Technical Report No. 214 036040-8-T THE DETECTION OF RANDOMLY OCCURRING EVENTS WITH RANDOM DURATIONS by Richard M. Heitmeyer Approved by: _ _ __ _ c Theodore G. Birdsall COOLEY ELECTRONICS LABORATORY Department of Electrical and Computer Engineering The University of-Michigan Ann Arbor, Michigan for Contract No. N00014-67-A-0181-0032 Office of Naval Research Department of the Navy Arlington, Virginia 22217 November 1971 Approved for public release; distribution unlimited.

ABSTRACT This report considers the detection of events for which the occurrence times and durations are random variables. In this context, two special cases are considered: (1) the continuous-action detectors that seek to respond at each time when an event is present and (2) the discrete-action detectors that seek to respond at only a single time during which each event is present. For both cases, the optimum detectors are specified in terms of recursion relations and examples of the numerical solutions of these relations are presented and compared. iii

TABLE OF CONTENTS Page ABSTRACT iii 1. INTRODUCTION 2. CONTINUOUS ACTION AND DISCRETE ACTION DETECTORS 3 3. THE OPTIMUM BAYES DETECTORS 8 4. SUMMARY AND CONCLUSIONS 22 APPENDIX 24 DISTRIBUTION LIST 29 FIGURES 1. Illustration of response functions 5 2. Canonical form for the optimum detectors 18 3. Illustrations of threshold functions (a) CAD (b) DAD 20 iv

1. INTRODUCTION Classical detection theory assumes that an observation extends over an interval of time at the end of which a decision is made to either accept or reject the hypothesis that a signal was present during that observation. This theory has found great application in modeling the decision process for those systems where it is possible to consider the occurrence of signals as being confined to well-defined observation intervals (e. g., active radar systems, active sonar systems, and digital communication systems). In many detection systems, however, both the times at which the signals begin and the durations of the signals are unknown. In these cases, the classical detection theory paradigm may no longer be applicable, and the decision theoretic model for the detection problem becomes more complicated. As a concrete example, consider a simple passive surveillance system for which the observation x(t) is given by x(t) = n(t) if no target is present at time t x(t) = n(t) + s(t) if a target is present at time t and the arrival time of each target and the length of time each target spends in the systems field of view are random variables. In these systems, it is often the case that an action must be taken each

2 time a target appears and while the target is still present. For example, the function of the system may be to take a continuous action, such as to sound an alarm when targets are present. Alternatively, its function may be to take a discrete action such as to allocate some resource to investigate the nature of the target further. In both of these cases, the system must operate in real time to detect the presence of randomly occurring events (target appearances) as they occur. In the continuous action system, the detector must decide at each point in time whether or not a target is present in order to control the alarm, whereas, in the discrete action system the detector must designate only a single time when each target is present in order to take the appropriate action. This distinction in the purpose of the systems detectors defines two different detection problems. In this report, the optimum Bayes detectors for each problem are determined.

3 2. CONTINUOUS ACTION AND DISCRETE ACTION DETECTORS In this section the continuous action detector (CAD) and the discrete action detector (DAD) are defined by specifying in terms of loss functions how the performance of each is to be measured. It is assumed that the observation x(t) is available at the input of the detector for t E [0, T]. The detector is to be a casual device that produces at its output the response function a(t) that extracts from x(t) the information about each event that is necessary to take the appropriate action. The loss function for each detector, therefore, depends on both the response function a(t) and the randomly occurring events present at the input. We consider the CAD case first. The objective of the continuous action detector is to take some action at each instant of time that an event is present. Thus, when no noise is present in the observation, the CAD response function may be defined by 1 if an event is present at time t a(t) = 0 otherwise The effect of noise on the performance of the CAD can be seen in Fig. 1. Figures l(a) to l(e) illustrate the times at which events are present in a typical event sequence, the corresponding noisy

