DirectionalSampling¶

class DirectionalSampling(*args)¶

Directional simulation.

Refer to Directional Simulation.

Available constructors:

DirectionalSampling(event=ot.Event())

DirectionalSampling(event, rootStrategy, samplingStrategy)

Parameters

eventRandomVector: Event we are computing the probability of.
rootStrategyRootStrategy: Strategy adopted to evaluate the intersections of each direction with the limit state function and take into account the contribution of the direction to the event probability. By default, rootStrategy = ot.RootStrategy(ot.SafeAndSlow()).
samplingStrategySamplingStrategy: Strategy adopted to sample directions. By default, samplingStrategy=ot.SamplingStrategy(ot.RandomDirection()).

Notes

Using the probability distribution of a random vector $\vect{X}$ , we seek to evaluate the following probability:

$P_f = \int_{\Rset^{n_X}} \mathbf{1}_{\{g(\ux,\underline{d}) \leq 0 \}}f_{\uX}(\ux)\di{\ux} = \Prob{g\left( \vect{X},\vect{d} \right) \leq 0}$

Here, $\vect{X}$ is a random vector, $\vect{d}$ a deterministic vector, $g(\vect{X},\vect{d})$ the function known as limit state function which enables the definition of the event $\cD_f = \{\vect{X} \in \Rset^n \, | \, g(\vect{X},\vect{d}) \le 0\}$ . $\mathbf{1}_{ \left\{ g(\vect{x}_i,\vect{d}) \leq 0 \right\} }$ describes the indicator function equal to 1 if $g(\vect{x}_i,\vect{d}) \leq 0$ and equal to 0 otherwise.

The directional simulation method is an accelerated sampling method. It implies a preliminary iso-probabilistic transformation, as for FORM and SORM methods; however, it remains based on sampling and is thus not an approximation method. In the transformed space, the (transformed) uncertain variables $\vect{U}$ are independent standard gaussian variables (mean equal to zero and standard deviation equal to 1).

Roughly speaking, each simulation of the directional simulation algorithm is made of three steps. For the $i^\textrm{th}$ iteration, these steps are the following:

Let $\cS = \big\{ \vect{u} \big| ||\vect{u}|| = 1 \big\}$ . A point $P_i$ is drawn randomly on $\cS$ according to an uniform distribution.
In the direction starting from the origin and passing through $P_i$ , solutions of the equation $g(\vect{X},\vect{d}) = 0$ (i.e. limits of $\cD_f$ ) are searched. The set of values of $\vect{u}$ that belong to $\cD_f$ is deduced for these solutions: it is a subset $I_i \subset \Rset$ .
Then, one calculates the probability $q_i = \Prob{ ||\vect{U}|| \in I_i }$ . By property of independent standard variable, $||\vect{U}||^2$ is a random variable distributed according to a chi-square distribution, which makes the computation effortless.

Finally, the estimate of the probability $P_f$ after $N$ simulations is the following:

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

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> 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))
>>> # We create a 'usual' RandomVector from the Distribution
>>> vect = ot.RandomVector(myDistribution)
>>> # We create a composite random vector
>>> output = ot.CompositeRandomVector(myFunction, vect)
>>> # We create an Event from this RandomVector
>>> myEvent = ot.ThresholdEvent(output, ot.Less(), -3.0)
>>> # We create a DirectionalSampling algorithm
>>> myAlgo = ot.DirectionalSampling(myEvent, ot.MediumSafe(), ot.OrthogonalDirection())
>>> myAlgo.setMaximumOuterSampling(150)
>>> myAlgo.setBlockSize(4)
>>> myAlgo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation
>>> myAlgo.run()
>>> print('Probability estimate=%.6f' % myAlgo.getResult().getProbabilityEstimate())
Probability estimate=0.169716

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.
`getId`()	Accessor to the object’s id.
`getMaximumCoefficientOfVariation`()	Accessor to the maximum coefficient of variation.
`getMaximumOuterSampling`()	Accessor to the maximum sample size.
`getMaximumStandardDeviation`()	Accessor to the maximum standard deviation.
`getName`()	Accessor to the object’s name.
`getResult`()	Accessor to the results.
`getRootStrategy`()	Get the root strategy.
`getSamplingStrategy`()	Get the direction sampling strategy.
`getShadowedId`()	Accessor to the object’s shadowed id.
`getVerbose`()	Accessor to verbosity.
`getVisibility`()	Accessor to the object’s visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`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 sample size.
`setMaximumStandardDeviation`(…)	Accessor to the maximum standard deviation.
`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.
`setShadowedId`(id)	Accessor to the object’s shadowed id.
`setStopCallback`(*args)	Set up a stop callback.
`setVerbose`(verbose)	Accessor to verbosity.
`setVisibility`(visible)	Accessor to the object’s visibility state.

__init__(*args)¶: Initialize self. See help(type(self)) for accurate signature.

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 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.

getEvent()¶

Accessor to the event.

Returns

eventRandomVector: Event we want to evaluate the probability.

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

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.

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 adopted.

getSamplingStrategy()¶

Get the direction sampling strategy.

Returns

strategySamplingStrategy: Direction sampling strategy adopted.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getVerbose()¶

Accessor to verbosity.

Returns

verbosity_enabledbool: If True, the computation is verbose. By default it is verbose.

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

hasVisibleNamebool: True if the name is not empty and not the default one.

run()¶

Launch simulation.

See also

setBlockSize, setMaximumOuterSampling, ResourceMap, SimulationResult

Notes

It launches the simulation and creates a SimulationResult, structure containing all the results obtained after simulation. It computes the probability of occurence 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 to use efficiently the distribution of the computation as well as it allows 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 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(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 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.

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 adopted.

setSamplingStrategy(samplingStrategy)¶

Set the direction sampling strategy.

Parameters

strategySamplingStrategy: Direction sampling strategy adopted.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

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)
>>> timer = ot.TimerCallback(0.1)
>>> algo.setStopCallback(timer)
>>> algo.run()

setVerbose(verbose)¶

Accessor to verbosity.

Parameters

verbosity_enabledbool: If True, make the computation verbose. By default it is verbose.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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