Complex deterministic and stochastic models are often used to
describe dynamic systems in climate science, ecology and biology. Inferring
unknown parameters of these models is of interest, both for understanding the
underlying scientific processes as well as for making valid predictions. Some
of the challenges typically involved in inference for these models are:
likelihood functions that are intractable or only implicitly specified by a
computer model; computationally expensive model simulations; and high-
dimensional, multivariate observations and model output.

I will outline computationally expedient Gaussian process-based inferential
approaches in the context of two very different models, a deterministic Earth-
system model used in climate science, and a stochastic spatial model for
infectious diseases. I will point out some of the common features between the
two, but also highlight significant differences in the modeling frameworks
and inferential goals.

This talk is based on joint work with K. Sham Bhat (Los Alamos National
Labs), Roman Jandarov (Dept. of Statistics, Penn State University [PSU]),
Roman Tonkonojenkov (Dept. of Geosciences, PSU), Klaus Keller (Dept. of
Geosciences, PSU), Ottar Bjornstad (Center for Infectious Disease
Dynamics, PSU), and Bryan Grenfell (Ecology and Evolutionary Biology, Princeton University)

If atmospheric, agricultural, and other environmental systems share one underlying theme it is complex spatial structures, being influenced by such features as topography and weather. For example, the air quality characteristics of cities are likely to be more similar than that of rural areas irrespective of their geographic proximity. Ideally we might model these effects directly; however, information on the underlying causes is often not routinely available. Hence, when modeling environmental systems there exists a need for a class of models that are more complex than those which rely on the assumption of stationarity.

In this talk, we propose a novel approach to modeling nonstationary spatial fields. The proposed method works by expanding the geographic plane over which these processes evolve into higher dimensional spaces, transforming and clarifying complex patterns in the physical plane. By combining aspects of multi-dimensional scaling, group lasso, and latent variables models, a dimensionally sparse projection is found in which the originally nonstationary field exhibits stationarity. Following a comparison with existing methods in a simulated environment, dimension expansion is studied on a classic test-bed data set historically used to study nonstationary models. Following this, we explore the use of dimension expansion in modeling air pollution in the United Kingdom, a process known to be strongly influenced by rural/urban effects, amongst others, which gives rise to a nonstationary field.