Threshold Considerations in Distributed Detection in a Network of Sensors.

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1 Threshold Considerations in Distributed Detection in a Network of Sensors. Gene T. Whipps 1,2, Emre Ertin 2, and Randolph L. Moses 2 1 US Army Research Laboratory, Adelphi, MD Department of Electrical & Computer Engineering, The Ohio State University, Columbus, OH Abstract We consider the problem of distributed sensing and detection using a network of sensor nodes, and the challenges that arise in fusing disparate data. Multiple sensors make local inferences on the state of nature (e.g., the presence of a signal), and those observations are then transmitted to a regional fusion center. The fusion center is tasked to make improved decisions. We develop methods to optimize those decisions. Interoperability between disparate sensor nodes can be addressed by combining similar types of parameters (e.g., direction of arrival and location estimates to better infer location), albeit with varying qualities. As an initial problem, we consider the case where each sensor makes a binary decision on the presence of a signal source and the fusion node combines these to make a more accurate decision. Even in this simple case, the cost of directly determining the global performance grows exponentially with the number of sensors and can be impractical for just tens of sensors. Instead, we use Chernoff bounds as surrogates for evaluating global performance. These bounds provide tractable methods for analysis and constrained optimization. We consider a lossy medium in which signals undergo a range-dependent propagation loss. We determine local thresholds that optimize a performance metric, including both constrained global detection performance and asymptotic error performance. We study the effect of propagation loss and sensor node density on these performance metrics. The performance metrics also provide indicators of the amount of value that each sensor contributes to the fusion task. 1. INTRODUCTION This paper considers the problem of distributed sensing and detection using a parallel network of sensor nodes. Specifically, we consider the problem in which multiple sensors make local binary decisions, and these decisions are further processed at a regional decision center. In this work, we consider the number and placement of sensors to be random. An example scenario is in the pre-deployment stage of a sensor network; the exact number and location of sensor nodes is not a priori knowledge. Over many possible realizations of a sensor network it seems natural to model the deployments probabilistically. Additionally, the source signal of interest is influenced by the propagation medium. As a result, the observations of the signal are not identically distributed. The goal is to determine the affects of varying the density of the sensor network. Figure 1 depicts a parallel network where individual sensors make local inferences, which are then shared with a fusion node to make a decision about the state of nature. Sensor 1 Z1 < Sensor 2 Z2 < H 1 H 0... Cluster Head nx H1 1(Zi ) R apple i=1 H0 H Sensor n Zn < Figure 1: Idealized parallel fusion network.

2 A recent study 1 considered optimizing the density of a sensor network under network constraints. There, each local decision strategy is fixed, based on Bayesian principles, and determined independently of the strategies of the other nodes and the decision strategy at the fusion node. It was also assumed that the sensor nodes always transmit to the fusion node (or cluster head) according to a random access protocol. This work considers a different design philosophy. First, while we do not assume a prior on the presence of a target, it is anticipated that the target is mostly absent. As a consequence, we only examine the problem from the Neyman-Pearson framework. Additionally, we assume reliable communication if the rate under the null hypothesis (i.e., no target) is below a desired level. In other words, the average communication rate is low, with infrequent periods of high network demand. It is clear that increasing the density of the sensor network with fixed local decision strategies will increase the load on the network. Here, the local decision strategies vary with sensor density to maintain reliable communications. In particular, sensors are made less sensitive as the density grows. Second, sensor nodes only transmit when a local detection occurs. This is a heuristic-based approach to save power at each sensor node. As noted in, 2 the power required in passive sensing can be negligible relative to the power needed to transmit a message. Additionally, the optimal fusion rule typically takes on the form of a weighted sum of the local decisions for conditionally independent sensor observations. 