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 without making any hypothesis on the dependence structure of , a -dimensional random vector. The covariance decomposition requires a functional decomposition of the model. Thus the model response is expanded as a sum of functions of increasing dimension as follows:
is the mean of . Each function represents, for any non empty set , the combined contribution of the variables to .
Using the properties of the covariance, the variance of can be decomposed into a variance part and a covariance part as follows:
The total part of variance of due to reads:
The variance formula described above enables to define each sensitivity measure as the sum of a (or ) part and a part such as:
where is the uncorrelated part of variance of due to :
and is the contribution of the correlation of with the other parameters:
As the computational cost of the indices with the numerical model 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 components . 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).
Then one may identify the component functions. For instance, for :
where is a set of degrees associated to the univariate polynomial .
Then the model response is evaluated using a sample 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.