A Bivariate Point Process Connected with Electronic Counters

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<ul><li><p>A Bivariate Point Process Connected with Electronic CountersAuthor(s): D. R. Cox and Valerie IshamSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 356, No. 1685 (Aug. 24, 1977), pp. 149-160Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/79374 .Accessed: 07/05/2014 08:34</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Mathematical and Physical Sciences.</p><p>http://www.jstor.org </p><p>This content downloaded from 169.229.32.136 on Wed, 7 May 2014 08:34:32 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/action/showPublisher?publisherCode=rslhttp://www.jstor.org/stable/79374?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>Proc. R. Soc. Lond. A. 356, 149-160 (1977) </p><p>Printed in Great Britain </p><p>A bivariate point process connected with electronic counters </p><p>BY D. R. Cox, F.R.S. AND VALERIE ISHAM </p><p>Department of Mathematics, Imperial College, London </p><p>(Received 21 January 1977) </p><p>Consider three independent Poisson processes of point events of rates A1, A2 and A12. There are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters have constant dead-time, i.e. following the recording of an event on a counter no further event can be recorded on that counter until the appropriate constant time has elapsed. Two ways of estimating A12 are via a coincidence rate, i.e. the rate of occurrence of pairs of events separated by less than a suitable small tolerance, and via the covariance of the numbers of events recorded on the two counters in a suitable time period. The theoretical values of these quantities are calculated allowing for dead-time. The techniques used illustrate the study of bivariate point processes. </p><p>1. INTRODUCTION </p><p>This paper deals with a counting problem in physics which can be described in general terms as follows. There are two counters and three independent Poisson processes of point events of rates respectively A1, A2 and A12. Events from the first process can be recorded only on the first counter. Events from the second process can be recorded only on the second counter. Events from the third process can be recorded virtually simultaneously on both counters. </p><p>This situation is equivalent to one in which there is a single Poisson process of rate A of events, all of which would be counted on both counters were the counters fully efficient. If the counters have efficiencies el and e2, then in the previous formulation </p><p>A1 = Ae1(l-62),A2 = Ae2(1-e1) and A12 = Aele2. </p><p>Both counters are subject to blocking. Following the recording of an event on the first counter, there is a constant dead-time r1 during which further events cannot be recorded on that counter. Similarly for the second counter there is a constant dead-time r2. </p><p>There is an extensive literature on the effect on a single counter of this and more complex forms of blocking (Feller 1948; Smith I958). Here we study aspects con- nected with the pair of counters, especially those concerned with the estimation of A12. There are in fact two broad techniques for such estimation, namely the counting of coincidences and the estimation of covariances. </p><p>For the first method, the sequences of recorded events on the two counters are </p><p>6 [149 ] Vol. 356. A. (24 August ,977) </p><p>This content downloaded from 169.229.32.136 on Wed, 7 May 2014 08:34:32 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>150 D. R. Cox and Valerie Isham </p><p>merged and, for some suitable small time span h, a count is made of the number of occasions on which there are two events less than h apart, one from each counter. Note that if h &lt; min (r1, r2) any two events less than h apart must indeed originate from different counters. A value of h is chosen in the light of the 'jitter' in the recording mechanism and of any real displacement between the virtually co- incident pairs of events in the third Poisson process. The mathematical problem then is to calculate the rate of occurrence of such coincidences, allowing for counter dead-time and including spurious coincidences. </p><p>The second method, which can be used when coincidence techniques are in- applicable for technical reasons, hinges on the fact that for the original Poisson processes, the covariance of the total numbers of events in a time t occurring on the two counters is equal to the variance of the number from the third Poisson process, which in turn is A12t. Therefore, in the absence of blocking, A12 can be estimated via the sample covariance computed from the numbers of events in the two counters as measured in a large number of independent intervals each of length t. The mathematical problem here is to calculate the theoretical value of this covariance for the numbers of events actually recorded, i.e. allowing for dead- time. For further discussion and references and for a semiempirical formula for the covariance, see Lewis, Smith &amp; Williams (I973). </p><p>Most of the present paper is devoted to the second problem, the calculation of cov {N1 (t), N2(t)}, where Nl(t) and N2(t) are the numbers of events recorded in time t on counters 1 and 2. In ? 8 we calculate the rate of occurrence of coincidences. </p><p>The results may have other applications. The system forms a very special example of a bivariate point process. While the arguments in the paper are entirely rigorous, they have been presented in a way avoiding mathematical technicalities. </p><p>2. SOME PRELIMINARY RESULTS </p><p>Consider the time interval (0, t) starting from statistical equilibrium. Then </p><p>rt NA7(t)= dNft(u) (i = 1,2). </p><p>Let Pi = A1 + A12 be the total occurrence rate for the first counter and PI be the equilibrium probability that the counter is open. Then the probability that an event is recorded on that counter in (u, u + Au) is PlPl Au + o(Au). Therefore, E(.) denoting mathematical expectation, we have that </p><p>E{N1(t)} = f pr {dNl (u) = 1 = f PlP du = plplt. (1) </p><p>Further it follows, by considering the trajectory of the first counter consisting of an alternation of constant dead-times of length -r, with open periods exponentially distributed with mean l/pl, that </p><p>Pi = (lIp,)I(lIp ?ri)= (+p1 -r1)-1. (2) </p><p>This content downloaded from 169.229.32.136 on Wed, 7 May 2014 08:34:32 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>A bivariate point process from electronic counters 151 </p><p>Similarly for the second counter </p><p>E{N2(t)} = p2P2t (3) </p><p>with P2 = A2 + A12,p2 (1 +P2r2)-1 To calculate c(t) = cov {Nl(t), N2(t)}, we consider </p><p>c(t) = E{Nl(t) N2(t)} - E{N1(t)} E{N2(t)} </p><p>= E (f dN(u) dN2(v)}-Pl P2 P1 p2t2 (4) </p><p>= f duj dvpr{dN1(u) = dN2(v) - 1}+ dv du o =t+0 0 t=V+0 </p><p>pr {dN1 (u) = dN2(v) 1}+ ftdupr{dNi(u) = dN2(u) = 1}-PI P2P1P2 t2, (5) </p><p>the region of integration in (4) having been split into three parts. Now, formally, </p><p>pr {dN,(u) = dN2(v) = 1} = pr {dN1(u) = 1} pr {dN2(v) = 1 i dN1(u) = 1} </p><p>=pipLduh12(v-u)dv (v &gt; u), (6) </p><p>pr{dN,(u) = dN2(v) = 1} = p2p2dvh21(u -v) du (u &gt; v), (7) </p><p>pr{dNL(u) = dN2(u) = 1} = A12P12du. (8) </p><p>Here P12 is the equilibrium probability that both counters are open simultaneously, and the second-order cross-intensity functions are defined, for example, by </p><p>pr (event recorded on counter 2 in (x, x + Ax) I event hI2(X) =im Mrecorded </p><p>on counter 1 at 0) h Ax-) O + Ax </p><p>forx &gt; 0. On substituting (6)-(8) into (5) and taking Laplace transforms, these being </p><p>denoted by an asterisk, we have that </p><p>c*(s) = Lc(t) e-Stdt </p><p>= Pi Pi 1h(s)/s2 +P2P2 h*(s)/82 + A12 P12/s2 -2Pi P2 Pl P2/83s (9) </p><p>This shows that in order to find c(t) we need to find P12, the equilibrium probability that both counters are open together, and the two cross-intensity functions. </p><p>3. Two COUNTERS AS A MARKOV PROCESS </p><p>To study the two counters in more detail, we represent them as a Markov process. For this we must define the state in such a way that the instantaneous transition probabilities are determined by the current state. Thus if a counter is blocked the </p><p>6-2 </p><p>This content downloaded from 169.229.32.136 on Wed, 7 May 2014 08:34:32 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>152 D. R. Cox and Valerie Isham </p><p>state description must specify for how long the counter has been blocked. The possible states and their equilibrium probabilities or probability densities are as follows: </p><p>(i) both counters are open, with probability P12; (ii) counter 1 has been blocked for time u and counter 2 is open, with probability </p><p>density q1(u) for 0 &lt; u &lt; -r; (iii) counter 1 is open and counter 2 has been blocked for time u, with probability </p><p>density q2(u) for 0 &lt; u &lt; r2; (iv) counter 1 has been blocked for time u1 and counter 2 for time u2, with </p><p>probability density q12(u1, u2) for 0 &lt; ui &lt; (i = 1, 2). For example, the probability that counter 1 is open and counter 2 has been </p><p>blocked for a time (u, u +Au) is q2(u) Au+ o(Au). Note that there is a non-zero probability that the two counters become blocked simultaneously and that then in case (iv) u1 = U2. It is thus convenient to put </p><p>q12(Ul, U2) =qs) (u1) 8(u1 - u2) + qj((u1 u2)j (10) </p><p>where 4(.) is the Dirac delta function and q(C2)(., .) is absolutely continuous. The equilibrium equations of the process can now be obtained; they represent </p><p>the balance between the probabilities of entering and leaving a particular state. Throughout the rest of this section and ?? 