## Transforming spatial point processes into Poisson processes using random superposition.(English)Zbl 1239.60034

C. J. Preston [“Spatial birth-and-death processes”, Bull. Int. Stat. Inst., Proc. of the 40th Session Warsaw 46, No. 2, 371–391 (1975; Zbl 0379.60082)] established the existence of a spatial birth-death process through a coupling to a nonexplosive birth-death process on the nonnegative integers which can be extended to a dominating spatial birth-death process. This coupling is particularly useful in connection with point processes that are locally stable, the latter being a property satisfied by most spatial point process models specified by a density.
This condition and other background material are presented in Section 2. In short, local stability implies that the Papangelou conditional intensity $$\lambda(x, u)$$ is bounded from above by an integrable function $$\beta(u)$$ defined on $$S$$, where $$S$$ denotes the state space of the points, $$x$$ is any finite point pattern (i.e., a finite subset of $$S$$), and $$u\in S\setminus x$$ is any point. To describe the coupling construction, consider a dominating birth-death process $$D_t$$ with birth rate $$\beta(u)$$ and death rate 1 so that its equilibrium distribution is a Poisson process on $$S$$ with intensity function $$\beta$$. It is possible to obtain a (target) birth-death process $$X_t$$ with birth rate $$\lambda(x, u)$$ and death rate 1 such that its distribution converges towards that of $$X$$ as time $$t$$ tends to $$\infty$$. Further details are given in Section 2.2.
In this paper, the authors study the so-called complementary birth-death process $$Y_t= D_t\setminus X_t$$, i.e., the points in the dominating process $$D_t$$ that are not included in the target birth-death process $$X_t$$. In Section 3, the authors establish that the bivariate birth-death process $$(X_t, Y_t)$$ converges towards a bivariate birth-death process $$(X, Y)$$. In general, it seems difficult to say anything detailed about this equilibrium distribution except in the spacial cases considered in Section 3 and Appendix A. Although the distribution of $$Y$$ conditional on $$X=x$$ seems complicated in general, it turns out to be simple to simulate from this conditional distribution. In Section 4.1, the authors present an algorithm which is both fast and easy to implemented. The algorithm can be used for model checking: given data $$x$$ and a model for the Papangelou intensity of the underlying spatial point process $$X$$, this model is used for generating a realization $$y$$ from the complementary process conditional on $$X=x$$. In Section 5.1, the authors prove that the resulting superposition $$x\cup y$$ is a Poisson process with intensity function $$\beta$$ if and only if the true Papangelou intensity is used. The above model checking procedure has some similarities to the approach considered by J. Møller and F. P. Schoenberg [“Thinning spatial point processes into Poisson processes”, Adv. Appl. Probab. 42, No. 2, 347–358 (2010; Zbl 1239.60035)].

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 62M99 Inference from stochastic processes 62M30 Inference from spatial processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 62H11 Directional data; spatial statistics

### Citations:

Zbl 0379.60082; Zbl 1239.60035

### Software:

spatstat; spatial
Full Text:

### References:

 [1] Baddeley, A. and Turner, R. (2005). Spatstat: an R package for analyzing spatial point patterns. J. Statist. Software 12 , 1-42. [2] Baddeley, A. and Turner, R. (2006). Modelling spatial point patterns in R. In Case Studies in Spatial Point Process Modeling (Lecture Notes Statist. 185 ), eds A. Baddeley et al. , Springer, New York, pp. 23-74. · Zbl 05243453 [3] Berthelsen, K. K. and Møller, J. (2002). A primer on perfect simulation for spatial point processes. Bull. Brazilian Math. Soc. 33, 351-367. · Zbl 1042.60028 [4] Besag, J. (1977). Some methods of statistical analysis for spatial data. Bull. Internat. Statist. Inst. 47, 77-91. [5] Carter, D. S. and Prenter, P. M. (1972). Exponential spaces and counting processes. Z. Wahrscheinlichkeitsth. 21, 1-19. · Zbl 0213.19301 [6] Ferrari, P. A., Fernández, R. and Garcia, N. L. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 63-88. · Zbl 1075.60583 [7] Fiksel, T. (1984). Simple spatial-temporal models for sequences of geological events. Elektron. Informationsverarb. Kypernet. 20, 480-487. · Zbl 0556.62100 [8] Geyer, C. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry (Monogr. Statist. Appl. Prob. 80 ), eds O. E. Barndorff-Nielsen, W. S. Kendall, and M. N. M. van Lieshout, Chapman and Hall/CRC, Boca Raton, FL, pp. 79-140. · Zbl 0809.62089 [9] Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns . John Wiley, Chichester. · Zbl 1197.62135 [10] Kelly, F. P. and Ripley, B. D. (1976). A note on Strauss’ model for clustering. Biometrika 63, 357-360. · Zbl 0332.60034 [11] Kendall, W. S. (1998). Perfect simulation for the area-interaction point process. In Probability Towards 2000 (Lecture Notes Statist. 128 ), eds L. Accardi and C. C. Heyde, Springer, New York, pp. 218-234. · Zbl 1045.60503 [12] Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844-865. · Zbl 1123.60309 [13] Møller, J. (1989). On the rate of convergence of spatial birth-and-death processes. Ann. Inst. Statist. Math. 41, 565-581. · Zbl 0702.60072 [14] Møller, J. and Schoenberg, R. P. (2010). Thinning spatial point processes into Poisson processes. Adv. Appl. Prob. 42, 347-358. · Zbl 1239.60035 [15] Møller, J. and Sørensen, M. (1994). Parametric models of spatial birth-and-death processes with a view to modelling linear dune fields. Scand. J. Statist. 21, 1-19. · Zbl 0790.62090 [16] Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes . Chapman and Hall/CRC, Boca Raton, FL. · Zbl 1044.62101 [17] Preston, C. J. (1977). Spatial birth-and-death processes. Bull. Internat. Statist. Inst. 46, 371-391. · Zbl 0379.60082 [18] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223-252. · Zbl 0859.60067 [19] Ripley, B. D. (1977). Modelling spatial patterns (with discussion). J. R. Statist. Soc. B 39, 172-212. · Zbl 0369.60061 [20] Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. London Math. Soc. 15, 188-192. · Zbl 0354.60037 [21] Stephens, M. (2000). Bayesian analysis of mixture models with an unknown number of components–an alternative to reversible jump methods. Ann. Statist. 28, 40-74. · Zbl 1106.62316 [22] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications , 2nd edn. John Wiley, Chichester. · Zbl 0838.60002 [23] Strauss, D. J. (1975). A model for clustering. Biometrika 62, 467-475. · Zbl 0313.62044
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