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.