Double Generalized Linear Models for spatio-temporal data

Abstract

Spatio-temporal data arises when a variable is measured at multiple locations over time. Modeling such data poses several challenges: In addition to spatial factors, such as the position of sensors or regions, and temporal trends, the data may also be subject to autoregressive dynamics, as is the case with infectious diseases. Furthermore, additional data characteristics, such as integer values or non-negativity, must be taken into account in the modeling process. This dissertation contributes to the state of research on the modeling of spatio-temporal data on a fixed spatial grid, with a particular focus on non-normal data. We adapt multivariate count data time series models for the spatio-temporal domain. Assuming marginal Poisson distributions for observations conditional on the past, the models describe the conditional expected value by past observations of the same and neighboring locations, a latent feedback process, and optional covariates. Building on this foundation, we investigate an approach to detect persistent level shifts within this framework. Such level shifts may be caused by political measures, campaigns, or other extraordinary events. Methods from univariate intervention analysis in count time series models are further developed to test for level shifts at unknown times or unknown locations. We propose an iterative procedure to detect multiple intervention effects simultaneously. Subsequently, the modeling framework is extended by using the idea of double generalized linear models (DGLMs), relaxing the assumption of a marginal Poisson distribution for the conditional observations in favor of arbitrary distributions from the exponential dispersion family. In addition to the conditional expectation, this allows the dispersion parameter to be described by a spatio-temporal model, thereby accommodating overdispersion and other data characteristics. The full framework is implemented in the R package glmSTARMA. The theoretical properties and usefulness of the models are studied in simulations. We illustrate the application using data examples.