A reduced rank model for spatial categorical data with many classes

TL;DR

This paper introduces a reduced-rank spatial multinomial model for categorical data with many classes, enabling scalable inference and joint predictions by representing class-specific effects through shared latent factors.

Contribution

It proposes a novel identifiable reduced-rank model that reduces parameter complexity and develops a Gibbs sampler with Laplace-approximation for efficient inference.

Findings

Effective dimension selection demonstrated in simulations

Laplace-approximation proposals improve MCMC efficiency

Application shows scalable inference for complex spatial data

Abstract

We develop an identifiable reduced-rank spatial multinomial model for categorical data with many classes. The model represents class-specific spatial effects through a low-dimensional set of shared latent factors, substantially reducing parameter dimension while preserving joint dependence across classes. Because standard conjugate and P\'olya-Gamma methods fail under this factorization, we propose a Gibbs sampler using Laplace-approximation proposals within Metropolis-Hastings updates. Simulation studies examine dimension selection and the accuracy of the Laplace proposals. An application to dominant tree species mapping in the Blue Ridge Mountains demonstrates scalable inference and flexible joint predictions for individual classes, class unions, and area-level summaries.