QuasiMonteCarlo¶

class
QuasiMonteCarlo
(*args)¶ QuasiMonte Carlo method.
 Available constructors:
 QuasiMonteCarlo(event=ot.Event(), lowDiscrepancySequence=ot.SobolSequence())
Parameters: event :
Event
Event we are computing the probability of.
lowDiscrepancySequence :
LowDiscrepancySequence
Lowdiscrepancy sequence used to generate the samples.
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 . describes the indicator function equal to 1 if and equal to 0 otherwise.
QuasiMonte Carlo method approximates the probability of failure using low
discrepancy sequences
instead of randomly generated sequences, as follows:In general, the integral of a function on can be approximated by using some low discrepancy sequence as follows:
The low discrepancy sequence is generated on according to the Lebesgue measure then may be transformed to any measure thanks to the inverse CDF technique in order to approximate the integral:
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 QuasiMonteCarlo algorithm >>> myAlgo = ot.QuasiMonteCarlo(myEvent) >>> myAlgo.setMaximumOuterSampling(50) >>> myAlgo.setBlockSize(4) >>> myAlgo.setMaximumCoefficientOfVariation(0.1) >>> # Perform the simulation >>> myAlgo.run() >>> print('Probability estimate=%.6f' % myAlgo.getResult().getProbabilityEstimate()) Probability estimate=0.125000
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. 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. 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: 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.

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,
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()
.

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 by a combination of blockSize and outerSampling.

setBlockSize
(blockSize)¶ Accessor to the block size.
Parameters: blockSize : int,
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,
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.

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.