4 times at which events are present (a) x(t) (c) a M a2(t) (d) a (t) 3 X X X (e) Fig. 1. Illustration of response functions

observation, and a typical response function. It is seen by comparing Fig. l(a) and Fig. l(c) that the time axis can be divided into four sets: detection times [a(t) = 1 and an event present], false alarm times [a(t) = 1 and an event present], miss times [a(t) = 0 and no event present], and correct rejection times [a(t) = 0 and no event present]. If we let TD TF, TM, and TC denote the total amount of detection times, false alarm times, miss time, and correct rejection time, respectively, then a reasonable choice for the CAD loss function is T T T T L =LTD + LFTF + LMTM + L (1) T T T T where LD, LF, LM, and LC are losses per unit detection time, etc. A more general definition for the CAD loss function is needed when it is necessary to show special concern for the possibility of missing an event altogether. For example in Fig. l(d) we have illustrated a second response function that results in the same values of TD TF I TM, and TC as the response function in Fig. l(c), but that detects only one of the three events. This distinction may be incorporated into the loss function by adding in the term LDND + LM N D M M

6 where ND and NM are the total number of detected pulses and missed pulses respectively, and LD and LM are the corresponding losses. The resulting general CAD loss function is T T T T L LD + F + LMTM + LTC + LN + L (2) Hereafter we shall refer to the CAD with the loss function in Eq. 1 as the "classical" CAD for reasons which will be apparent later. Next we define the loss function for the discrete action detector. The objective of the DAD is to take a single discrete action for each event that occurs. When no noise is present in the observation, this task can be accomplished if the response function a(t) is defined as 1 if an event begins at time t a(t) = 0 otherwise The effect of noise on the DAD performance can be seen by examining the typical response function in Fig. l(e). It is seen that some false alarms are made and some events are missed altogether. When an event is detected, the initial detection occurs some time after the arrival of the event, and finally some events are detected more than once. Let NF and NX denote the number of false alarms and extra F X detections respectively, and let LF and LX be the corresponding losses. Let T1 be the total amount of "late detection time. "

7 Specifically, TLD is the total amount of elapsed time until each event is detected or the duration of the event if it is not detected. Let LLD be the loss per unit late detection time. Then the performance of the DAD can be characterized in terms of the loss function L = LDND + LMNM + N LFN+ L + F LDTLD (3) where ND, NM, L, and LM are as defined above.

3. THE OPTIMUM BAYES DETECTORS In this section the optimum Bayes detectors are specified in terms of a recursion relationship. In order to minimize the mathematical difficulties involved in the derivation, the following assumptions are made. (1) Both the continuous action and the discrete action detectors make decisions only at the discrete decision times t e, = {tk =k, k = 1, 2,..., N} where N = T/A. This assumption states that the detectors determine the response functions a(-) only through the values a(tk). In order to interpret this assumption in the context of Section 2 we define the response function for the DAD by a(tk tk) tke a(t) = 0 otherwise and a(t) = a(tk) tE [tkl, tk) for the CAD. Thus, assumption (1) restricts the DAD to respond only at the times tk c 5d and restricts the CAD to respond either at every time in the interval [tk 1, tk) or at no time in the interval 8

9 I tk-1' tk) The decision separation time A can be interpreted as the shortest possible duration of an event as a result of the next assumption: (2) Events begin and end at the decision times tk 6, in the sense that if an event is present at any time in the interval (tk 1, tk], then it is present at every time in that interval. As a consequence of assumption (2), the occurrence of events may N be completely specified by the sequence of random variables {Hk}k=1 where 1 if an event is present in the interval (tk-1, tk] Hk = k 0 otherwise The specification of the prior probability law on the occurrence of events constitutes the third assumption: (3) The sequence {Hk} is a stationary two-state Markov chain with P[Hk =1 Hk-l = 0] = P[Hk=O | Hk-l=l] = 6 and

