Spatial modeling using graphical Markov models and wavelets

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1997
Authors
Huang, Hsin-Cheng
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Noel Cressie
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Altmetrics
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Abstract

Graphical Markov models use graphs to represent possible dependencies among random variables. This class of models is extremely rich and includes inter alia causal Markov models and Markov random fields. In this dissertation, we develop a very efficient optimal-prediction algorithm for graphical Markov models. The algorithm is a generalization of the Kalman-filter algorithm for temporal processes, and it can in principle be applied to any Gaussian undirected graphical model and any Gaussian acyclic directed graphical model;We also propose a new class of multiscale models for stochastic processes in terms of scale-recursive dynamics defined on acyclic directed graphs. The models are an extension of multiscale tree-structured models. The optimal prediction can be obtained using the newly developed generalized Kalman-filter algorithm referred to above, and the parameters can be estimated by maximum likelihood via the EM algorithm. A subclass of these models are multiscale wavelet models, for which we show that the optimal predictors of hidden state variables can be obtained by a level-dependent (scale-dependent) wavelet shrinkage rule;In a series of papers, D. Donoho and I. Johnstone develop wavelet shrinkage methods to solve statistical problems. We propose a new rationale for wavelet shrinkage, based on the assumption that the underlying process can be decomposed into a large-scale deterministic trend plus a small-scale Gaussian process. Our approach has several advantages over current shrinkage methods. It takes the dependencies of empirical wavelet coefficients, both within scales and across scales, into account. Moreover, it does not rely on asymptotic properties for its justification so that it is also appropriate when the sample size is small;Finally, we introduce partially ordered Markov models, which are acyclic directed graphical models for spatial problems. The model can be regarded as a Markov random field with neighborhood structures derivable from an associated partially ordered set. We use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for partially ordered Markov models. We prove that the maximum (composite) likelihood estimators are consistent, asymptotically normal, and also asymptotically efficient under checkable conditions.

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Wed Jan 01 00:00:00 UTC 1997