Statistical inference and simulation for spatial point processes.

*(English)*Zbl 1044.62101
Monographs on Statistics and Applied Probability 100. Boca Raton, FL: Chapman and Hall/CRC (ISBN 1-58488-265-4/pbk; 978-0-203-49693-0/ebook). xv, 300 p. (2004).

The book is concerned with simulation-based statistical inference for spatial point processes, with an emphasis on Markov chain Monte Carlo (MCMC) methods. It collects and unifies recent theoretical advances in this area and shows examples of their applications. Chapter 1 introduces examples of spatial point patterns which are used throughout the book for illustrative purposes. Chapter 2 gives an informal introduction to point and marked point processes. The most important properties of Poisson point processes are studied in Chapter 3. Chapter 4 introduces the most commonly used so-called summary statistics, which express second order characteristics and distributions of interpoint distances for point processes and multitype point processes. Nonparametric estimation of summary statistics is considered. Chapters 5 and 6 deal with the main properties of Cox processes and Markov point processes, respectively. The authors consider these two classes of processes as the most useful model classes. Chapter 7 is concerned with the Metropolis-Hastings algorithm for simulation of spatial point processes with an unnormalised density. Some background material on Markov chain theory is included. Chapter 8 introduces how MCMC methods are used in approximate likelihood-based inference. Simulation-based inference for Markov point processes and Cox processes is studied in Chapter 9 and 10, respectively. The main approaches here are maximum likelihood, pseudo likelihood and minimum contrast estimation. Chapter 11 deals with recent advances of perfect simulation for spatial point processes. Some technical topics are collected in the Appendices.

The book is well and clearly written. As is seen from the contents, it contains all the background material. The authors give proofs of almost all mathematical results. This makes the book fairly self-contained. However, to make the book accessible to a wide audience, the authors restrict attention to point processes on a Euclidean space and confine measure theoretical details to Appendices. The book will be useful not only to experienced statisticians and applied probabilists but also to students specialising in statistics.

The book is well and clearly written. As is seen from the contents, it contains all the background material. The authors give proofs of almost all mathematical results. This makes the book fairly self-contained. However, to make the book accessible to a wide audience, the authors restrict attention to point processes on a Euclidean space and confine measure theoretical details to Appendices. The book will be useful not only to experienced statisticians and applied probabilists but also to students specialising in statistics.

Reviewer: Alexander Gushchin (Moskva)

##### MSC:

62M30 | Inference from spatial processes |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

65C40 | Numerical analysis or methods applied to Markov chains |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

65C05 | Monte Carlo methods |

60J22 | Computational methods in Markov chains |

60D05 | Geometric probability and stochastic geometry |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |