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Directed Markov Point Processes as Limits of Partially Ordered Markov Models

Author

Listed:
  • Noel Cressie

    (The Ohio State University)

  • Jun Zhu

    (Iowa State University)

  • Adrian J. Baddeley

    (The University of Western Australia)

  • M. Gopalan Nair

    (Curtin University of Technology)

Abstract

In this paper, we consider spatial point processes and investigate members of a subclass of the Markov point processes, termed the directed Markov point processes (DMPPs), whose joint distribution can be written in closed form and, as a consequence, its parameters can be estimated directly. Furthermore, we show how the DMPPs can be simulated rapidly using a one-pass algorithm. A subclass of Markov random fields on a finite lattice, called partially ordered Markov models (POMMs), has analogous structure to that of DMPPs. In this paper, we show that DMPPs are the limits of auto-Poisson and auto-logistic POMMs. These and other results reveal a close link between inference and simulation for DMPPs and POMMs.

Suggested Citation

  • Noel Cressie & Jun Zhu & Adrian J. Baddeley & M. Gopalan Nair, 2000. "Directed Markov Point Processes as Limits of Partially Ordered Markov Models," Methodology and Computing in Applied Probability, Springer, vol. 2(1), pages 5-21, April.
    • Handle: RePEc:spr:metcap:v:2:y:2000:i:1:d:10.1023_a:1010095300231
    DOI: 10.1023/A:1010095300231
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    References listed on IDEAS

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    1. Cressie, Noel & Davidson, Jennifer L., 1998. "Image analysis with partially ordered markov models," Computational Statistics & Data Analysis, Elsevier, vol. 29(1), pages 1-26, November.
    2. Ivanoff, B.Gail & Merzbach, Ely, 1990. "Intensity-based inference for planar point processes," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 269-281, February.
    3. Yosihiko Ogata & Masaharu Tanemura, 1989. "Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(3), pages 583-600, September.
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