10 P[Hk 1] = /(+ 6) Assumption (3) asserts that the number of decision times between successive events is geometrically distributed with parameter v, and the number of decision times that each event is present is geometrically distributed with parameter 6. It follows that the average duty (percentage of time when an event is present), D, is equal to v/(6 + v), and the rate (number of events per second) is 6Dh-1 6DA1. The final assumption specifies the probability law on the observation process. Specifically, let xk denote the "current" observation pertaining to the interval [tk_ 1, tk), and let Xk denote the "total past" observation pertaining to the interval [ 0, tk) so that Xk = (Xk-1'k) Then, it is assumed that (4) The observation XN is continually independent and the current observation xk is described by the densities p(XkIS) = p(Xk|Hk = 1) p(Xk|N) = p(xk Hk = 0) It should be noted that as a consequence of assumption (4) the conditional density of xk depends only on whether or not an event is

11 present at time tk and not on the arrival time of that event. As a first consequence of the above assumptions, an updating formula for the a posteriori probability that an event is present at time tk may be obtained. Denoting this probability by Pk = Pr[ Hk =1 Xk], it is easily shown that, k+ = P[ Pk' Xk+ 1 (4) = [Pk(1 - 6) + (1 - Pk) ] p(xk+ S)/(xk+1 Xk) where P(xk+ I Xk) (5) =[P (1- 6) + (1- Pk) ] P(xk+ l S) + [Pk6+(1- Pk)( - )] P(Xk+l N) Next, we develop an updating formula for the a posteriori probability that any event present at time tk has been previously detected. This is done as follows. Let Tk { 1, 2,...} denote the elapsed number of decision times since the most recent response as measured from time tk. Let dk(Tk) be the a posteriori probability that a previously detected event is present at time t. Then, dk(Tk) = Pr[Hk =1, Hkl =1,., T l x] and the a posteriori probability that any event present at time tk has

12 been previously detected is dk(Tk) = Pr[Hk = 1,Hk_1 = 1,. Hk-T =1 Ik Hk ] dk(7k)/ k With these definitions, it is easy to show that dk+1(Tk+l) = dk+l[Pk, dk] (6) (1 - 6) Pk dk(Tk+l - 1) v(1- Pk) + (1- 6)Pk where dk(O) A 1 With the above results at hand, it is possible to characterize the optimum Bayes detectors. The basic result is contained in the following theorem. Theorem I. The optimum Bayes detector determines a(tk) according to:

13 l1 tif Rk(Pk, dk) < Wk(Pk, dk) a(tk) =(8) 0 otherwise where Pk(Pk,dk) (-Pk) - [Adk+ B(1- dk)] Pk + Wk(Pk'dk ) (9) Wk(Pk dk) = f k+l[Pk+l(Pkxk+l), dk+l(Pk,dk)] P(xk+1 lXk)dxk+ (10) and 0 k = N Vk+ (Pk+ldk+) = 0 (11) min [ Rk+l(Pk, dk), Wk(Pk dk)] The quantities A and B in Eq. 9 are "normalized losses" given by A = (12) (LT LT F C LM M - LD B =A + T- ) O, 0< A<B< oo (13) (L - L) for the continuous-action detectors, and

14 A= (14) LF LM LD LLD B = LF + L -o< < O < B< oo (15) F F for the discrete- action detectors. Note from Eqs. 12 and 13 that for the classical CAD, A = B. The proof of the above theorem is obtained by interpreting the loss functions from the preceding sections in terms of the loss at a particular decision time and then applying standard dynamic programming techniques to obtain the recursion relation for Vk(Pk, dk). The details of the argument are found in the appendix. The use of Theorem I to obtain an explicit expression for the Bayes detectors involves first computing the decision rule at the terminal time tN and then proceeding recursively through decreasing indices to determine the decision rule at time tk < tN. In the case of the classical CAD, the solution can be obtained immediately by setting B = A and noting that the dependence on dk in Eqs. 8-11 vanishes. The resulting decision rule is, |1 if Pk> 1+A( a(tk) (16) 0 otherwise

