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1991 | Book

Point and Counting Processes: Introduction and Preliminaries Read chapter Read first chapter

Random Point Processes in Time and Space

Authors: Donald L. Snyder, Michael I. Miller

Publisher: Springer New York

Book Series : Springer Texts in Electrical Engineering

Included in: Professional Book Archive

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About this book

This book is a revision of Random Point Processes written by D. L. Snyder and published by John Wiley and Sons in 1975. More emphasis is given to point processes on multidimensional spaces, especially to pro­ cesses in two dimensions. This reflects the tremendous increase that has taken place in the use of point-process models for the description of data from which images of objects of interest are formed in a wide variety of scientific and engineering disciplines. A new chapter, Translated Poisson Processes, has been added, and several of the chapters of the fIrst edition have been modifIed to accommodate this new material. Some parts of the fIrst edition have been deleted to make room. Chapter 7 of the fIrst edition, which was about general marked point-processes, has been eliminated, but much of the material appears elsewhere in the new text. With some re­ luctance, we concluded it necessary to eliminate the topic of hypothesis testing for point-process models. Much of the material of the fIrst edition was motivated by the use of point-process models in applications at the Biomedical Computer Labo­ ratory of Washington University, as is evident from the following excerpt from the Preface to the first edition. "It was Jerome R. Cox, Jr. , founder and [1974] director of Washington University's Biomedical Computer Laboratory, who ftrst interested me [D. L. S.

Table of Contents

Frontmatter
Chapter One. Point and Counting Processes: Introduction and Preliminaries
Abstract
A random point process is a mathematical model for a physical phenomenon characterized by highly localized events distributed randomly in a continuum. Each event is represented in the model by an idealized point to be conceived of as identifying the position of the event. If X denotes the continuum space, then a realization of a random point process on X is a set of points in X. The number and variety of phenomena for which this type of stochastic process provides a reasonable mathematical model is surprisingly large. Here are ten examples indicating some of the forms X may have.
Donald L. Snyder, Michael I. Miller
Chapter Two. Poisson Processes
Abstract
The Poisson process is the simplest process associated with counting random numbers of points. We begin our study of these processes when the space where the points occur is a one-dimensional, semiinfinite, real line. While there is no mathematical reason to do so, we refer to this space as “time” because temporal phenomena seem to predominate in applications. The study of temporal Poisson-processes permits many of the properties of Poisson processes to be exhibited, but Poisson processes on multidimensional spaces are also important in applications. These are developed in Sec. 2.5.
Donald L. Snyder, Michael I. Miller
Chapter Three. Translated Poisson-Processes
Abstract
The point processes we consider in this chapter are useful as models for measured data acquired about an underlying, unobservable point-process when the measurements are imperfect and in the form of a point process. Such a measurement is illustrated in Fig. 3.1. Point of the underlying process, called the input point-process, occur on a space X.
Donald L. Snyder, Michael I. Miller
Chapter Four. Compound Poisson-Processes
Abstract
One motivation for the model we develop in this chapter is provided by the atmospheric-noise data shown in Fig. 1.3. It is evident that a point process model can account for the occurrence times of the pulses. However, these times alone do not reflect all of the significant features. The amplitudes of the pulses exhibit wide variation and have a strong influence on a radio receiver operating at low frequencies. Even a first-approximation model for low-frequency atmospheric noise should, therefore, include the amplitude as well as occurrence time of each pulse. It is this procedure of endowing each temporal point with an ancillary variable, an amplitude in this instance, which characterizes the models of this chapter.
Donald L. Snyder, Michael I. Miller
Chapter Five. Filtered Poisson-Processes
Abstract
There are numerous physical phenomena that can be modeled as a response to the points of a marked point process. The simplest models occur when the response can be expressed as a superposition of separate responses to each marked point. These models are developed in Sec. 5.2. Later, in Sec. 5.3, we remove the superposition requirement but impose the additional structure of Markov processes.
Donald L. Snyder, Michael I. Miller
Chapter Six. Self-Exciting Point Processes
Abstract
A temporal point process must be both orderly and without aftereffects to be a Poisson process. The orderliness restriction that points be isolated from one another is relaxed in Ch. 4 where a generalized Poisson process is defined and studied. The restriction that the process be without aftereffects is removed in this chapter about self-exciting point processes.
Donald L. Snyder, Michael I. Miller
Chapter Seven. Doubly Stochastic Poisson-Processes
Abstract
The self-exciting point processes of Ch. 6 were obtained from the Poisson process by allowing its intensity to become causally dependent on the point process itself. In this way, the intensity is transformed into a random process having paths that are known exactly given the point process. Point processes in this chapter are also obtained by randomizing the intensity of the Poisson process. Here, however, this randomization is not through self excitation but, rather, by an “outside” process so that the paths of the resulting intensity process are not known given only the point process.
Donald L. Snyder, Michael I. Miller
Backmatter
Metadata
Title
Random Point Processes in Time and Space
Authors
Donald L. Snyder
Michael I. Miller
Copyright Year
1991
Publisher
Springer New York
Electronic ISBN
978-1-4612-3166-0
Print ISBN
978-1-4612-7821-4
DOI
https://doi.org/10.1007/978-1-4612-3166-0