DirectionalSampling¶

class DirectionalSampling(*args)¶

Directional simulation.

Available constructors:

DirectionalSampling(event=ot.Event())

DirectionalSampling(event, rootStrategy, samplingStrategy)

Parameters:

event : Event: Event we are computing the probability of.
rootStrategy : RootStrategy: 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()).
samplingStrategy : SamplingStrategy: 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.RandomVector(myFunction, vect)
>>> # We create an Event from this RandomVector
>>> myEvent = ot.Event(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:	level : float, optional The probability convergence is drawn at this given confidence length level. By default level is 0.95.
Returns:	graph : a `Graph` probability convergence graph

getBlockSize()¶

Accessor to the block size.

Returns:	blockSize : int 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_name : str The object class name (object.__class__.__name__).

getConvergenceStrategy()¶

Accessor to the convergence strategy.

Returns:	storage_strategy : `HistoryStrategy` Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.

getEvent()¶

Accessor to the event.

Returns:	event : `Event` Event we want to evaluate the probability.

getId()¶

Accessor to the object’s id.

Returns:	id : int Internal unique identifier.

getMaximumCoefficientOfVariation()¶

Accessor to the maximum coefficient of variation.

Returns:	coefficient : float Maximum coefficient of variation of the simulated sample.

getMaximumOuterSampling()¶

Accessor to the maximum sample size.

Returns:	outerSampling : int Maximum number of groups of terms in the probability simulation estimator.

getMaximumStandardDeviation()¶

Accessor to the maximum standard deviation.

Returns:	sigma : float, $\sigma > 0$ Maximum standard deviation of the estimator.

getName()¶

Accessor to the object’s name.

Returns:	name : str The name of the object.

getResult()¶

Accessor to the results.

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

getRootStrategy()¶

Get the root strategy.

Returns:	strategy : `RootStrategy` Root strategy adopted.

getSamplingStrategy()¶

Get the direction sampling strategy.

Returns:	strategy : `SamplingStrategy` Direction sampling strategy adopted.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:	id : int Internal unique identifier.

getVerbose()¶

Accessor to verbosity.

Returns:	verbosity_enabled : bool If True, the computation is verbose. By default it is verbose.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:	visible : bool Visibility flag.

hasName()¶

Test if the object is named.

Returns:	hasName : bool True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:	hasVisibleName : bool True if the name is not empty and not the default one.

run()¶

Launch simulation.

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:	blockSize : int, $blockSize \geq 1$ 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_strategy : `HistoryStrategy` 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:	coefficient : float Maximum coefficient of variation of the simulated sample.

setMaximumOuterSampling(maximumOuterSampling)¶

Accessor to the maximum sample size.

Parameters:	outerSampling : int Maximum number of groups of terms in the probability simulation estimator.

setMaximumStandardDeviation(maximumStandardDeviation)¶

Accessor to the maximum standard deviation.

Parameters:	sigma : float, $\sigma > 0$ Maximum standard deviation of the estimator.

setName(name)¶

Accessor to the object’s name.

Parameters:	name : str The name of the object.

setProgressCallback(*args)¶

Set up a progress callback.

Parameters:	callback : callable Takes a float as argument as percentage of progress.

setRootStrategy(rootStrategy)¶

Set the root strategy.

Parameters:	strategy : `RootStrategy` Root strategy adopted.

setSamplingStrategy(samplingStrategy)¶

Set the direction sampling strategy.

Parameters:	strategy : `SamplingStrategy` Direction sampling strategy adopted.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:	id : int Internal unique identifier.

setStopCallback(*args)¶

Set up a stop callback.

Parameters:	callback : callable Returns an int deciding whether to stop or continue.

setVerbose(verbose)¶

Accessor to verbosity.

Parameters:	verbosity_enabled : bool If True, make the computation verbose. By default it is verbose.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:	visible : bool Visibility flag.

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