Skip to main content
SpringerLink
Log in

Estimating Second-Order Characteristics of Inhomogeneous Spatio-Temporal Point Processes

Influence of Edge Correction Methods and Intensity Estimates

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

Non-parametric estimates of the K-function and the pair correlation function play a fundamental role for exploratory and explanatory analysis of spatial and spatio-temporal point patterns. These estimates usually require information from outside of the study region, resulting to the so-called edge effects which have to be corrected. They also depend on first-order characteristics, which have to be estimated in practice. In this paper, we extend classical edge correction methods to the spatio-temporal setting and compare the performance of the related estimators for stationary/non-stationary and/or isotropic/anisotropic point patterns. Further, we explore the influence of the estimated intensity function on these estimators.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Allard D, Brix A, Chadoeuf J (2001) Testing local independence between two point processes. Biometrics 57:508–517

    Article  MATH  MathSciNet  Google Scholar 

  • Arbia G, Espa G, Giuliani D, Mazzitelli A (2012) Clusters of firms in an inhomogeneous space: the high-tech industries in milan. Econ Model 29(1):3–11

    Article  Google Scholar 

  • Baddeley A (1999) Spatial sampling and censoring. In: Barndorff-Nielsen O, Kendall W, van Lieshout M (eds) Stochastic geometry: likelihood and computation. Chapman and Hall/CRC, London, pp 449–461

    Google Scholar 

  • Baddeley A, Turner R (2000) Practical maximum pseudolikelihood for spatial point patterns. Aust NZ J Stat 42:283–322

    Article  MATH  MathSciNet  Google Scholar 

  • Baddeley A, Moyeed R, Howard C, Boyde A (1993) Analysis of a three-dimensional point pattern with replication. Appl Stat 42(4):641–668

    Article  MATH  MathSciNet  Google Scholar 

  • Baddeley A, Møller J, Waagepetersen R (2000) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat Neerl 54(3):329–350

    Article  MATH  Google Scholar 

  • Berman M, Diggle P (1989) Estimating weighted integrals of the second-order intensity of a spatial point process. J Roy Stat Soc B 51:81–92

    MATH  MathSciNet  Google Scholar 

  • Berman M, Turner T (1992) Approximating point process likelihoods with GLIM. Appl Stat 41:31–38

    Article  MATH  Google Scholar 

  • Brix A, Senoussi R, Couteron P, Chadoeuf J (2001) Assessing goodness of fit of spatially inhomogeneous Poisson processes. Biometrika 88(2):487–497

    Article  MATH  MathSciNet  Google Scholar 

  • Cressie N (1993) Statistics for spatial data, revised edn. Wiley, New York

    Google Scholar 

  • Cressie N, Wikle C (2011) Statistics for spatio-temporal data, 1st edn. Wiley, New York

    MATH  Google Scholar 

  • Cronie O, Särkka A (2011) Some edge correction methods for marked spatio-temporal point process models. Comput Stat Data An 55(7):2209–2220

    Article  Google Scholar 

  • Daley D, Vere-Jones D (2003) An introduction to the theory of point processes, vol I: Elementary theory and methods, 2nd edn. Springer, New York

    Google Scholar 

  • Diggle P (1979) On parameter estimation and goodness-of-fit testing for spatial point patterns. Biometrics 35:87–101

    Article  MATH  Google Scholar 

  • Diggle P (1985) A kernel method for smoothing point process data. Appl Stat 34(2):138–147

    Article  MATH  Google Scholar 

  • Diggle P (2003) Statistical analysis of spatial point patterns, 2nd edn. Edward Arnold, London

    MATH  Google Scholar 

  • Diggle P (2006) 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. Chapman and Hall/CRC, London, pp 1–45

    Chapter  Google Scholar 

  • Diggle P, Gabriel E (2010) Spatio-temporal point processes. In: Gelfand A, Diggle P, Fuentes M, Guttorp P (eds) Handbook of spatial statistics. Chapman and Hall/CRC, London, pp 449–461

    Chapter  Google Scholar 

  • Diggle P, Lange N, Benes F (1991) Analysis of variance for replicated spatial point patterns in clinical neuroanatomy. J Am Stat Assoc 86:618–625

    Article  Google Scholar 

  • Diggle P, Chetwynd A, Häggkvist R, Gooding S (1995) Second-order analysis of space-time clustering. Stat Methods Med Res 4:124–136

    Article  Google Scholar 

  • Diggle P, Gómez-Rubio V, Brown P, Chetwynd A, Gooding S (2007) Second-order analysis of inhomogeneous spatial point processes using case-control data. Biometrics 63(2):550–557

    Article  MATH  MathSciNet  Google Scholar 

  • Doguwa S (1990) On edge-corrected kernel-based pair-correlation function estimators for point processes. Biom J 32:95–106

    Article  Google Scholar 

  • Gabriel E (2013) Estimating second-order characteristics of inhomogeneous spatio-temporal point processes: influence of edge correction methods and intensity estimates. ArXiv:e-prints1304.7178

