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:
(1)¶
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.
API:
See
ANCOVA
Examples:
References: