Sensitivity analysis using Sobol’ indices from polynomial chaos expansion¶
In this page, we introduce the method to compute Sobol’ sensitivity indices from a polynomials chaos expansion. Sobol’ indices are introduced in Sensitivity analysis using Sobol’ indices and polynomial chaos expansion (PCE) is introduced in Functional Chaos Expansion.
Introduction¶
Sobol’Hoeffding is the decomposition of a function on a basis made of orthogonal functions. Since the PCE expansion is also an orthogonal decomposition, the Sobol’ decomposition of a function can be expressed depending on its PCE (see [knio2010] page 139). As a result, Sobol’ indices can be obtained analytically from the coefficients of the PCE (see [sudret2006], [sudret2008]).
Consider the input random vector and the output random variable of the physical model:
Variance and part of variance of a PCE¶
Let be the dimension of the input random vector. Let be the number of coefficients in the functional basis. Let the set of multiindices up to the index . Depending on the way the coefficients are computed, the set of multiindices is the consequence of the choice of the polynomial degree, the enumeration rule, and, if necessary, the selection method (e.g. the LARS selection method). Let be the polynomial chaos expansion:
where is the standardized input random vector, are the coefficients and are the functions in the functional basis.
The variance of the polynomial chaos expansion is:
In the previous expression, let us emphasise that the variance is a sum of squares, excepted the coefficient. If the polynomial basis is orthonormal, the expression is particularly simple (see [legratiet2017] eq. 38.43 page 1301):
The part of variance of the multiindex is:
The sum of the part of variances of all multiindices is equal to 1:
Hence, we can identify the multiindices which contribute more significantly to the variance of the output by sorting the multiindices by decreasing order of their part of variance. This result is printed by the str representation of the FunctionalChaosSobolIndices class and is accessed by the print function: see an example of this below.
All the Sobol’ indices that we introduce in this section depend on a specific set of multiindices which are presented in the next section.
Sets of multiindices¶
Let a subset of the multiindices involved in the polynomial chaos expansion. Let be the function of the coefficients associated to the multiindices , defined by:
Then any Sobol’ index can be defined by the equation:
If the polynomial basis is orthonormal, therefore:
Hence, in the methods presented below, each Sobol’ index is defined by its corresponding set of multiindices.
First order Sobol’ index of a single variable¶
See First order Sobol’ sensitivity index of a variable for the mathematical definition of this sensitivity index. Let the index of an input variable. Let the set of multiindices such that and the other components of the multiindices are zero (see [legratiet2017] eq. 38.44 page 1301):
Therefore, the first order Sobol’ index of the variable is:
Total Sobol’ index of a single variable¶
See Total sensitivity index of a variable for the mathematical definition of this sensitivity index. Let the set of multiindices such that (see [legratiet2017] eq. 38.45 page 1301):
Therefore, the total Sobol’ index is:
Interaction Sobol’ index of a group of variables¶
See Interaction sensitivity index of a variable for the mathematical definition of this sensitivity index. Let the list of variable indices in the group. Let the set of multiindices:
Therefore, the interaction (high order) Sobol’ index is:
Total interaction Sobol’ index of a group of variables¶
See Total interaction sensitivity index of a group of variables for the mathematical definition of this sensitivity index. Let the set of multiindices:
Therefore, the total interaction (high order) Sobol’ index is:
Closed first order Sobol’ index of a group of variables¶
See First order closed sensitivity index of a group of variables for the mathematical definition of this sensitivity index. Let the set of multiindices such that each component of is contained in the group :
Therefore, the first order (closed) Sobol’ index is:
Total Sobol’ index of a group of variables¶
See Total sensitivity index of a group of variables for the mathematical definition of this sensitivity index. Let the set of multiindices:
Therefore, the total Sobol’ index is:
Summary¶
The next table presents the multiindices involved in each Sobol’ index.
Single variable or group 
Sensitivity Index 
Multiindices 

One single variable 
First order 

Total 

Interaction of a group 
First order 

Total interaction 

Group (closed) 
First order (closed) 

Total 
Table 1. Multiindices involved in the first order and total Sobol’ indices of a single variable or a group .