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.