A Geostatistical Framework for Categorical Spatial Data Modeling

This dissertation presents a general geostatistical framework for modeling categorical spatial data, an all important information source in many scientific fields. Due to the non-linear and non-Gaussian characteristics of categorical variables and complex spatial patterns in categorical fields, statistical modeling of such data has long been considered as one of the most fundamental and challenging problems in both geostatistics and geography. In the proposed framework, transiogram models, a recently proposed set of spatial transition probabilities diagrams, are used as spatial continuity measures. The properties of transiograms, such as their connections with compactness measures of shape, and their eligibility as models for indicator random fields are investigated herein. A non-parametric regression method is also proposed for efficient transiogram modeling. More importantly, the class occurrence probability (multi-point) for (target) locations with unknown class labels given observed class labels at sample (source) locations is then decomposed into a weighted combination of two-point spatial interactions in two different approaches, while accounting for complex spatial interdependencies. In the first approach, two-point spatial interactions are measured directly by transiograms, and the sought-after multi-point class occurrence probability is approximated based on a general paradigm ( Tau model ) for integrating knowledge from interdependent diverse information sources while accounting for information redundancy between such sources. In the second approach, geostatistical modeling of categorical spatial data is set in the framework of generalized linear mixed models (GLMMs), where intermediate, latent (unobservable) spatially correlated Gaussian variables (random effects) are assumed for the observable non-Gaussian responses to account for spatial correlation. Instead of using Markov Chain Monte Carlo sampling to infer the assumed latent variables, an approach which is computationally expensive and associated with convergence issues, an ad-hoc method is proposed in this dissertation to approximate the analytically intractable posterior probability of the latent variables. The connections of these two proposed models with other methods, such as indicator variants of the kriging family (indicator kriging and indicator cokriging), spatial Markov Chain model and Bayesian Maximum Entropy are discussed in detail. The advantages of the new proposed framework are analyzed and highlighted through real and synthetic cases studies involving the generation of spatial patterns via sequential indicator simulation and interpolation or estimation of categorical spatial data.