  Conventional kernel adaptive filtering (KAF) uses a prescribed, positive
definite, nonlinear function to define the Reproducing Kernel Hilbert Space
(RKHS), where the optimal solution for mean square error estimation is
approximated using search techniques. Instead, this paper proposes to embed the
full statistics of the input data in the kernel definition, obtaining the first
analytical solution for nonlinear regression and nonlinear adaptive filtering
applications. We call this solution the Functional Wiener Filter (FWF).
Conceptually, the methodology is an extension of Parzen's work on the
autocorrelation RKHS to nonlinear functional spaces. We provide an extended
functional Wiener equation, and present a solution to this equation in an
explicit, finite dimensional, data-dependent RKHS. We further explain the
necessary requirements to compute the analytical solution in RKHS, which is
beyond traditional methodologies based on the kernel trick. The FWF analytic
solution to the nonlinear minimum mean square error problem has better accuracy
than other kernel-based algorithms in synthetic, stationary data. In real world
time series, it has comparable accuracy to KAF but displays constant complexity
with respect to number of training samples. For evaluation, it is as
computationally efficient as the Wiener solution (with a larger number of
dimensions than the linear case). We also show how the difference equation
learned by the FWF from data can be extracted leading to system identification
applications, which extend the possible applications of the FWF beyond optimal
nonlinear filtering.
