NAIS¶

class NAIS(*args)¶

Nonparametric Adaptive Importance Sampling (NAIS) algorithm.

Parameters:

eventRandomVector: Event we are computing the probability of.
quantileLevelfloat $0<quantileLevel<1$: 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, $N$ the number of independent samples generated with $h$ 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 the quantile level $\rho \in [0,1]$
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 quantile of level $\rho$ $q_k=\max(T,y^{(k)}_{\left \lfloor \rho N \right\rfloor})$
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.
`getInputSample`(*args)	Input sample accessor.
`getMaximumCoefficientOfVariation`()	Accessor to the maximum coefficient of variation.
`getMaximumOuterSampling`()	Accessor to the maximum sample size.
`getMaximumStandardDeviation`()	Accessor to the maximum standard deviation.
`getMaximumTimeDuration`()	Accessor to the maximum duration.
`getName`()	Accessor to the object's name.
`getOutputSample`(*args)	Output sample accessor.
`getQuantileLevel`()	Accessor to the intermediate quantile level.
`getResult`()	Accessor to the results.
`getStepsNumber`()	Subset steps number accessor.
`getThresholdPerStep`()	Threshold accessor.
`hasName`()	Test if the object is named.
`run`()	Launch simulation.
`setBlockSize`(blockSize)	Accessor to the block size.
`setConvergenceStrategy`(convergenceStrategy)	Accessor to the convergence strategy.
`setKeepSample`(keepSample)	Sample storage accessor.
`setMaximumCoefficientOfVariation`(...)	Accessor to the maximum coefficient of variation.
`setMaximumOuterSampling`(maximumOuterSampling)	Accessor to the maximum sample size.
`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.
`setQuantileLevel`(quantileLevel)	Accessor to the intermediate quantile level.
`setStopCallback`(*args)	Set up a stop callback.

__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.

getInputSample(*args)¶

Input sample accessor.

Parameters:

stepint

Iteration index

selectint, optional

Selection flag:

EVENT0 : points not realizing the event are selected
EVENT1 : points realizing the event are selected
BOTH : all points are selected (default)

Returns:

inputSampleSample: Input sample.

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.

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.

getOutputSample(*args)¶

Output sample accessor.

Parameters:

stepint

Iteration index

selectint, optional

Selection flag:

EVENT0 : points not realizing the event are selected
EVENT1 : points realizing the event are selected
BOTH : all points are selected (default)

Returns:

outputSampleSample: Output sample.

getQuantileLevel()¶

Accessor to the intermediate quantile level.

Returns:

quantileLevelfloat: Intermediate quantile level.

getResult()¶

Accessor to the results.

Returns:

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

getStepsNumber()¶

Subset steps number accessor.

Returns:

nint: Number of subset steps, including the initial Monte Carlo sampling.

getThresholdPerStep()¶

Threshold accessor.

Returns:

thresholdPoint: Threshold values at each step.

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 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.

setKeepSample(keepSample)¶

Sample storage accessor.

Parameters:

keepsamplebool: Whether to keep the working samples at each iteration.

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.

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()

setQuantileLevel(quantileLevel)¶

Accessor to the intermediate quantile level.

Parameters:

quantileLevelfloat $0<quantileLevel<1$: Intermediate quantile level.

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¶

Non parametric Adaptive Importance Sampling (NAIS)

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NAIS¶

Examples using the class¶