15 A more familiar expression for the response condition is obtained in terms of the a posteriori odds ration, k = Pk/(1- Pk) The updating equations for Ok and dk are obtained immediately from applying the definition to Eqs. 4 and 6. The results are k- l(1 - 6) + v Ok = kk 0 5k 6+ (1- ) and Ok-_(1- 6)dkk k- 6+(1- v) In terms of Ok, the response condition for the classical CAD is LT T a(t) = 1 iff 0> C a(tk) 1 iff Ok-> A = k k A T T LM- L M D It is noted that this is precisely the same response condition as would be obtained in the problem of testing the hypothesis that an event is present in the interval [tk_ 1 tk) against the alternative that no event is present during [ tk_ 1' tk)' The decision rules for the other detectors have been obtained using a digital computer to solve the equations in Theorem I. The

16 major results of these computations are stated as two properties. The first of the two properties states that, as in the classical CAD case, the optimum detector responds iff Pk exceeds a threshold. The threshold, however, depends on the current values of dk, the a posteriori probability that any event that is present has been previosly detected. Stated formally: Property I (The Threshold Property). For each optimum detector there exists a sequence of functions Kk( ) k = 1,..., N, such that a(tk) = iff Pk > Kk(dk) or, stated in terms of the odds ration, a(tk) = 1 iff Ok Tk(dk) 1 1 - Kk(dk) Tk(dk) - + 00 It follows from Property I that the general form of the optimum Bayes detectors is as indicated in Fig. 2. The operation of the optimum detector is clear from Fig. 2. At time tl, O_1, dk_l and ak-l are all stored in memory.

g _ _ } p -^o-I'2 1 Elo-1^ ^ —^ ---- "~ -----' -g g _IXIo o- _P i l 10-1 u4LL Figure 9. Cylindrical chamber characteristics, P = 10-2, 13, andTP "0" torr. 1-0-13=^15 ~11 15

18 Then at time tk, the current likelihood ratio Ak along with Ok 1 are used to determine Ok. Simultaneously, dk is calculated from Ok 1 and d'k 1' where d'k 1 can be considered as the result of setting dkl equal to one whenever the detector responds. The statistic dk is then used to determine the value of the threshold Tk(dk). Finally the detector responds iff Ok exceeds Tk(Ok) and Ok, dk and a are stored for use at the next decision time tk+ k k'k k+k It should be pointed out that only that portion of the general optimum detector enclosed in dotted lines is necessary in the classical CAD. The second of the two computational properties states that, except for decision times tk that are close to the terminal decision time tN, the structure of the optimum detector does not depend on time. More precisely: Property II (Asymptotic Stationarity Property). For each optimum detector, the sequence of threshold functions {Kk( )} converges through decreasing indices k to a limiting threshold function K(). As a consequence of Property II, the threshold functions Tk( ) in the general system model can be replaced by the limiting threshold function K(d)/[ 1 - K(d)] K(d) < 1 Ti(d) = (17) +co K(d) = +00

19 except for decision times near the terminal decision times. We complete the description of the optimum Bayes detectors by illustrating some specific examples of the threshold functions for both the continuous action and the discrete action detectors. For simplicity, we illustrate the thresholds K( ) with the understanding that T(d) can be obtained by the monotonic relationship in Eq. 17. The CAD case is considered first. Figure 3(a) illustrates three different threshold functions for continuous-action detectors. [The values of the losses A = (LT M - L )/ (LF - LC) and B =A + (LM - LD)/ (L- L)A corresponding to each threshold function appear next to the curve as the couple (A, B). ] The three different threshold functions in Fig. 3(a) were computed for the same value of A. The threshold function for (A, B) = (0. 5, 0. 5), corresponding to the classical CAD, is constant at the value 1/(1+A) = 2/3 as expected from Eq. 16. The other two threshold functions both lie below the classical CAD threshold except at the point d = 1 where they are all equal. [ It can be shown that K(d = 1) = 1/(1 + A) for all continuous-action detectors. ] It is noted that the threshold function for the larger value of B is smaller than the threshold function for the intermediate value of B. This is reasonable since a large value of B reflects a large value of the net loss of a missed pulse, (LM - LD), and thus the optimum detector

