Skip to main content

Advertisement

SpringerLink
Log in

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

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • 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):443–450

    Article  Google Scholar 

  • 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 9:23–35

    Article  Google Scholar 

  • Beneš V, Bodlák K, Møller J, Waagepetersen R (2011) A case study on point process modelling in disease mapping. Image Anal Stereol 24(3):159–168

    Article  Google Scholar 

  • Brix A, Diggle PJ (2001) Spatiotemporal prediction for log-Gaussian Cox processes. J R Stat Soc Ser B (Stat Methodol) 63(4):823–841

    Article  Google Scholar 

  • Cox DR (1955) Some statistical methods connected with series of events. J R Stat Soc Ser B (Methodol) 17(2):129–164

    Google Scholar 

  • 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):560–574

    Article  Google Scholar 

  • Davies TM, Hazelton ML (2013) Assessing minimum contrast parameter estimation for spatial and spatiotemporal log-Gaussian Cox processes. Statistica Neerlandica 67(4):355–389

    Article  Google Scholar 

  • De Cesare L, Myers D, Posa D (2002) Fortran programs for space-time modeling. Comput Geosci 28(2):205–212

    Article  Google Scholar 

  • De Iaco S, Myers DE, Posa D (2002) Nonseparable space-time covariance models: some parametric families. Math Geol 34(1):23–42

    Article  Google Scholar 

  • 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 1–45

    Google Scholar 

  • Diggle PJ (2013) Statistical analysis of spatial and spatio-temporal point patterns. CRC Press, Boca Raton

    Book  Google Scholar 

  • Diggle P, Rowlingson B, Tl Su (2005) Point process methodology for on-line spatio-temporal disease surveillance. Environmetrics 16(5):423–434

    Article  Google Scholar 

  • 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):542–563

    Article  Google Scholar 

  • Gabriel E (2014) Estimating second-order characteristics of inhomogeneous spatio-temporal point processes. Methodol Comput Appl Prob 16(2):411–431

    Article  Google Scholar 

  • Gabriel E, Diggle PJ (2009) Second-order analysis of inhomogeneous spatio-temporal point process data. Statistica Neerlandica 63(1):43–51

    Article  Google Scholar 

  • 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):1–29

    Article  Google Scholar 

  • Gelfand AE, Diggle P, Guttorp P, Fuentes M (2010) Handbook of spatial statistics. CRC Press, Boca Raton

    Book  Google Scholar 

  • 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 19:125–135

    Article  Google Scholar 

  • 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 151

    Google Scholar 

  • Guan Y (2007) A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Stat Prob Lett 77(18):1722–1729

    Article  Google Scholar 

  • Guan Y (2009) A minimum contrast estimation procedure for estimating the second-order parameters of inhomogeneous spatial point processes. Stat Interface 2(1):91–99

    Article  Google Scholar 

  • Guan Y, Sherman M (2007) On least squares fitting for stationary spatial point processes. J R Stat Soc Ser B (Stat Methodol) 69(1):31–49

    Article  Google Scholar 

  • Hawkes A, Adamopoulos L (1973) Cluster models for erthquakes-regional comparison. Bull Int Stat Inst 45(3):454–461

    Google Scholar 

  • Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, New York

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Møller J (2003) Shot noise Cox processes. Adv Appl Prob 35(3):614–640

    Article  Google Scholar 

  • 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):2–25

    Article  Google Scholar 

  • Møller J, Ghorbani M (2012) Aspects of second-order analysis of structured inhomogeneous spatio-temporal point processes. Statistica Neerlandica 66(4):472–491

    Article  Google Scholar 

  • Møller J, Toftaker H (2014) Geometric anisotropic spatial point pattern analysis and Cox processes. Scand J Stat 41(2):414–435

    Article  Google Scholar 

  • Møller J, Syversveen AR, Waagepetersen RP (1998) Log-Gaussian Cox processes. Scand J Stat 25(3):451–482

    Article  Google Scholar 

  • Porcu E, Fassò A, Barrientos S, Catalán PA (2017) Seismomatics. Stoch Environ Res Risk Assess 31(7):1577–1582

    Article  Google Scholar 

  • Prokešová M, Dvořák J (2014) Statistics for inhomogeneous space-time shot-noise Cox processes. Methodol Comput Appl Prob 16(2):433–449

    Article  Google Scholar 

  • Prokešová M, Dvořák J, Jensen EBV (2017) Two-step estimation procedures for inhomogeneous shot-noise Cox processes. Ann Inst Stat Math 69(3):513–542

    Article  Google Scholar 

  • 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

  • Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Prob 13:255–266

    Article  Google Scholar 

  • 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):553–567

    Article  Google Scholar 

  • 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):93–101

    Article  Google Scholar 

  • 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):1–25

    Article  Google Scholar 

  • Schoenberg FP (2005) Consistent parametric estimation of the intensity of a spatial-temporal point process. J Stat Plan Inference 128(1):79–93

    Article  Google Scholar 

  • 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):531–563

    Article  Google Scholar 

  • 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):683–690

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

  • Tamayo-Uria I, Mateu J, Diggle PJ (2014) Modelling of the spatio-temporal distribution of rat sightings in an urban environment. Spat Stat 9:192–206

    Article  Google Scholar 

  • 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):1–40

    Article  Google Scholar 

  • 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):1–48

    Article  Google Scholar 

  • Waagepetersen R, Guan Y (2009) Two-step estimation for inhomogeneous spatial point processes. J R Stat Soc Ser B (Stat Methodol) 71(3):685–702

    Article  Google Scholar 

  • 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

Download references

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.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Economiche, Aziendali e Statistiche, Università degli studi di Palermo, Palermo, Italy

    Marianna Siino & Giada Adelfio

  2. Centro Nazionale Terremoti, Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy

    Giada Adelfio

  3. Department of Mathematics, Universitat Jaume I, Castellón de la Plana, Castellón, Spain

    Jorge Mateu

Authors
  1. Marianna Siino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giada Adelfio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jorge Mateu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giada Adelfio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Siino, M., Adelfio, G. & Mateu, J. Joint second-order parameter estimation for spatio-temporal log-Gaussian Cox processes. Stoch Environ Res Risk Assess 32, 3525–3539 (2018). https://doi.org/10.1007/s00477-018-1579-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-018-1579-0

Keywords

Search

Navigation