# ProbabilitySimulationAlgorithm¶

class ProbabilitySimulationAlgorithm(*args)

Iterative sampling methods.

Available constructor:

ProbabilitySimulationAlgorithm(event, experiment, convergenceStrategy=ot.Compact())

ProbabilitySimulationAlgorithm(event, convergenceStrategy=ot.Compact())

Parameters:
eventRandomVector

The event we are computing the probability of, must be composite.

experimentWeightedExperiment

Sequential experiment

convergenceStrategyHistoryStrategy, optional

Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.

Notes

Using the probability distribution of a random vector , we seek to evaluate the following probability:

Here, is a random vector, a deterministic vector, the function known as limit state function which enables the definition of the event

If we have the set of independent samples of the random vector , we can estimate as follows:

where describes the indicator function equal to 1 if and equal to 0 otherwise; the idea here is in fact to estimate the required probability by the proportion of cases, among the samples of , for which the event occurs.

By the law of large numbers, we know that this estimation converges to the required value as the sample size tends to infinity.

The Central Limit Theorem allows one to build an asymptotic confidence interval using the normal limit distribution as follows:

with , and is the -quantile of the standard normal distribution.

A ProbabilitySimulationAlgorithm object makes sense with the following sequential experiments:

The estimator built by Monte Carlo method is:

where describes the indicator function equal to 1 if and equal to 0 otherwise; the idea here is in fact to estimate the required probability by the proportion of cases, among the samples of , for which the event occurs.

By the law of large numbers, we know that this estimation converges to the required value as the sample size tends to infinity.

The Central Limit Theorem allows one to build an asymptotic confidence interval using the normal limit distribution as follows:

with , and is the -quantile of the standard normal distribution.

The estimator built by Importance Sampling method is:

where:

• is the total number of computations,

• the random vectors are independent, identically distributed and following the probability density function .

Examples

Estimate a probability by Monte Carlo

>>> 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
>>> event = ot.ThresholdEvent(output, ot.Less(), -3.0)
>>> # We create a Monte Carlo algorithm
>>> experiment = ot.MonteCarloExperiment()
>>> algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
>>> algo.setMaximumOuterSampling(150)
>>> algo.setBlockSize(4)
>>> algo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation
>>> algo.run()
>>> print('Probability estimate=%.6f' % algo.getResult().getProbabilityEstimate())
Probability estimate=0.140000


Estimate a probability by Importance Sampling

>>> ot.RandomGenerator.SetSeed(0)
>>> # assume we obtained a design point from FORM
>>> standardSpaceDesignPoint = [-0.0310363,0.841879,0.445462,-0.332318]
>>> standardEvent = ot.StandardEvent(event)
>>> importanceDensity = ot.Normal(standardSpaceDesignPoint, ot.CovarianceMatrix(4))
>>> experiment = ot.ImportanceSamplingExperiment(importanceDensity)
>>> algo = ot.ProbabilitySimulationAlgorithm(standardEvent, experiment)
>>> algo.setMaximumOuterSampling(150)
>>> algo.setBlockSize(4)
>>> algo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation
>>> algo.run()
>>> print('Probability estimate=%.6f' % algo.getResult().getProbabilityEstimate())
Probability estimate=0.153315


Estimate a probability by Quasi Monte Carlo

>>> ot.RandomGenerator.SetSeed(0)
>>> experiment = ot.LowDiscrepancyExperiment()
>>> algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
>>> algo.setMaximumOuterSampling(150)
>>> algo.setBlockSize(4)
>>> algo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation
>>> algo.run()
>>> print('Probability estimate=%.6f' % algo.getResult().getProbabilityEstimate())
Probability estimate=0.141667


Estimate a probability by Randomized Quasi Monte Carlo

>>> ot.RandomGenerator.SetSeed(0)
>>> experiment = ot.LowDiscrepancyExperiment()
>>> experiment.setRandomize(True)
>>> algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
>>> algo.setMaximumOuterSampling(150)
>>> algo.setBlockSize(4)
>>> algo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation
>>> algo.run()
>>> print('Probability estimate=%.6f' % algo.getResult().getProbabilityEstimate())
Probability estimate=0.160000


Estimate a probability by Randomized LHS

>>> ot.RandomGenerator.SetSeed(0)
>>> experiment = ot.LHSExperiment()
>>> experiment.setAlwaysShuffle(True)
>>> algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
>>> algo.setMaximumOuterSampling(150)
>>> algo.setBlockSize(4)
>>> algo.setMaximumCoefficientOfVariation(0.1)
>>> # Perform the simulation
>>> algo.run()
>>> print('Probability estimate=%.6f' % algo.getResult().getProbabilityEstimate())
Probability estimate=0.140000


Methods

 Draw the probability convergence at a given level. Accessor to the block size. Accessor to the object's name. Accessor to the convergence strategy. Accessor to the event. Accessor to the experiment. Accessor to the object's id. Accessor to the maximum coefficient of variation. Accessor to the maximum sample size. Accessor to the maximum standard deviation. Accessor to the object's name. Accessor to the results. Accessor to the object's shadowed id. Accessor to the object's visibility state. Test if the object is named. Test if the object has a distinguishable name. Launch simulation. setBlockSize(blockSize) Accessor to the block size. setConvergenceStrategy(convergenceStrategy) Accessor to the convergence strategy. setExperiment(experiment) Accessor to the experiment. Accessor to the maximum coefficient of variation. setMaximumOuterSampling(maximumOuterSampling) Accessor to the maximum sample size. Accessor to the maximum standard deviation. setName(name) Accessor to the object's name. Set up a progress callback. Accessor to the object's shadowed id. setStopCallback(*args) Set up a stop callback. setVisibility(visible) Accessor to the object's visibility state.
 getVerbose setVerbose
__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:
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.

getExperiment()

Accessor to the experiment.

Returns:
experimentWeightedExperiment

The experiment that is sampled at each iteration.

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,

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().

Accessor to the object’s shadowed id.

Returns:
idint

Internal unique identifier.

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.

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_strategyHistoryStrategy

Storage strategy used to store the values of the probability estimator and its variance during the simulation algorithm.

setExperiment(experiment)

Accessor to the experiment.

Parameters:
experimentWeightedExperiment

The experiment that is sampled 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.

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()


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()

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:
visiblebool

Visibility flag.

## Examples using the class¶

Create a process from random vectors and processes

Create a process from random vectors and processes

Estimate a probability with Monte Carlo

Estimate a probability with Monte Carlo

Use a randomized QMC algorithm

Use a randomized QMC algorithm

Specify a simulation algorithm

Specify a simulation algorithm

Use the Importance Sampling algorithm

Use the Importance Sampling algorithm

Estimate a probability with Monte-Carlo on axial stressed beam: a quick start guide to reliability

Estimate a probability with Monte-Carlo on axial stressed beam: a quick start guide to reliability

Exploitation of simulation algorithm results

Exploitation of simulation algorithm results

Create a domain event

Create a domain event

Time variant system reliability problem

Time variant system reliability problem

Create unions or intersections of events

Create unions or intersections of events

Axial stressed beam : comparing different methods to estimate a probability

Axial stressed beam : comparing different methods to estimate a probability

Using the FORM - SORM algorithms on a nonlinear function

Using the FORM - SORM algorithms on a nonlinear function

Estimate a process-based event probability

Estimate a process-based event probability

Control algorithm termination

Control algorithm termination