Inference for cluster point processes with over- or under-dispersed cluster sizes. (English) Zbl 1452.62431
Summary: Cluster point processes comprise a class of models that have been used for a wide range of applications. While several models have been studied for the probability density function of the offspring displacements and the parent point process, there are few examples of non-Poisson distributed cluster sizes. In this paper, we introduce a generalization of the Thomas process, which allows for the cluster sizes to have a variance that is greater or less than the expected value. We refer to this as the cluster sizes being over- and under-dispersed, respectively. To fit the model, we introduce minimum contrast methods and a Bayesian MCMC algorithm. These are evaluated in a simulation study. It is found that using the Bayesian MCMC method, we are in most cases able to detect over- and under-dispersion in the cluster sizes. We use the MCMC method to fit the model to nerve fiber data, and contrast the results to those of a fitted Thomas process.
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 65C05 | Monte Carlo methods |
Keywords:
Bayesian estimation; generalized Poisson distribution; Markov chain Monte Carlo; minimum contrast estimation; Neyman-Scott point process; Thomas processReferences:
| [1] | Andersson, C., Rajala, T., Särkkä, A.: A Bayesian hierarchical point process model for epidermal nerve fiber patterns. Math. Biosci. 313, 48-60 (2019). 10.1016/j.mbs.2019.04.010 · Zbl 1425.92037 |
| [2] | Baddeley, A.; Rubak, E.; Turner, R., Spatial Point Patterns: Methodology and Applications with R (2015), London: Chapman and Hall/CRC Press, London |
| [3] | Consul, PC; Jain, GC, A generalization of the Poisson distribution, Technometrics, 15, 4, 791-799 (1973) · Zbl 0271.60020 · doi:10.1080/00401706.1973.10489112 |
| [4] | Diggle, PJ; Gratton, RJ, Monte Carlo methods of inference for implicit statistical models, J. R. Stat. Soc. Ser. B, 46, 2, 193-227 (1984) · Zbl 0561.62035 |
| [5] | Dvořák, J.; Prokešová, M., Moment estimation methods for stationary spatial Cox processes—a comparison, Kybernetika, 48, 5, 1007-1026 (2012) · Zbl 1297.62201 |
| [6] | Eddelbuettel, D., Seamless R and C++ Integration with Rcpp (2013), New York: Springer, New York · Zbl 1283.62001 |
| [7] | Eddelbuettel, D.; François, R., Rcpp: seamless R and C++ integration, J. Stat. Softw., 40, 8, 1-18 (2011) · doi:10.18637/jss.v040.i08 |
| [8] | Guan, Y., A composite likelihood approach in fitting spatial point process models, J. Am. Stat. Assoc., 101, 476, 1502-1512 (2006) · Zbl 1171.62348 · doi:10.1198/016214506000000500 |
| [9] | Guttorp, P.; Thorarinsdottir, TL; Porcu, E.; Montero, J-M; Schlather, M., Bayesian inference for non-markovian point processes, Advances and Challenges in Space-time Modelling of Natural Events, 79-102 (2012), New York: Springer, New York |
| [10] | Illian, J.; Penttinen, A.; Stoyan, H.; Stoyan, D., Statistical Analysis and Modelling of Spatial Point Patterns (2008), New York: Wiley, New York · Zbl 1197.62135 |
| [11] | Joe, H.; Zhu, R., Generalized poisson distribution: the property of mixture of poisson and comparison with negative binomial distribution, Biom. J., 47, 2, 219-229 (2005) · Zbl 1442.62431 · doi:10.1002/bimj.200410102 |
| [12] | Lund, A.: Spatio-temporal modeling of neuron fields, Ph.D. thesis, Copenhagen. ISBN: 978-87-7078-934-9 (2017) |
| [13] | Matérn, B.: Spatial variation, Meddelanden från statens skogsforskningsinstitut 4(5). Statens Skogsforskningsinstitut, Stockholm (1960) |
| [14] | Matérn, B.: Spatial Variation, Volume 36 of Lecture Notes in Statistics (1986) · Zbl 0608.62122 |
