Distributed Weak Signal Detection and Asymptotic Relative Eciency in Dependent Noise Hakan Delic Signal and Image Processing Laboratory (BUSI) Department of Electrical and Electronics Engineering Bogazici University Bebek 80815 Istanbul, Turkey Tel.: +90 (1) 63-1540 (x1859) Fax: +90 (1) 87-465 E-mail: delic@busim.ee.boun.edu.tr Abstract This paper considers discrete-time distributed detection of a constant, weak signal in dependent noise. A number of sensors transmit binary decisions to a central processor, which assimilates them to produce a nal verdict. Asymptotic relative eciency (ARE) performance measure is used to compare the decentralized and centralized detection schemes. The precise denitions of ARErobustness and asymptotic ARE-robustness are introduced, and utilized to understand better the trade-os involved in multiple-sensor detection in the presence of dependent noise. Keywords: Data fusion, m-dependent noise, asymptotic relative eciency, robustness.
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 1 Introduction Decentralized processing of information in multiple-sensor detection systems is necessary, because of the cost of transmitting large amounts of data. In general, it is assumed that the sensors send their binary decisions, regarding the hypotheses, to the central processor. This scheme is called decision fusion, and it is suboptimal due to the information loss caused by the R N -to-f0; 1g transformation where N is the data size per sensor. A distributed decision network (DDN) consists of a number of sensors that transmit their binary outputs to a central processor where the numbers are combined to declare the nal decision according to some fusion rule. The problem of nding the best fusion rule in a DDN, in the most general setting, involves simultaneous optimization of the local detectors and the fusion rule. Unfortunately, for more than two sensors, the computations become NP-complete [11], and one has to make certain assumptions, such as temporal and spatial independence among the sensors, in order to make the solution tractable. These assumptions are not realistic when the sensors sample at high rates and/or are geographically close to each other. In this paper we consider decision fusion in the presence of m-dependent sensor noise. The following denition of m-dependence is taken from [8]: The stationary random process fy n g 1 n=1 is called m-dependent if there is a nonnegative integer m such that the sequences fy n g t n=1 and fy n g 1 n=s are statistically independent for all t s 1 satisfying t? s > m. Examples of m-dependent sequences are those obtained by sampling Gaussian processes whose autocorrelation functions have nite support, or nite-order moving-average processes. The more general class of -mixing processes [6] include arkov processes satisfying Doeblin's condition [1], as well as m-dependent sequences. The results of the paper extend to symmetrically -mixing noise under some mild requirements, when we let m! 1 in the model described in Section [3]. We utilize an asymptotic relative eciency (ARE) performance measure to compare the optimal centralized and decentralized detection structures. The deployment of ARE and the ensuing asymptotic computations lead naturally to the concept of ARE-robustness, which was informally introduced in [] and [5]. In this work, we derive the ARE expressions, and infer performance and robustness-related conclusions for distributed detection in m-dependent noise. The organization of the paper is as follows: Section presents the system and noise model. Section 3 nds the asymptoptic relative eciency of distributed detection in m-dependent noise. The concepts of ARE-robustness and asymptotic ARE-robustness are introduced in Section 4. Section 5 has concluding remarks. Throughout the paper, capital letters will denote random variables and lower case letters will denote their realizations. System and Noise odel We consider discrete-time detection with sensors. Sensor j receives a data string of length N, denoted by x N j = fx ji g N i=1, j = 1; : : : ;. The objective is to choose the acting hypothesis between the alternatives X ji = + V ji ; under hypothesis H 1 ; X ji = V ji ; under hypothesis H 0 ; (1a) (1b)
