AdaptiveDirectionalSampling¶

class
AdaptiveDirectionalSampling
(*args)¶ Adaptative directional simulation.
Parameters: event :
Event
Event we are computing the probability of.
rootStrategy :
RootStrategy
, optionalStrategy 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
, optionalStrategy adopted to sample directions. Set to
RandomDirection
by default.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
(*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. getGamma
()Gamma accessor. 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. getMaximumStratificationDimension
()Maximum stratification dimension accessor. getName
()Accessor to the object’s name. getPartialStratification
()Partial stratification accessor. getQuadrantOrientation
()Quadrant orientation accessor. getResult
()Accessor to the results. getRootStrategy
()Get the root strategy. getSamplingStrategy
()Get the direction sampling strategy. getShadowedId
()Accessor to the object’s shadowed id. getTStatistic
()T statistic accessor. 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. setGamma
(gamma)Gamma accessor. setMaximumCoefficientOfVariation
(...)Accessor to the maximum coefficient of variation. setMaximumOuterSampling
(maximumOuterSampling)Accessor to the maximum sample size. setMaximumStandardDeviation
(...)Accessor to the maximum standard deviation. setMaximumStratificationDimension
(...)Maximum stratification dimension accessor. setName
(name)Accessor to the object’s name. setPartialStratification
(partialStratification)Partial stratification accessor. setProgressCallback
(*args)Set up a progress callback. setQuadrantOrientation
(quadrantOrientation)Quadrant orientation accessor. setRootStrategy
(rootStrategy)Set the root strategy. setSamplingStrategy
(samplingStrategy)Set the direction sampling strategy. 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.

getGamma
()¶ Gamma accessor.
The computational budget per step .
Returns: gamma :
NumericalPoint
Gamma value.

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.

getMaximumStratificationDimension
()¶ Maximum stratification dimension accessor.
Returns: max : int
Maximum stratification dimension.

getName
()¶ Accessor to the object’s name.
Returns: name : str
The name of the object.

getPartialStratification
()¶ Partial stratification accessor.
Returns: partialStratification : bool
Partial stratification.

getQuadrantOrientation
()¶ Quadrant orientation accessor.
Returns: orientation :
NumericalPoint
Quadrant orientation.

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

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

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

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

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

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.

setGamma
(gamma)¶ Gamma accessor.
The computational budget per step .
Parameters: gamma : sequence of float
Gamma value.

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.

setMaximumStratificationDimension
(maximumStratificationDimension)¶ Maximum stratification dimension accessor.
Parameters: max : int
Maximum stratification dimension.

setName
(name)¶ Accessor to the object’s name.
Parameters: name : str
The name of the object.

setPartialStratification
(partialStratification)¶ Partial stratification accessor.
Parameters: partialStratification : bool
Partial stratification.

setProgressCallback
(*args)¶ Set up a progress callback.
Parameters: callback : callable
Takes a float as argument as percentage of progress.

setQuadrantOrientation
(quadrantOrientation)¶ Quadrant orientation accessor.
Parameters: orientation : sequence of float
Quadrant orientation.

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

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

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.
