A canonical correlations approach to multiscale stochastic realization

We develop a realization theory for a class of multiscale stochastic processes having white-noise driven, scale-recursive dynamics that are indexed by the nodes of a tree. Given the correlation structure of a 1-D or 2-D random process, our methods provide a systematic way to realize the given correlation as the finest scale of a multiscale process. Motivated by Akaike's use of canonical correlation analysis to develop both exact and reduced-order models for time-series, we too harness this tool to develop multiscale models. We apply our realization scheme to build reduced-order multiscale models for two applications, namely linear least-squares estimation and generation of random-field sample paths. For the numerical examples considered, least-squares estimates are obtained having nearly optimal mean-square errors, even with multiscale models of low order. Although both field estimates and field sample paths exhibit a visually distracting blockiness, this blockiness is not an important issue in many applications. For such applications, our approach to multiscale stochastic realization holds promise as a valuable, general tool.