Year of Award


Document Type


Degree Type

Doctor of Philosophy (PhD)

Degree Name


Department or School/College

Department of Mathematical Sciences

Committee Chair

Jon M. Graham

Commitee Members

John Bardsley, Solomon Harrar, Dave Patterson, Jesse Johnson


Autologistic, Block Generalized Pseudolikelihood, Generalized Pseudolikelihood, Markov Chain Monte Carlo Maximum Likelihood, Pseudolikelihood, Strong Consistency


University of Montana


A regular lattice of spatially dependent binary observations is often modeled using the autologistic model. It is well known that likelihood-based inference methods cannot be employed in the usual way to estimate the parameters of the autologistic model due to the intractability of the normalizing constant for the corresponding joint likelihood. Two popular and vastly contrasting approaches to parameter estimation for the autologistic model are maximum pseudolikelihood (PL) and Markov Chain Monte Carlo Maximum Likelihood (MCMCML). Two newer and less understood approaches are maximum generalized pseudolikelihood (GPL) and maximum block generalized pseudolikelihood (BGPL). Both of these newer methods represent varying degrees of compromise between maximum pseudolikelihood and MCMCML. The research performed in this dissertation focuses on these four estimation methods, with particular emphasis given to GPL and BGPL, and incorporates theoretical, simulation-based, and application-based components. The theoretical components of this dissertation are three-fold. First, when employing GPL or BGPL, the need to distinguish between types of neighbors within the neighborhood set of a lattice site is formally developed. Such a distinction ultimately affects the functional forms of the generalized pseudolikelihood and block generalized pseudolikelihood functions. Second, extensions of generalized and block generalized pseudolikelihood for use in the space-time domain are proposed. As GPL and BGPL were initially only developed for use in the spatial domain, these extensions are the first of their kind. Third, and finally, the basic asymptotic property of strong consistency is established for the estimates obtained via maximum generalized and maximum block generalized pseudolikelihood.

In addition to the aforementioned theoretical components, this dissertation also includes two simulation studies. In particular, a large scale purely spatial simulation study using the autologistic model was conducted comparing the performances of PL, MCMCML, GPL, and BGPL. This was the first such study to simultaneously compare GPL and BGPL, and it was also the first such study to simultaneously incorporate a covariate, spatial anisotropy, and higher-order neighborhood systems. The results of this study indicate that GPL tends to outperform BGPL, and that both of these newer methods tend to achieve their intended performance-based compromise between PL and MCMCML, particularly in situations where estimation is notoriously difficult. Additionally, a small-scale space-time simulation study using the spatio-temporal autologistic model was conducted comparing the performances of PL, MCMCML, and the proposed space-time extensions of GPL and BGPL. Such a space-time simulation study has never before been conducted. The results of this study suggest that the proposed extensions of GPL and BGPL are indeed appropriate, and that the relative performances of the four estimation methods in the space-time domain are largely analogous to those from the purely spatial domain.

The final component of this dissertation is application-based. More specifically, fire occurrence data from Oregon and Washington state are modeled, while accounting for the Departure from Average fire potential metric as a covariate, using the spatio-temporal autologistic model. All four estimation methods (PL, MCMCML, GPL, and BGPL) are employed to estimate the parameters of several proposed spatio-temporal autologistic models, and a Monte Carlo-based sum of absolute error (SAE) statistic is used as a model selection criterion.



© Copyright 2012 Jordan Earl Purdy