Sensivity analysis with correlated inputs

The ANCOVA (ANalysis of COVAriance) method, is a variance-based method generalizing the ANOVA (ANalysis Of VAriance) decomposition for models with correlated input parameters.

Let us consider a model Y = h(\vect{X}) without making any hypothesis on the dependence structure of \vect{X} = \{X^1, \ldots, X^{n_X}\}, a n_X-dimensional random vector. The covariance decomposition requires a functional decomposition of the model. Thus the model response Y is expanded as a sum of functions of increasing dimension as follows:

(1)h(\vect{X}) = h_0 + \sum_{u\subseteq\{1,\dots,n_X\}} h_u(X_u)

h_0 is the mean of Y. Each function h_u represents, for any non empty set u\subseteq\{1, \dots, n_X\}, the combined contribution of the variables X_u to Y.

Using the properties of the covariance, the variance of Y can be decomposed into a variance part and a covariance part as follows:

    Var[Y] &=& Cov\left[h_0 + \sum_{u\subseteq\{1,\dots,n_X\}} h_u(X_u), h_0 + \sum_{u\subseteq\{1,\dots,n\}} h_u(X_u)\right] \\
           &=& \sum_{u\subseteq\{1,\dots,n_X\}} Cov\left[h_u(X_u), \sum_{u\subseteq\{1,\dots,n_X\}} h_u(X_u)\right] \\
           &=& \sum_{u\subseteq\{1,\dots,n_X\}} \left[Var[h_u(X_u)] + Cov[h_u(X_u), \sum_{v\subseteq\{1,\dots,n_X\}, v\cap u=\varnothing} h_v(X_v)]\right]

The total part of variance of Y due to X_u reads:

S_u = \frac{Cov[Y, h_u(X_u)]}{Var[Y]}

The variance formula described above enables to define each sensitivity measure S_u as the sum of a \mathit{physical} (or \mathit{uncorrelated}) part and a \mathit{correlated} part such as:

S_u = S_u^U + S_u^C

where S_u^U is the uncorrelated part of variance of Y due to X_u:

S_u^U = \frac{Var[h_u(X_u)]}{Var[Y]}

and S_u^C is the contribution of the correlation of X_u with the other parameters:

S_u^C = \frac{Cov[h_u(X_u), \displaystyle \sum_{v\subseteq\{1,\dots,n_X\}, v\cap u=\varnothing} h_v(X_v)]}{Var[Y]}

As the computational cost of the indices with the numerical model h can be very high, it is suggested to approximate the model response with a polynomial chaos expansion. However, for the sake of computational simplicity, the latter is constructed considering \mathit{independent} components \{X^1,\dots,X^{n_X}\}. Thus the chaos basis is not orthogonal with respect to the correlated inputs under consideration, and it is only used as a metamodel to generate approximated evaluations of the model response and its summands in (1).

Y \simeq \hat{h} = \sum_{j=0}^{P-1} \alpha_j \Psi_j(x)

Then one may identify the component functions. For instance, for u = \{1\}:

h_1(X_1) = \sum_{\alpha | \alpha_1 \neq 0, \alpha_{i \neq 1} = 0} y_{\alpha} \Psi_{\alpha}(\vect{X})

where \alpha is a set of degrees associated to the n_X univariate polynomial \psi_i^{\alpha_i}(X_i).

Then the model response Y is evaluated using a sample X=\{x_k, k=1,\dots,N\} of the correlated joint distribution. Finally, the several indices are computed using the model response and its component functions that have been identified on the polynomial chaos.