and the sufficient statistic is shown to be a weighted sum of the local decisions collected by the DFC within the observation interval. The weights are functions of ...
I.
Asynchronous Distributed Detection
NOMENCLATURE
a(.) Normal distribution: @(XI
= l/&J:w
exp(-y2/2) dy
yf Variable used to describe the sign argument
of Gaussian function in P J t ] Variable used to describ? the sign argument of Gaussian function in P, [ t ] q DFC decision threshold X Sum of local Poisson parameters under 7ij hypothesis H, : X j = Poisson parameter of detector i under qj hypothesis Hj (a,b) Open interval between a and b on the real line DFC Data fusion center E [ . ] Statistical expectation operator Hj jth hypothesis L(. : .) Discrimination function used to measure the information (cf. [ l ] ) k f ) Total number of decisions transmitted to the DFC by the ith local detector during the interval [O,t) N Number of local detectors O(.)Asymptotic analysis symbol: f ( x ) = O ( $ ( x ) )( x + CO) means that there exist real numbers a and A such that If(x)l 5 AI$(x)l whenever a < x < CO o( .) Asymptotic analysis symbol: f ( x ) = o(g(x)) ( x CO) means that f ( x ) / g ( x ) tends to 0 when x + CO PI;] Probability measure Pdg Global probability of detection at time instant t with known {kf)}El Pdg[t] Global probability of detection at time instant t pdi Detection probability of the ith local detector :,)et] Global probability of error at time instant t P e [ t ] Approximation to the global probability of error at time t PFj Global probability of false alarm at time instant t with known { k f ) } E l P f g [ t ]Global probability of false alarm at time instant t Pfi False alarm probability of the ith local detector t Time instant U - l [ . ] Unit step function: (Pd
WEI CHANG, Student Member, IEEE MOSAE KAM, Senior Member, IEEE Drexel University
xEl
Binary parallel distributed-detection architectures employ a bank of local detectors to observe a co-n
volume of
surveluance, and form binary local decisions about the existence or nonexistence of a target in that volume. The local decisions are transmitted to a central detector, the data fusion center (DFC), whlch integrates them to a global targel or 110 target decision. Most studies of distributeddetection system assume that the local detectors an synchronized. In practice local decisions are made asynchronously, and the DFC has to update its global decision continually. In this study the number of local decisions observed by the central detector within any observation period is Poisson distributed An optimal fusion rule is developed,
and the sufficient statistic is shown to be a weighted sum of the local decisions collected by the DFC within the observation interval. The weights are functions of the individual local detector performance probabilities (Le., probabilities of false alarm and detection). In this respect the decision rule is similar to the one developed by Chair and Varshney for the synchronized system.
Unlike the Chair-Varshmy rule, however, the DFC's decision threshold in the asynchronous system is time varying. Exact expressions and asymptotic approximations are developed for the detection performance with the optimal rule. These expressions allow performance prediction and assessment of tradeoffs in realistic decision fusion architectures which operate over modem "munication networks.
Manuscript received June 17,1992; revised July 6, 1992. IEEE Log NO. T-AESBOLW6639.
----)
This work was supported by the National Science Foundation through Grant ECS 9057587. Authors' current addresses: Dept. of Electrical and Computer Engineering, Drexel University, Philadelphia, PA 19104.
ut)
ui 41
0018-9251/94/$4.00 @ 1994 IEEE 818
Decision of the DFC at time instant t Decision of the ith local detector Set of observations collected by the ith local detector
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1
Fig. 1. Decentralized detection structure.
II.
