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Spatiotemporal prediction for log‐Gaussian Cox processes

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  • Anders Brix
  • Peter J. Diggle

Abstract

Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment‐based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.

Suggested Citation

  • Anders Brix & Peter J. Diggle, 2001. "Spatiotemporal prediction for log‐Gaussian Cox processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(4), pages 823-841.
    • Handle: RePEc:bla:jorssb:v:63:y:2001:i:4:p:823-841
    DOI: 10.1111/1467-9868.00315
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    1. Michaela Prokešová & Jiří Dvořák, 2014. "Statistics for Inhomogeneous Space-Time Shot-Noise Cox Processes," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 433-449, June.
    2. Matthew J. Heaton & Stephan R. Sain & Andrew J. Monaghan & Olga V. Wilhelmi & Mary H. Hayden, 2015. "An Analysis of an Incomplete Marked Point Pattern of Heat-Related 911 Calls," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 123-135, March.
    3. Nicoletta D’Angelo & Marianna Siino & Antonino D’Alessandro & Giada Adelfio, 2022. "Local spatial log-Gaussian Cox processes for seismic data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 633-671, December.
    4. R. Lechnerová & K. Helisová & V. Beneš, 2008. "Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes," Methodology and Computing in Applied Probability, Springer, vol. 10(3), pages 315-335, September.
    5. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392, April.
    6. Christopher Wikle & Mevin Hooten, 2010. "A general science-based framework for dynamical spatio-temporal models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 417-451, November.
    7. Renshaw, Eric & Mateu, Jorge & Saura, Fuensanta, 2007. "Disentangling mark/point interaction in marked-point processes," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 3123-3144, March.
    8. Katharina Parry & David P. Watling & Martin L. Hazelton, 2016. "A new class of doubly stochastic day-to-day dynamic traffic assignment models," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 5-23, March.
    9. Reis, Edna A. & Gamerman, Dani & Paez, Marina S. & Martins, Thiago G., 2013. "Bayesian dynamic models for space–time point processes," Computational Statistics & Data Analysis, Elsevier, vol. 60(C), pages 146-156.
    10. Chen, Jiaxun & Micheas, Athanasios C. & Holan, Scott H., 2022. "Hierarchical Bayesian modeling of spatio-temporal area-interaction processes," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).
    11. Benjamin M. Taylor & Ricardo Andrade‐Pacheco & Hugh J. W. Sturrock, 2018. "Continuous inference for aggregated point process data," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 181(4), pages 1125-1150, October.
    12. Fangpo Wang & Anirban Bhattacharya & Alan E. Gelfand, 2018. "Rejoinder on: Process modeling for slope and aspect with application to elevation data maps," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 783-786, December.
    13. D'Angelo, Nicoletta & Adelfio, Giada & Mateu, Jorge, 2023. "Locally weighted minimum contrast estimation for spatio-temporal log-Gaussian Cox processes," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    14. Jesper Møller & Carlos Díaz‐Avalos, 2010. "Structured Spatio‐Temporal Shot‐Noise Cox Point Process Models, with a View to Modelling Forest Fires," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 2-25, March.
    15. van Lieshout, M.N.M., 2016. "Likelihood based inference for partially observed renewal processes," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 190-196.
    16. Tilman M. Davies & Martin L. Hazelton, 2013. "Assessing minimum contrast parameter estimation for spatial and spatiotemporal log‐Gaussian Cox processes," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 67(4), pages 355-389, November.
    17. Jiří Dvořák & Michaela Prokešová, 2016. "Parameter Estimation for Inhomogeneous Space-Time Shot-Noise Cox Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 939-961, December.
    18. Yehua Li & Yongtao Guan, 2014. "Functional Principal Component Analysis of Spatiotemporal Point Processes With Applications in Disease Surveillance," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1205-1215, September.
    19. Waagepetersen, Rasmus, 2004. "Convergence of posteriors for discretized log Gaussian Cox processes," Statistics & Probability Letters, Elsevier, vol. 66(3), pages 229-235, February.
    20. Li, Yehua & Qiu, Yumou & Xu, Yuhang, 2022. "From multivariate to functional data analysis: Fundamentals, recent developments, and emerging areas," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    21. Athanasios Kottas, 2018. "Discussion of paper “nonparametric Bayesian inference in applications” by Peter Müller, Fernando A. Quintana and Garritt L. Page," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(2), pages 219-225, June.
    22. Michele Nguyen & Almut E. D. Veraart, 2017. "Spatio-temporal Ornstein–Uhlenbeck Processes: Theory, Simulation and Statistical Inference," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 46-80, March.
    23. Markéta Zikmundová & Kateřina Staňková Helisová & Viktor Beneš, 2012. "Spatio-Temporal Model for a Random Set Given by a Union of Interacting Discs," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 883-894, September.
    24. Tang, Jinjun & Zhao, Chuyun & Liu, Fang & Hao, Wei & Gao, Fan, 2022. "Analyzing travel destinations distribution using large-scaled GPS trajectories: A spatio-temporal Log-Gaussian Cox process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 599(C).
    25. Sarkka, Aila & Renshaw, Eric, 2006. "The analysis of marked point patterns evolving through space and time," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1698-1718, December.

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