Continuous auto-regressive moving average random fields on ℝn

Peter J. Brockwell, Yasumasa Matsuda

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


We define an isotropic Lévy-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 Lévy 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 Lévy 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.

Original languageEnglish
Pages (from-to)833-857
Number of pages25
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Issue number3
Publication statusPublished - 2017 Jun


  • Compound Poisson process
  • Continuous auto-regressive moving average random field
  • Convolution
  • Gibbs sampling
  • Knot selection
  • Lévy noise
  • Lévy sheet
  • Matérn class

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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