INTRODUCTION
Binary parallel distributed-detection architectures (Fig. 1) employ a bank of local detectors which observe a common volume of surveillance, and form binary local decisions about the existence or nonexistence of a target in the surveyed volume. The local decisions are transmitted to a central detector, the data fusion center (DFC), which integrates these decisions to a global target or no target decision. Many localdetector and objective-functioncombinations have been studied over the years: Bayesian (e.g., R M and ~ Sandell [ll], Chair and Varshney [2], Reibman and Nolte [9], Hoballah and Varshney [6], Kam, et al. [7),Neyman-Pearson (e.g., Thomopoulus, et al. [12]), minimax (e.g., Geraniotis and Chau [3]), Bayesian-sequential (e.g., Hashemi and Rhodes [4], Rneketzis [lo]), max-divergence (e.g., Lee and Chao [SI) and information theoretic (e.g., Hoballah and Varshney [5]). Almost all studies assume that the local detectors are synchronized, namely the local decisions are transmitted to the DFC in common predetermined time instants. This assumption is made even in some studies of decentralized sequential detection systems (e.g. [41). In the configuration studied here, each local detector employs a fixed decision rule and makes its decisions asynchronously and independently of all other detectors. Every local detector that reaches a decision transmits it to the central detector instantaneously and proceeds to collect new observations for a subsequent decision. During any interval of observation, the number of decisions that each dectector makes is Poisson distributed. Consequently, the total number of local decisions received by the DFC is also Poisson distributed. We develop the sufficient statistic of the DFC for an optimal threshold test using the Bayes risk criterion and show that it is a weighted sum of the local decisions collected by the DFC over the observation period. The weights are functions of the local detector performance probabilities, namely the probability of false alarm and the probability of detection. The sufficient statistic is compared with a (time-varying) threshold which depends on the total number of local decisions received, the local-decision performance probabilities, the decision-rate (Poisson) parameters, CHANG & KAM: ASYNCHRONOUS DISTRIBUTED DETECTION
~
the hypothesis a priori probabilities, the Bayesian costs, and time. In addition we find exact expressions for the global performance probabilities, and numerically efficient tight approximations for them. In Section I11 we define the asynchronous distributed detection system and develop the DFC's optimal fusion rule for a global Bayes risk criterion. In Section IV exact expressions for the global probabilities of false alarm and detection as function of time are obtained. Asymptotic approximations to these probabilities are developed in terms of the localdetector performance and the parameters of the Poisson distribution which describe the local decision rates. The approximations become more accurate as the observation period becomes longer. Specifically we show that when the global objective function is the probability of error, this probability tends asymptotically to zero with time. Ill.
DISTRIBUTED DETECTION WITH ASYNCHRONOUS LOCAL DETECTORS
The asynchronous distributed-detection system (Fig. 1) comprises N local detectors and a DFC. Each local detector makes a decision about a binary hypothesis (Ho,H I ) :the ith local detector collects the observations xi,and declares ui = 0 if it accepts Ho on the basis of its observations vi,or ui = 1 if it accepts HI. During an observation interval, each local detector makes several (possibly zero) binary decisions, and sends each one of them to the DFC immediately after it was made.' The DFC combines these local decisions into global decisions about the hypotheses, by continually updating its global decision ut).ut) = 0 means that the DFC decides in favor of HOat time instant t; ut) = 1 means that the DFC decides in favor of HI at time t. The number of decisions that the ith local detector makes within any observation period is assumed to be Poisson distributed with parameter ri0 (> 0) under hypothesis Ho, and 7i1(> 0) under hypothesis H I . Moreover, each decision of a local detector is assumed to be conditionally statistically independent of all other decisions (made by the same detector or by other detectors) under hypothesis Hj (J = 0,l). The conditional probability that the ith local detector makes kf) (2 0) decisions during the interval [O,t] is given by
for i = l , ...,N
and J = O , l .
The variable P[.] is a probability measure. The set (j) k!') ~ =l { U!'), U?),
{ ui } j
.. .,uy!'))} contains kf) local
'We assume that the true hypothesis did not change during the observation time interval.
