View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CiteSeerX Point processes, temporal David R. Brillinger, Peter M. Guttorp & Frederic Paik Schoenberg Volume 3, pp 1577–1581 in Encyclopedia of Environmetrics (ISBN 0471 899976) Edited by Abdel H. El-Shaarawi and Walter W. Piegorsch John Wiley & Sons, Ltd, Chichester, 2002 is an encyclopedic account of the theory of the Point processes, temporal subject. A temporal point process is a random process whose Examples realizations consist of the times fjg, j 2 , j D 0, š1, š2,... of isolated events scattered in time. A Examples of point processes abound in the point process is also known as a counting process or environment; we have already mentioned times a random scatter. The times may correspond to events of floods. There are also times of earthquakes, of several types. fires, deaths, accidents, hurricanes, storms (hale, Figure 1 presents an example of temporal point ice, thunder), volcanic eruptions, lightning strikes, process data. The figure actually provides three dif- tornadoes, power outages, chemical spills (see ferent ways of representing the timing of floods Meteorological extremes; Natural disasters). on the Amazon River near Manaus, Brazil, dur- ing the period 1892–1992 (see Hydrological ex- tremes)[7]. Questions The formal use of the concept of point process has a long history going back at least to the life tables The questions that scientists ask involving point of Graunt [14]. Physicists contributed many ideas in process data include the following. Is a point pro- the first half of the twentieth century; see, for ex- cess associated with another process? Is the associ- ample, [23]. The book by Daley and Vere-Jones [11] ation between two point processes actually causal? Is there a change or trend in time (see Trend, detecting)? Does the structure change (see Change, 20 detecting)? Are the times clustered? Are the times 15 repelled from each other? What is the predicted behavior? What is the risk (probability) of some 10 event of negative consequence occurring at some 5 future time (see Risk assessment, probabilistic)? How does one learn or describe the relationship of (a) 1900 1920 1940 1960 1980 such processes? How does one carry out system Time (year) identification? 2.0 1.5 Representations 1.0 0.5 A number of methods are used for the representation of point processes and of point process data. The 0.0 figure shows three types of displays. The represen- 1900 1920 1940 1960 1980 (b) Time (year) tations include: 20 ž step function 15 Nt D #f0 <j Ä tg 1 10 ž generalized function (involving the Dirac delta 5 function) 5101520 dNt (c) Index D υt j2 dt j Figure 1 Floods on the Amazon River near Manaus, Brazil, during the years 1892–1992. (a) Amazon floods – ž counting measure cumulative count; (b) dates of floods; (c) intervals between floods NI D #fj 2 Ig 3 2 Point processes, temporal ž binary time series functions. In the stationary case these functions will not depend on t. The first two generalize to 1, if some point in (t, t C dt] Ð D dNt D 4 product densities pK , K 1, 2,..., giving the 0, otherwise relative probabilities with which the points of interest (one might also write Ndt or Nt, t C dt] here) are distributed at prespecified locations in time. Specifically ž interevent intervals fXjg PrfdNt1 D 1,...,dNtK D 1g Xj D jC1 j 5 D pKt1,...,tK dt1 ...dtK 10 assuming jC1 j are on non-negative. In these expressions #fAg refers to the number of elements for the tk distinct and K D 1, 2,.... in the set A. Under weak conditions, including being orderly, a point process is characterized by its conditional or complete intensity function, Ð,asin Distinctions PrfdNt D 1jHtgDtjHt dt11 There are a variety of distinctions that may be made concerning types of point processes. A process may where Ht is the history Ht Dfj Ä tg. be either deterministic or stochastic. In the determin- Another general way to define an (orderly) point process is via its zero probability function, that is istic case the values j are fixed. In the latter case (see Stochastic process) the process is determined I D PrfNI D 0g for bounded I12 by a consistent collection of probabilities such as PrfNI1 D n1,...,NIK D nKg, Specific Point Processes K D 1, 2,... 