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 3 where fv ji g 1 i=1 is a zero-mean, stationary and m-dependent noise sequence corresponding to sensor j, and is a constant positive signal. We will assume that the density functions f Vj, j = 1; : : : ;, of the noise processes are symmetric around zero, continuously dierentiable, strictly positive on the entire real line, with nite Fisher information. The local processing will be limited to memoryless detection; that is, sensors will deploy the following decision procedure: where j (N) = j (N; g; x N j ) = T g = T g (N; x N j ) = 1 N 8 >< >: NX i=1 1 if T g > p if T g = 0 if T g < ; g(x ji ); j = 1; : : : ; : () g is assumed to be a measurable, memoryless and nonlinear function for all j, and j (N) is the probability of sensor j deciding on H 1, given x N j, j = 1; : : : ;. The threshold and the randomization constant p [0; 1] are determined such that a prespecied error probability is achieved. We will limit ourselves to continuous distributions so that p = 0. Notice that due to the identical nature of the noise characteristics across all sensors, the same detector structure is employed by all local processors. oreover, due to the stationarity of fv ji g 1 i=1, the memoryless nonlinearity assumption is valid without signicant loss of generality. In fact, when the dependence among samples is too weak, it is well-justied to keep the independent sampling structure of a conventional detector [7]. In the sequel we shall take advantage of the following functional central limit theorem for m- dependent processes [8], which is actually a corollary to an earlier theorem developed for the more general -mixing class [1]. Theorem Suppose that fy n g 1 n=1 is a stationary m-dependent sequence and that g is a measurable function satisfying Efg(Y 1 )g = 0 and Efg (Y 1 )g < 0. Dene = Efg (Y 1 )g + Then, if > 0, the sum 1 p N NX i=1 mx i=1 g(y i )?! G(0; 1) Efg(Y 1 )g(y i+1 )g: in distribution where G(a; b) stands for the Gaussian distribution with mean a and variance b. The nonlinearity g, will be restricted by the following set of assumptions: jefg(x 1 )jh 1 gj < 1; 8 ; Efg(X 1 )jh 0 g = 0; (3) varfg(x 1 )jh 1 g < 1; 8 ; 1(g) > 0;
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 4 where, for k = 0; 1, k(g) = varfg(x 1 )jh k g + mx i=1 covfg(x 1 ); g j (X i+1 )jh k g: (4) The optimal nonlinearity g employed by each sensor is asymptotically the most ecient. In other words, among the class of memoryless detectors, it achieves a given power and false alarm rate with the least number of samples when the hypotheses are close to each other (i.e., hard to distinguish as! 0) and N! 1. Under some mild regularity conditions this criterion corresponds to maximizing the ecacy EF F (g) of the detector. Hence, g is such that EF F (g) = [@Efg(X 1)jH 1 g=@] =0 0 (g) is maximized, and the solution of this maximization problem involves numerical evaluation of Fredholm equations of the second kind [4, 7]. 3 Decision Fusion in Dependent Noise We consider decision fusion, where each sensor transmits its decision about the hypotheses in (1) to the central processor. The observations received by each sensor are temporally m-dependent, and we assume that there is spatial independence among the sensors. Let us also suppose that the sensors are identical so that the optimal fusion rule becomes the K-out-of- rule; that is, the central processor declares H 1 only if at least K sensors (0 K ) support his decision. Letting u j ; j = 1; : : : ;, be the decision of the jth sensor and u CP = u CP (u 1 ; : : : ; u ) denote the decision of the central processor upon receiving the sensor outputs, we have u CP = 8 >< >: 1 with probability one; if P j=1 u j > K 1 with probability r; if P j=1 u j = K 0; otherwise where K and r [0; 1] are such that a certain error probability performance is attained. The block diagram in Figure 1 depicts the structure of the distributed detection system under consideration. Our goal is to compare the performance of the optimal distributed detection system to that of the optimal centralized scheme through an appropriately dened asymptotic relative eciency. In particular, we will consider an ARE due to Pitman [10], where the sample size is asymptotically large and the distributions of the hypotheses are close. For the location parameter case in (1), a weak signal assumption will be made by letting = A= p N! 0 (as N! 1), which is consistent with the philosophy behind relative eciencies. Lemma 1 Consider a centralized detection scheme where the noise is modeled to be m-dependent. Suppose that the detector utilizes the optimal nonlinearity g in its decision making. Let also! 0 such that asymptotically many observations (N! 1) are needed to achieve a prespecied false alarm rate,, and power rate,. The number of data, N c (; ), that is required to meet the (; ) performance level is ( ) 1 (g )?1 ()? 0 (g )?1 () N c (; ) = (6) EfT g jh 1 g where T g and k (g ); k = 0; 1, are as dened in () and (4), respectively. (5)
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 5 x x x 1 g (.) g (.) g(.) Accumulator Accumulator Accumulator 1/ N 1/ N 1/ N Decision Decision Decision Device Device Device u u u 1 Σ Decision Device Central Processor u CP Figure 1: Distributed memoryless detection in m-dependent noise. Decision devices perform the threshold comparison operations. Lemma Consider mutually independent and identical sensors. Let the noise process be m-dependent for all sensors, and suppose that each sensor deploys an optimal nonlinearity g. Furthermore, let be nite and! 0 so that a prespecied (; ) performance level can only be attained with asymptotically many observations per sensor. The overall number of data (across all sensors), N d (; ; ), required to attain and is given by the following expression: ( ) inf(x; y) D(; ) [ 1 (g )?1 (y)? 0 (g )?1 (x)] N d (; ; ) = EfT g jh 1 g where D(; ) = = X k=k+1 8 < : (x; y): 0 < x < y 1; = X k=k+1 k x k (1? x)?k + r K x K (1? x)?k ; 9 k y k (1? y)?k + r K = y K (1? y)?k ; : (7) The following asymptotic relative eciency denition is in close coherence with that of Pitman's: ARE(; ; ) = N d(; ; ) : N c (; ) Thus ARE(; ; ) gives the factor of increase in the number of observations required of multiplesensor detection with data fusion, to perform as good as the optimal centralized scheme. Combining the results in Lemmata 1 and, we obtain ( ) inf(x; y) D(; ) [ 1 (g )?1 (y)? 0 (g )?1 (x)] ARE(; ; ) = 1 (g )?1 ()? 0 (g : (8) )?1 ()
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 6 ARE(; ; ) was rst introduced in [] and [4] as a performance measure for distributed detection schemes, and its utility is well-established [5]. In particular, ARE(; ; ) allows for making an assessment as to what size data set has to be collected by the local sensors to reach the performance of a comparable centralized system. 3.1 Distributed Detection with a Large Number of Sensors In this section, we further investigate the expression for ARE(; ; ) in equation (8) for a large number of sensors. (The spatial independence assumption becomes questionable with asymptotically many sensors sharing a nite space; but it is nevertheless valuable as a worst case scenario since any exploitation of the dependence will improve the performance.) Of particular interest is determining how ARE(; ; ) varies with and the best ARE performance the distributed detection scheme can attain. As a rst step towards the goals above, we start with the following proposition which is a consequence of Deoivre-Laplace Theorem (see [5] for the proof). Proposition Suppose that the conditions of Lemma hold. Under the assumption that 1, ( ) inf(x; y) F(; ) [ 1 (g )?1 (y)? 0 (g )?1 (x)] ARE(; ; ) = 1 (g )?1 ()? 0 (g )?1 () where F(x; y) = ( (x; y): 0 < x < y 1; " 1 q q # y(1? y) + x(1? x)?1 (1? ) = y? x?1 () p ) :?1 () To simplify the analysis further, assume = 1? which will still supply us with the inputs we are searching for. When = 1?, the optimization problem becomes inf [ 1 (g )?1 (y)? 0 (g )?1 (x)] (9) (x; y) F(x; y) subject to the constraints and 0 < x < y 1 (10) 1 q q y? x [ y(1? y) + x(1? x)] = and its solution is given below. p?1 (1? ) ; (11) Lemma 3 For the weak signal condition (! 0), the solution of the optimization problem formulated by equations (9), (10) and (11) is as follows: inf [ 1 (g )?1 (y)? 0 (g )?1 (x)] = [ 1 (g ) + 0 (g 1 )]?1 (x; y) F(x; y) + 1 p 1 + A where A = p =?1 (1? ). The inmum is attained for y = 1 + 1 p 1 + A ; x = 1? 1 p 1 + A : Lemma 3 allows for the computation of ARE(; ; 1?) for large values of. This is expressed in Lemma 4 below.