(1)
decisions generated by detector i during the interval [O,tI. Let Pf;and Pdi (0 < Pfi,Pdi < 1) be the (stationary) probability of false alarm and probability of detection, respectively, of the ith local detector (i.e., ~ f = i ~ [ u j k=) 1 I ~ 0 1 Pdj , = ~ [ u j k=) 1 I H I ] for all k = 1,2,. . .,ky)). The global performance criterion is the Bayes risk at time instant t ,
P(t) = c,P[u:'
+
= O,Ho] c l l P [ u p = 1,HlI
synchronous case and for minimum probability of error. Indeed, Chair and Varshney's rule can be obtained from (3) with the appropriate costs, kf) = 1 and q0 = ~i~ for i = 1,. ..,N . We also note that the threshold, and hence the decision rule (3), is time varying, i.e., the DFC's global decision may change even if no new local decisions have arrived. The global probability of false alarm at time t is defined as
P f J t ] = P[U!' = 1 1 HO]
+ clOP[up = l,Ho] + colP[up = O,H1]
( j ) kr)
= E[P[ur) = 1 1 Ho, {ui } j = l , i = 1,...,NI]
where the costs Cij (i,j = 0 , l ) are specified. It specializes to the global probability of error when C ~= OCol = 1 and CW= C11 = 0, namely,
(53)
+
Pe[t]= PoP[uy = 1 I Ho] P1(1- P["p = 1 IHl]) = POPf,[t] + P1(1- Pdg[t])
(2b)
and the global probability of detection at time t is defined as
P&] = P[up = 1 1 HI]
where P, is the a priori probability of hypothesis H, ( j = 0,l). The DFC integration rule is derived in the following theorem.
THEOREM 1 A n asynchronous distributed detection system with the objective of minimizing the global Bayes risk (see (2a)) comprises N local detectors and a DFC. The number of decisions that the ith local detector makes within its observation period is Poisson distributed with parameter rij (> 0) under hypothesis H, ( j = 0,l). If all local decisions made up to time t are known to the DFC, and are conditionally statistically independent, then the optimal DFC decision rule is
=Ep[u:)=lIH1,
( j ) k!')
{U; }j:l,
i = l ,...,NI] (5b)
where E[.] is the expectation function over all possible ( j ) k!')
{ ui } j:l for all i = 1,. ..,N . The global probabilities of false alarm and detection at time instant t thus can be expressed explicitly as (see Appendix B for details)
where where A j = CLl~ij for j = 0,l
with q = (Po(C10- C,)/P1(Col unit step function:
{ o,
1,
U-l[Xl=
- C ~ I ) )U-l[.] , is
if
x
the
20
if x < o
7
and k;) (20 ) is the number of decisions that the ith local detector made during the interval [O,t].
PROOF. See Appendix A. The fusion rule (3) is a generalization of the rule obtained by Chair and Varshney [2] for the 820
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO. 3 JULY 1994
-
and the summations
EHare performed over all
possible 2'~lk-f' binary tuples in H = { { h ~ ) } ~ ! ~ } ~ l with hy' E {0,1} (H corresponds to all possible combinations of local decisions that can be transmitted to the DFC during the time interval [O,t]). Equations (6a)-(6d) require only that the DFC know the performance probabilities (Pfi, pdi) and the Poisson parameters ( q j ) of each local detector. In general, equations (6) are difficult to evaluate. They are easier to compute and approximate when all local detectors are identical, namely, all detectors have the same probabilities of false alarm and detection: Pfi = Pf and pdi = p d , i = 1,2,. ..,N; and all detectors have the same Poisson parameters: T ; ~= TI and T ; ~= TO, i = 1,2,. ..,N . Next, we concentrate on this case.
w=o
To simphfy (S), we use the following theorem.
THEOREM2 Let two independent random variables X and Y have Poisson distributions with Parameters A, and A , respectiveb. Let 2 = (ax bY + c)/t, where U, b and c E (-00, 00) are @ed and t E (0,m) Then for
+
each p e d z E (-00,00)
P
IV. GLOBAL PERFORMANCE WITH IDENTICAL LOCAL DETECTORS
.. .
