Advertisement

Joint second-order parameter estimation for spatio-temporal log-Gaussian Cox processes

  • Marianna Siino
  • Giada Adelfio
  • Jorge Mateu
Original Paper
  • 91 Downloads

Abstract

We propose a new fitting method to estimate the set of second-order parameters for the class of homogeneous spatio-temporal log-Gaussian Cox point processes. With simulations, we show that the proposed minimum contrast procedure, based on the spatio-temporal pair correlation function, provides reliable estimates and we compare the results with the current available methods. Moreover, the proposed method can be used in the case of both separable and non-separable parametric specifications of the correlation function of the underlying Gaussian Random Field. We describe earthquake sequences comparing several Cox model specifications.

Keywords

Earthquakes Log-Gaussian Cox processes Minimum contrast method Non-separable covariance function Spatio-temporal pair correlation function 

Notes

Acknowledgements

We would like to thank the Associate Editor and the two anonymous Referees for their suggestions and comments, that considerably improved the manuscript. This paper has been supported by the national grant of the Italian Ministry of Education University and Research (MIUR) for the PRIN-2015 program (Progetti di ricerca di Rilevante Interesse Nazionale), “Prot. 20157PRZC4— Research Project Title Complex space-time modeling and functional analysis for probabilistic forecast of seismic events. PI: Giada Adelfio”. J. Mateu has been partially founded by Grants P1-1B2015-40 and MTM2016-78917-R.