3 Others have adopted, as we do here, a counting rule, 4, 5 where the fusion rule is simply counting the number of detections. The trade-off is performance for simplicity. For binary observations (the local decisions), reliable communications, and a fusion rule in the form of a weighted sum, the non-detects are not required at the fusion node. In this preliminary work of a random sensor network we do not consider correlated observations. One would expect the observations to become correlated as the sensor density grows. A number of works have considered correlated observations. 6 8 In, 8 it is observed that the simple counting rule performs reasonably well for low correlation. These works only considered a deterministic sensor network. Here, we assume the sensor observations are independent conditioned on each hypothesis. In this case, that the sensor decision and fusion rules are likelihood ratio tests is well-established. 9 Also, we consider the case where each sensor node is configured identically. More specifically, each share a common threshold. While potentially suboptimal, this greatly simplifies the design of the sensor network. Finding the set of optimal thresholds is not trivial, particularly for a large parallel network. For instance, for the two-sensor case see. 10 It has been observed that there is little to no loss in optimality by assuming identical decision strategies. 11 Additionally, it was shown in 12 that nodes with identical decision strategies are asymptotically optimal in the sense of minimizing total error (Bayesian approach) or Type II errors (Neyman- Pearson framework). The remainder of this work is outlined as follows. In the next section, we present the general system model. In Sec. 3, we analyze the performance as a function of sensor density. We also consider two specific cases of chi-squared and normally distributed observations. Concluding remarks are given in Sec SIGNAL MODEL AND DETECTION PROBLEM In this section, we present the general system model. First, we define the observation model at the sensor node level. The outputs of the sensor nodes serve a inputs to the fusion node. We then describe the fusion rule and resulting provide performance metrics. 2.1 Sensor Node Level For now, consider a set of n randomly-located sensors in region R. The sensors are indexed by V n = {1, 2,..., n}. The sensors collect noisy measurements of the environment for the purpose of detecting whether or not a source signal is present in the scene. Under hypothesis H 0, when no signal is present, sensors measure only noise. Under hypothesis H 1, in which a source is present, the sensor measures a signal component in addition to noise. Thus, the measurement Z i is modelled by H j : Z i = js(r i ) + Y i i V n, j {0, 1} (1)

3 where each Y i f Y is i.i.d. additive noise across the sensors, and s(r i ) is a positive, deterministic signal given the distance r i. We restrict Y i to be a continuous-type random variable with support [a, b]. This implies that f Y is continuous on [a, b], and so its cumulative distribution function F Y is differentiable in (a, b). Correspondingly, the exceedance of Y i (or complementary CDF), defined by Q Y (y) = 1 F Y (y), is strictly decreasing and differentiable in (a, b). The source signal is affected by the medium as it propagates. We assume a homogeneous medium that attenuates the signal with distance. In other words, s(r) is a monotonically decreasing function of distance r between the source and a sensor. Additionally, we assume s(r) is positive and finite for all r 0. As a result, the signal-to-noise ratio is also positive and finite. For the current problem, it is well-known that the local tests in a parallel fusion system are likelihood ratio tests, if the local observations are conditionally independent. In this case, the local tests simplify to testing z i (a realization of Z i ) at a sensor node against a threshold. We assume each sensor shares a common threshold τ. Local decisions are then made by testing z i against τ. For notational purposes, the probability of declaring H 1 is true under H j, j = 0, 1, to each sensor are assigned according to p j,i = Pr{Z i > τ H j, R i = r i } = Q Y (τ js(r i )), (2) for some τ (a, b). If r 1 < r 2 <, it is clear that 0 < p 0,i < p 1,2 p 1,1 1. The local decisions are shared with a fusion node. The fusion node then combines these binary decisions to make a final decision as to the presence or absence of a source. The fusion process is discussed in the following section. 2.2 Fusion Node Level We adopt the counting rule as the fusion process. This design choice, along with common local thresholds, greatly simplifies the design of the sensor network. It is also a practical choice since location information about the sensors and signal source are not a priori knowledge. Given location information and the local decision rules, it was shown previously that the optimal fusion rule is a weighted sum of the (conditionally independent) binary decisions, where the weights are functions of the local likelihoods. 3 However, location information must be known to form the local likelihoods. Here, we assume location-independent weights. More specifically, all the weights are set to unity. Let the local threshold τ be specified and the number of sensors is N = n. The observation at the fusion node is then given by X = n 1(Z i τ), (3) i=1 where 1( ) is the indicator function. Given the true hypothesis and distances between the source and sensors, the conditional moment generating function (MGF) of the sum (3) is then Φ X (s H j, r 1,..., r n, n) = n (1 + (e s 1)p j,i ). (4) If p j,i for each sensor were identical, this MGF would represent that of a binomial random variable. This is not the case under H 1 due to the lossy propagation medium. As a side note, the MGF for the optimal rule is given by (4) with s replaced by w i s, where w i is a function of the local likelihoods for sensor i. From (2), the dependence of w i on r i and τ is clear. For the counting rule in (3), it is clear that the support of X is {0, 1,..., n}. For fixed τ and given {r 1, r 2,..., r n }, it can be shown that the support set for the weighted-sum rule can take on up to 2 n points. And, the cardinality of and values in the support set change with τ. Thus, the difficulty in working with the optimal fusion rule. i=1

4 Random Number and Location of Sensor Nodes For the pre-deployment scenario, we consider the number and placement of sensors to be random. The exact number and location of sensor nodes is not a priori knowledge. Suppose the number of nodes is chosen from a large cache of sensor nodes and deployed over some region. The actual nodes, their quantity, and location vary with each deployment. Over many possible realizations of a sensor network it seems natural to model the deployments probabilistically. To that end, let each sensor node be randomly and independently located with distances to the signal source described by some probability distribution f R, identical to each sensor. For the moment, let the number of nodes be conditioned on N = n, as before, and independent of the true hypothesis. Marginalizing the distances in (4), the resulting MGF of X, conditioned on H j and N = n, is Φ X (s H j, n) = (1 + (e s 1)E Rl (p j,l )) n, (5) for j {0, 1} and for any l V n. The MGF in (5) is that of a binomial random variable. To simplify notation, we can write E R (p j ) = E Rl (p j,l ) since these average probabilities are identical to each sensor. Notice that the expectation drops out under H 0. Now, let N be random and Poisson distributed with mean µ = λa with both λ > 0 and A > 0. Here, λ represents the density of sensors in a region R, and A is the finite area of that region. Poisson and binomially 1, 4, 5 distributed numbers of sensors in distributed detection were also considered in. Given the true hypothesis, the conditional MGF of X, after some simplification, becomes Φ X (s H j ) = e λae R(p j)(e s 1), (6) for j {0, 1}. Here, Φ X (s H j ) is the MGF of a Poisson random variable with mean µ j = λae R (p j ). The conditional distributions of X are completely characterized by their respective means. These means also specify the expected arrival rate of messages into the network. Since it is assumed the target is rarely present, the average arrival rate is approximated by the mean of X under H 0, given by µ 0 = λap 0. So, for a given density of sensor nodes, a constraint on the arrival rate is equivalent to a constraint on the local false alarm probability. If C > 0 is the maximum desired arrival rate, then the maximum local false alarm probability is α s = min{c/(λa), 1}. We assume α s (0, 1), and typically α s 1. The local threshold is then found directly by τ = Q 1 Y (α s). Fusion Node Level Performance Analytically, the probabilities of false alarm (Type I error) and missed detection (Type II error) of the statistic X are given by P fa = e µ0 µ i 0 + γ µ κ κ 0 and P m = e µ1 µ i 1 γ µ κ 1, (7) i! κ! i! κ! i= κ +1 where κ R + is the global threshold and γ [0, 1] is the randomization parameter since X is a discrete RV. We refer to these as the global error probabilities. The goal is then to find κ and γ that satisfy P fa = α f, where α f is the maximum desired false alarm probability at the fusion node. Alternatively, these system errors can be bounded by the Chernoff Bounds. 13 Without going through all the details, it is straightforward to show that the bounds are given by i=0 P fa { ( e (µ 0 κ) µ 0 ) κ κ, if κ > µ0, 1, else, and P m { ( e (µ 1 κ) µ 1 ) κ κ, if κ < µ1, 1, else. (8) It can be shown that the bound on P fa is strictly decreasing with respect to κ > µ 0. Similarly, the bound on P m is strictly increasing for κ (0, µ 1 ). Whereas, the actual error probabilities in (7) have a stair-step behavior. Both bounds are trivial otherwise. Thus, the benefit of the upper bounds in this case is that they are differentiable everywhere except at κ = µ j for j = 0, 1. Thus, finding κ so that the bound on P fa meets α f with equality can be done by standard optimization techniques. It is, however, fairly straightforward to find κ and γ recursively based on the expression for P fa in (7) to meet the desired α f with equality.

5 3. PERFORMANCE VERSUS DENSITY OF SENSORS The system performance is characterized by a small number of parameters: the density of sensor nodes λ, the local threshold τ (or maximum local false alarm probability α s ), and the maximum desired false alarm probability α f at the fusion node. The goal is to understand the system performance with respect to these parameters. The performance is also dependent on a number of other factors such as the signal-to-noise ratio, the propagation loss model, the location of the target relative to the sensors, and the coverage area of the sensor network. These additional factors are assumed to be fixed in this work. However, the following results do not depend on the actual form or value of these factors, given mild constraints. For example, it is assumed the coverage area A is positive and finite; consequently the maximum distance between the signal source and any sensor node is finite. What remains is to study the effect of increasing the density of sensors while fixing the global false alarm probability P fa. While increasing the density, a fixed global false alarm probability can be achieved by duty cycling the sensor nodes or by desensitizing each sensor node. By duty cycling, we mean that each sensor node decides to transmit its detections, independent of the true hypothesis, according to a Bernoulli RV with the probability of transmission p t. By desensitizing, we mean the local threshold is increased. One could also consider a combination of the two. We limit the discussion to just the two separate approaches. In either case, α f remains fixed, as do κ and γ, in what follows. Then, it is clear by inspecting (7) that fixing µ 0 is equivalent to fixing P fa. Subsequently, the analysis in following sections focuses on the conditional means of X. Duty Cycle Sensor Nodes Here, the local threshold is fixed. From (6), the statistic X at the fusion node is distributed as a Poisson RV. This statistic corresponds to the case of p t = 1; meaning all sensor nodes transmit their detections. By duty cycling with p t < 1, it is straightforward to show that the resulting conditional means of X become µ j = λae R (p j )p t. In order to maintain P fa = α f, the probability of transmission p t must decrease inversely proportional to the sensor density λ. Thus, the product λp t remains constant. As a result, the mean of X under H 0 is constant as λ increases. However, the mean under H 1 also remains constant. So, in terms of Type II error, the system performance remains unchanged as sensor density increases. In other words, there is no benefit by increasing density when duty cycling. Desensitize Sensor Nodes Here, the probability of transmission is p t = 1, and the sensor nodes are desensitized as the sensor density increases. To hold µ 0 = λap 0 constant, the local probability p 0 must decrease while increasing the sensor density. Equivalently, the local threshold τ must increase. Hence, the sensor nodes must be desensitized. Since X is Poisson distributed under both hypothesis, the CDF of X is decreasing in its mean. Subsequently, we only need to understand the behavior of µ 1 while fixing µ 0. An increasing µ 1 implies that the Type II error is decreasing. For incremental changes in sensor density the change in the means of X are given by dµ i = AE R (Q Y (τ is(r)))dλ λae R (f Y (τ is(r)))dτ. (9) Setting (9) equal to zero under H 0 we can solve for the rate of change of the threshold. Thus, we have dτ dλ = 1 Q Y (τ) > 0, (10) λ f Y (τ) for τ (a, b). As expected, this shows that the local threshold increases with the density of sensor nodes. Also, the rate of change of the threshold depends exclusively on the noise-only observation model. The rate of change in the mean of X under H 1 is then dµ 1 dλ = AQ Y (τ) [ ER (Q Y (τ s(r))) Q Y (τ) E ] R(f Y (τ s(r))). (11) f Y (τ) It remains to show that dµ 1 /dλ is non-negative. The term AQ Y (τ) in (11) is non-negative. For many distributions on Y, this term is strictly positive for τ (a, b). For example, if Y X 2 (n), then Q Y (τ) > 0 for all τ (0, ). In general, AQ Y (τ) 0, so it is sufficient to show the term in brackets on the right-hand-side of (11) is non-negative.

6 If τ = Q 1 Y (α s) is such that E R (p 1 ) = 1, then clearly p 1 = Q Y (τ s(r)) = 1 for any finite r 0, which implies that f Y (τ s(r)) = 0. Since Q Y (τ) = α s > 0 and f Y (τ) > 0, it follows that (11) is strictly positive. More specifically, dµ 1 /dλ = A > 0 in this case. The case where α s = 0 is also trivial, and there is nothing to prove. Now, restrict α s (0, 1) such that E R (p 1 ) (0, 1). Call this interval I. This is the region of most interest. Recall that the coverage area is finite, so that the maximum distance between the source and any sensor is finite. Define this maximum distance as ρ. It follows that 0 < s(r) s(r) s(0) < for any r [0, ρ]. For a given distance r between any sensor and the source, the (local) detection probability as a function of α s is given by g r (α s ) = Q Y (Q 1 Y (α s) s(r)). (12) It is known that g r, for fixed r, is a non-negative, monotonically increasing, concave function in (0, 1). Since s > 0 on [0, ρ], we have g r (α s ) > α s. This means that the local detector is better than a random coin flip with success probability 1 α s, but how much better depends on the signal-to-noise ratio and the severity of the signal propagation loss. The detection probability without conditioning is then g(α s ) = E R (g r (α s )). The expectation preserves properties of g r, so it follows that g is a non-negative, monotonically increasing, concave function in (0, 1). For τ = Q 1 Y (α s), it is clear that g(α s ) = E R (p 1 ). To shorten the proof, we simplify the problem by assuming g is differentiable in I. The complete proof is similar, but extended to include points where g may not be differentiable. For any function h that is concave in (a, b) and differentiable at u (a, b), we have the inequality: h(u) h(t) + (u t)h (u), (13) for a < t < u < b. Put h = g and limit the points so that t, u I. Since g(t) > 0 and g is differentiable in I (thus strictly increasing), we have g(α s ) > α s g (α s ). (14) Without much effort, it can be shown that g (α s ) = E R (f Y (τ s(r)))/f Y (τ). Recalling that Q Y (τ) = α s, it follows that dµ 1 /dλ is positive. Thus, the Type II error of the fusion node is decreasing with sensor density while desensitizing the sensor nodes. This result can be interpreted graphically. The function g represents the receiver operating characteristics (ROC) for the Neyman-Pearson detector at the sensor level. The first term in brackets in (11) is seen as the slope of the line segment from the origin intersecting the ROC at g(α s ). Since Y is assumed to be a continuous-type random variable, g (α s ) equates to the likelihood ratio of Y for the corresponding threshold τ. The value g (α s ) also represents the slope of the ROC curve at α s. This slope corresponds to the second term in the brackets of (11). Figure 2 graphically depicts the tangent of the ROC curve at α s = 0.4 and the line segment between the origin and (0.4, g(0.4)). For this example, Y X 2 (n), with n = 10 degrees of freedom and s(r) = 4, regardless of r. We have established that the policy of desensitizing sensors, according to (10), while increasing sensor density is no worse than duty cycling. In most cases of interest, this policy results in increased system performance in terms of Type II errors while achieving a network constraint. In the remainder of this work, it is assumed that this policy is employed as the sensor network is scaled. 4. ASYMPTOTIC PERFORMANCE The upper bounds in (8) provide insight into the behaviour of the system errors. Alternatively, we can consider the asymptotic performance as opposed to bounded. In addition, we also find the asymptotic performance assuming additional information is known. In particular, we find the best achievable error exponent of Type II error with and without location information.

7 g(α) α Figure 2: ROC curve ( ) for chi-squared observations, tangent (+) at α s = 0.4, and line segment ( ) between the origin and α s = 0.4. The slope of the tangent is the value of the likelihood ratio corresponding to α s = Case 1: Local Decisions with Unknown Locations Suppose the number of sensor nodes N are chosen according to a Poisson distribution with mean µ = λa, and the nodes are deployed i.i.d. according to the distribution f R. Let the sensor observations be modeled as in Sec. 2. The nodes only report their local decisions from the set U = {0, 1}. Denote these observations as U i U for each i N 0. Marginalizing for the random sensor positions and noting each sensor shares a common threshold, then each U i are i.i.d. Bernoulli with P j (U i = 1) = E R (Pr{U i = 1 R, H j }) = E R (Q(τ js(r))), (15) given H j is the true state. Note that the expectation in (15) drops out for j = 0 since U i is independent of R given H 0. From the collection of samples {U i } N i=1, it is up to the fusion center to choose between H 0 and H 1. We appeal to the Chernoff-Stein Lemma, which shows that the best achievable exponent of one of the system errors goes as the relative entropy between the distributions of each hypothesis when the other error is held fixed. 14 We leave out the details of the lemma and its proof, and just state that the conditions of the lemma are met. Before providing the result, we briefly introduce additional notation. Let the probability of false alarm and missed detection be P µ fa = Pr{Γ(U 1, U 2,..., U N ) = 1, N H 0 }, and P µ m = Pr{Γ(U 1, U 2,..., U N ) = 0, N H 1 }, (16) where Γ is any decision function such that Γ(U 1, U 2,..., U n ) = j implies H j is chosen. Here, the goal is to minimize P m µ subject to P µ fa < ɛ, where ɛ (0, 1/2) is held fixed. Denote this minimum achievable miss probability by Pm. 1 The Chernoff-Stein Lemma tells us that best achievable error exponent is given by 14 1 lim µ µ log P m 1 = D(P 0 (U) P 1 (U)), (17) where D(p q) is the relative entropy between probability mass functions p and q. Essentially, this relates the amount of information from a single sensor to the global performance of the counting rule in terms of missed detection probability for a fixed false alarm probability. Moreover, this represents the best achievable error exponent for the fusion rule discussed in Sec. 2. Here, the local thresholds are also common and fixed to achieve a network constraint. Additionally, marginalizing for the spatial randomness of the sensor network leads to the counting rule being the optimal rule. So, (17) describes the asymptotic behavior of the Type II error of threshold testing on samples of the count statistic X. 4.2 Case 2: Local Decisions and Known Locations Let the setup be the same as in Case 1. In addition, for each realization, the sensor nodes also report their (exact) locations relative to the signal source. With the locations belonging to the set R = [0, R], denote the

8 observations as W i W = U R for each i N 0. Then each W i are i.i.d. with distribution P j (W i = [1, r] T ) = Pr{U i = 1 R = r, H j }f R (r) = Q(τ js(r))f R (r), (18) given H j is the true state. Let the probability of false alarm and missed detection be P µ fa = Pr{Γ(W 1, W 2,..., W N ) = 1, N H 0 }, and P µ m = Pr{Γ(W 1, W 2,..., W N ) = 0, N H 1 }, (19) where Γ is any decision function such that Γ(W 1, W 2,..., W n ) = j implies H j is chosen. Here, the goal is to minimize P m µ subject to P µ fa < ɛ, where ɛ (0, 1/2) is held fixed. Denote this minimum achievable miss probability by Pm. 2 The best achievable error exponent, for this case, is given by 1 lim µ µ log P m 2 = D(P 0 (U R) P 1 (U R)). (20) In this case, the optimal fusion rule is not specified, and is not likely to be a pure count. Instead, the fusion rule would tend to give more preference to sensors closer to the signal source due to the loss of information with distance. In a lossy propagation medium, sensor nodes farther away from the source essentially become uninformative, and should be weighted accordingly. It can be shown that, for any given threshold τ, the inequality λad(p 0 (U) P 1 (U)) λad(p 0 (U R) P 1 (U R)), (21) holds. This inequality follows by first noting that U is independent of R given H 0 and by an application of Jensen s Inequality. Intuitively, one would expect this relation since the fusion center in Case 2 takes advantage of error-free location information on a per-realization basis, while the fusion center does not have such information in Case 1. It is reasonable then to conjecture that the performance of a system that uses (noisy) estimates of location will fall somewhere between these two error exponents. Equation (17) represents the best achievable error exponent of the system described in Sec. 2. We have seen, from a non-asymptotic analysis, that P m of that system decreases for a fixed P fa while desensitizing sensors according to (10). This implies that λad(p 0 (U) P 1 (U)) is increasing with density, which in turn implies λad(p 0 (U R) P 1 (U R)) is increasing with density by the relation (21). It follows then that system performance, even with access to location information, improves when sensors are desensitized to maintain a constant P fa. We know that relative entropy is non-negative, and is zero if and only if the distributions are identical, assuming they have the same support. Here, P 0 and P 1, with and without conditioning on R, represent Bernoulli distributions. If we increase the threshold τ with λ in order to maintain a constant P fa, the distributions P 0 (U) and P 1 (U) both tend toward a degenerate distribution (i.e., P j (U = 0) 1 for j = 0, 1 as λ ). Similarly, this behavior also holds for P 0 (U R) and P 1 (U R) as λ grows. This implies that both D(P 0 (U) P 1 (U)) and D(P 0 (U R) P 1 (U R)) tend to zero as sensor density tends to infinity. So, in an large sensor network, one could infer that having location information has diminishing return as the network grows. 5. SIMULATION EXAMPLE In this section, we look at two simulation examples. The first example considers the system performance versus sensor density for a given range-dependent signal propagation loss model. Comparisons are made with a system that has perfect location information. The second example considers the system performance versus sensor density for various propagation loss models. In each case, it is assumed the H 0 observations are i.i.d. according to a standard normal distribution and the SNR at the signal source under H 1 is 30 db. It is assumed the sensor nodes are i.i.d. spatially with the distance to the source distributed as f R (r) = { 2πr A, 0 r ρ <, 0, elsewhere. (22)