...
k=O
[%5
z] converges to @(z) as t approaches
+
00
+
where pz = (ax, bX, + c)/t, flz = (a2X, b2Ay)/t2 For identical local detectors, alternative expressions @ ( x ) = l/fiJ:OOexp(-y2/2)dy. and for the global probability of false alarm (Pfg[t] in (6a)) and the global probability of detection (Pdg[t] in (a)) can be obtained by using the well-known multiple PROOF. See Appendix C. Poisson distribution formula. The probabilities are Using Theorem 2, the global probability of false alarm in (sa) can be expressed as
The random variables X and Y are independent and Poisson distributed, with parameters AotPf and Xot(1- Pf), respectively; a = ln(XIPd/XoPf), b = ln(Xl(1- pd)/.\O(l- Pf)), c = -In7 and z = A 1 - Ao. Similarly, the global probability of detection in (Sb) can be expressed as
with the random variables X and Y being independent and Poisson distributed with parameters AltPd and Xlt(1- pd), respectively. From Theorem 2, the global probabilities of false alarm (6a) and detection (6b) (7b) tend asymptotically to where m = N k f ) . Moreover, by interchanging indices m and k, and setting w = m - k, equations (7) can be rewritten as
k=O
W=O
where
1
I
- -In~-(X1-XO) t
(sa)
CHANG & KAM: ASYNCHRONOUS DISTRIBUTED DETECTION
~~
~
-
821
~~
~-
-
The asymptotic behavior of P,[t] is then determined by the signs of (Pf and (Pd, namely,3
< o and (Pd < 0,
if yf
then P,[t] approaches
Oast-+oo
> 0 and (Pd < 0,
if (Pf
Po as t if (Pf
-+
For large t, the global probability of error P,[t] in (2b) can now be approximated by
(12) In (12) the (approximated) global probability of error of the asynchronous system PJt] is completely characterized by the means (pf and Pd) and variances (c$ and U:) of the two Gaussian functions. The moments p f , Pd, U;, and CT: are in turn functions of the parameters Pf,pd, XO, XI, q, and t (cf. (lla)-(lld)). It is of course of interest to analyze the asymptotic behavior of P,[t] as t CO. Next, we show that P,[t] -+ 0 as t + CO. From examination of the arguments of the Gaussian functions in ?,[t] ((12)) it is easy to verify that2
if $of
then P,[t] approaches
00
(15b)
< 0 and (Pd > 0, 1- Po as t
(1%
( W
-+
then
P,[[t] approaches ( W
00
> 0 and (Pd > 0, then ij,[t] approaches 1 as t -+ CO.
(154
Using properties of the discrimination function (cf. [l]), we show in Appendix D that (15a) is the only one of practical interest. Consequently, we obtain the following theorem for the asymptotic performance of the asynchronous system as t + CO.
THEOREM 3 The global probability of error for the asynchronous distributed detection system converges asymptoticalIy to zero a s t + CO with probability one. PROOF. See Appendix D.
+
Using @ ( X I = 1- (1/G;)exp(-x2/2) CO), the global probability O ( ~ - ~ e x p ( - x ~ / 2 )( x) of error P,[t] in (12) can be approximated further by -+
-+
-'= p f d ( l + o(1))
Pf
Uf
(t + C O ) ~ (13a)
~
2 f ( ~ =) o(g(x))(x
822
-+
CO)
(16)
where c = m i n { p ~ , p ~ } . Fig. 2 shows the global probability of error (12) versus the observation time t for a decentralized asynchronous detection system with three identical local detectors (LDs) and with a DFC that minimizes the global probability of error. The parameters of the LDs are shown in a b l e I. In Fig. 2, values of PJt] for large t (t 2 50) were calculated using approximation (16). Comparing case I to case I1 we see that the rate of decrease in the global probability of error for case I1 is much faster than that for case I, because system I1 has better LDs. Comparing case I1 to case I11 we see that the rate of decrease in the global probability of error for case I11 is faster than that for case 11. This is because
where
X".
+ 0(t-~/~exp(-ct/2))(t
-, CO) means that f ( x ) / g ( x ) tends to zero as
3The cases 'pf = 0 and/or 'pd = 0 are analyzed in Appendix D. It is shown there that these are nongeneric cases of no practical interest. 4 f ( ~ )= O ( @ ( x ) )( x + CO) means that there exist real numbers a and A such that I f ( x ) l 5 Al@(x)lwhenever a < x < 03.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO. 3 JULY 1994
TABLE I Parameters of LDMs in Three Asynchronous Decision Fusion Architectures
Note: Roman characters here correspond to italic figures in text.
asymptotic approximations for the DFC performance probabilities are calculated. The approximations allow the assessment and prediction of DFC performance for large observation intervals, and can be used to assess parameter tradeoffs in the design of asynchronous distributed detection architectures.
APPENDIX A. 0
20
10
Lo
LOO 110 I10 16D obswvath tire 80
PROOF OF THEORM 1
110 200
Consider an asynchronous distributed detection
Fig. 2. Global probability of error versus observation time for
system with N local detectors. The set {
asynchronous systems described in 'Ihble I.
U Y ) } ~ =: ~
(1 )
{U!'), U?), ..,,U? I}contains the k r ) decisions made by the ith local detector and received by the DFC during the observation interval [O,t]. The number of decisions, k r ) , is Poisson distributed (see (1))with parameter 7ij under hypothesis Hj ( j = 0,l). Let q be the global threshold corresponding to the Bayes risk (?a), namely 77 = (Po(Clo - C ~ ) / P 1 ( C o l Cll)). Then the optimal fusion rule at the DFC is given by a likelihood ratio test:
system I11 gets a higher volume of reliable mformation than system I1 when the rate of local decision making under hypothesis Ho is increased (from 0.8 to 3.0): Pf = 0.1 for HObut P,,,= 1- Pd = 0.3 for H I . Expression (12), along with the asymptotic expression (16), allow the generation of performance graphs (such as the ones in Fig. 2) for asynchronous systems with similar detectors. When the local detectors are not similar, it is at least possible to get upper bounds on expected performance assuming that all local detectors posses the worst detector performance probabilities. Asymptotic global performance of asynchronous systems can be predicted, and asynchronous distributed detection and fusion models can be compared. V.
Using Bayes rule on the left-hand side, we obtain
CONCLUSION
A model of asynchronous decision making in distributed-detection systems is studied, when the number of local decisions during any observation interval is Poisson distributed. The optimal Bayesian fusion decision rule is developed, and the sufficient statistic is shown to be a weighted sum of local decisions collected by the DFC during the observation period. The weights are functions of the local-detector performance probabilities. The DFC decision rule for the asynchronous system is a generalization of the rule obtained by Chair and Varshney for the synchronized system. The main quantitative difference is the decision threshold in the synchronous scheme is time invariant, whereas in the asynchronous scheme the threshold is time varying. Exact expressions and
Using (l),(18) is reduced to
Assuming the conditional statistical independence of the local-detector decisions u y ) E {O,l}, j = 1,2,... ,kf), and expanding each term in the left-hand side of (19) over U?), the log-likelihood of the ith local 823
CHANG 8t KAM: ASYNCHRONOUS DISTRIBUTED DETECTION
~~
~~
_____
-
___
detector becomes
equation (23) can be rewritten as
(20) The optimal fusion rule (see (3)) is then obtained by combining (19) and (20).
where H = { { h j " } F ! y } z lwith h y ) E {0,1}. By the Poisson distribution assumption for { k f ) } E 1in (22) and (25), the global probability of false alarm in (21) can now be expressed as c o c a
APPENDIX B.
DERIVATION OF EQUATIONS (6)
Pfg[t] =
...
k(')=O kt)=O
00
k,$)=O
e - y N- p(rLt)k? P ( t ) kf)! fg' i=l
We derive (6a);(6b) follows similarly. By the definition of the global false alarm probability at time t
Pf&] = P[up = 1I Ho].
(21)
(26) APPENDIX C.
PROOF OF THEOREM 2
Assume that two independent random variables
Equivalently,
X and Y have Poisson distribution with parameters
y'
k!)