References

  1. Adelfio G, Chiodi M (2015) Alternated estimation in semi-parametric space-time branching-type point processes with application to seismic catalogs. Stoch Environ Res Risk Assess 29(2):443–450CrossRefGoogle Scholar
  2. Ahn J, Johnson TD, Bhavnani D, Eisenberg JN, Mukherjee B (2014) A space-time point process model for analyzing and predicting case patterns of diarrheal disease in northwestern ecuador. Spat Spatiotemporal Epidemiol 9:23–35CrossRefGoogle Scholar
  3. Beneš V, Bodlák K, Møller J, Waagepetersen R (2011) A case study on point process modelling in disease mapping. Image Anal Stereol 24(3):159–168CrossRefGoogle Scholar
  4. Brix A, Diggle PJ (2001) Spatiotemporal prediction for log-Gaussian Cox processes. J R Stat Soc Ser B (Stat Methodol) 63(4):823–841CrossRefGoogle Scholar
  5. Cox DR (1955) Some statistical methods connected with series of events. J R Stat Soc Ser B (Methodol) 17(2):129–164Google Scholar
  6. D’Alessandro A, Luzio D, Martorana R, Capizzi P (2016) Selection of time windows in the horizontal-to-vertical noise spectral ratio by means of cluster analysis. Bull Seismol Soc Am 106(2):560–574CrossRefGoogle Scholar
  7. Davies TM, Hazelton ML (2013) Assessing minimum contrast parameter estimation for spatial and spatiotemporal log-Gaussian Cox processes. Statistica Neerlandica 67(4):355–389CrossRefGoogle Scholar
  8. De Cesare L, Myers D, Posa D (2002) Fortran programs for space-time modeling. Comput Geosci 28(2):205–212CrossRefGoogle Scholar
  9. De Iaco S, Myers DE, Posa D (2002) Nonseparable space-time covariance models: some parametric families. Math Geol 34(1):23–42CrossRefGoogle Scholar
  10. Diggle PJ (2007) Spatio-temporal point processes: methods and applications. In: Finkenstadt B, Held L, Isham V (eds) Statistical methods for spatio-temporal systems. Monographs on statistics and applied probability, vol 107. Chapman & Hall, Boca Raton, pp 1–45Google Scholar
  11. Diggle PJ (2013) Statistical analysis of spatial and spatio-temporal point patterns. CRC Press, Boca RatonCrossRefGoogle Scholar
  12. Diggle P, Rowlingson B, Tl Su (2005) Point process methodology for on-line spatio-temporal disease surveillance. Environmetrics 16(5):423–434CrossRefGoogle Scholar
  13. Diggle PJ, Moraga P, Rowlingson B, Taylor BM (2013) Spatial and spatio-temporal log-Gaussian Cox processes: extending the geostatistical paradigm. Stat Sci 28(4):542–563CrossRefGoogle Scholar
  14. Gabriel E (2014) Estimating second-order characteristics of inhomogeneous spatio-temporal point processes. Methodol Comput Appl Prob 16(2):411–431CrossRefGoogle Scholar
  15. Gabriel E, Diggle PJ (2009) Second-order analysis of inhomogeneous spatio-temporal point process data. Statistica Neerlandica 63(1):43–51CrossRefGoogle Scholar
  16. Gabriel E, Rowlingson BS, Diggle PJ (2013) stpp: an R package for plotting, simulating and analyzing spatio-temporal point patterns. J Stat Softw 53(2):1–29CrossRefGoogle Scholar
  17. Gelfand AE, Diggle P, Guttorp P, Fuentes M (2010) Handbook of spatial statistics. CRC Press, Boca RatonCrossRefGoogle Scholar
  18. Giorgi E, Kreppel K, Diggle PJ, Caminade C, Ratsitorahina M, Rajerison M, Baylis M (2016) Modeling of spatio-temporal variation in plague incidence in Madagascar from 1980 to 2007. Spat Spatiotemporal Epidemiol 19:125–135CrossRefGoogle Scholar
  19. Gneiting T, Genton MG, Guttorp P (2006) Geostatistical space-time models, stationarity, separability, and full symmetry. Monographs on statistics and applied probability, vol 107. Chapman & Hall, Boca Raton, p 151Google Scholar
  20. Guan Y (2007) A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Stat Prob Lett 77(18):1722–1729CrossRefGoogle Scholar
  21. Guan Y (2009) A minimum contrast estimation procedure for estimating the second-order parameters of inhomogeneous spatial point processes. Stat Interface 2(1):91–99CrossRefGoogle Scholar
  22. Guan Y, Sherman M (2007) On least squares fitting for stationary spatial point processes. J R Stat Soc Ser B (Stat Methodol) 69(1):31–49CrossRefGoogle Scholar
  23. Hawkes A, Adamopoulos L (1973) Cluster models for erthquakes-regional comparison. Bull Int Stat Inst 45(3):454–461Google Scholar
  24. Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, New YorkGoogle Scholar
  25. Lombardo L, Opitz T, Huser R (2018) Point process-based modeling of multiple debris flow landslides using inla: an application to the 2009 messina disaster. Stoch Environ Res Risk Assess.  https://doi.org/10.1007/s00477-018-1518-0 CrossRefGoogle Scholar
