Root strategy used to evaluate the frontiers of the event along each direction in the standard space.

getSamplingStrategy()¶

Get the direction sampling strategy.

Returns:

strategySamplingStrategy: Direction sampling strategy used to sample directions in the standard space.

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()

setRootStrategy(rootStrategy)¶

Set the root strategy.

Parameters:

strategyRootStrategy: Root strategy to evaluate the frontiers of the event along each direction in the standard space.

setSamplingStrategy(samplingStrategy)¶

Set the direction sampling strategy.

Parameters:

strategySamplingStrategy: Direction sampling strategy used to sample directions in the standard space.

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 Directional Sampling Algorithm

Axial stressed beam : comparing different methods to estimate a probability

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

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DirectionalSampling¶

Examples using the class¶