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 $\vect{X} = \left( X_1,\ldots,X_{n_X} \right)$ and the output random variable $Y$ of the physical model:

$Y = \operatorname{g}(\vect{X}).$

Variance and part of variance of a PCE¶

Let $n_X$ be the dimension of the input random vector. Let $P \in \Nset$ be the number of coefficients in the functional basis. Let $\mathcal{J}_P \subseteq \Nset^{n_X}$ the set of multi-indices up to the index $P$ . Depending on the way the coefficients are computed, the set of multi-indices 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 $\tilde{h}(\bdZ)$ be the polynomial chaos expansion:

$\tilde{h}(\bdZ) &= \sum_{\idx \in \mathcal{J}_P} a_\idx \psi_\idx(\bdZ)$

where $\bdZ$ is the standardized input random vector, $\{a_\idx\}_{\idx \in \mathcal{J}_P}$ are the coefficients and $\{\psi_\idx\}_{\idx \in \mathcal{J}_P}$ are the functions in the functional basis.

The variance of the polynomial chaos expansion is:

$\Var{\tilde{h}(\bdZ)} = \sum_{\idx \in \mathcal{J}_P} a_\idx^2 \|\psi_\idx\|^2.$

In the previous expression, let us emphasise that the variance is a sum of squares, excepted the $a_0$ coefficient. If the polynomial basis is orthonormal, the expression is particularly simple (see [legratiet2017] eq. 38.43 page 1301):

$\Var{\tilde{h}(\bdZ)} = \sum_{\idx \in \mathcal{J}_P} a_\idx^2.$

The part of variance of the multi-index $\idx$ is:

$\operatorname{PoV}_\idx = \frac{a_\idx^2 \|\psi_\idx\|^2}{\Var{\tilde{h}(\bdZ)}}.$

The sum of the part of variances of all multi-indices is equal to 1:

$\sum_{\idx \in \mathcal{J}_P} \operatorname{PoV}_\idx = 1.$

Hence, we can identify the multi-indices which contribute more significantly to the variance of the output by sorting the multi-indices 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 multi-indices which are presented in the next section.

Sets of multi-indices¶

Let $\mathcal{J}^S \subseteq \mathcal{J}_P$ a subset of the multi-indices involved in the polynomial chaos expansion. Let $\operatorname{S}^{PCE}$ be the function of the coefficients associated to the multi-indices $\mathcal{J}^S$ , defined by:

$\operatorname{S}^{PCE}\left(\mathcal{J}^S\right) = \frac{\sum_{\idx \in \mathcal{J}^S} a_\idx^2 \|\psi_\idx\|^2}{\Var{\tilde{h}(\bdZ)}}.$

Then any Sobol’ index $S$ can be defined by the equation:

$S = \operatorname{S}^{PCE}\left(\mathcal{J}^S\right).$

If the polynomial basis is orthonormal, therefore:

$\operatorname{S}^{PCE}\left(\mathcal{J}^S\right) = \frac{\sum_{\idx \in \mathcal{J}^S} a_\idx^2}{\Var{\tilde{h}(\bdZ)}}.$

Hence, in the methods presented below, each Sobol’ index is defined by its corresponding set of multi-indices.

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 $i \in \{0, ..., n_X - 1\}$ the index of an input variable. Let $\mathcal{J}_i^S$ the set of multi-indices such that $\alpha_i > 0$ and the other components of the multi-indices are zero (see [legratiet2017] eq. 38.44 page 1301):

$\mathcal{J}_i^S =\left\{\idx=(0, ..., 0, \alpha_i, 0, ..., 0) \in \mathcal{J}_P, \quad \alpha_i > 0 \right\}.$

Therefore, the first order Sobol’ index $S_i$ of the variable $X_i$ is:

$S_i = \operatorname{S}^{PCE}\left(\mathcal{J}_i^S\right).$

Total Sobol’ index of a single variable¶

See Total sensitivity index of a variable for the mathematical definition of this sensitivity index. Let $\mathcal{J}_i^T$ the set of multi-indices such that $\alpha_i > 0$ (see [legratiet2017] eq. 38.45 page 1301):

