A Canonical Correlations Approach to Multiscale Stochastic Realization

Abstract

Abstract-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 reducedorder model 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.

Multiply-rooted multiscale models for large-scale estimation

“…To be sure, a variety of state reductions have been proposed for Markov-like random fields in the multiscale setting 1) allowing neighboring regions to overlap [19] to reduce the effect of artifacts caused by residual state-to-state correlations; 2) subsampling the pixels along the state boundary [23], as illustrated in Fig. 3; 3) taking averages or wavelet transforms of the boundary pixels [21]; 4) determining from the statistics of the boundary pixels the optimum linear functionals [18], [19] which maximize the decorrelation. However, none of these methods change the asymptotic behavior of the computational complexity, since the state dimension at the root of the decomposition is, in each case, .…”

“…• existence of efficient estimation [7] and likelihood [22] algorithms; • existing base of multiscale models to represent the statistics of the process being modeled; • a stochastic realization theory [19]; • ability to represent both local and nonlocal measurements; • desirable asymptotic properties, based on the proposed model of this paper. We need to define models; to make the model structure as regular as possible, we will select some scale and develop models, such that model represented on tree estimates the region associated with the th multiscale state on scale .…”

“…Each model is uniquely defined by the statistics of the process and the nature of the state [19], [23]. We propose to define the state at tree node as a subsampling of the union of the boundaries of the children of node .…”

“…These are the best linear estimates based on all of the measurements on the whole tree, but computed locally as (19) For model on tree , we are interested only in the estimates within ; that is, the set of estimates from which we wish to mosaic. Therefore the downwards pass is relevant only on and on , the ancestors of .…”

“…An approximate model can affect the reliability of the estimates and estimation error statistics; however in many circumstances a greater con- cern revolves around the introduction of estimation artifacts at the hierarchical boundaries [18], [19], [22], due to an inadequate decorrelation in (1). For example, the two pixels and in Fig.…”