PhysicalSpaceCrossEntropyImportanceSampling¶
- class PhysicalSpaceCrossEntropyImportanceSampling(*args)¶
Physical Space Cross-Entropy Importance Sampling.
Warning
This class is experimental and likely to be modified in future releases. To use it, import the
openturns.experimental
submodule.- Parameters:
- event
ThresholdEvent
Event we are computing the probability of.
- activeParameterssequence of integers
List of active parameters indices for the auxiliary distribution.
- initialAuxiliaryDistributionParameterssequence of floats
Initial value of active parameters of the auxiliary distribution.
- bounds
Interval
Bounds on the active parameters of the auxiliary distribution.
- auxiliaryDistribution
Distribution
Auxiliary distribution for the Cross Entropy Importance Sampling algorithm.
- quantileLevelfloat
Intermediate quantile level. The default number can be tweaked with the CrossEntropyImportanceSampling-DefaultQuantileLevel key from
ResourceMap
.
- event
See also
Notes
The following explanations are given for a failure event defined as with a random vector following a joint PDF , a threshold and a limit state function, without loss of generality.
The Importance Sampling (IS) probability estimate is given by:
with the auxiliary PDF of Importance Sampling, the number of independent samples generated with and the indicator function of the failure domain.
The optimal density minimizing the variance of the estimator is defined as:
with the failure probability which is inaccessible in practice since this probability is the quantity of interest and unknown.
The Physical Space Cross-Entropy Importance Sampling algorithm [rubinstein2017] uses a parametric auxiliary distribution in order to optimize its parameters to compute the probability of interest with accuracy.
It involves an auxiliary optimization problem to find the auxiliary distribution parameters minimizing the Kullback-Leibler divergence with respect to . The following algorithm is used:
, set the quantile level and
Generate the population according to the PDF , apply the function in order to have
Compute the empirical -quantile using the floor of .
If : go to Step 7
Estimate the auxiliary distribution parameters:
, go to Step 2
Estimate the probability .
Default solver for the optimization of auxiliary distribution is LD_LBFGS from
NLopt
.Examples
>>> import openturns as ot >>> import openturns.experimental as otexp >>> ot.RandomGenerator.SetSeed(0) >>> distribution_R = ot.LogNormalMuSigma(300.0, 30.0, 0.0).getDistribution() >>> distribution_F = ot.Normal(75e3, 5e3) >>> marginals = [distribution_R, distribution_F] >>> distribution = ot.JointDistribution(marginals) >>> # create the model >>> model = ot.SymbolicFunction(['R', 'F'], ['R-F/(pi_*100.0)']) >>> # create the event >>> vect = ot.RandomVector(distribution) >>> G = ot.CompositeRandomVector(model, vect) >>> event = ot.ThresholdEvent(G, ot.Less(), 0.0) >>> distribution_marginR = ot.LogNormalMuSigma().getDistribution() >>> distribution_marginF = ot.Normal() >>> aux_marginals = [distribution_marginR, distribution_marginF] >>> aux_distribution = ot.JointDistribution(aux_marginals) >>> # definition of auxiliary distribution parameters to be optimized >>> active_parameters = [0,1,2,3,4] >>> ### WARNING : native parameters of distribution have to be considered >>> bounds = ot.Interval([3,0.09,0.,50e3,2e3], [7,0.5,0.5,100e3,10e3])# bounds on the active parameters >>> initial_theta= [5.70,0.1,0.,75e3,5e3] # initial value of the active parameters >>> my_IS_physical = otexp.PhysicalSpaceCrossEntropyImportanceSampling(event,aux_distribution,active_parameters,initial_theta,bounds,0.3) >>> my_IS_physical.run() >>> myResult = my_IS_physical.getResult()
Methods
drawProbabilityConvergence
(*args)Draw the probability convergence at a given level.
Accessor to the block size.
Accessor to the object's name.
Accessor to the convergence strategy.
getEvent
()Accessor to the event.
getInputSample
(*args)Input sample accessor.
Accessor to the maximum coefficient of variation.
Accessor to the maximum sample size.
Accessor to the maximum standard deviation.
Accessor to the maximum duration.
getName
()Accessor to the object's name.
Accessor to the optimization algorithm used to update auxiliary distribution parameters.
getOutputSample
(*args)Output sample accessor.
Accessor to the intermediate quantile level.
Accessor to the intermediate quantile level.
Subset steps number accessor.
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.
Accessor to the maximum coefficient of variation.
setMaximumOuterSampling
(maximumOuterSampling)Accessor to the maximum sample size.
Accessor to the maximum standard deviation.
setMaximumTimeDuration
(maximumTimeDuration)Accessor to the maximum duration.
setName
(name)Accessor to the object's name.
setOptimizationAlgorithm
(solver)Accessor to the optimization algorithm used to update auxiliary distribution parameters.
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
- grapha
- 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_strategy
HistoryStrategy
Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.
- storage_strategy
- getEvent()¶
Accessor to the event.
- Returns:
- event
RandomVector
Event we want to evaluate the probability.
- event
- 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:
- inputSample
Sample
Input sample.
- inputSample
- 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,
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.
- getOptimizationAlgorithm()¶
Accessor to the optimization algorithm used to update auxiliary distribution parameters.
- Returns:
- algorithm
OptimizationAlgorithm
- algorithm
- 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:
- outputSample
Sample
Output sample.
- outputSample
- getQuantileLevel()¶
Accessor to the intermediate quantile level.
- Returns:
- quantileLevelfloat
Result of Cross Entropy algorithm.
- getResult()¶
Accessor to the intermediate quantile level.
- Returns:
- CrossEntropyResult
CrossEntropyResult
Contains all the results of the Cross Entropy algorithm simulation.
- CrossEntropyResult
- getStepsNumber()¶
Subset steps number accessor.
- Returns:
- nint
Number of subset steps, including the initial Monte Carlo sampling.
- hasName()¶
Test if the object is named.
- Returns:
- hasNamebool
True if the name is not empty.
- run()¶
Launch simulation.
See also
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 by a combination of blockSize and outerSampling.
- setBlockSize(blockSize)¶
Accessor to the block size.
- Parameters:
- blockSizeint,
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_strategy
HistoryStrategy
Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.
- storage_strategy
- 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,
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.
- setOptimizationAlgorithm(solver)¶
Accessor to the optimization algorithm used to update auxiliary distribution parameters.
- Parameters:
- algorithm
OptimizationAlgorithm
- algorithm
- 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
Intermediate quantile level, must belong to the interval (0,1).
- 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¶
Cross Entropy Importance Sampling