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 52
... origin , say t = 0 , to the first subsequent point . Because the development of the process in the region t > 0 is independent of Ho , it is immaterial whether there is a point at t = 0 . Then the probability density function ( p.d.f. ) ...
... origin at xị . The functional equation ( 1.5 ) , together with the initial condition FX ( 0 ) = 1 , implies ( 1.4 ) . Further , the above definition of the Poisson process implies that , starting from an arbitrary time origin ...
... origin , the intervals X1 , X2 , ... between successive points are independently exponentially distributed with parameter p , which we call an interval specification ; ( c ) by the above specification of the joint 6 POINT PROCESSES.
... origin falls in an interval of length in ( z , 2 + 8 ) is proportional to zg ( z ) 8 + 0 ( 8 ) and hence , on normalization , the density of the length of that interval is zg ( 2 ) / Hy , where lx E ( X ) is the mean of the density g ...
... origin to the next point is uniformly distributed over ( 0 , 2 ) , so that finally the density of X , is 1 zg ( z ) ... origin from those in which the time origin is arbitrary , i.e. in which the process started in the remote past . It is ...
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 |