Point ProcessesThere 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. |
From inside the book
Results 1-5 of 64
... given positive constant P , with dimensions [ time ] ยป !, the Poisson process of rate p is defined by the requirements that for all t , as 8 + 0 + , pr { N ( t , t + ) = 11H , } = p8 +0 ( 8 ) , ( 1.1 ) pr { N ( t , t + 8 ) > 11H ; } = 0 ...
... ) / Hy , where lx E ( X ) is the mean of the density g . This type of sampling is known as length - biased sampling . Given that 1 > Z the origin falls in an interval of length [ 1.2 ] INTRODUCTION 7 (ii) Renewal processes.
... given by ( 1.9 ) , thus obtaining what is called an equilibrium renewal process . It can be shown directly from ( 1.9 ) that the equilibrium renewal process and the corresponding ordinary renewal process are identical if and only if g ...
... given any realization H1,50 that for such a process ( 1.10 ) is effectively uniquely defined and determines the probability structure of the point process . Condition ( 1.11 ) ensures that there are essentially no multiple simultaneous ...
... given that a point occurs at the origin . This is specified by the conditional intensity function h ( t ) = lim dz'pr { N ( t , t + 82 ) > 0 N ( -81,0 ) > 0 } . ( 1.19 ) 81,82 -0+ For the renewal process , this gives the probability ...
Contents
1 | |
2 Theoretical framework | 21 |
3 Special models | 45 |
4 Operations on point processes | 97 |
5 Multivariate point processes | 117 |
6 Spatial processes | 143 |
References | 173 |
Author index | 182 |
Subject index | 184 |