Monte Carlo Likelihood Estimation for Conditional Autoregressive Models with Application to Sparse Spatiotemporal Data

Spatiotemporal modeling is increasingly used in a diverse array of fields, such as ecology, epidemiology, health care research, transportation, economics, and other areas where data arise from a spatiotemporal process. Spatiotemporal models describe the relationship between observations collected from different spatiotemporal sites. The modeling of spatiotemporal interactions arising from spatiotemporal data is done by incorporating the space-time dependence into the covariance structure. A main goal of spatiotemporal modeling is the estimation and prediction of the underlying process that generates the observations under study and the parameters that govern the process. Furthermore, analysis of the spatiotemporal correlation of variables can be used for estimating values at sites where no measurements exist. In this work, we develop a framework for estimating quantities that are functions of complete spatiotemporal data when the spatiotemporal data is incomplete. We present two classes of conditional autoregressive (CAR) models (the homogeneous CAR (HCAR) model and the weighted CAR (WCAR) model) for the analysis of sparse spatiotemporal data (the log of monthly mean zooplankton biomass) collected on a spatiotemporal lattice by the California Cooperative Oceanic Fisheries Investigations (CalCOFI). These models allow for spatiotemporal dependencies between nearest neighbor sites on the spatiotemporal lattice. Typically, CAR model likelihood inference is quite complicated because of the intractability of the CAR model's normalizing constant. Sparse spatiotemporal data further complicates likelihood inference. We implement Monte Carlo likelihood (MCL) estimation methods for parameter estimation of our HCAR and WCAR models. Monte Carlo likelihood estimation provides an approximation for intractable likelihood functions. We demonstrate our framework by giving estimates for several different quantities that are functions of the complete CalCOFI time series data.