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1 On the Estimation of the Intensity Function of a Stationary Point Process Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 2 (1965), pp Published by: Wiley for the Royal Statistical Society Stable URL: Accessed: :52 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Royal Statistical Society, Wiley are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series B (Methodological)
2 332 [No. 2, On the Estimation of the Intensity Function of a Stationary Point Process By D. R. Cox Birkbeck College, University of London [Received December Revised February 1965] SUMMARY The intensity function of a stationary point process gives as a function of x the conditional probability, given an event at time t, of an event in the interval (t+ x, t+ x+ Ax). A very nearly unbiased estimate of the grouped intensity function is constructed. Its sampling properties are investigated when the data are generated by a Poisson process. 1. INTRODUCTION CONSIDER a stationary process of point events in which there is zero probability that two or more events occur simultaneously. Let p be the mean rate of occurrence, so that as At-> 0 prob {one event in (t, t + At)} = pat + o(at), (1) for all t. The second-order properties of the process in continuous time are determined by the intensity function m(x), defined for x > 0, by m(x) = im prob {event in (t + x, t + x+ Ax) I event at t} (2 Ax-+>O Ax When the stationary point process is an equilibrium renewal process, the function m(x) is the renewal density associated with the corresponding ordinary renewal process; see, for instance, Cox (1962) for the definitions of these terms. For a gener stationary point process without multiple occurrences, the covariance density (Bartlett, 1955, p. 166) is defined for x >0 by y(x) = lim cov{no. events in (t, t + A1), no. events in (t+ x, t+ x+ A2)} AI,A2->O ^A1 A2 This is related to the intensity function m(x) by y(x) = p{m(x) - p}, (3) from which a spectral density can be defined (Bartlett, 1963). If the definitions are extended to x = 0, contributions proportional to Dirac delta functions have to be added. For a Poisson process, m(x) = p. For a renewal process it is known that under very weak conditions on the distribution of intervals m(x) -> p as x - cc, where p is the reciprocal of the mean interval between events. In general, it follows easily from (1) and the stationarity that the expected number of events in an interval of length t
3 1965] Cox - On the Estimation of the Intensity Function 333 starting from an arbitrary time point is pt, whereas, from (2), the corresponding expectation in an interval starting from an arbitrary event is I m(x) dx. It can be seen from this correspondence that, for a broad class of processes without long-term after-effects and in which m(x) tends to a limit as x -- oo, that limit is p. In the present note we are concerned, however, not with the general probabilistic properties of m(x), but with its estimation from a realization. A point estimate is obtained. Then some properties of the estimate are given when the data come from a Poisson process. We do not consider here the distribution of the estimates under other models, nor do we examine extensions to point processes in two or more dimensions, or to processes with several types of event. 2. A POINT ESTIMATE Suppose that the process is observed over an interval (0, to), starting from an arbitrary time. Let N(t) be the sample counting function, having N(O) = 0 and jumping by one at each event; denote by n the total number of events observed. Now the estimation of m(x) raises the same problems as does the estimation of other density functions, for example probability or spectral density functions. Some smoothing is required for graphical presentation and for the derivation of estimates with reasonably small variances; see Parzen (1962) for a formal procedure for the estimation of a probability density function and for references to earlier work. Here, however, we consider only a simple procedure directly analogous to the formation of a histogram. For a grouping interval -r, we estimate the grouped intensities 1 *r7+r mg(r7 +17-)- m(x) dx (r=0,1,...). (4) T Jr+0 In practical work it is usually advisable to examine a number of values of -i; for formal study of limiting properties it would be reasonable to allow -i to tend slowly to zero as to-> oo. Here, however, we regard -r as fixed. It is natural to base an estimate of (4) on the total number Sr of pairs of events separated by an interval between r-r and r-r+-r. That is we take, in principle, the in(n -1) (positive) intervals between all possible ordered pairs of events and form them into a histogram with groups of width r. The formal expression for the number Sr is I rto rrz +, Sr= dnu dnu+x to-rr-t r4tr +-rr to-u = ffoirr~ frrdnu dnu+x + L dnu dnu+ x (5) u=o =rr J =to-rr-tr s=rr To evaluate the expectation of (5), we write (1) as E(dNu dnu+x) = pdu E(dNu+x j event at u) = pm(x) du dx. (6) It follows easily from (6) that the expectation of the first double integral in (5) is exactly p7-(to - r- ) mg(r7r + 17-).
