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**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)].

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)].

Reviewer: Viktor Ohanyan (Erevan)

### 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 |

### Keywords:

complementary point process; coupling; local stability; Papangelou conditional intensity; spatial birth-death process; Strauss process
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\textit{J. Møller} and \textit{K. K. Berthelsen}, Adv. Appl. Probab. 44, No. 1, 42--62 (2012; Zbl 1239.60034)

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