# 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: