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Identification and estimation of superposed Neyman–Scott spatial cluster processes

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Abstract

This paper proposes an estimation method for superposed spatial point patterns of Neyman–Scott cluster processes of different distance scales and cluster sizes. Unlike the ordinary single Neyman–Scott model, the superposed process of Neyman–Scott models is not identified solely by the second-order moment property of the process. To solve the identification problem, we use the nearest neighbor distance property in addition to the second-order moment property. In the present procedure, we combine an inhomogeneous Poisson likelihood based on the Palm intensity with another likelihood function based on the nearest neighbor property. The derivative of the nearest neighbor distance function is regarded as the intensity function of the rotation invariant inhomogeneous Poisson point process. The present estimation procedure is applied to two sets of ecological location data.

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Acknowledgments

The authors are grateful to Koichi Katsura for his generous help in programming matters and to Jancang Zhuang for his helpful advice related to the graphical software R. This study is partially supported by Grant-in-Aid 23240039 for Scientific Research (A), Ministry of Education, Science, Sports and Culture. Y.O. was supported by the Aihara Innovative Mathematical Modelling Project and the Japan Society for the Promotion of Science (JSPS) through the ‘Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)’, initiated by the Council for Science and Technology Policy (CSTP).

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Authors and Affiliations

  1. Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Ushio Tanaka

  2. Institute of Industry Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan

    Yosihiko Ogata

  3. The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan

    Yosihiko Ogata

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  1. Ushio Tanaka
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  2. Yosihiko Ogata
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Correspondence to Ushio Tanaka.

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Tanaka, U., Ogata, Y. Identification and estimation of superposed Neyman–Scott spatial cluster processes. Ann Inst Stat Math 66, 687–702 (2014). https://doi.org/10.1007/s10463-013-0431-z

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  • DOI: https://doi.org/10.1007/s10463-013-0431-z

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