Point Processes

D.R. Cox, Valerie Isham

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351423854
• Edition: 1
• Topic: Mathematik
• Subtopic: Wahrscheinlichkeitsrechnung & Statistiken

There has been much recent research on the theory of point processes, i.e., on random systems consisting of point events occurring in space or time. Applications range from emissions from a radioactive source, occurrences of accidents or machine breakdowns, or of electrical impluses along nerve fibres, to repetitive point events in an individual's medical or social history. Sometimes the point events occur in space rather than time and the application here raneg from statistical physics to geography. The object of this book is to develop the applied mathemathics of point processes at a level which will make the ideas accessible both to the research worker and the postgraduate student in probability and statistics and also to the mathemathically inclined individual in another field interested in using ideas and results. A thorough knowledge of the key notions of elementary probability theory is required to understand the book, but specialised "pure mathematical" coniderations have been avoided.

CHAPTER 1

Introduction

1.1 Preliminary remarks

This book deals with a particular kind of random process. The central idea is to study random collections of point occurrences. For the most part we consider the points as occurring along a time axis, although later we do allow other possibilities, for example that the points occur in some region of space.

A few outline examples illustrate the breadth of potential applications:

(i) emissions from a radioactive source occur in an irregular sequence in time, each emission defining a time instant;

(ii) Fig. 1.1 shows a small section of a time series of electrical energy in a nerve fibre. The occurrences of peaks define a sequence of points in time. If attention is concentrated on this sequence of time points, rather than on the magnitudes of the peak signals, we have a point process derived from a more complicated process;

Fig. 1.1 Electrical signal in nerve fibre. ↑, points ofassociated point process.

(iii) Fig. 1.2 shows part of the sequence of dates of coal-mining disasters in Great Britain for 1851-1975, a disaster being defined as involving the death of 10 or more men;

Fig. 1.2 Coal mining disasters in U.K. killing 10 or more. Extractfrom larger set of data (Jarrett, 1979). ↑, times of disasters.

(iv) in road traffic studies, we may consider the sequence of time points at which vehicles pass a reference point. Alternatively, if we examine a length of road at one fixed time instant and regard the position each vehicle as specified by a point, e.g. by the position of its front wheels, we have a point process in one-dimensional space, rather than in time;

(v) many of the stochastic problems of operational research involve a point process. The instants of arrival of customers in a queue, the instants of withdrawal of items from a store, and the instants of failure of a component in some system are all examples;

(vi) in several of these examples, each point may be classified into one of several different classes or types. For example, in (iv), we may distinguish two classes, cars and lorries. In a queueing problem, (v), there may be several types of customer; alternatively, we may wish to consider both customer arrivals and customer departures as points, but as points of two different types. A point process like this in which several classes of point are distinguished is called multivariate;

(vii) point processes occurring along a single dimension in space rather than in time occur, for example, in plant ecology, when a line transect is taken in a field and the position along it of plants of a particular species is noted. Such applications give rise also in an obvious way to point processes in spaces of two or more dimensions.

There are three broad aspects to the theoretical study of point processes:

(a) the general theory, with particular stress on generality of formulation and on the exploration of conditions for existence and uniqueness of the various functions associated with the process;

(b) the study of special processes of potential importance in applications, including the derivation of techniques for investigating such processes assuming reasonably ‘good’ behaviour of all properties studied;

applications, including the derivation of techniques for investigating such processes assuming reasonably ‘good’ behaviour of all properties studied;

(c) the development of techniques for the statistical analy...