# Geostatistical Modeling of Positive Definite Matrices: An Application to Diffusion Tensor Imaging

## TL;DR

This paper introduces a novel spatial Wishart process for geostatistical modeling of positive definite matrices, specifically applied to diffusion tensor imaging (DTI), enabling valid spatial dependence modeling and improved inference in neuroimaging.

## Contribution

It extends geostatistical modeling to positive definite matrices using the spatial Wishart process and develops approximation methods for practical inference in DTI analysis.

## Key findings

- The proposed model accurately captures spatial dependence in DTI data.
- Approximation methods enable feasible computation despite the lack of closed-form density.
- Simulation and real data show improved inference and performance.

## Abstract

Geostatistical modeling for continuous point-referenced data has been extensively applied to neuroimaging because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging characterizing the brain structure produces a positive definite (p.d.) matrix for each voxel. Current geostatistical modeling has not been extended to p.d. matrices because introducing spatial dependence among positive definite matrices properly is challenging. In this paper, we use the spatial Wishart process, a spatial stochastic process (random field) where each p.d. matrix-variate marginally follows a Wishart distribution, and spatial dependence between random matrices is induced by latent Gaussian processes. This process is valid on an uncountable collection of spatial locations and is almost surely continuous, leading to a reasonable means of modeling spatial dependence. Motivated by a DTI dataset of cocaine users, we propose a spatial matrix-variate regression model based on the spatial Wishart process. A problematic issue is that the spatial Wishart process has no closed-form density function. Hence, we propose approximation methods to obtain a feasible working model. A local likelihood approximation method is also applied to achieve fast computation. The simulation studies and real data analysis demonstrate that the working model produces reliable inference and improved performance compared to other methods.