Hierarchical General Quadratic Nonlinear Models for Spatio-Temporal Dynamics

Christopher K. Wikle
Department of Statistics
University of Missouri

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. Although descriptive models that approach this problem from the second-order (covariance) perspective are important, many real-world processes are dynamic, and it can be more efficient in such cases to characterize the associated spatio-temporal dependence by the use of dynamic models. The challenge with the specification of such dynamical models has been related to the curse of dimensionality and the specification of realistic dependence structures. Even in fairly simple linear/Gaussian settings, spatio-temporal 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 and science-based parameterizations.  The problems associated with linear dynamic models are compounded in the case of nonlinear models, yet these are the processes that govern environmental and physical science.  Here, we present some recent results for accommodating realistic parametric nonlinear structure in hierarchical spatio-temporal models in terms of a class of general quadratic models. We also discuss dimension reduction of the parameter space, the state space, as well as computationally efficient strategies to deal with the curse of dimensionality in these models.  Illustrative examples from the environmental and ecological sciences will be included.