Continuous ARMA random fields [by Yasumasa Matsuda]

[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 {{\mathbb{R}}^n} by extending the kernel representation of a class of non-causal continuous time auto-regressive and moving average (CARMA) processes on {{\mathbb R}}. (Causality has no natural analogue for data indexed by {{\mathbb{R}}^n} with {n>1}.) Suppose that {a_*(z)=\prod_{i=1}^p(z-\lambda_i)} and {b_*(z)=\prod_{i=1}^q(z-\mu_i)} are polynomials with real coefficients and {q<p}. Suppose also that each of the zeros, {\lambda_1,\ldots,\lambda_p}, of {a} has multiplicity one and strictly negative real part and that the zeros, {\mu_1,\ldots,\mu_q}, of {b} have negative real parts. The strictly stationary CARMA{(p,q)} process with autoregressive polynomial {a_*}, moving average polynomial {b_*}, and driven by the Lévy process {L}, is the strictly stationary solution of the (suitably interpreted) formal stochastic differential equation,

\displaystyle a_*(D)Y(t)=b_*(D)DL(t),

where {D} denotes differentiation with respect to {t}. If {E[\max(0,\log |L(1)|)]<\infty}, the solution (see Brockwell and Lindner (2009)) is

\displaystyle Y(t)=\int_{\mathbb{R}} g(t-u)dL(u)\ \ \ \ \ (1)

where

\displaystyle g(t)=\sum_{i=1}^p \frac{b_*(\lambda_i)}{a_*^{'}(\lambda_i)}e^{\lambda_i t}{\bf 1}_{(0,\infty)}(t).

If we introduce new polynomials {a(z)=\prod_{i=1}^p(z^2-\lambda_i^2)} and {b(z)=\prod_{i=1}^q(z^2-\mu_i^2)}, then the strictly stationary non-causal CARMA{(2p,2q)} process with autoregressive polynomial {a}, moving average polynomial {b}, and driven by the Lévy process {L}, is given by (1) with

\displaystyle g(t)=\sum_{i=1}^p \frac{b(\lambda_i)}{a^{'}(\lambda_i)}e^{\lambda_i t}{\bf 1}_{(0,\infty)}(t) - \sum_{i=1}^p \frac{b(-\lambda_i)}{a^{'}(-\lambda_i)}e^{-\lambda_i t}{\bf 1}_{(-\infty,0)}(t) =\sum_{\lambda:a(\lambda)=0}\frac{b(\lambda)}{a^{'}(\lambda)}e^{\lambda |t|}.

This {(p+q)}-parameter non-causal CARMA kernel extends naturally to a kernel on {{\mathbb{R}}^n} by replacing {|t|, t\in{\mathbb{R}}}, by {||{t}||, {t}\in\mathbb{R}^n}, the Euclidean norm of {{t}}. This leads to the definition of an isotropic CARMA random field {S} on {{\mathbb R}^n} as

\displaystyle S_n(t)=\int_{{\mathbb R}^n}g_n(t-u)dL(u), ~t\in{\mathbb{R}}^n,

where

\displaystyle g_n(t)=\sum_{\lambda:a(\lambda)=0}\frac{b(\lambda)}{a^{'}(\lambda)}e^{\lambda ||{t}||}, t\in{\mathbb{R}}^n,\ \ \ \ \ (2)

and {L} is a Lévy sheet on {{\mathbb{R}}^n} . An anisotropic CARMA random field is obtained by replacing the kernel (2) in the definition of {S_n} by

\displaystyle h_n(t):=g_n(\Lambda R t), t\in{\mathbb{R}}^n,\ \ \ \ \ (3)

where {R} is an {n\times n} orthogonal matrix with determinant 1 and Λ is a strictly positive definite {n\times n} diagonal matrix. Under the assumption that the Lévy sheet has finite second moments, we determine the mean, spectral density and covariance function of {S_n}. As in the case {n=1}, the parameterization by {\lambda_1,\ldots,\lambda_p} and {\mu_1,\ldots,\mu_q} permits a very wide range of possible correlation functions for {S_n} 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 {a} and {b} a trivial matter and, from a second-order point of view, incurs no loss of generality. In this case {S_n} can be expressed as

\displaystyle S_n(t)=\sum_{i=1}^\infty g_n(t-X_i)Y_i,\ \ \ \ \ (4)

where {X_i} denotes the location of the {i^{\rm th}} unit point mass of a Poisson random measure on {{\mathbb{R}}^n} and {\{Y_i\}} is a sequence of independently and identically distributed random variables independent of {\{X_i\}}. For estimation based on observations at irregularly spaced locations in {\mathbb R^n} we use Bayesian MCMC with a compound Poisson sheet to provide the prior distribution for the knot locations {X_i}. Kriging is performed in the course of the MCMC procedure.

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