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Statistical Depth in Spatial Point Process

Author

Listed:
  • Xinyu Zhou

    (Department of Statistics, Florida State University, Tallahassee, FL 32306, USA)

  • Wei Wu

    (Department of Statistics, Florida State University, Tallahassee, FL 32306, USA)

Abstract

Statistical depth is widely used as a powerful tool to measure the center-outward rank of multivariate and functional data. Recent studies have introduced the notion of depth to the temporal point process, which exhibits randomness in the cardinality as well as distribution in the observed events. The proposed methods can well capture the rank of a point process in a given time interval, where a critical step is to measure the rank by using inter-arrival events. In this paper, we propose to extend the depth concept to multivariate spatial point process. In this case, the observed process is in a multi-dimensional location and there are no conventional inter-arrival events in the temporal process. We adopt the newly developed depth in metric space by defining two different metrics, namely the penalized metric and the smoothing metric, to fully explore the depth in the spatial point process. The mathematical properties and the large sample theory, as well as depth-based hypothesis testings, are thoroughly discussed. We then use several simulations to illustrate the effectiveness of the proposed depth method. Finally, we apply the new method in a real-world dataset and obtain desirable ranking performance.

Suggested Citation

  • Xinyu Zhou & Wei Wu, 2024. "Statistical Depth in Spatial Point Process," Mathematics, MDPI, vol. 12(4), pages 1-20, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:4:p:595-:d:1340338
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    References listed on IDEAS

    as
    1. López-Pintado, Sara & Romo, Juan, 2009. "On the Concept of Depth for Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 718-734.
    2. Jun Li & Juan A. Cuesta-Albertos & Regina Y. Liu, 2012. "DD -Classifier: Nonparametric Classification Procedure Based on DD -Plot," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 737-753, June.
    3. Rasmus Waagepetersen & Yongtao Guan, 2009. "Two‐step estimation for inhomogeneous spatial point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 685-702, June.
    4. Tatjana Lange & Karl Mosler & Pavlo Mozharovskyi, 2014. "Fast nonparametric classification based on data depth," Statistical Papers, Springer, vol. 55(1), pages 49-69, February.
    5. Yongtao Guan, 2008. "A KPSS Test for Stationarity for Spatial Point Processes," Biometrics, The International Biometric Society, vol. 64(3), pages 800-806, September.
    6. Frederic Paik Schoenberg, 2004. "Testing Separability in Spatial-Temporal Marked Point Processes," Biometrics, The International Biometric Society, vol. 60(2), pages 471-481, June.
    7. Mark Berman, 1986. "Testing for Spatial Association between a Point Process and Another Stochastic Process," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 35(1), pages 54-62, March.
    Full references (including those not matched with items on IDEAS)

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