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| V.8. Sensitivity Indices based on HSIC | ||
|---|---|---|
 | Chapter V. Sensitivity analysis |  |
The sensivity measures based on Hilbert-Schmidt independence criterion (HSIC) [Gretton05,DaVeiga15] aims at analyzing the influence
of input variables 
on the output variables 
determining the dependance computing the dissimilary between the joint distribution 
and the product of the marginal distributions 
.
on the output variables
determining the dependance computing the dissimilary between the joint distribution and the product of the marginal distributions .
We associate respectively to 
and
Reproducing Kernel Hilbert Space (RKHS)
and
associating projection functions
and
defined by characteristic kernels
and
. Data are projected in a characteristic space using the kernel
trick operation.
The advantage of this approach is that the projection has not to be known, we just need to arbitrary define the kernel. Using the kernel trick operation, the HSIC measure, defined as Hilbert-Shmidt of the cross covariance operator, is expressed as
which is estimated as
with
where 
is the Kronecker operator
Sensitivity indices are therefore defined such as
These indices are mainly used for screening or ranking the parameters.
| |  | |
| V.7. The Johnson relative weight |  | Chapter VI. Dealing with optimisation issues |