Computationally efficient stochastic realization for internal multiscale autoregressive models

In this paper we develop a stochastic realization theory for multiscale aut oregressive (MAR) processes that leads to computationally efficient realiza tion algorithms. The utility of MAR processes has been limited by the fact that the previously known general purpose realization algorithm, based on c anonical correlations, leads to model inconsistencies and has complexity qu artic in problem size. Our realization theory and algorithms addresses thes e issues by focusing on the estimation-theoretic concept of predictive effi ciency and by exploiting the scale-recursive structure of so-called interna l MAR processes. Our realization algorithm has complexity quadratic in prob lem size and with an approximation we also obtain an algorithm that has com plexity linear in problem size.