Sammendrag
Spatio-temporal statistical models are increasingly being used 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 dynamic models. The chief challenge with the specification of such dynamical models has been related to the curse of dimensionality and the specification of realistic dependence. 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. In addition, this framework has allowed for the specification of science-based 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. Most of the focus for the application of such models in statistics has been in the linear case. The problems mentioned above with linear dynamic models are compounded in the case of nonlinear models, yet these are the processes that govern environmental science. In this sense, the need for coherent and sensible model parameterizations is not only helpful, it is essential. Here, we present some recent results for accommodating realistic “science-based” nonlinear structure as well as some discussion of model selection and approximation.