  26. Møller J (2003) Shot noise Cox processes. Adv Appl Prob 35(3):614–640CrossRefGoogle Scholar
  27. Møller J, Díaz-Avalos C (2010) Structured spatio-temporal shot-noise Cox point process models, with a view to modelling forest fires. Scand J Stat 37(1):2–25CrossRefGoogle Scholar
  28. Møller J, Ghorbani M (2012) Aspects of second-order analysis of structured inhomogeneous spatio-temporal point processes. Statistica Neerlandica 66(4):472–491CrossRefGoogle Scholar
  29. Møller J, Toftaker H (2014) Geometric anisotropic spatial point pattern analysis and Cox processes. Scand J Stat 41(2):414–435CrossRefGoogle Scholar
  30. Møller J, Syversveen AR, Waagepetersen RP (1998) Log-Gaussian Cox processes. Scand J Stat 25(3):451–482CrossRefGoogle Scholar
  31. Porcu E, Fassò A, Barrientos S, Catalán PA (2017) Seismomatics. Stoch Environ Res Risk Assess 31(7):1577–1582CrossRefGoogle Scholar
  32. Prokešová M, Dvořák J (2014) Statistics for inhomogeneous space-time shot-noise Cox processes. Methodol Comput Appl Prob 16(2):433–449CrossRefGoogle Scholar
  33. Prokešová M, Dvořák J, Jensen EBV (2017) Two-step estimation procedures for inhomogeneous shot-noise Cox processes. Ann Inst Stat Math 69(3):513–542CrossRefGoogle Scholar
  34. R Development Core Team (2005) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org. ISBN 3-900051-07-0
  35. Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Prob 13:255–266CrossRefGoogle Scholar
  36. Rodrigues A, Diggle PJ (2010) A class of convolution-based models for spatio-temporal processes with non-separable covariance structure. Scand J Stat 37(4):553–567CrossRefGoogle Scholar
  37. Rodrigues A, Diggle PJ (2012) Bayesian estimation and prediction for inhomogeneous spatiotemporal log-Gaussian Cox processes using low-rank models, with application to criminal surveillance. J Am Stat Assoc 107(497):93–101CrossRefGoogle Scholar
  38. Schlather M, Malinowski A, Menck PJ, Oesting M, Strokorb K (2015) Analysis, simulation and prediction of multivariate random fields with package RandomFields. J Stat Softw 63(8):1–25CrossRefGoogle Scholar
  39. Schoenberg FP (2005) Consistent parametric estimation of the intensity of a spatial-temporal point process. J Stat Plan Inference 128(1):79–93CrossRefGoogle Scholar
  40. Serra L, Saez M, Mateu J, Varga D, Juan P, Díaz-Ávalos C, Rue H (2014) Spatio-temporal log-Gaussian Cox processes for modelling wildfire occurrence: the case of Catalonia, 1994–2008. Environ Ecol Stat 21(3):531–563CrossRefGoogle Scholar
  41. Sheather SJ, Jones MC (1991) A reliable data-based bandwidth selection method for kernel density estimation. J R Stat Soc Ser B (Methodol) 53(3):683–690Google Scholar
  42. Siino M, Adelfio G, Mateu J, Chiodi M, D’Alessandro A (2016) Spatial pattern analysis using hybrid models: an application to the hellenic seismicity. Stoch Environ Res Risk Assess.  https://doi.org/10.1007/s00477-016-1294-7 CrossRefGoogle Scholar
  43. Siino M, D’Alessandro A, Adelfio G, Scudero S, Chiodi M (2018) Multiscale processes to describe the eastern sicily seismic sequences. Ann Geophys 61(2). https://doi.org/10.441/ag-7711
  44. Tamayo-Uria I, Mateu J, Diggle PJ (2014) Modelling of the spatio-temporal distribution of rat sightings in an urban environment. Spat Stat 9:192–206CrossRefGoogle Scholar
  45. Taylor BM, Davies TM, Rowlingson BS, Diggle PJ (2013) lgcp: an R package for inference with spatial and spatio-temporal log-Gaussian Cox processes. J Stat Softw 52(4):1–40CrossRefGoogle Scholar
  46. Taylor BM, Davies TM, Rowlingson BS, Diggle PJ (2015) Bayesian inference and data augmentation schemes for spatial, spatiotemporal and multivariate log-Gaussian Cox processes in R. J Stat Softw 63(7):1–48CrossRefGoogle Scholar
  47. Waagepetersen R, Guan Y (2009) Two-step estimation for inhomogeneous spatial point processes. J R Stat Soc Ser B (Stat Methodol) 71(3):685–702CrossRefGoogle Scholar
  48. Wand M (2015) KernSmooth: functions for Kernel smoothing supporting Wand & Jones (1995). https://CRAN.R-project.org/package=KernSmooth, r package version 2.23-15

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Scienze Economiche, Aziendali e StatisticheUniversità degli studi di PalermoPalermoItaly
  2. 2.Centro Nazionale TerremotiIstituto Nazionale di Geofisica e VulcanologiaRomeItaly
  3. 3.Department of MathematicsUniversitat Jaume ICastellón de la PlanaSpain

Personalised recommendations