SobolIndicesAlgorithm¶
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- class SobolIndicesAlgorithm(*args)¶
Sensitivity analysis.
Notes
This method analyzes the influence of each component of an input random vector on an output random vector by computing Sobol’ indices (see [sobol1993]). It computes, for every output random variable (), the part of its variance due to each input component () of .
Let be the physical model such as . Let us first consider the case where . In that case, we denote by . Let us write the variance of as a function of . For any subset , define .
Using the Hoeffding decomposition, we get:
where:
For the sake of conciseness, for any integers , define and . We have:
for .
First and second order Sobol’ indices are defined as follows:
for .
The first order Sobol’ index measures the part of the variance of explained by alone. The second order Sobol’ index measures the part of the variance of explained by the interaction of and .
For any , let us define
Total order Sobol’ indices are defined as follows:
for .
The total order Sobol’ index quantifies the part of the variance of that is due to the effect of and its interactions with all the other input variables. It can also be viewed as the part of the variance of that cannot be explained without .
In practice, to estimate these quantities, Sobol’ proposes to use numerical methods that rely on two independent realizations of the random vector . Let and be two independent samples (of size ) of :
Each line is a realization of the random vector .
We are now going to mix these two samples to get an estimate of the sensitivity indices.
For the sake of stability, computations will be performed with centered output. Let be the mean of the combined samples and . Let be the empirically centered function defined, for any , by:
To estimate the total variance , we use the
computeVariance()
method of theSample
.Several estimators of , and are provided by the
SobolIndicesAlgorithm
implementations:SaltelliSensitivityAlgorithm
for the Saltelli method,JansenSensitivityAlgorithm
for the Jansen method,MauntzKucherenkoSensitivityAlgorithm
for the Mauntz-Kucherenko method,MartinezSensitivityAlgorithm
for the Martinez method.
Specific formulas for , and are given in the corresponding documentation pages.
The estimator of is the same for all these classes:
Notice that the value of the second order conditional variance depends on the estimators and which are method-dependent. This implies that the value of the second order indices may depend on the specific Sobol’ estimator we use.
For multivariate outputs i.e. when , we compute the Sobol’ indices with respect to each output variable. In this case, the methods
getFirstOrderIndices()
andgetTotalOrderIndices()
return the Sobol’ indices of the first output, but the index of the output can be specified as input argument. Moreover, the indices can be aggregated [gamboa2013]. Let be the (first order) variance of the conditional expectation of the k-th output :for and . Similarily, let be the total variance of the conditional expectation of for and .
The indices can be aggregated with the following formulas:
for .
Aggregated indices can be retrieved with the
getAggregatedFirstOrderIndices()
andgetAggregatedTotalOrderIndices()
methods.Notice that the distribution of the estimators of the first and total order indices can be estimated thanks to the
getFirstOrderIndicesDistribution()
andgetTotalOrderIndicesDistribution()
methods. This is done either through bootstrapping or using an asymptotic estimator. TheResourceMap
key SobolIndicesAlgorithm-DefaultUseAsymptoticDistribution stores a boolean that decides the default behavior, but it can be overridden by the methodsetUseAsymptoticDistribution()
.Corresponding confidence intervals are provided by the methods
getFirstOrderIndicesInterval()
andgetTotalOrderIndicesInterval()
. Their confidence level can be adjusted withsetConfidenceLevel()
. The default confidence level is stored in theResourceMap
and can be accessed with the SobolIndicesAlgorithm-DefaultConfidenceLevel key.Indices estimates can be slightly outside of [0,1] if the estimator has not converged. For the same reason some first order indices estimates can be greater than the corresponding total order indices estimates.
The asymptotic estimator of the distribution requires an asymptotic estimate of its variance, which is computed using the [janon2014] delta method, as expained in the technical report [pmfre01116].
Methods
DrawCorrelationCoefficients
(*args)Draw the correlation coefficients.
DrawImportanceFactors
(*args)Draw the importance factors.
DrawSobolIndices
(*args)Draw the Sobol' indices.
draw
(*args)Draw sensitivity indices.
Get the evaluation of aggregated first order Sobol indices.
Get the evaluation of aggregated total order Sobol indices.
Get the number of bootstrap sampling size.
Accessor to the object's name.
Get the confidence interval level for confidence intervals.
getFirstOrderIndices
([marginalIndex])Get first order Sobol indices.
Get the distribution of the aggregated first order Sobol indices.
Get interval for the aggregated first order Sobol indices.
getId
()Accessor to the object's id.
Accessor to the underlying implementation.
getName
()Accessor to the object's name.
getSecondOrderIndices
([marginalIndex])Get second order Sobol indices.
getTotalOrderIndices
([marginalIndex])Get total order Sobol indices.
Get the distribution of the aggregated total order Sobol indices.
Get interval for the aggregated total order Sobol indices.
Select asymptotic or bootstrap confidence intervals.
setBootstrapSize
(bootstrapSize)Set the number of bootstrap sampling size.
setConfidenceLevel
(confidenceLevel)Set the confidence interval level for confidence intervals.
setDesign
(inputDesign, outputDesign, size)Sample accessor.
setName
(name)Accessor to the object's name.
Select asymptotic or bootstrap confidence intervals.
- __init__(*args)¶
- static DrawCorrelationCoefficients(*args)¶
- Draw the correlation coefficients.
As correlation coefficients are considered, values might be positive or negative.
- Available usages:
DrawCorrelationCoefficients(correlationCoefficients, title=’Correlation coefficients’)
DrawCorrelationCoefficients(values, names, title=’Correlation coefficients’)
- Parameters
- correlationCoefficients
PointWithDescription
Sequence containing the correlation coefficients with a description for each component. The descriptions are used to build labels for the created graph. If they are not mentioned, default labels will be used.
