Introduction

Spatio-temporal modeling

Spatio-temporal statistical models are being used increasingly across a wide variety of scientific disciplines to describe and predict spatially explicit processes that evolve over time. Correspondingly, in recent years there has been a significant amount of research on new statistical methodology for such models. Although descriptive models that approach the problem from the second-order (covariance) perspective are important, and innovative work is being done in this regard, many real-world processes are dynamic, and it can be more efficient in some cases to characterize the associated spatio-temporal dependence by the use of dynamical models. The chief challenge with the specification of such dynamical models has been related to the curse of dimensionality. Even in fairly simple linear, first-order Markovian, Gaussian error settings, statistical models are often over parameterized. Hierarchical models have proven invaluable in their ability to deal to some extent with this issue by allowing dependency among groups of parameters (Cressie et al. 2009). In addition, this framework has allowed for the specification of science-based (i.e. based on knowledge and hypotheses about the ecological system) parameterizations (and associated prior distributions) in which classes of deterministic dynamical models [e.g. partial differential equations (PDEs), integro-difference equations (IDEs), matrix models, and agent-based models] are used to guide specific parameterizations (Wikle and Hooten 2010).

Therefore, two of the main questions to ask in early stages of building a statistical model for a spatio-temporal ecological process are:

• Does enough a priori scientific information exist about the process to specify an intelligent mechanistic model that can mimic it?

• If the answer to the above question is “yes”, then do sufficient data exist (or can they be collected) to fit such a model?

If the answer to either of these questions is “no”, then perhaps a more naïve statistical model is warranted – one that is still spatially and temporally explicit, but that is sufficiently parsimonious to enable statistical learning. In that situation, because the actual model structure is limited, it is critical to consider potential lurking sources of latent autocorrelation, both spatially and temporally.