6 There are a number of important point processes that arise in both theory and practice. where the I are Borel sets of the real line. k The renewal process has the property that the A process may be stationary, i.e. the time or intervals between successive points are independent space origin does not matter. A process may be and identically distributed positive random variables. mixing, i.e. distant values are only weakly related The Poisson process has a variety of definitions. probabilistically. Points of the processes may be One is that the conditional intensity function is clumped together, i.e. clustered, or they may be constant. Another is that the counts NI ,...,NI repelled. In many cases a process is orderly, i.e. the 1 K of points in disjoint intervals I are independent points occurring are isolated. k Poisson variates with consistent expected values K D 2, 3,.... A Poisson process is characterized by its rate Parameters function. For the doubly stochastic Poisson process a non- A variety of parameters provide useful descriptors of negative random rate process in continuous time is stochastic point processes. These include moments first realized. Then a Poisson process with that rate such as the rate function is generated. EfdNtg For a cluster process there is a sequence of cluster 7 dt centers fjg, then further point processes fujk,k D 1, 2,...g are generated for each j. The cluster process the auto-intensity then consists of the times fj C ujkg (see Poisson EfdNt C u dNtg cluster process). 8 dt du The Neyman–Scott and Bartlett–Lewis processes are particular cases of the cluster process. In the and the conditional rate former the ujk are independent and identically dis- PrfdNt C u D 1jdNt D 1g tributed. In the latter the fujkg are renewal processes 9 du having the j as points of origin. Point processes, temporal 3 Operations on Point Processes Inference There is a calculus or algebra for manipulating point There is now a fairly extensive literature concerning processes. This involves functions of realizations of inference for point processes. One may refer to the basic processes. The operations may be applied by various books listed at the end of this entry. nature or by an analyst. One might consider, for In particular large sample properties of histogram example, a linear functional of a point process such as type estimates of product densities are developed in [4]. Nearest neighbor methods are studied in [13]. log t dNt 13 In the case of the conditional cross rate function PrfdNt C u D 1jdMt D 1g 17 for some function Ð. du In the operation of superposition several processes a histogram type estimate is provided by are involved. In the superposed process the identity of each process is ignored and the times retained. If #fj uj <b/2g k j 18 there are two processes M and N, then the superposed bMT process is M C N with the count of points in the set I given by MI C NI. where the terms MT j come from the process M, A point process may be thinned. In this operation the j come from N,andb is a binwidth parameter. points are deleted randomly. This estimate may be computed exceedingly rapidly. Time substitutions are useful. What is involved Its distribution is approximately proportional to a is that a process M is converted to a process N Poisson when the point process (M, N) is stationary by writing Nt D M[t] for some nondecreasing, and mixing. possibly random, function Ð. Through such a There are a variety of useful statistical models. substitution, general processes may be derived from One may mention the Hawkes process where the a homogeneous Poisson process. conditional intensity function is given by [16] Another operation is random translation. Here the 1 points of the process are shifted tjHt D C at u dNu 19 0 fjg!fj C εjg 14 There are models containing explanatory variables. The latter are useful for dealing with nonstationary with the fεjg taken as random. processes for example. One means of constructing There are point process systems where a point them is by multiplying an elementary conditional process input is carried into a point process output. intensity function by a function of some given The mechanism is typically stochastic; see [5]. Ran- functional form. dom translation as illustrated by (14) provides an The likelihood may be set down given an example. Another example is provided by a model expression for the conditional intensity function. The satisfying result is PrfdNt D 1jMg T D C at u dMu 15 L D j exp tj dt 20 dt j j 0 when the input point process is M. This provides a where the available data values are the points f g point process analog of the linear model. j observed in the time interval (0, T] and where  is The expression (13) is the basis of the probability an unknown parameter. Large sample properties of generating functional. This is defined as estimates obtained by maximizing L are developed in [12]. G[ ] D Eexp log t dNt 16 There are residual analyses; see [19] and [20].