- valuessequence of float
Correlation coefficients.
- namessequence of str
Variables’ names used to build labels for the created the graph.
- titlestr
Title of the graph.
- correlationCoefficients
- Returns
- static DrawImportanceFactors(*args)¶
Draw the importance factors.
- Available usages:
DrawImportanceFactors(importanceFactors, title=’Importance Factors’)
DrawImportanceFactors(values, names, title=’Importance Factors’)
- Parameters
- importanceFactors
PointWithDescription
Sequence containing the importance factors with a description for each component. The descriptions are used to build labels for the created Pie. If they are not mentioned, default labels will be used.
- valuessequence of float
Importance factors.
- namessequence of str
Variables’ names used to build labels for the created Pie.
- titlestr
Title of the graph.
- importanceFactors
- Returns
- static DrawSobolIndices(*args)¶
Draw the Sobol’ indices.
- Parameters
- Returns
- graph
Graph
For each variable, draws first and total indices
- graph
- draw(*args)¶
Draw sensitivity indices.
- Usage:
draw()
draw(marginalIndex)
With the first usage, draw the aggregated first and total order indices. With the second usage, draw the first and total order indices of a specific marginal in case of vectorial output
- Parameters
- marginalIndex: int
marginal of interest (case of second usage)
- Returns
- graph
Graph
A graph containing the aggregated first and total order indices.
- graph
Notes
If number of bootstrap sampling is not 0, and confidence level associated > 0, the graph includes confidence interval plots in the first usage.
- getAggregatedFirstOrderIndices()¶
Get the evaluation of aggregated first order Sobol indices.
- Returns
- indices
Point
Sequence containing aggregated first order Sobol indices.
- indices
- getAggregatedTotalOrderIndices()¶
Get the evaluation of aggregated total order Sobol indices.
- Returns
- indices
Point
Sequence containing aggregated total order Sobol indices.
- indices
- getBootstrapSize()¶
Get the number of bootstrap sampling size.
- Returns
- bootstrapSizeint
Number of bootstrap sampling
- getClassName()¶
Accessor to the object’s name.
- Returns
- class_namestr
The object class name (object.__class__.__name__).
- getConfidenceLevel()¶
Get the confidence interval level for confidence intervals.
- Returns
- confidenceLevelfloat
Confidence level for confidence intervals
- getFirstOrderIndices(marginalIndex=0)¶
Get first order Sobol indices.
- Parameters
- iint, optional
Index of the output marginal of the function, equal to by default.
- Returns
- indices
Point
Sequence containing first order Sobol indices.
- indices
- getFirstOrderIndicesDistribution()¶
Get the distribution of the aggregated first order Sobol indices.
- Returns
- distribution
Distribution
Distribution for first order Sobol indices for each component.
- distribution
- getFirstOrderIndicesInterval()¶
Get interval for the aggregated first order Sobol indices.
- Returns
- interval
Interval
Interval for first order Sobol indices for each component. Computed marginal by marginal (not from the joint distribution).
- interval
- getId()¶
Accessor to the object’s id.
- Returns
- idint
Internal unique identifier.
- getImplementation()¶
Accessor to the underlying implementation.
- Returns
- implImplementation
A copy of the underlying implementation object.
- getName()¶
Accessor to the object’s name.
- Returns
- namestr
The name of the object.
- getSecondOrderIndices(marginalIndex=0)¶
Get second order Sobol indices.
- Parameters
- iint, optional
Index of the marginal of the function, equals to by default.
- Returns
- indices
SymmetricMatrix
Tensor containing second order Sobol indices.
- indices
- getTotalOrderIndices(marginalIndex=0)¶
Get total order Sobol indices.
- Parameters
- iint, optional
Index of the output marginal of the function, equal to by default.
- Returns
- indices
Point
Sequence containing total order Sobol indices.
- indices
- getTotalOrderIndicesDistribution()¶
Get the distribution of the aggregated total order Sobol indices.
- Returns
- distribution
Distribution
Distribution for total order Sobol indices for each component.
- distribution
- getTotalOrderIndicesInterval()¶
Get interval for the aggregated total order Sobol indices.
- Returns
- interval
Interval
Interval for total order Sobol indices for each component. Computed marginal by marginal (not from the joint distribution).
- interval
- getUseAsymptoticDistribution()¶
Select asymptotic or bootstrap confidence intervals.
- Returns
- useAsymptoticDistributionbool
Whether to use bootstrap or asymptotic intervals
- setBootstrapSize(bootstrapSize)¶
Set the number of bootstrap sampling size.
Default value is 0.
- Parameters
- bootstrapSizeint
Number of bootstrap sampling
- setConfidenceLevel(confidenceLevel)¶
Set the confidence interval level for confidence intervals.
- Parameters
- confidenceLevelfloat
Confidence level for confidence intervals
- setDesign(inputDesign, outputDesign, size)¶
Sample accessor.
Allows one to estimate indices from a predefined Sobol design.
- Parameters
- inputDesign
Sample
Design for the evaluation of sensitivity indices, obtained thanks to the SobolIndicesAlgorithmImplementation.Generate method
- outputDesign
Sample
Design for the evaluation of sensitivity indices, obtained as the evaluation of a Function (model) on the previous inputDesign
- Nint
Base size of the Sobol design
- inputDesign
- setName(name)¶
Accessor to the object’s name.
- Parameters
- namestr
The name of the object.
- setUseAsymptoticDistribution(useAsymptoticDistribution)¶
Select asymptotic or bootstrap confidence intervals.
Default value is set by the SobolIndicesAlgorithm-DefaultUseAsymptoticDistribution key.
- Parameters
- useAsymptoticDistributionbool
Whether to use bootstrap or asymptotic intervals