Spectral Analysis of Qualitative Time Series
Christopher K. Wikle, Mevin B. Hooten · 2016
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 350 of 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.