4 334 Cox - On the Estimation of the Intensity Function [No. 2, When </(to - r-r) is small, as would always be the case in practice, th in (5) is only a small correction term and can be approximated by assuming m(x) to be constant over the relevant range and equal to mg(rr+j-r). Then the expectatio of the second integral is With this approximation, IpT2 mg(rr + I-). E(Sr) = p-r(to - r-r - I-r) mg(rt + IT). (7) Since A = nlto has expectation p, we are led to define A ~~~~~sr to mg(rt + IT) = n(t-rt-j) (8) The qualitative reason for the factor to/(to - r series decreases with increasing r. It is of interest to note a sample result corresponding to the probabilistic property, noted in Section 1, that m(x) -* p as x oo. Since the total number of pairs of intervals is in(n - 1), we have that E Sr = n(n - 1), (9) r=0 where ro = (to/-r) -1 and to avoid unimportant end effects we suppose that to/l is an integer. On combining (8) and (9), we have that m w7m(rt+ j) n-i A n Ewr to-2t+t2to =t (10) where wr = (to- r-r - Ji)/t0. Equation (10 fluctuating around an average independent of r, that average must be nearly p. 3. PROPERTIES FOR A POISSON PROCESS When we use the estimates (8) to interpret possible systematic departures from a Poisson process, we are interested in the sampling properties of (8) under the null hypothesis of a Poisson process. Under this null hypothesis, the estimates (8) all have expectations exactly p, but rather than examine the sampling deviations from p, it is more useful to examine the deviations from p. Alternatively, we argue that to obtain sampling properties of the Poisson process independent of the nuisance parameter p, we should consider the distribution of S, conditionally on the observed value of n, the total number of events. In this conditional distribution, we can regard the positions of events as independently rectangularly distributed over (0, to). We therefore consider the following problem. Let U1,..., U. be independently rectangularly distributed over (0, t0) and let Then, in the notation of Section 2, V s)= (1 if rtiui -UjI<rr+ -r, otherwise. Sr = V. (12)
5 1965] Cox - On the Estimation of the Intensity Function 335 The mean and variance of (12) are best obtained from the conditional mean of V given Ui, namely that if 2rT + T < to, E(Vr) IO?, Ui < r) = -lto, (13) E(V(j)Ir'(Uj = ui<r+r) = + ui t-' (14) to to E( V) I rt + -r< Ui < to -r-r - r) = 27/to, (15) with expressions analogous to (14) and (13) in the ranges to -r - ' < U* < to0- r and to- r-r < U, < to. These follow directly from (11), since Uj is distributed independently of U*. The unconditional mean of VW is obtained from (13)-(15) by integrating with respect to the uniform distribution of U* and is '-r i-i 2 fr+rj7- u-r 2-rd(to -2r-r - 2r) E(VW ) = 2--+-X (-+ ) du + (to t to to t o to to (16) Thus and, from (8), E(Sr) = n(n-l1) -(to-r-r--r) (17) E{mg(rr + -)} = n-i (18) to To find var (Sr), we have first that since V( is a (0, 1) variable, its expected square is given by (16). Further, conditionally on Ui, the random variables V., V. (i#?) are independent. Hence E(V(0 V(r) I lj = u*) can be obtained directly from (13)-(15), and the unconditional expectation is then found to be E(V--1) 1(-r) 4T2 6r'3 10n3 (19) Finally 00 and VI(n) are independent when i,j, 1, n are shows that E(Sr2) = 1 n(n -1) E(V W) whence, on using (16), (17) and (19), we have that + n(n -1) (n -2) E( VW r V(r)) + {n(n - 1) (n -2) (n -3) {E(V.j)1)}2, (20) var(sr) = n(n-)-+ n(n-1) (2rn + 3 t3 -n(n- 1)(4n-6) (r+ 2t4; (21) 0
