Explicit recursivity into reproducing kernel Hilbert spaces

This paper presents a methodology to develop recursive filters in reproducing kernel Hilbert spaces (RKHS). Unlike previous approaches that exploit the kernel trick on filtered and then mapped samples, we explicitly define model recursivity in the Hilbert space. The method exploits some properties of functional analysis and recursive computation of dot products without the need of pre-imaging. We illustrate the feasibility of the methodology in the particular case of the gamma filter, an infinite impulse response (IIR) filter with controlled stability and memory depth. Different algorithmic formulations emerge from the signal model. Experiments in chaotic and electroencephalographic time series prediction scenarios demonstrate the potentiality of the approach.