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 7 Lemma 4 Let and hence = 1? be given. For the weak signal condition and [?1 (1? )], ARE(; ; 1? ) = : Thus, almost twice more data are required for distributed memoryless detection with data fusion in m-dependent noise, in comparison to using a single detector. It is certainly reasonable to expect = 1? to approximately hold as the receiver operating point, and the result in Lemma 4 can be regarded as a general asymptotic relationship. 4 ARE-Robustness Consider mutually independent and identical sensors, where for each sensor, the hypotheses H 1 and H 0 are generated by stationary and memoryless processes with distributions f 1 and f 0, respectively. Suppose that the distributions are bounded, analytic almost everywhere and such that j 1? 0 j < "; 1 0 < 1 where k and k denote the mean and variance of f k, k = 0; 1, respectively. " > 0 is some arbitrarily small constant. Let be nite, and let the false alarm and power rates, and be given. Thus, the hypotheses are close to each other in an appropriate sense related to the Kullback-Leibler distance [5], and asymptotically many data are required to attain some given (; ) point. Pursuing Neyman- Pearson detector optimization gives ( ) inf(x; y) D(; ) [?1 (y)??1 (x)] ARE(; ; ) = ; (1)?1 ()??1 () where ARE(; ; ) and D(; ) are dened as in equations (8) and (7), respectively [5]. When the problem is detecting a constant signal in memoryless noise using a number of independent and identical sensors, ARE(; ; ) does not depend on the distribution of the hypotheses. As explained in [5], this quality makes the distributed detection topologies with nite number of sensors robust to variations in noise characteristics. Robustness is understood here in the sense that the asymptotic number of data required by the decentralized system to achieve the same performance level as the optimal centralized detector is invariant to the underlying distributional models. We call this property ARE-robustness. For an ARE-robust multiple sensor system, the data sizes determined through the ARE calculations of the previous sections can be used universally, and the engineer's decision as to how many observations are needed (i.e., how long the observed volume should be monitored) to reach a certain receiver operating point becomes independent of the statistical characteristics of the environment. When the noise process has memory, the distributed detection scheme is not ARE-robust, since, by equation (8), ARE(; ; ) is then a function of the second-order moments of the local test function. However, for large number of sensors ( [?1 (1? )] ), ARE(; ; ) converges to the constant = for low-amplitude signals. Hence, distributed detection of weak signals in dependent noise is an asymptotically ARE-robust operation.
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 8 5 Conclusions We have considered discrete-time memoryless detection of a constant signal in dependent noise using multiple sensors. For decision fusion, we have utilized the asymptotic relative eciency performance measure to extend the results reported in [5]. Distributed detection with nite sensors collecting large amounts of data loses its ARE-robustness property, when the system operates in dependent noise. But under the weak signal assumption, the decentralized scheme becomes asymptotically ARE-robust for suciently large number of sensors. The ARE-robustness concept is of practical signicance since it helps determine the monitoring time duration for optimal performance in multiple sensor systems. It is noteworthy that regardless of whether the noise process is memoryless or possesses m- dependent distribution, for the weak signal case, ARE(; ; 1? ) converges to = when the system consists of many sensors. We conjecture that keeping the same assumptions, a distributed detection scheme with correlated sensors (as modeled by m-dependence) collecting independent data should also be at least asymptotically ARE-robust with ARE(; ; 1? )! = as! 1 and N! 1. We base the latter conclusion on the observation that the collective data ow into the distributed system is equivalent when there is m-dependence across either the sensors or the data only. 6 Appendix This section furnishes the proofs of the lemmata expressed in the paper. Proof of Lemma 1 The centralized decision, u C, is such that u C = ( 1 if Tg 0 if T g <, (13) for some appropriately chosen threshold. Then, as N! 1, by the functional central limit theorem in Section, under H k ; k = 0; 1, T g (N; x N )! G EfT g jh k g; k (g )= p N : (14) From (13) and (14), asymptotically,? EfTg jh 1 gp (N) = 1? N 1 (g ; (15) ) where (N) is the power probability at time N; that is, the probability of deciding in favor of H 1, given that H 1 is true. Similarly, recalling (3), p (N) = 1? N 0 (g : (16) ) Setting (N) = in (16), and then solving for, we get = 0 (g )?1 (1? )= p N: (17) Finally, setting (N) = in (15), and using (17), we obtain the result in (6).
