A canonical correlations approach to multiscale stochastic realization

We develop a realization theory for a class of multiscale stochastic proces ses 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 rando m process, our methods provide a systematic way to realize the given correl ation as the finest scale of a multiscale process. Motivated by Akaike's us e of canonical correlation analysis to develop both exact and reduced-order model for time-series, we too harness this tool to develop multiscale mode ls. We apply our realization scheme to build reduced-order multiscale model s for two applications, namely linear least-squares estimation and generati on of random-field sample paths. For the numerical examples considered, lea st-squares estimates are obtained having nearly optimal mean-square errors, even with multiscale models of low order. Although both field estimates an d field sample paths exhibit a visually distracting blockiness, this blocki ness is not an important issue in many applications. For such applications, our approach to multiscale stochastic realization holds promise as a valua ble, general tool.