Sensitivity analysis using Sobol’ indices¶
Consider the input random vector
and let
be the output of the physical model:
We consider the output for any index
.
Sobol’ indices measure the influence of the input
to the output
.
The method considers the part of the variance of the output
produced by
the different inputs
.
In the first part of this document, we introduce the Sobol’ indices of a
scalar output .
Hence, the model is simplified to:
In the second part of the document, we consider the general case where the output is multivariate. In this case, aggregated Sobol’ indices can be used [gamboa2013].
The Sobol’ decomposition is described more easily when the domain of the input
is the unit interval .
It can be easily extended to any input domain using expectations, variances
and variance of conditional expectations.
We assume that the input marginal variables are
independent.
This restrictive hypothesis implies that the only copula of the input
random vector
for which the
Sobol’ indices are easy to interpret is the independent copula.
If the input variables are dependent, then the Sobol’ indices can be defined,
but some of their properties are lost.
Partition of the input¶
For any , let
be
the vector made of components of
which
indices are different from
.
Hence, if
, then:
Consider the function defined by the equation:
where .
With this notation, we can partition the input of
:
The goal of sensitivity analysis is to measure the sensitivity of the variance
of the output depending on the variable
.
This may take into account the dependence of the output to the interactions
of
and
through the function
.
More generally, let be a group of
variables.
Therefore:
The goal of sensitivity analysis is to measure the sensitivity of the variance
of the output depending on the group of variables
.
This may take into account the dependence of the output to the interactions
of
and
through the function
.
Sobol’ decomposition¶
In this section, we introduce the Sobol’-Hoeffding decomposition [sobol1993].
If can be integrated in
, then there is a unique
decomposition:
where is a constant and the functions of the decomposition satisfy the equalities:
for any and any indices
and
.
Extension to any input distribution with independent marginals¶
In this section, we extend the previous definitions to an input random vector
that is not necessarily defined on the input unit cube .
To do this, we define the functions
using conditional
expectations.
The functions satisfy the equality:
for any group of variables with
size lower or equal to
, where
is the
cardinal of the subset
.
The functions
can be defined recursively, using groups of
variables of lower dimensionality:
where denotes a proper subset.
Let
be a point and
let
be a group of variables.
Therefore:
The Möbius inversion formula implies (see [daveiga2022] Theorem 3.3 page 49):
The previous equation is a consequence of the Möbius inversion formula [rota1964] (also called the exclusion-inclusion principle).
Decomposition of the variance¶
The variance of the function can be
decomposed into:
where the interaction variances are:
More generally, the interaction variance of a group of variables is:
for any .
Using the Hoeffding decomposition, we get:
The Möbius inversion formula implies (see [daveiga2022] corollary 3.5 page 52):
Interaction sensitivity index of a variable¶
The first order interaction sensitivity indices are equal to:
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
.
More generally, the first order interaction Sobol’ index of a group
of variables is:
where is the function of the input variables in the group
of the functional Sobol’-Hoeffding ANOVA decomposition of the physical model.
This index measures the sensitivity of the variance of the output explained
by interactions within the group.
Total interaction sensitivity index of a group of variables¶
The total interaction sensitivity index of the group
is (see [liu2006] eq. 8 page 714 where it is named “superset importance”):
This index measures the sensitivity of the variance of the output explained by interactions within the group and groups of variables containing it.
First order Sobol’ sensitivity index of a variable¶
The first order Sobol’ sensitivity index is equal to the corresponding
interaction index of the group :
for .
The first order Sobol’ index
measures the sensitivity of the
output variance explained by the effect of
alone.
We can alternatively define the first order Sobol’ sensitivity index using
the variance of a conditional expectation.
The first order Sobol’ sensitivity index satisfies the equation:
for .
Total sensitivity index of a variable¶
The total Sobol’ sensitivity index is:
for .
The total Sobol’ sensitivity index can be equivalently defined in terms
of the variance of a conditional expectation.
The total Sobol’ sensitivity index satisfies the equation:
for .
For any
, let us define
Total Sobol’ indices satisfy the equality:
for .
The total Sobol’ index measures the part of the variance
of
explained by
and its interactions with other input variables.
It can also be viewed as the part of the variance of
that cannot
be explained without