ABSTRACT
Inthischapter, we are concerned with discrete-valued spatio-temporal processes. To facilitate presentation, we will restrict our attention to such processes in discrete time, yet allow space to be continuous or discrete in principle. In spatio-temporal statistics, it is common to consider such models from a generalized linear mixed-model perspective (e.g., see Cressie and Wikle, 2011 and Holan and Wikle [2015; Chapter 15 in this volume]). This is a “top-down”approach where by the spatio-temporalproperties of the system are modeled in terms of a latent Gaussian spatio-temporal dynamical process (e.g., Wikle, 2002). Alternatively, one may consider such processes as Markov random fields (MRFs) using one of the classes of spatio-temporal “auto” models (e.g., spatio-temporal auto-logistic) as described in Zhu and Zheng (2015; Chapter 17 in this volume). The MRF approach is a local specification where relationships between neighbors are specified conditionally in a way to guarantee a valid joint distribution (see Section 16.2). In this chapter, we discuss an alternative “bottom-up” modeling strategy for discrete-valued spatio-temporal dynamical processes, which is agent based. Such agent-based models (ABMs) are prevalent in epidemiology and social sciences (e.g., Filatova et al., 2013; Gilbert, 2008; Keeling and Rohani, 2008; Sattenspiel, 2009) and also are called individual-based models in the ecological sciences (e.g., Grimm and Railsback, 2005), multi-agent models in engineering (e.g., Olfati-Saber, 2006), and cellular automata in the physical and mathematical sciences (e.g., Wolfram, 1984). All of these paradigms are characterized by autonomous agents (or individuals) that take on one of a discrete number 350of states (hence, discrete valued) that vary with time and space depending on a set of deterministic or probabilistic “rules.” As discussed later, from a stochastic perspective, ABMs can sometimes be linked to MRF-based models through Markov network properties.
