NAIS¶

class NAIS(*args)¶

Non Adaptive Importance Sampling (NAIS) algorithm.

Parameters

eventRandomVector: Event we are computing the probability of.
rhoQuantilefloat $0<\rho<1$: Intermediate quantile level.

See also

SubsetSampling

Notes

The following explanations are given for a failure event defined as $g(\mathbf{X})<T$ with $\mathbf{X}$ a random vector following a joint PDF $f_\mathbf{X}$ , $T$ a threshold and $g$ a limit state function, without loss of generality.

The Importance Sampling (IS) probability estimate $\widehat{P}^\text{IS}$ is given by:

$\widehat{P}^\text{IS}=\frac{1}{N} \sum_{i=1}^{N} {\mathbf{1}}_{g(\mathbf{x}_i)<T} \frac{h_0(\mathbf{x}_i)}{h(\mathbf{x}_i)},$

with $h_0 = f_\mathbf{X}$ the PDF of $\mathbf{X}$ , $h$ the auxiliary PDF of Importance Sampling, $N$ and ${\mathbf{1}}_{g(\mathbf{x}_i)<T}$ the indicator function of the failure domain.

The optimal density minimizing the variance of the estimator $h_{opt}$ is defined as:

$h_{opt}=\frac{{\mathbf{1}}_{g(x)<T}h_0}{P},$

with $P$ the failure probability which is inaccessible in practice since this probability is the quantity of interest and unknown.

The objective of Non parametric Adaptive Importance Sampling (NAIS) [morio2015] is to approximate the IS optimal auxiliary density $h_{opt}$ from the preceding equation with a kernel density function (e.g. Gaussian kernel). Its iterative principle is described by the following steps.

$k=1$ and set $\rho \in [0,1]$
Generate the population $\mathbf{x}_1^{(k)},...,\mathbf{x}_N^{(k)}$ according to the PDF $h_{k-1}$ , apply the function $g$ in order to have $y_1^{(k)}=g(\mathbf{x}_1^{(k)}),...,y_N^{(k)} = g(\mathbf{x}_N^{(k)})$
Compute the empirical $\rho$ -quantile $q_k=\max(T,y^{(k)}_{\left \lfloor \rho N \right\rfloor})$
Estimate $I_k= \frac{1}{kN} \displaystyle \sum_{j=1}^{k}\sum_{i=1}^{N} {\mathbf{1}}_{g(\mathbf{x}_i^{(j)}) \leq q_k} \frac{h_0(\mathbf{x}_i^{(j)})}{h_{j-1}(\mathbf{x}_i^{(j)})}$
Update the Gaussian kernel sampling PDF with:

$h_{k}(\mathbf{x})=\frac{1}{k N I_k \det\left(B_{k+1}\right)}\sum_{j=1}^{k}\sum_{i=1}^{N} w_{j}(\mathbf{x}_i^{(j)})K_d\left(B_{k+1}^{-1}\left(\mathbf{x}-\mathbf{x}_i^{(j)}\right)\right)$

where $K_d$ is the PDF of the standard $d$ -dimensional normal distribution, $B_{k+1}=\text{diag}(b^1_{k+1},...,b^d_{k+1})$ and $w_j={\mathbf{1}}_{g(\mathbf{x}_i^{(j)}) \leq q_k} \frac{h_0(\mathbf{x}_i^{(j)})}{h_{j-1}(\mathbf{x}_i^{(j)})}$ . The coefficients of the matrix $B_{k+1}$ can be approximated (Silverman Rule) or postulated according to the AMISE (Asymptotic Mean Integrated Square Error) criterion for example.
If $q_k>T$ , $k\leftarrow k+1$ , go to Step 2
Estimate the probability $\widehat{P}^{NAIS}(g(\mathbf{\mathbf{X}}<T))=\frac{1}{N}\displaystyle \sum_{i=1}^{N} \mathbf{1}_{g(\mathbf{x}_i^{(k)})<T} \frac{h_0(\mathbf{x}_i^{(k)})}{h_{k-1}(\mathbf{x}_i^{(k)})}$

The NAIS algorithm with the Silverman rule is implemented in the current NAIS class.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> # We create the function defining the limit state
>>> myFunction = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3*E*I)'])
>>> # We define a joint PDF of interest 
>>> 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(), -10.0)
>>> # We create a NAIS algorithm
>>> algo = ot.NAIS(myEvent, 0.1)
>>> # Perform the simulation
>>> algo.run()

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.
`getRhoQuantile`()	Accessor to the intermediate quantile level.
`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.
`setRhoQuantile`(rhoQuantile)	Accessor to the intermediate quantile level.
`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)¶

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().

getRhoQuantile()¶

Accessor to the intermediate quantile level.

Returns

rhoQuantile: float: Intermediate quantile level.

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 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 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 one 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()

setRhoQuantile(rhoQuantile)¶

Accessor to the intermediate quantile level.

Parameters

rhoQuantile: float $0<\rho<1$: Intermediate quantile level.

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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