# 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. 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.