20 K(d) 1. (0. 5, 0. 5) 0. 5 (0.5,10.) 0 | I d 0. 5 1. 1., K(d) (-2., 1.) 1(-0.5,1.) 0. 5 (-2.,10) (-0.5,10.) __ ____________l________________ d 0.5 2.0 Fig. 3. Illustrations of threshold functions (a) CAD (b) DAD

21 would be expected to respond more often. Figure 3(b) illustrates four different threshold functions for discrete-action detectors. It is noted that for each threshold function there is a d* such that when dk > d*, K(dk) equals 1, and it is impossible for the detector to respond. The value of the "inhibit threshold" d* depends on both the losses A = -LX/LF and B = (LM + LLDA)/LF. For constant values of B, an increase in the loss for an extraneous detection LX results in a decrease in d* [compare the threshold functions for (-2, 1) and (-0. 5, 1)]. On the other hand, for fixed A, an increase in either the loss for a missed pulse LM or the loss per unit detection delay LLD causes an increase in d* [compare the threshold functions for (-0. 5, 1) and (-0. 5, 10)]. The threshold functions for the CAD and the DAD depend not only on the normalized losses A and B, but on the parameters of the prior probability law and on the signal-to-noise as well. The threshold functions in Fig. 3 were calculated for a pulse rate R = 0. 05 and a duty D = 0. 5 and a signal-to-noise ratio S/N = 1. Other calculations indicate that although tha a e basic form of the threshold functions do not vary with changes in R, D and SN, increases in either R, D or SN result in a decrease in the threshold functions.

4. SUMMARY AND CONCLUSIONS In the previous sections we have considered two classes of detectors both of which function to detect randomly occurring events as they occur. One class, the continuous-action detectors, seek to respond at each point in time that an event is present whereas the other class, the discrete-action detectors, seek to respond at only one time while each event is present. It has been seen that in both cases, the optimum detectors have the same canonical form (Fig. 2). Both calculate the odds ratio Ok and the previous detection probability dk at each decision time tk and respond iff Ok T(dk) The difference in the optimum for the two cases appears in the form of the threshold function T(-). For the continuous action detectors, T(-), is everywhere finite. In the special case of the classical CAD, T(-) is equal to a constant (L - L )/(L - LT), independent of F C m D dk, a result that could have been derived using classical detection theory. For the other continuous action detectors T(-) is an innp rr~ T creasing function of dk, bounded above by (LF - LT)/(LT - L ). The threshold function for the discrete action detector equals +oo for dk greater than some inhibiting threshold d*. For dk < d*, 22

23 the discrete-action threshold function is an increasing function of dk.

APPENDIX Theorem I must be proved for the CAD and the DAD separately. Continuous-Action Detector We begin with the following observations: (1) Since each event that occurs must be either detected or missed, NE = total number of events = ND + N D M (2) Since at each instant that an event is present, either a detection or a miss occurs, TS = total time when events are present TD + TM OD M (3) Since at each instant that no event is present, either a false alarm or a correct rejection occurs, T = total time when no event is present - T + T F C 24

25 Substitute for, NM, TM and TC from the above into Eq. 2 to obtain L [(LD- L N + (L - L) TD + (L- LC)TF] T T + [LMNE + LTS + LCTN] The second bracketed term does not depend on the decision process and hence can be neglected. Denoting the first bracketed term by L and noting by assumptions (1) and (2) in Section 3, that TD = A(N + NX) TF = ANF we may write '^D-1^^-1^ 1^ [T T T t =LD - LM + (L - LM) A] ND + [(LD - L) A NX + [(L- - L) A] NF Now, since (LF - LT ) is assumed positive, it is sufficient to minimize the expected value of T T L = L/(LF- L) A -BND- ANX + NF

26 where T(T T T A =(LT - L)/(L- LT ) B = A + (LM- LD)/(L - L But L may also be written in the form N L= Z Lk k=l where L = 0 if a(t) = k =0 and 1 if no event is present at time tk and a(tk) = 1 Lk = -B if an undetected event is present at time tk and a(tk) = 1 -A if a detected event is present at time tk and a(tk) = 1 Equations 8 through 15 then follow immediately after noting that 1 - Pk' (1 - dk)Pk and dkPk are the a posteriori probabilities that an event is present at time tk an undetected event is present at time tk and a detected event is present at time tk, respectively.

