The aim of this book is to present some statistical methods in a way that may also be understood by non-mathematicians, in particular by materials scientists, geologists, environmental researchers and biologists. We assume that the reader has a basic knowledge of mathematics and statistics, although some concepts and methods that may be unfamiliar to non-mathematicians are explained in the appendices. Our aim was to write a clear and popular text that is nevertheless mathematically correct. Although many parts of the book may interest applied mathmaticians or statisticians, these readers have to accept that this book does not contain proofs - it merely outlines the mathematical ideas. We treat three different subjects: fractals, random shapes and point fields (processes). In discussing these we always restrict attention to planar structures. From the reactin to the book Stochastic Geometry and its Application by Stoyan, Kendall and Mecke, we know that many applied researchers are deeply interested in the first two topics. Part I gives an introduction to the theory of fractals. This should familiarise the reader with the methods of measuring fractal dimensions. These are used to describe extremely irregular geometric structures. Furthermore, important mathematical models involving fractals are explained, including some of a stochastic nature. We explain the notion of fractal dimension in more detail than is customary for applied mathematicians. Thus Part I has some difficult passages. However, a reader interested only in applications is led quickly to the measurement techniques. In Part II we recount important modern methods for the statistical analysis of random shapes. Random shapes are studied in such diverse areas as biology and praticle science. Biological shapes result from growth process, so that often the geometries correspond to life functions; typically such ojects have points on their contours or in their interiors that play specific biological roles. Such points do not usually occur in particles resulting from geological or technological processes. We consider three approaches: (1) describing objects by their contours and using methods for the analysis of functions; (2) considering them as random compact sets; and (3) describing them as k-tuples of points. These points, frequently called landmarks, are characteristic points on the boundaries or in the interiors of the objects. The mathematics behind these methods is not simple, and hence some expositions are given in outline only. Those interested should consult the relevant references. On the other hand, we hope that our treatment will encourage non-mathematicians to sue these statistical methods. Finally, Part III presents an introduction to the statistical theory of point fields , with and without marks. An important area of application is the analysis of homogeneous systems of particles where the "points" are particle centres and the "marks" particle characteristics such as size or orientation. We try to give the theory in an elementary form, with emphasis on the aspects of analysis currently of greatest interest. We also discuss some important classes of point field models, in particular Gibbsian processes. Here some quite modern results are presented. Furthermore, Part III contains, in condensed form, an exposition of the theory of correlations of marked point fields and their statistics, which has been developed during the last decade.