$\mathcal{J}_i^T =\left\{\idx = (\alpha_0,...,\alpha_i,...,\alpha_{n_X - 1}) \in \mathcal{J}_P, \quad \alpha_i > 0 \right\}.$

Therefore, the total Sobol’ index $S^T_i$ is:

$S^T_i = \operatorname{S}^{PCE}\left(\mathcal{J}_i^T\right).$

Interaction Sobol’ index of a group of variables¶

See Interaction sensitivity index of a variable for the mathematical definition of this sensitivity index. Let $\bdu \subseteq \{0, ..., n_X - 1\}$ the list of variable indices in the group. Let $\mathcal{J}_\bdu$ the set of multi-indices:

$\mathcal{J}_\bdu =\left\{\idx \in \mathcal{J}_P, \quad \idx \neq \boldsymbol{0}, \quad \alpha_i > 0 \textrm{ if } i \in \bdu, \quad \alpha_i = 0 \textrm{ if } i \not \in \bdu, \quad i = 1, \ldots, n_X \right\}.$

Therefore, the interaction (high order) Sobol’ index $S_\bdu$ is:

$S_\bdu = \operatorname{S}^{PCE}\left(\mathcal{J}_\bdu\right).$

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 $\mathcal{J}_\bdu$ the set of multi-indices:

$\mathcal{J}_\bdu^{T, i} = \left\{\boldsymbol{\alpha} \in \mathcal{J}_P, \quad \boldsymbol{\alpha} \neq \boldsymbol{0}, \quad \alpha_i > 0 \textrm{ if } i \in \bdu, \quad i = 1, \ldots, n_X \right\}.$

Therefore, the total interaction (high order) Sobol’ index $S_\bdu$ is:

$S_\bdu^{T, i} = \operatorname{S}^{PCE}\left(\mathcal{J}_\bdu^{T, i}\right).$

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 $\mathcal{J}_\bdu^{S, \operatorname{cl}}$ the set of multi-indices such that each component of $\idx$ is contained in the group $\bdu$ :

$\mathcal{J}_\bdu^{S, \operatorname{cl}} = \left\{\idx \in \mathcal{J}_P, \quad \idx \neq \boldsymbol{0}, \quad \alpha_i = 0 \quad \textrm{ if } \quad i \not \in \bdu, \quad i = 1, \ldots, n_X \right\}.$

Therefore, the first order (closed) Sobol’ index $S^{\operatorname{cl}}_\bdu$ is:

$S^{\operatorname{cl}}_\bdu = \operatorname{S}^{PCE}\left(\mathcal{J}_\bdu^{S, \operatorname{cl}}\right).$

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 $\mathcal{J}_\bdu^T$ the set of multi-indices:

$\mathcal{J}_\bdu^T = \left\{\idx \in\mathcal{J}_P, \quad \idx \neq \boldsymbol{0}, \quad \exists i \in \{1, \ldots, n_X\} \quad \textrm{s.t.} \quad i \in \bdu \textrm{ and } \alpha_i > 0 \right\}.$

Therefore, the total Sobol’ index $S^T_\bdu$ is:

$S^T_\bdu = \operatorname{S}^{PCE}\left(\mathcal{J}_\bdu^T\right).$

Summary¶

The next table presents the multi-indices involved in each Sobol’ index.

Single variable or group	Sensitivity Index	Multi-indices
One single variable $i$	First order	$\alpha_j > 0 \textrm{ if } j = i, \textrm{ and } \alpha_j = 0 \textrm{ if } j \neq i, \quad j=1, \ldots, n_X$
	Total	$\alpha_i > 0$
Interaction of a group $\bdu$	First order	$\alpha_i > 0 \textrm{ if } i \in \bdu, \quad \alpha_i = 0 \textrm{ if } i \not \in \bdu$
	Total interaction	$\alpha_i >0 \textrm{ if } i \in \bdu$
Group (closed) $\bdu$	First order (closed)	$\alpha_i = 0 \textrm{ if } i \not\in \bdu, \quad i = 1, \ldots, n_X$
	Total	$\exists i\in\{1,\ldots, n_X\} \quad \textrm{s.t.} \quad i \in \bdu \textrm{ and } \alpha_i > 0$

Table 1. Multi-indices involved in the first order and total Sobol’ indices of a single variable $i$ or a group $\bdu$ .

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