A theory for multiscale stochastic realization
Describes a generalisation theory for a class of multiscale stochastic processes. This realisation problem represents a non-trivial variation on the standard time series stochastic realisation problem. The authors generalize some Gaussian Markov random fields representation results, thereby developing a fuller understanding of what types of stochastic phenomena can be usefully captured by the multiscale framework and how the corresponding models can be built. The authors' theory for multiscale stochastic realization is based on a novel application of the techniques of canonical correlation analysis. These techniques, which rely heavily on the singular value decomposition, constitute a standard analytical tool in both multivariate statistics and in the field of reduced-order modeling of linear systems. Using canonical correlation theory, the authors demonstrate that any given l-D or 2-D Gaussian random process, Markov or otherwise, can be represented by some multiscale model. More importantly, the authors also introduce a family of approximate representations.<<ETX>>