Internal multiscale autoregressive processes, stochastic realization, and covariance extension

Abstract

The focus of this thesis is on the identification of multiscale autoregressive (MAR) models for stochastic processes from second-order statistical characterizations. The class of MAR processes constitutes a rich and powerful stochastic modeling framework that admits efficient statistical inference algorithms. To harness the utility of MAR processes requires that the phenomena of interest be effectively modeled in the framework. This thesis addresses this challenge and develops MAR model identification theory and algorithms that overcome some of the limitations of previous approaches (e.g., model inconsistency and computational complexity) and that extend the breadth of applicability of the framework. One contribution of this thesis is the resolution of the problem of model inconsistency. This is achieved through a new parameterization of so-called internal MAR processes. This new parameterization admits a computationally efficient, scale-recursive approach to model realization. The efficiency of this approach stems from both its scale-recursive structure and from a novel application of the estimation-theoretic concept of predictive efficiency. Another contribution of this thesis is to provide a unification of the MAR and wavelet frameworks. This unification leads to wavelet-based stochastic models that are fundamentally different from conventional ones. A limitation of previous MAR model identification approaches is that they require a complete second-order characterization of the process to be modeled. Relaxing this assumption leads to the problem of covariance extension in which unknown covariance elements are inferred from known ones. This thesis makes two contributions in this area. First, the classical covariance extension algorithm (Levinson's algorithm) is generalized to address a wider range of extension problems. Second, this algorithm is applied to the problem of designing a MAR model from a partially known covariance matrix. The final contribution of this thesis is the development of techniques for incorporating nonlocal variables (e.g., multiresolution measurements) into a MAR model. These techniques are more powerful than those previously developed and lead to computational efficiencies in model realization and statistical inference.