27 Discrete-Action Detector By using the first observation above to substitute for NM in Eq. 3 and by dividing through by LF > 0, we may reduce the problem of minimizing L to the problem of minimizing L = [(LD-LM)/LF]ND + [LX/LF]NX + (LLDA/LF)NLD + NF where NLD = TLD/A is the number of times an undetected pulse is missed. Alternatively we may write, N L = Lk k=1 where LLD A/LF if a(tk) = 0 and undetected event present (LD - LM)/LF if a(tk) = 1 and undetected event present Lk = LX/LF if a(tk) = 1 and previously detected event present 1 if a(t) = 1 and no event present

28 Then by taking expected values, we have V(P, dk) = min {Rk(Pk, dk), Wk(Pk dk)} where Rk(Pk,dk 1- Pk + {[LX/LF] dk + [(LD - LM)/LF] (1 - dk)} Pk + S Vk+l[Pk+l dk+l(Pk' dk = 1)] P(k+ll Xk)dk k(Pk,dk) = LLDA/F(1- dk)Pk + S Vk+l[Pk+1' dk+l(Pk'dk)] P(xk+1) from which Eqs. 8 through 15 follow with Rk(Pkdk)- Rk(Pkdk) + (LLD /LF) (-dk)Pk Wk(Pkdk) = Wk(Pk,dk) + (LLD/LF) dk) Pk and A -LX/LF B = (LM-L)/ LF + LLDA/LF

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DOCUMENT CONTROL DATA. R & 0 (Sr'.,riyff rlfieallin of tile, ho)' of ahefrnrft rnd indknd nfnoiallnln hnlm e enterf d whn Ithe overall repoi fi cnlasellled). I. O0RIGNA TIlNO ACTIVITY (Corporate8 esehor) 2a.REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory Unclassified The University of Michigan 2h. GROU Ann Arbor, Michigan 48105 3. REPORT TITLI THE DETECTION OF RANDOMLY OCCURRING EVENTS WITH RANDOM DURATIONS 4. bDESCtIPTIVE' NOT (Type of report and ineleive date.) C. E. L. Technical Report No. 214 - November 1971 S. AU TS-H IS) WFrstr naime, middle initial, las name) Richard M. Heitmeyer. REPORT OATE 7. TOTAL NO. OF PAGES 7b. NO. OF REFS November 1971 42 )4. CNTIRACT OR GRANT NO. 9O. ORIGINATOR'S REPORT NUMBER(S) N000 14- 67-A- 0181- 0032 0360408 b. PROJECT NO. 036040-8-T c. b6. OTHER REPORT NO(S) (Any other numbers that may be aseigned this report) d. TR214 10. OISTRIBUTION STATEMENT Approved for public release; distribution unlimited. II. SUPPLEMKENTARY NOT9ES 1 2. S9ORING MILITARY ACTIVITY OiSce of Naval Research Department of the Navy Arlington, Virginia 22217 I 3. ABSTRAC T This report considers the detection of events whose occurrence times and durations are random variables. In this context, two special cases are considered: (1) the continuous-action detectors that seek to respond at each time when an event is present and (2) the discrete-action detectors that seek to respond at only a single time during which each event is present. For both cases, the optimum detectors are specified in terms of recursion relations and examples of the numerical solutions of these relations are presented and compared. DD,.OR. 1473 Security Classification

Security Classification *14~~~~.~ LLNK A LINK s LINK C KEY WORDS.. LNK _ ROLE W T ROL WT[ HT O L' WT Random variable occurrence times Continuous-action detectors Discrete-action detectors Optimum Bayes detectors Illl- - [ - l I l I1tITn" t[t [ -

UNIVERSITY OF MICHIGAN 3 9015 03026 7820