##
**Statistics for spatial data.**
*(English)*
Zbl 0799.62002

Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons, Inc. (ISBN 0-471-84336-9/hbk). xx, 900 p. (1991).

The book is divided into three parts dealing with statistics for geostatistical data, lattice data, point patterns and other geometrical objects, respectively. The basic stochastic model of the first part (Chapters 2 through 5) is that of a vector-valued random field \(\{\mathbb{Z} (s) : s \in D\}\) where \(D\) is a non-random subset of \(\mathbb{R}^ d\) that contains a \(d\)-dimensional rectangle of positive volume, i.e. \(\mathbb{Z} (s)\) describes the random state at location \(s \in D\). Mainly, second-order stationary random fields are considered, i.e. the expectation \(E(\mathbb{Z}(s))\) does not depend on \(s \in D\), and the covariance of \(\mathbb{Z} (s_ 1)\) and \(Z(s_ 2)\) depends on the difference \(s_ 1 - s_ 2\) only. Important characteristics of such random fields are the notions of variogram and covariogram. It is discussed how they can be estimated from the data and how a variogram model can be fitted.

Chapter 3 then deals with questions of spatial prediction and, in particular, with several variants of kriging. Simulation techniques for random fields are also discussed. Substantial applications of kriging are given in Chapter 4, e.g. for Wolfcamp-aquifer data, wheat-yield data, and acid-deposition data. In Chapter 5, some special topics in statistics for spatial data are considered, like nonlinear geostatistics, problems auf change-of-support, stability of the kriging predictor and kriging variance, spatial sampling design, nearest-neighbor analysis.

The second part of the book (Chapters 6 and 7) is concerned with the situation that the index set \(D\) is a lattice, i.e. a countable collection of locations which are connected by edges. First, various stochastic models on lattices are introduced like conditionally and simultaneously specified spatial Gaussian and non-Gaussian models, Markov random fields, models for data on regular and irregular lattices, regression models with spatial autoregressive-moving-average errors, space-time models for lattice data. Then the basic tools of inference for lattice models are given, including parameter estimation and related statistical properties. The fundamentals of statistical image analysis and remote sensing are presented which are an important application of Markov random fields. Methods of handling epidemiological data on irregular lattices are also discussed. Different methods of regional mapping are compared to each other.

The third part (Chapters 8 and 9) is devoted to statistical analysis of spatial patterns. Formally the stochastic models considered in these two chapters can again be understood as a random field \(\{\mathbb{Z}(s) : s \in D\}\) where the index set \(D\) is random. In Chapter 8 special emphasis is given to the case when \(D\) is a random point process, i.e. the realizations of \(D\) are spatial point patterns. An illustrative spatial data analysis of longleaf pines introduces into these topics where such notions like complete spatial randomness, regularity and clustering of point patterns are discussed.

Next a mathematically elegant approach to spatial point processes is presented which is based on random (counting) measures. Several point- process models are discussed like Poisson processes, Cox processes, Poisson cluster processes, Markov point processes, stationary and isotropic point processes, marked point processes. The basic characteristics of point processes are surveyed including the notions of moment measures, generating functionals, Palm distributions, nearest- neighbor distribution functions. It is discussed how the parameters of point-process models can be fitted from an observed point pattern. Algorithms for simulating point processes are given.

The final Chapter 9 deals with the more general model of random closed sets. Non-random set models (fractal sets, fuzzy sets) are also mentioned. However, particular attention receives the Boolean model which can be seen as the union of random grains which are located at the points of a homogeneous Poisson process. Main properties of the Boolean model are discussed, including a discussion of parameter estimation and inference. Applications to tumor-growth data are given.

The book surveys the approaches and the enormous diversity of problems connected with statistical analysis of spatial data. It assumes that the reader is familiar with basic notions and techniques in probability and mathematical statistics. Some additional knowledge on stochastic processes helps in reading the more theoretical sections of the book. Most chapters begin with an application. All data sets for the various spatial statistical analyses are given. The book is equipped with a comprehensive list of over 1300 references. An author and a subject index are contained as well.

Chapter 3 then deals with questions of spatial prediction and, in particular, with several variants of kriging. Simulation techniques for random fields are also discussed. Substantial applications of kriging are given in Chapter 4, e.g. for Wolfcamp-aquifer data, wheat-yield data, and acid-deposition data. In Chapter 5, some special topics in statistics for spatial data are considered, like nonlinear geostatistics, problems auf change-of-support, stability of the kriging predictor and kriging variance, spatial sampling design, nearest-neighbor analysis.

The second part of the book (Chapters 6 and 7) is concerned with the situation that the index set \(D\) is a lattice, i.e. a countable collection of locations which are connected by edges. First, various stochastic models on lattices are introduced like conditionally and simultaneously specified spatial Gaussian and non-Gaussian models, Markov random fields, models for data on regular and irregular lattices, regression models with spatial autoregressive-moving-average errors, space-time models for lattice data. Then the basic tools of inference for lattice models are given, including parameter estimation and related statistical properties. The fundamentals of statistical image analysis and remote sensing are presented which are an important application of Markov random fields. Methods of handling epidemiological data on irregular lattices are also discussed. Different methods of regional mapping are compared to each other.

The third part (Chapters 8 and 9) is devoted to statistical analysis of spatial patterns. Formally the stochastic models considered in these two chapters can again be understood as a random field \(\{\mathbb{Z}(s) : s \in D\}\) where the index set \(D\) is random. In Chapter 8 special emphasis is given to the case when \(D\) is a random point process, i.e. the realizations of \(D\) are spatial point patterns. An illustrative spatial data analysis of longleaf pines introduces into these topics where such notions like complete spatial randomness, regularity and clustering of point patterns are discussed.

Next a mathematically elegant approach to spatial point processes is presented which is based on random (counting) measures. Several point- process models are discussed like Poisson processes, Cox processes, Poisson cluster processes, Markov point processes, stationary and isotropic point processes, marked point processes. The basic characteristics of point processes are surveyed including the notions of moment measures, generating functionals, Palm distributions, nearest- neighbor distribution functions. It is discussed how the parameters of point-process models can be fitted from an observed point pattern. Algorithms for simulating point processes are given.

The final Chapter 9 deals with the more general model of random closed sets. Non-random set models (fractal sets, fuzzy sets) are also mentioned. However, particular attention receives the Boolean model which can be seen as the union of random grains which are located at the points of a homogeneous Poisson process. Main properties of the Boolean model are discussed, including a discussion of parameter estimation and inference. Applications to tumor-growth data are given.

The book surveys the approaches and the enormous diversity of problems connected with statistical analysis of spatial data. It assumes that the reader is familiar with basic notions and techniques in probability and mathematical statistics. Some additional knowledge on stochastic processes helps in reading the more theoretical sections of the book. Most chapters begin with an application. All data sets for the various spatial statistical analyses are given. The book is equipped with a comprehensive list of over 1300 references. An author and a subject index are contained as well.

Reviewer: V.Schmidt (Ulm)

### MSC:

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

62H11 | Directional data; spatial statistics |

62M30 | Inference from spatial processes |

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

86A32 | Geostatistics |

62M40 | Random fields; image analysis |

60G57 | Random measures |