Third Year Candidacy Presentation: Fast Direct H-likelihood Estimation Using Selected Inverse for Irregular Spatial Data

Spatial models with sparse precision matrices are widely used across various statistical domains. However, irregular spatial structures pose challenges for traditional REML estimation and statistical inference due to methodological constraints and computational inefficiencies. To address these issues, we introduce a fast direct h-likelihood computation for irregular spatial data. In the linear mixed model setup, the h-likelihood method leverages two interconnected generalized linear models: a Gaussian regression to estimate fixed effects and predict Gaussian spatial effects, and a gamma regression to estimate precision parameters. This approach eliminates the need to integrate out random effects, as required in REML. However, computing key h-likelihood components, such as the diagonal of the hat matrix in Gaussian regression, has long been a major implementation challenge. To overcome this, we employ sparse Cholesky factorization and selected inverse algorithms to compute these diagonals efficiently, circumventing costly full matrix inversions. We further discuss how this approach enhances model diagnostics, inference, and prediction. We apply the h-likelihood framework to analyze micronutrient concentrations in crops from Ethiopia and Malawi, where irregular sample domains and sampling frames necessitate spatial linear mixed-effects models based on irregular Gaussian Markov random fields. Through this application, we demonstrate how the h-likelihood framework produces efficient estimates, quantifies uncertainties, and supports critical inferential tasks.

Advisor: Debashis Mondal