×

Statistics for inhomogeneous space-time shot-noise Cox processes. (English) Zbl 1305.62338

Summary: In the paper we introduce a flexible inhomogeneous space-time shot-noise Cox process model and derive a two-step estimation procedure for it. In the first step the inhomogeneity is estimated by means of a Poisson score estimating equation and in the second step we use minimum contrast estimation based on second order properties to obtain estimates of the clustering parameters. The suggested model is not separable but it has a special interaction structure which enables to use the spatial and temporal projections of the process for parameter estimation. Efficiency of the introduced method is investigated by means of a simulation study and it is compared to a previously used method.

MSC:

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

Software:

spatstat
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baddeley AJ, Møller J, Waagepetersen R (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Stat Neerlandica 54:329-350 · Zbl 1018.62027 · doi:10.1111/1467-9574.00144
[2] Baddeley AJ, Turner R (2005) Spatstat: an R package for analyzing spatial point patterns. J Stat Softw 12:1-42
[3] Brix A (1999) Generalized gamma measures and shot-noise Cox processes. Adv Appl Probab 31:929-953 · Zbl 0957.60055 · doi:10.1239/aap/1029955251
[4] Brix A, Diggle PJ (2001) Spatiotemporal prediction for log-Gaussian Cox processes. J R Stat Soc B 63:823-841 · Zbl 0996.62076 · doi:10.1111/1467-9868.00315
[5] Brix A, Møller J (2001) Space-time multi type log Gaussian Cox processes with a view to modelling weed data. Scand J Statist 28:471-488 · Zbl 0981.62079 · doi:10.1111/1467-9469.00249
[6] Daley DJ, Vere-Jones D (2003) An introduction to the theory of point processes. Elementary Theory and Methods, vol 1. Springer, New York · Zbl 1026.60061
[7] Daley DJ, Vere-Jones D (2008) An introduction to the theory of point processes. General Theory and Structure, vol 2. Springer, New York · Zbl 1159.60003 · doi:10.1007/978-0-387-49835-5
[8] Diggle PJ, Chetwynd AG, Häggkvist R, Morris S (1995) Second-order analysis of space-time clustering. Stat Methods Med Res 4:124-136 · doi:10.1177/096228029500400203
[9] Diggle P (2003) Statistical analysis of spatial point patterns, 2nd edn. Oxford University Press, New York · Zbl 1021.62076
[10] Diggle, P.; Finkenstädt, B. (ed.); Held, L. (ed.); Isham, V. (ed.), Spatio-temporal point processes: methods and applications (2007), Boca Raton · Zbl 1121.62080
[11] Dvořák J, Prokešová M (2012) Moment estimation methods for stationary spatial Cox processes—a comparison. Kybernetika 48:1007-1026 · Zbl 1297.62201
[12] Gabriel E, Diggle P (2009) Second-order analysis of inhomogeneous spatio-temporal point process data. Stat Neerlandica 63:43-51 · doi:10.1111/j.1467-9574.2008.00407.x
[13] Guan Y (2009) A minimum contrast estimation procedure for estimating the second-order parameters of inhomogeneous spatial point processes. Stat Interface 2:91-99 · Zbl 1245.62101 · doi:10.4310/SII.2009.v2.n1.a9
[14] Hellmund G, Prokešová M, Vedel Jensen EB (2008) Lévy-based Cox point processes. Adv Appl Probab 40:603-629 · Zbl 1149.60031 · doi:10.1239/aap/1222868178
[15] Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, Chichester · Zbl 1197.62135
[16] Møller J, Syversveen AR, Waagepetersen RP (1998) Log-Gaussian Cox processes. Scand J Statist 25:451-482 · Zbl 0931.60038 · doi:10.1111/1467-9469.00115
[17] Møller J (2003) Shot noise Cox processes. Adv Appl Probab 35:614-640 · Zbl 1045.60007 · doi:10.1239/aap/1059486821
[18] Møller J, Waagepetersen RP (2003) Statistical inference and simulation for spatial point processes. Chapman and Hall/CRC, Florida · doi:10.1201/9780203496930
[19] Møller J, Ghorbani M (2012) Aspects of second-order analysis of structured inhomogeneous spatio-temporal point processes. Stat Neerlandica 66:472-491 · doi:10.1111/j.1467-9574.2012.00526.x
[20] Prokešová M (2010) Inhomogeneity in spatial Cox point processes—location dependent thinning is not the only option. Image Anal Ster 29:133-141 · Zbl 1219.60051 · doi:10.5566/ias.v29.p133-141
[21] Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields: methods of geometrical statistics. Wiley, Chichester · Zbl 0828.62085
[22] Thomas M (1949) A generalization of poisson’s binomial limit for use in ecology. Biometrika 36:18-25 · doi:10.1093/biomet/36.1-2.18
[23] Waagepetersen RP, Guan Y (2009) Two-step estimation for inhomogeneous spatial point processes. J R Stat Soc B 71:685-702 · Zbl 1250.62047 · doi:10.1111/j.1467-9868.2008.00702.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.