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 9 Proof of Lemma Since! 0, any given pair of false alarm and power rates (x; y); x < y, is attained by each of the sensors for asymptotically many data, N(x; y). Directly from (6), we have ( ) 1 (g )?1 (y)? 0 (g )?1 (x) N(x; y) = : EfT g jh 1 g With the fusion rule in (5), the central processor attains a false alarm probability of (N) and a power probability of (N), at time N, where (see [] and [9]) and (N) = (N) = X k=k+1 X k=k+1 k k x k (1? x)?k + r K x K (1? x)?k y k (1? y)?k + r K y K (1? y)?k : Setting (N) = and (N) =, we conclude that the smallest number of data needed per sensor to attain the (; ) level is inf (x; y) D(; ) N(x; y). Considering the total number of data needed by all the sensors, we nd the result in Lemma. Proof of Lemma 3 For A as in the Lemma, dene c = y? x; and d = ca. Then, from the constraints in (10) and (11), we nd that s s x = 1? c d 1 c + d? 1; y = 1 + c d 1 c + d? 1; subject to c + d x = 1? c 1 [5]. Utilizing the denitions of c and d, equivalently, s s A 1 1 + A? c ; y = 1 + c A 1 1 + A? c ; for 0 < c 1= p 1 + A. Next, using the equality?1 (1? x) =??1 (x), 1 (g 1 + c )?1 1 (g 1 + c )?1? 0 (g 1? c )?1 = + 0 (g 1 + c )?1 where = A p 1=(1 + A )? c =. Hence, the search now becomes for inf c B 1 (g 1 + c )?1 + where B = fc : 0 < c 1= p 1 + A g. The function 1 (g 1 + c )?1 + 0 (g 1 + c )?1? (18) + + 0 (g 1 + c )?1? ; c B; (19) is minimized for = 0, or equivalently, c = 1= p 1 + A. Furthermore, for c B, the equation (19) is also minimized for = 0 as long as 1 (g ) 0 (g ), which is true under the weak signal assumption. The result in Lemma 3 is obtained by proper substitution of c in equation (18).
H. Delic: Distributed detection and asymptotic relative eciency in dependent noise 10 Proof of Lemma 4 Substituting the result of Lemma 3 into (8) and setting = 1?, we nd?1 1 +?1 (1?) p +[ ARE(; ; 1? ) =?1 (1?)] : (0) [?1 (1? )] For [?1 (1? )], we have?1 (1? )= q + [?1 (1? )] 1: Using Taylor expansion and rst order approximation,!?1 1 + p?1 (1? ) + [?1 (1? )]?1 (1? ) (0) p + [?1 (1? )] The result in the lemma follows by substituting (1) into (0). References?1 (1? ) (0) p r =?1 (1? ): (1) [1] P. Billingsley, Convergence of Probability easures, New York, New York: Wiley, 1968. [] H. Delic, Robustness and Asymptotic Performance Studies in Distributed Detection, Ph.D. Dissertation, University of Virginia, Charlottesville, Virginia, ay 199. [3] H. Delic and D. Kazakos, \On distributed detection in dependent noise", Proceedings of the 6th Annual Conference on Information Sciences and Systems, Princeton, New Jersey, arch 199, pp. 989-994. [4] H. Delic and P. Papantoni-Kazakos, \Robust decentralized detection by asymptotically many sensors", Signal Processing, Vol. 33, No., pp. 3-33, August 1993. [5] H. Delic, P. Papantoni-Kazakos and D. Kazakos, \Fundamental structures and asymptotic performance criteria in decentralized binary hypothesis testing", IEEE Transactions on Communications, Vol. 43, No. 1, pp. 3-43, January 1995. [6] D. R. Halverson and G. L. Wise, \Discrete-time detection in -mixing noise", IEEE Transactions on Information Theory, Vol. 6, No., pp. 189-198, arch 1980. [7] H. V. Poor, \Signal detection in the presence of weakly dependent noise-part I: optimum detection", IEEE Transactions on Information Theory, Vol. 8, No. 5, pp. 735-744, September 198. [8] H. V. Poor and J. B. Thomas, \emoryless discrete-time detection of a constant signal in m- dependent noise", IEEE Transactions on Information Theory, Vol. 5, No. 1, pp. 54-61, January 1979.
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