AdaptiveDirectionalSampling¶

class
AdaptiveDirectionalSampling
(*args)¶ Adaptative directional simulation.
 Parameters
 event
RandomVector
Event we are computing the probability of.
 rootStrategy
RootStrategy
, optional Strategy adopted to evaluate the intersections of each direction with the limit state function and take into account the contribution of the direction to the event probability. Set to
SafeAndSlow
by default. samplingStrategy
SamplingStrategy
, optional Strategy adopted to sample directions. Set to
RandomDirection
by default.
 event
See also
Notes
Let denote the failure domain defined as , where are realization of the random vector and is the limitstate function as defined elsewhere in the documentation.
The purpose of the ADS2 algorithm and its variants is to estimate the following probability:
Principles
The ADS2 method [munoz2011] combines the stratified and directional sampling concepts. Stratified sampling consists in splitting the support of the random vector into mutually exclusive and collectively exhaustive subsets. Here, ADS2 splits the standard space into quadrants, where is the dimension of the random vector . Stratified sampling is often run in two steps: (i) a learning step is used for polling the input space and detect the subsets that contribute most to the probability and (ii) an estimation step is used for estimating the probability by weighted sampling (some subsets are more sampled than the others). Directional sampling uses the spheric symmetry of the standard space for estimating the failure probability as the average of conditional probabilities calculated on directions drawn at random in the standard space.
The learning step uses an a priori number of random directions that is uniformly distributed over the quadrants, meaning the weights are as follows:
Directional sampling is used for estimating the failure probability in each quadrant:
and the corresponding estimation variances are denoted as . These probabilities are estimated using the same number of random directions per quadrant as told by the uniform weights distribution.
The probability of interest is then computed as a weighted average of the previously defined conditional probabilities:
where is the conditional probability estimator in the ith quadrant. The corresponding variance of the stratified estimator reads:
where is the variance of the conditional probability estimator in the ith quadrant.
At the end of the learning step, the weights are updated so as to minimize the stratified estimator variance. Indeed, it can be shown that the updated weights:
minimize the final estimation variance in eqref{eq:pf_est_sda2_var}. Note that some weights might be zero (due to a somewhat arbitrary rounding of the conditional probabilities’ estimation variance). The quadrants associated with a zeroweight will not be sampled in the estimation step.
Eventually, the estimation step proceeds in essentially the same way as the learning step with different weights for the quadrants though. eqref{eq:pf_est_sda2} and eqref{eq:pf_est_sda2_var} are used for evaluating the final probability probability estimate and its variance.
The computational budget per step is parametrized by a fraction of the total budget , such that . The number of directions sampled in quadrant at step is then defined as follows:
The number of evaluation of the limitstate function is of course greater than the total budget since directional sampling is used.
Variants
The ADS2+ variant performs a dimension reduction step after the learning step for reducing the number of stratified quadrants. The statistic aggregates the sensitivity of expectation along dimension . It is defined as follows:
It is used for ranking the contributions of the quadrants. Then, only the most influential variables according to are stratified, leaving the remaining variables simulated without stratification. The corresponding quadrants will not be sampled.
The DPADS2 variant combines the ADS method with a rotation of the quadrants. The idea is to get a possible design point (available e.g. after a preliminary FORM analysis) on the bisector of one of the quadrants to make the stratification even more efficient and thus save some evaluations of the model.
This 2step algorithm can be generalized to steps by adding more than one learning step. For now, only ADS2 is implemented.
Methods
drawProbabilityConvergence
(self, \*args)Draw the probability convergence at a given level.
getBlockSize
(self)Accessor to the block size.
getClassName
(self)Accessor to the object’s name.
getConvergenceStrategy
(self)Accessor to the convergence strategy.
getEvent
(self)Accessor to the event.
getGamma
(self)Gamma accessor.
getId
(self)Accessor to the object’s id.
Accessor to the maximum coefficient of variation.
getMaximumOuterSampling
(self)Accessor to the maximum sample size.
Accessor to the maximum standard deviation.
Maximum stratification dimension accessor.
getName
(self)Accessor to the object’s name.
getPartialStratification
(self)Partial stratification accessor.
getQuadrantOrientation
(self)Quadrant orientation accessor.
getResult
(self)Accessor to the results.
getRootStrategy
(self)Get the root strategy.
getSamplingStrategy
(self)Get the direction sampling strategy.
getShadowedId
(self)Accessor to the object’s shadowed id.
getTStatistic
(self)T statistic accessor.
getVerbose
(self)Accessor to verbosity.
getVisibility
(self)Accessor to the object’s visibility state.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
run
(self)Launch simulation.
setBlockSize
(self, blockSize)Accessor to the block size.
setConvergenceStrategy
(self, convergenceStrategy)Accessor to the convergence strategy.
setGamma
(self, gamma)Gamma accessor.
setMaximumCoefficientOfVariation
(self, …)Accessor to the maximum coefficient of variation.
setMaximumOuterSampling
(self, …)Accessor to the maximum sample size.
setMaximumStandardDeviation
(self, …)Accessor to the maximum standard deviation.
setMaximumStratificationDimension
(self, …)Maximum stratification dimension accessor.
setName
(self, name)Accessor to the object’s name.
setPartialStratification
(self, …)Partial stratification accessor.
setProgressCallback
(self, \*args)Set up a progress callback.
setQuadrantOrientation
(self, quadrantOrientation)Quadrant orientation accessor.
setRootStrategy
(self, rootStrategy)Set the root strategy.
setSamplingStrategy
(self, samplingStrategy)Set the direction sampling strategy.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setStopCallback
(self, \*args)Set up a stop callback.
setVerbose
(self, verbose)Accessor to verbosity.
setVisibility
(self, visible)Accessor to the object’s visibility state.