4 and 5 we deal with the special case of equal counter dead-times, T1- = = r. We write p Al1+ A2 + A12 for the total rate of occurrence of events of all kinds. Then </p><p>PP12 = q (T) + q2(T) + q(s)(T), q, (?) A1 P12, q2(0) = A2 P12, q(s2)(0) = A12 p12, (11) </p><p>q(c) 0) P2 q(u), q2)(0, u) = p q2(U); </p><p>q'(u)-2q(f = q (j2) (u, -r) (12) q2(u) = -p1q2(U) + q1(j(T, u); J </p><p>Dq12(U1, U2)/Du1 +? qI2(u1, U2)/au2 =0. (13) </p><p>Finally there is a normalizing condition that the total probability is unity. This system is readily solved. Thus (13) shows that q12(ul, u2) is a function of </p><p>l- '62* Therefore we can write q(s)(u) = q2I a constant, and by the last part of (11) </p><p>(e) </p><p>[P2 q1(ul- u2) (U1 </p><p>&gt; U2), q12 (U,t2) = </p><p>pjq2(u2-U1) (U1 &lt; U2). </p><p>Then the equations (12) become </p><p>qj(u) +p2q (2)q p1q2(T-U),1 (14) </p><p>q2(u) +?pq2(U) = p2q,(T-u). </p><p>This content downloaded from 169.229.32.136 on Wed, 7 May 2014 08:34:32 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>A bivariate point process from electronic counters 153 </p><p>By examining solutions of the form qi(u) = a exiu, it is easily shown that we must have Xi = -X2 and that Xl(Xl - PI + P2) = O. It can thereby be shown that provided that P, * P2 </p><p>q1(u) = Ap1 + B e-Plr e(Pl-P2)U, q2(U) = Ap2+Be-P2r e(P2P) U. (15) </p><p>We now substitute into ( 1) to find </p><p>A = P12 (A1 e T - A2 eP2r) (16) (p1 ePlr-P2 eP2r) </p><p>1 </p><p>B i </p><p>A12 P12(P2 - PI) e(Pl+P2) 1 (p, ePl'r- P2 eP2r) </p><p>and finally apply the normalizing condition to show that </p><p>P12 - PI P2(P1 ePlT -P2 eP2T) (18 (A1 ePlr-A2eP2r) </p><p>Note that were the two processes {N1(t)} and {N2(t)} statistically independent, i.e. A12 = 0 and pi = Ai (i = 1, 2), we would have P12 = P1P2* The final factor in (18) expresses the association between the states of the two counters; the factor is strictly greater than unity unless A12 = 0, when the factor is obviously equal to unity. It would be possible to base the estimation of A12 directly on an experi- mental determination of P12. </p><p>4. THE CROSS-INTENSITY FUNCTIONS </p><p>On the basis of the results of ? 3 we now calculate the cross-intensity function hl2(x). In the definition, the condition that an event is recorded on counter 1 at time 0, implies that counter 1 is open at 0-. Therefore the state of counter 2 at time 0- has the following conditional probabilities: </p><p>(i) open, with probability P12/Pl; (ii) closed, having been closed for time u, with probability density q2(U)/pl. In case (ii), the state of counter 2 is unchanged at time 0 +, but in case (i) there </p><p>are two possibilities depending on whether the originating event on counter 1 comes from the third or first Poisson processes. Thus at time 0+, we have three possibilities for the state of counter 2, namely: </p><p>(i)' just beginning a blocked period, with probability (p12AI2)/(p1pI); (i)" open, with probability (P12A1)/(p1P1); (ii) as above. </p><p>For each of these cases the process of recorded events on counter 2 forms a renewal process with the interval between successive events having the density P2 e-P2(X-) (x &gt; T). For (i)', the renewal process is ordinary (i.e. starts with one of the above intervals), in (i)n it is modified by starting with an interval with </p><p>This content downloaded from 169.229.32.136 on Wed, 7 May 2014 08:34:32 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>154 D. R. Cox and Valerie Isham </p><p>exponential density P2 e-P2X (X &gt; 0) and in (ii) it is modified by starting with an interval with density P2 e-P2(x-r+U) (x &gt; - u). It follows that </p><p>P12;A12 pA A P1+ P2hx-y h12(x) = 2h(x; T, p2) + P2 eP P2 e x-yT,P2) dy pip' ip </p><p>+p q2(u) h(x + u; T P2) du, (19) </p><p>where h(x; , P2) is the ordinary renewal density for process (i)'. Further </p><p>hT(s; , P2) P2{(P2 + s) es,- P2-1 (20) </p><p>From (19), (20) and (15) we can now calculate the Laplace transform hz (s); in dealing with the last term of (19), note that h(y; T, P2) = (0 &lt; y &lt; T). It follows that </p><p>PI 2A1p2 P2 [p22A12 p12AkP2 PIPI(P2 +s) (P2+s) esr P2 L PlPl P1P1(P2 + s) </p><p>{ esr-1 A12(p2- p1) ePi 7 (e(s-P1+P2) 1)}] (21 8s (A, ePlr 2eP2r) (8-pl + p2) </p><p>with the symmetrical expression for h2*(s). </p><p>5. THE COVARIANCE </p><p>By substituting (21) and (18) into the formula (9) for the Laplace transform of c(t) = cov {N1(t), N2(t)}, we obtain an explicit expression for c*(s) in our special case </p><p>j= 2= -. It is easily shown that c*(s) has a double pole at s = 0 and that the remaining singularities have negative real parts. That is </p><p>c*(s) - k/s2 + I/s + c(s), (22) </p><p>where cO*(s) is analytic in the half-plane Re (s) &gt; - yo, with yo &gt; 0; in fact elemen- </p><p>tary but tedious calculations give </p><p>k = A12plp2p12, (23) </p><p>21 = 2A122 (PIPi + P2 P2) + 2P1P2A12 P12(P1 eP1-P2 eP2r)-1 </p><p>x {r2(eP2T - ePlr) (p2 ?p2) + 2(p2 ePlr-p2 eP2r) </p><p>+ 2(pl - p2)' (ePl r + eP2...</p></li></ul>