Explicit Recursive and Adaptive Filtering in Reproducing Kernel Hilbert Spaces

This brief presents a methodology to develop recursive filters in reproducing kernel Hilbert spaces. Unlike previous approaches that exploit the kernel trick on filtered and then mapped samples, we explicitly define the model recursivity in the Hilbert space. For that, we exploit some properties of functional analysis and recursive computation of dot products without the need of preimaging or a training dataset. We illustrate the feasibility of the methodology in the particular case of the γ-filter, which is an infinite impulse response filter with controlled stability and memory depth. Different algorithmic formulations emerge from the signal model. Experiments in chaotic and electroencephalographic time series prediction, complex nonlinear system identification, and adaptive antenna array processing demonstrate the potential of the approach for scenarios where recursivity and nonlinearity have to be readily combined.

I. Introduction

Recursive estimation algorithms play a crucial role for many problems in adaptive control, adaptive signal processing, system identification, and general model building and monitoring problems [1]. Efficient implementation of such algorithms has been paid a great attention in the signal processing literature. Linear autoregressive models require relatively few parameters and allow closed-form analysis, while ladder or lattice implementation of linear filters has long been studied in signal theory. However, when the system generating the data is driven by nonlinear dynamics, the model specification and parameter estimation problems increase their complexity.