LHS¶
- class LHS(*args)¶
Latin Hypercube Sampling (LHS) method.
- Available constructors:
LHS(event=ot.Event())
- Parameters
- event
RandomVector
Event we are computing the probability of.
- event
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.
LHS or Latin Hypercube Sampling is a sampling method enabling to better cover the domain of variations of the input variables, thanks to a stratified sampling strategy. This method is applicable in the case of independent input variables. The sampling procedure is based on dividing the range of each variable into several intervals of equal probability. The sampling is undertaken as follows:
Step 1: The range of each input variable is stratified into isoprobabilistic cells,
Step 2: A cell is uniformly chosen among all the available cells,
Step 3: The random number is obtained by inverting the Cumulative Density Function locally in the chosen cell,
Step 4: All the cells having a common strate with the previous cell are put apart from the list of available cells.
The estimator of the probability of failure with LHS is given by:
where the sample of is obtained as described previously.
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.CompositeRandomVector(myFunction, vect) >>> # We create an event from this RandomVector >>> myEvent = ot.ThresholdEvent(output, ot.Less(), -3.0) >>> # We create a LHS algorithm >>> myAlgo = ot.LHS(myEvent) >>> myAlgo.setMaximumOuterSampling(150) >>> myAlgo.setBlockSize(4) >>> myAlgo.setMaximumCoefficientOfVariation(0.1) >>> # Perform the simulation >>> myAlgo.run() >>> print('Probability estimate=%.6f' % myAlgo.getResult().getProbabilityEstimate()) Probability estimate=0.151667
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.
getId
()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.
getName
()Accessor to the object's name.
Accessor to the results.
Accessor to the object's shadowed id.
Accessor to verbosity.
Accessor to the object's visibility state.
hasName
()Test if the object is named.
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.
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.
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
- 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
- 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
- results
SimulationResult
Structure containing all the results obtained after simulation and created by the method
run()
.
- results
- 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.
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
- 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()
- 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.