×
Coeurjolly, Jean-François; Møller, Jesper

Variational approach for spatial point process intensity estimation. (English) Zbl 1400.62208

Bernoulli 20, No. 3, 1097-1125 (2014).
Summary: We introduce a new variational estimator for the intensity function of an inhomogeneous spatial point process with points in the \(d\)-dimensional Euclidean space and observed within a bounded region. The variational estimator applies in a simple and general setting when the intensity function is assumed to be of log-linear form \(\beta+\theta^{\top}z(u)\) where \(z\) is a spatial covariate function and the focus is on estimating \(\theta\). The variational estimator is very simple to implement and quicker than alternative estimation procedures. We establish its strong consistency and asymptotic normality. We also discuss its finite-sample properties in comparison with the maximum first order composite likelihood estimator when considering various inhomogeneous spatial point process models and dimensions as well as settings were \(z\) is completely or only partially known.

Cited in 5 Documents

MSC:

62M30 Inference from spatial processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

asymptotic normality; composite likelihood; estimating equation; inhomogeneous spatial point process; strong consistency; variational estimator

Software:

spatstat
PDFBibTeX XMLCite
\textit{J.-F. Coeurjolly} and \textit{J. Møller}, Bernoulli 20, No. 3, 1097--1125 (2014; Zbl 1400.62208)
Full Text: DOI arXiv Euclid

References:

[1] Almeida, M.P. and Gidas, B. (1993). A variational method for estimating the parameters of MRF from complete or incomplete data. Ann. Appl. Probab. 3 103-136. · Zbl 0771.60091 · doi:10.1214/aoap/1177005510
[2] Baddeley, A. (2010). Modeling strategies. In Handbook of Spatial Statistics (A.E. Gelfand, P.J. Diggle, P. Guttorp and M. Fuentes, eds.). Chapman & Hall/CRC Handb. Mod. Stat. Methods 339-369. Boca Raton, FL: CRC Press. · doi:10.1201/9781420072884-c20
[3] Baddeley, A. and Dereudre, A. (2013). Variational estimators for the parameters of Gibbs point process models. Bernoulli 19 905-930. · Zbl 1273.62203 · doi:10.3150/12-BEJ419
[4] Baddeley, A. and Turner, R. (2000). Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42 283-322. · Zbl 0981.62078 · doi:10.1111/1467-842X.00128
[5] Baddeley, A. and Turner, R. (2005). Spatstat: An R package for analyzing spatial point patterns. J. Statist. Softw. 12 1-42.
[6] Berman, M. and Turner, R. (1992). Approximating point process likelihoods with GLIM. J. Appl. Stat. 41 31-38. · Zbl 0825.62614 · doi:10.2307/2347614
[7] Besag, J. (1977). Some methods of statistical analysis for spatial data. Bulletin of the International Statistical Institute 47 77-92.
[8] Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047-1050. · Zbl 0496.60020 · doi:10.1214/aop/1176993726
[9] Daley, D.J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I : Elementary Theory and Methods , 2nd ed. Probability and Its Applications ( New York ). New York: Springer. · Zbl 1026.60061
[10] Diggle, P.J. (2003). Statistical Analysis of Spatial Point Patterns , 2nd ed. London: Arnold. · Zbl 1021.62076
[11] Diggle, P.J. (2010). Nonparametric methods. In Handbook of Spatial Statistics (A.E. Gelfand, P.J. Diggle, P. Guttorp and M. Fuentes, eds.). Chapman & Hall/CRC Handb. Mod. Stat. Methods 299-316. Boca Raton, FL: CRC Press. · doi:10.1201/9781420072884-c18
[12] Doukhan, P. (1994). Mixing : Properties and Examples. Lecture Notes in Statistics 85 . New York: Springer. · Zbl 0801.60027
[13] Guan, Y. (2006). A composite likelihood approach in fitting spatial point process models. J. Amer. Statist. Assoc. 101 1502-1512. · Zbl 1171.62348 · doi:10.1198/016214506000000500
[14] Guan, Y., Jalilian, A. and Waagepetersen, R. (2011). Optimal estimation of the intensity function of a spatial point process. Research Report R-2011-14, Dept. Mathematical Sciences, Aalborg Univ.
[15] Guan, Y. and Loh, J.M. (2007). A thinned block bootstrap variance estimation procedure for inhomogeneous spatial point patterns. J. Amer. Statist. Assoc. 102 1377-1386. · Zbl 1332.62108 · doi:10.1198/016214507000000879
[16] Guan, Y. and Shen, Y. (2010). A weighted estimating equation approach for inhomogeneous spatial point processes. Biometrika 97 867-880. · Zbl 1204.62149 · doi:10.1093/biomet/asq043
[17] Guyon, X. (1995). Random Fields on a Network : Modeling , Statistics , and Applications. Probability and Its Applications ( New York ). New York: Springer. · Zbl 0839.60003
[18] Heinrich, L. (1992). On existence and mixing properties of germ-grain models. Statistics 23 271-286. · Zbl 0811.60034 · doi:10.1080/02331889208802375
[19] Hörmander, L. (2003). The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis . Berlin: Springer. Reprint of the second (1990) edition. · Zbl 0712.35001
[20] Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables . Groningen: Wolters-Noordhoff. · Zbl 0219.60027
[21] Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Statistics in Practice . Chichester: Wiley. · Zbl 1197.62135
[22] Jensen, J.L. (1993). Asymptotic normality of estimates in spatial point processes. Scand. J. Stat. 20 97-109. · Zbl 0814.62063
[23] Jensen, J.L. and Künsch, H.R. (1994). On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes. Ann. Inst. Statist. Math. 46 475-486. · Zbl 0820.62083
[24] Jensen, J.L. and Møller, J. (1991). Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1 445-461. · Zbl 0736.60045 · doi:10.1214/aoap/1177005877
[25] Karácsony, Z. (2006). A central limit theorem for mixing random fields. Miskolc Math. Notes 7 147-160. · Zbl 1120.41301
[26] Møller, J., Syversveen, A.R. and Waagepetersen, R.P. (1998). Log Gaussian Cox processes. Scand. J. Stat. 25 451-482. · Zbl 0931.60038 · doi:10.1111/1467-9469.00115
[27] Møller, J. and Waagepetersen, R.P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100 . Boca Raton, FL: Chapman & Hall/CRC. · Zbl 1044.62101
[28] Møller, J. and Waagepetersen, R.P. (2007). Modern statistics for spatial point processes. Scand. J. Stat. 34 643-684. · Zbl 1157.62067
[29] Politis, D.N., Paparoditis, E. and Romano, J.P. (1998). Large sample inference for irregularly spaced dependent observations based on subsampling. Sankhyā Ser. A 60 274-292. · Zbl 1058.62549
[30] Prokešová, M. and Jensen, E.B.V. (2013). Asymptotic Palm likelihood theory for stationary point processes. Ann. Inst. Statist. Math. 65 387-412. · Zbl 1440.62343 · doi:10.1007/s10463-012-0376-7
[31] Schoenberg, F.P. (2005). Consistent parametric estimation of the intensity of a spatial-temporal point process. J. Statist. Plann. Inference 128 79-93. · Zbl 1058.62069 · doi:10.1016/j.jspi.2003.09.027
[32] Stoyan, D., Kendall, W.S. and Mecke, J. (1995). Stochastic Geometry and Its Applications , 2nd ed. Chichester: Wiley. · Zbl 0838.60002
[33] Waagepetersen, R. and Guan, Y. (2009). Two-step estimation for inhomogeneous spatial point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 685-702. · Zbl 1250.62047 · doi:10.1111/j.1467-9868.2008.00702.x
[34] Waagepetersen, R.P. (2007). An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63 252-258, 315. · Zbl 1122.62073 · doi:10.1111/j.1541-0420.2006.00667.x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.