I U l II
=exp(s:)
.exp{A, [exp(sq) -I]}
Equation (22) is obtained using Bayes' rule. In (22), let
fg
=
1':
= ',
{'i
I Ho; { k f ) } z l l
(28)
- 11
The first equality UeS the independence of X and Y,and the second is a consequence of X and Y being Poisson random variables with m(d'/t)(s)= exp{Ax [exp(s(a/t)) - 11) and m(bY/f)(s) = exp{A,[exp(s(b/t)) - l]}. Using a Bylor expansion of exp(z) = 1+ z + z2/2! O(z3)(z -+ 0), (28) can be rewritten as
tuy)}jIl k?)
+
ti)k?)
tui
>.
.exp{~, [exp(s:)
( j ) k? '(')
ljl,
m z ( s ) = exp (23)
+(
Moreover, since
+ O(t-3)(t
---f
41.
t2 (29)
Thus, from (29), the moment-generating function of the random variable Z converges to a normal generating function exp(pu,s c7;(s2/2)) as t -+oc) with the mean pz = (ax, + bA, + c)/t and the variance c7; = (a2A, b2A,)/t2. Hence, for each fixed z E
+
(24)
824
+
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VOL. 30, NO. 3 JULY 1994
P[(Z - pz)/oz 5 21 converges to @(z) as t
(-00,00),
REFERENCES
approaches 00.
Blahut, R. E. (1987)
APPENDIX D.
Principles and Practice of Information Theory. Reading, MA: Addison-Wesley, 1987. Chair, Z., and Varshney, P. K. (1986)
PROOF OF THEOREM 3
Optimal data fusion in multiple sensor detection systems.
With some algebra, expression (14a) becomes
IEEE Transactions on Aerospace and Electronic Systems,
where
141
AES-22, 1 (1986), 98-101. Geraniotis, E., and Chau, Y. A. (1990) Robust data fusion for multisensor detection systems. IEEE Transactions on Information Theory, 36, 6 (1990), 1265-1279. Hashemi, H. R., and Rhodes, I. B. (1989) Decentralized sequential detection. IEEE Transactions on Information Theory, 35, 3 (1989),
509-520. Cf
=
If(In3)
- pd))
+ (1 - pf)(h Xo(1-
Pf)
1'
151 1'2
> 0 for all values of Pf,Pd, XO, and XI. In (30) we recognize the term g(Pf,Pd)is the negative of the expected value of the log likelihood ratio P(uj I Hl)/P(uiI Ho) with respect to hypothesis Ho. Therefore, g(Pf,Pd) is the negative of the discriminationfunction (cf. [l,107-1131), and as such satisfies g(pf,Pd)5 0. (31)
IEEE Transactions on Aerospace and Electronic Systems,
Furthermore, using the inequality 1 1- - < lnx 5 x - 1 x -
[91
(32)
25, 4 (1989), 536-543. Reibman, A. R., and Nolte, L. W. (1987) Optimal detection and performance of distributed sensor systems.
IEEE Transactions on Aerospace and Electronic Systems, AES-23, 1 (1987), 24-30,
(33) From (31) and (33), it follows that for all values of Pf, pd, XO, and XI,
Pfln-pd Pf
Hoballah, I. Y., and Varshney, P. K. (1989) An information theoretic approach to the distributed detection problem. IEEE Transactions on Information 7iieory, 35, 5 (1989), 988-994. Hoballah, I. Y., and Varshney, P. K. (1989) Distributed Bayesian signal detection. IEEE Transactions on Information Theory, 35, 5 (1989), 995-1000. Kam, M., Chang, W., and Zhu, Q. (1991) Hardware complexity of binary distributed detection systems with isolated local Bayesian detectors. IEEE Transactions on Systems, Man, and Cybernetics, 21, 3 (1991), 565-571. Lee, C. C., and Chao, J. J. (1989) Optimum local decision space partitioning for distributed detection.
l-pd XO + (1 - Pf)ln-1-Pf