6 336 Cox - On the Estimation of the Intensity Function [No. 2, the terms have been collected in powers of -rlto, which would n Finally {A var (Sr) var {mg(r7r + 27T)} = 2T(0 - (22) For many purposes an adequate approximation to (22) is n-i (23) ntr(to - rtr + -1T) (3 A rough approach to the distribution theory, likely to be valid whe is to neglect the end effects in (13)-(15), i.e. to suppose that (15) is always applicable. To this order of approximation all V(>) are independent and hence Sr has a binomial distribution corresponding to ln(n - 1) trials each with probability of "success" 2r/to. That is, E(Sr) =n(n ), var(sr) n(n-1) (24) to to These are indeed the first terms of an expansion of (17) and (21). This argument, and more generally the approximate equality of (17) and (21), suggest that Sr should have nearly a Poisson distribution. We shall not examine this approximation further. As n -- oo, the random variables Sr are asymptotically normally distributed (Hoeff 1948), because they are so-called U-statistics. Thus it will help in interpreting a graph of mlig(rt + j-) against r-r + -r to put approximate "control limits" at taking say k = 2.? + k lvar (miig), (25) 4. COVARIANCE OF Two ESTIMATES We can apply the arguments of Section 3 to examine the covariance of Sri and S for r1 < r2. We have then that E(V(.,) V(j2)) = 0, (26) since it is certain that at least one of V&i,) and V.>2) is zero. When i, j, 1, m are all different, Vis ) and VIMr2) are independent, so that the only essentially new calculation needed is of E(V(.,) V.p2) (jr2 ). (27) The derivation is an extension of (13) E(Vrij) V(.r2) 1? < Ui < r1 7-) = 0 E(Vri1) V(r2)jr1-i-Uj = ui<r1i+-) = (t + u t) t (28) E(Vij,) Vn2) j rl r + -r < Ui < r_ -r) = 2-r 2/t2, etc.
7 1965] Cox - On the Estimation of the Intensity Function 337 We have that cov (S, Sr) =-n(n - 1) +n(n-1)(2r1n+n-2r1+2r2) t3 -n(n-1)(n-3)(2r? + 1)(2r2+1) r4. (29) 2 t~~~~~04 Note that, in accordance with the discussion of (24), the leading term of (29) is the covariance in a multinomial distribution with index In(n -1) between the numbers of occurrences in two cells, each with probability 2T/to. To this order of approximation the correlation coefficient between any two of the estimates m^,g(rr + 1T) is -2T/to. A useful partial check on the correctness of (21) and (29) is obtained by taking r1 even. Then var(sri +S,+) can be calculated from (21) and (29) and also directly from (21) with T- replaced by 2r and r by lrl. We shall not examine in detail the construction of tests based on combinations of estimates m', (rr+ I1T). To a first approximation s-1 ~~n-l ~2 E M gr- +1 r) - UL var {uh g(r-r + r=ox 2 to 2 2 ( is distributed as chi-squared with s d the standard error of linear combinations of the estimates (8) can be calculated. 5. DisCUSSION A number of estimated intensity functions for empirical series are given by Cox and Lewis (1966). In a general way, the analysis of the intensity function is equivalent to that of the spectrum of the point process (Bartlett, 1963). Substantial practical experience would be desirable before reaching a firm conclusion about the relative merits of the two approaches. The referee has made the very interesting suggestion of estimating the intensity function by first converting the point process into a real-valued process, by summing impulse functions attached to each event. Then standard techniques for estimating autocorrelation functions and spectral density functions are available and yield estimates of certain weighted averages of the intensity function or spectral density function of the point process. A comparison with the method of the present paper will not be attempted; however, the estimate (8) is very simply related to the original data and this must be an advantage in interpretation. REFERENCES BARTLETT, M. S. (1955), Introduction to the Theory of Stochastic Processes. Cambridge University Press. (1963), "The spectral analysis of point processes", J. R. statist. Soc. B, 25, Cox, D. R. (1962), Renewal Theory. London: Methuen. - and LEWIS, P. A. W. (1966), Statistical Analysis of Series of Events. London: Methuen. To appear. HOEFFDING, W. (1948), "A class of statistics with asymptotically normal distributions", Ann. math. Statist., 19, PARZEN, E. (1962), "On estimation of a probability density function and mode", Ann. math. Statist., 33,
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