__init__
(self, \*args)¶ Initialize self. See help(type(self)) for accurate signature.

drawProbabilityConvergence
(self, \*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
(self)¶ 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
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getConvergenceStrategy
(self)¶ 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
(self)¶ Accessor to the event.
 Returns
 event
RandomVector
Event we want to evaluate the probability.
 event

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getMaximumCoefficientOfVariation
(self)¶ Accessor to the maximum coefficient of variation.
 Returns
 coefficientfloat
Maximum coefficient of variation of the simulated sample.

getMaximumOuterSampling
(self)¶ Accessor to the maximum sample size.
 Returns
 outerSamplingint
Maximum number of groups of terms in the probability simulation estimator.

getMaximumStandardDeviation
(self)¶ Accessor to the maximum standard deviation.
 Returns
 sigmafloat,
Maximum standard deviation of the estimator.

getMaximumStratificationDimension
(self)¶ Maximum stratification dimension accessor.
 Returns
 maxint
Maximum stratification dimension.

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getPartialStratification
(self)¶ Partial stratification accessor.
 Returns
 partialStratificationbool
Partial stratification.

getQuadrantOrientation
(self)¶ Quadrant orientation accessor.
 Returns
 orientation
Point
Quadrant orientation.
 orientation

getResult
(self)¶ Accessor to the results.
 Returns
 results
SimulationResult
Structure containing all the results obtained after simulation and created by the method
run()
.
 results

getRootStrategy
(self)¶ Get the root strategy.
 Returns
 strategy
RootStrategy
Root strategy adopted.
 strategy

getSamplingStrategy
(self)¶ Get the direction sampling strategy.
 Returns
 strategy
SamplingStrategy
Direction sampling strategy adopted.
 strategy

getShadowedId
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getTStatistic
(self)¶ T statistic accessor.
The statistic aggregates the sensitivity of expectation.
 Returns
 gamma
Point
T statistic value.
 gamma

getVerbose
(self)¶ Accessor to verbosity.
 Returns
 verbosity_enabledbool
If True, the computation is verbose. By default it is verbose.

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

run
(self)¶ 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
(self, 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 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
(self, 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

setGamma
(self, gamma)¶ Gamma accessor.
The computational budget per step .
 Parameters
 gammasequence of float
Gamma value.

setMaximumCoefficientOfVariation
(self, maximumCoefficientOfVariation)¶ Accessor to the maximum coefficient of variation.
 Parameters
 coefficientfloat
Maximum coefficient of variation of the simulated sample.

setMaximumOuterSampling
(self, maximumOuterSampling)¶ Accessor to the maximum sample size.
 Parameters
 outerSamplingint
Maximum number of groups of terms in the probability simulation estimator.

setMaximumStandardDeviation
(self, maximumStandardDeviation)¶ Accessor to the maximum standard deviation.
 Parameters
 sigmafloat,
Maximum standard deviation of the estimator.

setMaximumStratificationDimension
(self, maximumStratificationDimension)¶ Maximum stratification dimension accessor.
 Parameters
 maxint
Maximum stratification dimension.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setPartialStratification
(self, partialStratification)¶ Partial stratification accessor.
 Parameters
 partialStratificationbool
Partial stratification.

setProgressCallback
(self, \*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()

setQuadrantOrientation
(self, quadrantOrientation)¶ Quadrant orientation accessor.
 Parameters
 orientationsequence of float
Quadrant orientation.

setRootStrategy
(self, rootStrategy)¶ Set the root strategy.
 Parameters
 strategy
RootStrategy
Root strategy adopted.
 strategy

setSamplingStrategy
(self, samplingStrategy)¶ Set the direction sampling strategy.
 Parameters
 strategy
SamplingStrategy
Direction sampling strategy adopted.
 strategy

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setStopCallback
(self, \*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
(self, verbose)¶ Accessor to verbosity.
 Parameters
 verbosity_enabledbool
If True, make the computation verbose. By default it is verbose.

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.