Geometrical objects and structures are studied in many branches of science and
engineering. Often these objects have non-regular random shapes or are randomly
scattered in space. If the number of the objects under study is very great then
statistical analysis makes sense and is indeed necessary. Many methods exist for
such analyses; some of the more recent attempt to consider spatial dependences or
other complicated correlations.
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 mathematicians 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 reaction to the book Stochastic Geometry and its Applications 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 mathe-
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 important modern methods for the statistical analysis of