arXiv subjects

J. Brandon Carter

Publications and source records attributed to J. Brandon Carter.

4 recordsLinked to original sources

arXiv2026

Markov Random Fields: Structural Properties, Phase Transition, and Response Function Analysis

This paper presents a focused review of Markov random fields (MRFs)--commonly used probabilistic representations of spatial dependence in discrete spatial domains--for categorical data, with an emphasis on models for binary-valued observations or latent variables. We examine core structural properties of these models, including clique factorization, conditional independence, and the role of neighborhood structures. We also discuss the phenomenon of phase transition and its implications for statistical model specification and inference. A central contribution of this review is the use of response functions, a unifying tool we introduce for prior analysis that provides insight into how different formulations of MRFs influence implied marginal and joint distributions. We illustrate these concepts through a case study of direct-data MRF models with covariates, highlighting how different formulations encode dependence. While our focus is on binary fields, the principles outlined here extend naturally to more complex categorical MRFs and we draw connections to these higher-dimensional modeling scenarios. This review provides both theoretical grounding and practical tools for interpreting and extending MRF-based models.

stat.ME↗

arXiv2024

Mixture of Directed Graphical Models for Discrete Spatial Random Fields

Current approaches for modeling discrete-valued outcomes associated with spatially-dependent areal units incur computational and theoretical challenges, especially in the Bayesian setting when full posterior inference is desired. As an alternative, we propose a novel statistical modeling framework for this data setting, namely a mixture of directed graphical models (MDGMs). The components of the mixture, directed graphical models, can be represented by directed acyclic graphs (DAGs) and are computationally quick to evaluate. The DAGs representing the mixture components are selected to correspond to an undirected graphical representation of an assumed spatial contiguity/dependence structure of the areal units, which underlies the specification of traditional modeling approaches for discrete spatial processes such as Markov random fields (MRFs). Notably, the MDGM is not proposed as an approximation to an MRF, but rather as an alternative that provides valid posterior inference while being computationally faster than exact MRF inference and more principled than the pseudo-likelihood approximation (aMRF) commonly used in practice. We introduce the concept of compatibility to show how an undirected graph can be used as a template for the dependencies between areal units to create sets of DAGs which, as a collection, preserve the dependencies represented in the template undirected graph. Lastly, we compare highlighted classes of MDGMs to MRFs and a popular Bayesian MRF model approximation used in high-dimensional settings in a series of simulations and an analysis of ecometrics data collected as part of the Adolescent Health and Development in Context Study.

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arXiv2023

Land-Use Filtering for Nonstationary Prediction of Collective Efficacy in an Urban Environment

Collective efficacy -- the capacity of communities to exert social control toward the realization of their shared goals -- is a foundational concept in the urban sociology and neighborhood effects literature. Traditionally, empirical studies of collective efficacy use large sample surveys to estimate collective efficacy of different neighborhoods within an urban setting. Such studies have demonstrated an association between collective efficacy and local variation in community violence, educational achievement, and health. Unlike traditional collective efficacy measurement strategies, the Adolescent Health and Development in Context (AHDC) Study implemented a new approach, obtaining spatially-referenced, place-based ratings of collective efficacy from a representative sample of individuals residing in Columbus, OH. In this paper, we introduce a novel nonstationary spatial model for interpolation of the AHDC collective efficacy ratings across the study area which leverages administrative data on land use. Our constructive model specification strategy involves dimension expansion of a latent spatial process and the use of a filter defined by the land-use partition of the study region to connect the latent multivariate spatial process to the observed ordinal ratings of collective efficacy. Careful consideration is given to the issues of parameter identifiability, computational efficiency of an MCMC algorithm for model fitting, and fine-scale spatial prediction of collective efficacy.

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arXiv2021

Cluster Analysis via Random Partition Distributions

Hierarchical and k-medoids clustering are deterministic clustering algorithms based on pairwise distances. Using these same pairwise distances, we propose a novel stochastic clustering method based on random partition distributions. We call our method CaviarPD, for cluster analysis via random partition distributions. CaviarPD first samples clusterings from a random partition distribution and then finds the best cluster estimate based on these samples using algorithms to minimize an expected loss. We compare CaviarPD with hierarchical and k-medoids clustering through eight case studies. Cluster estimates based on our method are competitive with those of hierarchical and k-medoids clustering. They also do not require the subjective choice of the linkage method necessary for hierarchical clustering. Furthermore, our distribution-based procedure provides an intuitive graphical representation to assess clustering uncertainty.

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