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

    Article  MathSciNet  Google Scholar 

  • Gabriel E, Wilson D, Leatherbarrow H, Cheesbrough J, Gee S, Bolton E, Fox A, Fearnhead P, Hart A, Diggle P (2010) Spatio-temporal epidemiology of campylobacter jejuni enteritis, in an area of northwest england, 2000–2002. Epidemiol Infect 138:1384–1390

    Article  Google Scholar 

  • Gabriel E, Rowlingson B, Diggle P (2013) stpp: an R package for plotting, simulating and analysing spatio-temporal point patterns. J Stat Softw 53(2):1:29

    Google Scholar 

  • Goreaud F, Pélissier R (1999) On explicit formulas of edge effect correction for Ripley’s K-function. J Veg Sci 10(3):433–438

    Article  Google Scholar 

  • Guan Y, Sherman M, Calvin J (2006) Assessing isotropy for spatial point processes. Biometrics 62:119–125

    Article  MATH  MathSciNet  Google Scholar 

  • Haase P (1995) Spatial pattern analysis in ecology based on Ripley’s K-function: introduction and methods of edge correction. J Veg Sci 6:575–582

    Article  Google Scholar 

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

  • Jafari-Mamaghani M, Andersson M, Krieger P (2010) Spatial point pattern analysis of neurons using Ripley’s K-function in 3D. Front Neuroinform 4:1–10

    Google Scholar 

  • Law R, Illian J, Burslem D, Gratzer G, Gunatilleke C, Gunatilleke I (2009) Ecological information from spatial patterns of plants: insights from point process theory. J Ecol 97(4):616–628

    Article  Google Scholar 

  • Li F, Zhang L (2007) Comparison of point pattern analysis methods for classifying the spatial distributions of spurce-fir stands in the north-east usa. Forestry 80(3):337–349

    Article  Google Scholar 

  • Liu D, Kelly M, Gong P, Guo Q (2007) Characterizing spatial-temporal tree mortality patterns associated with a new forest disease. For Ecol Manag 253:220–231

    Article  Google Scholar 

  • Møller J (2003) Shot-noise Cox processes. Adv Appl Probab 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. Scan 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. Stat Neerl 66:472–491

    Article  MathSciNet  Google Scholar 

  • Møller J, Toftager H (2012) Geometric anisotropic spatial point pattern analysis and Cox processes. Tech. Rep. R-2012-01, Department of Mathematical Sciences, Aalborg University

  • Møller J, Waagepetersen R (2003) Statistical inference and simulation for spatial point processes. Monographs on statistics and applied probability. Chapman and Hall/CRC, London

    Book  Google Scholar 

  • Mrkvička T, Muška M, Kubečka J (2012) Two step estimation for Neyman-Scott point process with inhomogeneous cluster centers. Stat Comp. doi:10.1007/s11222-012-9355-3

    Google Scholar 

  • Ohser J (1983) On estimators for the reduced second moment measure of point processes. Math Oper Stat 14:63–71

    MATH  MathSciNet  Google Scholar 

  • Pommerening A, Stoyan D (2006) Edge-correction needs in estimating indices of spatial forest structure. Can J For Res 36:1723–1739

    Article  Google Scholar 

  • Prokešová M (2010) Inhomogeneity in spatial Cox point processes—location dependent thinning is not the only option. Image Anal Stereol 29:133–141

    Article  MATH  MathSciNet  Google Scholar 

  • R Development Core Team (2012) R: a language and environment for statistical computing. R foundation for statistical computing. Vienna Austria. http://www.r-project.org

  • Ripley B (1977) Modelling spatial patterns (with discussion). J Roy Stat Soc B 39:172–212

    MathSciNet  Google Scholar 

  • Ripley B (1988) Statistical inference for spatial processes. Cambrige University Press, Cambridge

    Book  Google Scholar 

  • Silverman B (1986) Density estimation for statistics and data analysis. Chapman and Hall/CRC, London

    Book  MATH  Google Scholar 

  • Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields: methods of geometrical statistics. Wiley

  • Vere-Jones D (2009) Some models and procedures for space-time point processes. Env Ecol Stat 16(2):173–195

    Article  MathSciNet  Google Scholar 

  • Yamada I, Rogerson P (2003) An empirical comparison of edge effect correction methods applied to K-function analysis. Geogr Anal 35(2):97–109

    Google Scholar 

  • Zhuang J, Ogata Y, Jones D (2002) Stochastic declustering of space-time earthquake occurrences. J Am Stat Assoc 97(458):369–380

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Avignon University, LMA EA 2151, 84000, Avignon, France

    Edith Gabriel

Authors
  1. Edith Gabriel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edith Gabriel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gabriel, E. Estimating Second-Order Characteristics of Inhomogeneous Spatio-Temporal Point Processes. Methodol Comput Appl Probab 16, 411–431 (2014). https://doi.org/10.1007/s11009-013-9358-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9358-3

Keywords

AMS 2000 Subject Classification

Search

Navigation