Nonparametric independent process analysis

Linear dynamical systems are widely used tools to model stochastic time processes, but they have severe limitations; they assume linear dynamics with Gaussian driving noise. Independent component analysis (ICA) aims to weaken these limitations by allowing independent, non-Gaussian sources in the model. Independent subspace analysis (ISA), an important generalization of ICA, has proven to be successful in many source separation applications. Still, the general ISA problem of separating sources with nonparametric dynamics has been hardly touched in the literature yet. The goal of this paper is to extend ISA to the case of (i) nonparametric, asymptotically stationary source dynamics and (ii) unknown source component dimensions. We make use of functional autoregressive (fAR) processes to model the temporal evolution of the hidden sources. An extension of the well-known ISA separation principle is derived for the solution of the introduced fAR independent process analysis (fAR-IPA) task. By applying fAR identification we reduce the problem to ISA. The Nadaraya-Watson kernel regression technique is adapted to obtain strongly consistent fAR estimation. We illustrate the efficiency of the fAR-IPA approach by numerical examples and demonstrate that in this framework our method is superior to standard linear dynamical system based estimators.