| [15] | Milne, R.; Westcott, M., Further results for Gauss-Poisson processes, Adv. Appl. Probab., 4, 1, 151-176 (1972) · Zbl 0244.60041 · doi:10.2307/1425809 |
| [16] | Møller, J., Shot noise Cox processes, Adv. Appl. Probab., 35, 3, 614-640 (2003) · Zbl 1045.60007 · doi:10.1239/aap/1059486821 |
| [17] | Møller, J.; Torrisi, GL, Generalised shot noise Cox processes, Adv. Appl. Probab., 37, 1, 48-74 (2005) · Zbl 1078.60012 · doi:10.1239/aap/1113402399 |
| [18] | Møller, J.; Waagepetersen, RP, Statistical Inference and Simulation for Spatial Point Processes (2003), Boca Raton: CRC Press, Boca Raton |
| [19] | Mrkvička, T., Distinguishing different types of inhomogeneity in Neyman-Scott point processes, Methodol. Comput. Appl. Probab., 16, 2, 385-395 (2014) · Zbl 1305.62337 · doi:10.1007/s11009-013-9365-4 |
| [20] | Mrkvička, T.; Soubeyrand, S., On parameter estimation for doubly inhomogeneous cluster point processes, Spat. Stat., 20, 191-205 (2017) · doi:10.1016/j.spasta.2017.03.005 |
| [21] | Mrkvička, T.; Muška, M.; Kubečka, J., Two step estimation for Neyman-Scott point process with inhomogeneous cluster centers, Stat. Comput., 24, 1, 91-100 (2014) · Zbl 1325.62077 · doi:10.1007/s11222-012-9355-3 |
| [22] | Mrkvička, T.; Myllymäki, M.; Hahn, U., Multiple Monte Carlo testing, with applications in spatial point processes, Stat. Comput., 27, 5, 1239-1255 (2017) · Zbl 1369.62094 · doi:10.1007/s11222-016-9683-9 |
| [23] | Myllymäki, M.; Mrkvicka, T.; Grabarnik, P.; Seijo, H.; Hahn, U., Global envelope tests for spatial processes, J. R. Stat. Soc. Ser. B (Stat. Methodol.), 79, 381-404 (2017) · Zbl 1414.62404 · doi:10.1111/rssb.12172 |
| [24] | Neyman, J., Scott, E.L.: Statistical approach to problems of cosmology. J. R. Stat. Soc. Ser. B (Methodol.) 1-43 (1958) · Zbl 0085.42906 |
| [25] | Neyman, J.; Scott, E., A theory of the spatial distribution of galaxies, Astrophys. J., 116, 144-163 (1952) · doi:10.1086/145599 |
| [26] | R Core Team: R: a Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2017). https://www.R-project.org/ |
| [27] | Ripley, BD, The second-order analysis of stationary point processes, J. Appl. Probab., 13, 2, 255-266 (1976) · Zbl 0364.60087 · doi:10.2307/3212829 |
| [28] | Robert, Ch; Casella, G., Monte Carlo Statistical Methods (2004), New York: Springer, New York · Zbl 1096.62003 |
| [29] | Scollnik, DP, On the analysis of the truncated generalized Poisson distribution using a Bayesian method, ASTIN Bull. J. IAA, 28, 1, 135-152 (1998) · Zbl 1168.60311 · doi:10.2143/AST.28.1.519083 |
| [30] | Tanaka, U.; Ogata, Y.; Stoyan, D., Parameter estimation and model selection for Neyman-Scott point processes, Biom. J., 50, 1, 43-57 (2008) · Zbl 1442.62645 · doi:10.1002/bimj.200610339 |
| [31] | Thomas, M., A generalization of Poisson’s binomial limit for use in ecology, Biometrika, 36, 18-25 (1949) · doi:10.1093/biomet/36.1-2.18 |
| [32] | van Lieshout, M.; Baddeley, AJ; Lawson, A.; Denison, D., Extrapolating and interpolating spatial patterns, Spatial Cluster Modelling, 61-86 (2001), Boca Raton: Chapman and Hall/CRC, Boca Raton |
| [33] | Waagepetersen, R.; Schweder, T., Likelihood-based inference for clustered line transect data, J. Agric. Biol. Environ. Stat., 11, 3, 264 (2006) · doi:10.1198/108571106X130557 |
| [34] | Wang, W.; Famoye, F., Modeling household fertility decisions with generalized Poisson regression, J. Popul. Econ., 10, 3, 273-283 (1997) · doi:10.1007/s001480050043 |
| [35] | Wickham, H., ggplot2: Elegant Graphics for Data Analysis (2016), New York: Springer, New York · Zbl 1397.62006 |
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.
