[This blog entry refers to Continuous auto-regressive moving average random fields on ℝn by Peter J. Brockwell and Yasumasa Matsuda]
We define a new family of random fields on by extending the kernel representation of a class of non-causal continuous time auto-regressive and moving average (CARMA) processes on . (Causality has no natural analogue for data indexed by with .) Suppose that and are polynomials with real coefficients and . Suppose also that each of the zeros, , of has multiplicity one and strictly negative real part and that the zeros, , of have negative real parts. The strictly stationary CARMA process with autoregressive polynomial , moving average polynomial , and driven by the Lévy process , is the strictly stationary solution of the (suitably interpreted) formal stochastic differential equation,
where denotes differentiation with respect to . If , the solution (see Brockwell and Lindner (2009)) is
If we introduce new polynomials and , then the strictly stationary non-causal CARMA process with autoregressive polynomial , moving average polynomial , and driven by the Lévy process , is given by (1) with
This -parameter non-causal CARMA kernel extends naturally to a kernel on by replacing , by , the Euclidean norm of . This leads to the definition of an isotropic CARMA random field on as
and is a Lévy sheet on . An anisotropic CARMA random field is obtained by replacing the kernel (2) in the definition of by
where is an orthogonal matrix with determinant 1 and Λ is a strictly positive definite diagonal matrix. Under the assumption that the Lévy sheet has finite second moments, we determine the mean, spectral density and covariance function of . As in the case , the parameterization by and permits a very wide range of possible correlation functions for and the choice of Lévy sheet allows for a very wide range of marginal distributions. The use of a compound Poisson sheet in particular makes simulation of a field with specified polynomials and a trivial matter and, from a second-order point of view, incurs no loss of generality. In this case can be expressed as
where denotes the location of the unit point mass of a Poisson random measure on and is a sequence of independently and identically distributed random variables independent of . For estimation based on observations at irregularly spaced locations in we use Bayesian MCMC with a compound Poisson sheet to provide the prior distribution for the knot locations . Kriging is performed in the course of the MCMC procedure.