Statistics for Spatio-Temporal Data (Wiley Series in Probability and Statistics) by Cressie Noel & Wikle Christopher K

Author:Cressie, Noel & Wikle, Christopher K. [Cressie, Noel]
Language: eng
Format: epub
Publisher: John Wiley and Sons
Published: 2011-06-16T21:00:00+00:00

Physical Statistical Models

When there is scientific knowledge about a process Y, it can often be described in terms of a partial differential equation (PDE), sometimes in time, sometimes in space, and sometimes in both. Heine (1955) considers such easier in two dimensions—in particular, the general second-order, linear, nonstochastic PDE:

He shows that by simple changes of variables, this PDE reduces to several special cases.

where α, β and γ are generic, nonnegative parameters.

A nonstochastic (or deterministic) PDE can be converted into a stochastic PDE by replacing 0 on the PDE’s right-hand side with a (mean-zero) stochastic term. In a remarkable early contribution to the Statistics literature, Hotelling (1927) investigated a number of one-dimensional (in time) stochastic PDEs. Heine (1955) did the same thing in two dimensions, based on the equations given above; here, we write the stochastic term in the PDE as δ(u,v), where δ(·, ·) is a mean-zero, two-dimensional stochastic process that models either (a) random impulses from smaller-order contributions or (b) uncertainty in the scientific relationship expressed through the PDE. The resulting equation is an example of a physical statistical model (Berliner, 2003). Clearly, the process Y(·, ·) is now a stochastic process, and its moments are of interest. Because E(δ(u, v)) ≡ 0, linearity of the PDE implies that E(Y(u, v)) ≡ 0. What is cov(Y(u,v),Y(p,q))?

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