Statistical spatial series modelling

A random spatial series is a collection of random variables F(xi , · · ·, xn ) depending on several spatial coordinates (x 1, · · ·, xn ). An attempt is made to construct a statistical second-order theory of such series when (x 1, · · ·, xn ) varies over a regular cartesian lattice. Using the properties of the linear (Hilbert) space associated with the series, the concepts of innovation and purely non-deterministic (p.n.d.) series are introduced. For a p.n.d. series F(x 1, · · ·, xn ) a unilateral representation is obtained in terms of a white innovations series Z(y 1, · · ·, yn ) where The representatation is specialized to the homogeneous case and we discuss spectral conditions for p.n.d. The familiar time-series condition ∫ log f(λ) dλ > –∞ on the spectral density f is necessary but not sufficient. A sufficient condition is stated. Motivated by the p.n.d. unilateral representation results we define unilateral arma (autoregressive-moving average) spatial series models. Stability and invertibility conditions are formulated in terms of the location of zero sets of polynomials relative to the unit polydisc in Cn , and a rigorous shift operator formalism is established. For autoregressive spatial series a Yule–Walker-type matrix equation is formulated and it is shown how this can be used to obtain estimates of the autoregressive parameters. It is demonstrated that under mild conditions the estimates are consistent and asymptotically normal.

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