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Continuous auto-regressive moving average random fields on n
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DOI：10.1111/rssb.12197.png)
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Abstract
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We define an isotropic Levy-driven continuous auto-regressive moving average CARMA(p,q) random field on Rn as the integral of a radial CARMA kernel with respect to a Levy sheet. Such fields constitute a parametric family characterized by an auto-regressive polynomial a and a moving average polynomial b having zeros in both the left and the right complex half-planes. They extend the well-balanced Ornstein-Uhlenbeck process of Schnurr and Woerner to a well-balanced CARMA process in one dimension (with a much richer class of autocovariance functions) and to an isotropic CARMA random field on Rn for n>1. We derive second-order properties of these random fields and extend the results to a larger class of anisotropic CARMA random fields. If the driving Levy sheet is compound Poisson it is trivial to simulate the corresponding random field on any bounded subset of Rn. A method for joint estimation of the CARMA kernel parameters and knot locations is proposed for compound-Poisson-driven fields and is illustrated by applications to simulated data and Tokyo land price data.
Keywords:
Compound Poisson process
Continuous auto-regressive moving average random field
Convolution
Gibbs sampling
Knot selection
Levy noise
Levy sheet
Matern class
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