9 P m P m λ λ (a) (b) Figure 3: Probability of missed detection versus sensor network density. Additionally, each case assumes the policy of desensitizing sensors with increasing density to maintain a constant false alarm probability of α f = 10 8 at the fusion node. The false alarm probability at each sensor is α s = 10 8 when the density of the network is λ = 10 2 sensors/m 2. To achieve a false alarm probability of 10 8 the global threshold and randomization parameter of the test are κ = 1 and γ 0.08, respectively. In the first example, the signal is attenuated as (1 + r) 2, where r is the distance between a sensor and signal source. Fig. 3a plots the probability of missed detection at the fusion node as a function of sensor density. The solid curve is the exact performance from (7) according to Poisson distributed count statistics. Under the above conditions, Figure 3a suggests that the sensor density should be between 1 and 2 sensors/m 2 to achieve a miss detection probability of P m = Also plotted in Fig. 3a are curves from the best achievable error without location information ( ) and with location information ( ). The benefit of perfect location information is clear. The sensor density needed to lower the miss probability to P m = 10 2 is approximately λ , two orders of magnitude less than not having/using location information. In practice, perfect location information is not known, but instead estimated. The resulting performance should fall somewhere between those shown in the figure. For the second example, the signal is attenuated as (1 + r) δ, for various values of δ, which define different signal propagation loss models. Fig. 3b plots the probability of missed detection at the fusion node as a function of sensor density using the exact expression from (7). The solid curve represents P m for δ = 2; so it is the same as in the previous example. The remaining curves represent propagation loss models with δ = 1.5 ( ) and δ = 1 ( ). As expected, the sensor density needed to achieve a desired P m is more favorable when the signal attenuation is less severe. In this example, as the attenuation increases by an order of magnitude, so does the sensor density required to achieve a certain P m.

10 6. CONCLUSION In this work we analyzed the performance of a distributed detection system where the local decisions are binaryvalued. A fusion node collects these local decisions and combines them to make a final decision on the presence or absence of a signal. The goal of this work was to understand how the performance of the system scaled with the density of sensor nodes. The difficulty of such scaling is the increased load on the communications network. Therefore, the performance was analyzed as a function of sensor density subject to a network constraint. When the number of nodes in the system is described by a Poisson distribution and the fusion rule is simply counting the number of detections in the sensor network, the count statistic is also Poisson distributed. Thus, the system performance, in terms of Type I and Type II errors, is completely characterized by the conditional means of the count statistic. We showed that the policy of desensitizing sensors was no worse than duty cycling sensor nodes while increasing density to maintain a desired false alarm probability at the fusion node. In many cases, Type II errors decrease under this policy. Additionally, we showed that this policy is also beneficial when location information is known, whatever the optimal fusion rule may be. REFERENCES [1] Kapnadak, V. and Coyle, E., Optimal density of sensors for distributed detection in single-hop wireless sensor networks, in [Information Fusion (FUSION), 2011 Proceedings of the 14th International Conference on], 1 8 (july 2011). [2] Raghunathan, V., Schurgers, C., Park, S., and Srivastava, M., Energy-aware wireless microsensor networks, Signal Processing Magazine, IEEE 19, (mar 2002). [3] Chair, Z. and Varshney, P., Distributed bayesian hypothesis testing with distributed data fusion, Systems, Man and Cybernetics, IEEE Transactions on 18, (sep/oct 1988). [4] Niu, R. and Varshney, P., Performance analysis of distributed detection in a random sensor field, Signal Processing, IEEE Transactions on 56, (jan. 2008). [5] Aldalahmeh, S., Ghogho, M., and Swami, A., Distributed detection of an unknown target in clustered wireless sensor networks, in [Signal Processing Advances in Wireless Communications (SPAWC), 2011 IEEE 12th International Workshop on], (june 2011). [6] Sundaresan, A., Varshney, P., and Rao, N., Copula-based fusion of correlated decisions, Aerospace and Electronic Systems, IEEE Transactions on 47, (january 2011). [7] Chamberland, J.-F. and Veeravalli, V., How dense should a sensor network be for detection with correlated observations?, Information Theory, IEEE Transactions on 52, (nov. 2006). [8] Unnikrishnan, J. and Veeravalli, V., Decentralized detection with correlated observations, in [Signals, Systems and Computers, ACSSC Conference Record of the Forty-First Asilomar Conference on], (nov. 2007). [9] Viswanathan, R. and Varshney, P., Distributed detection with multiple sensors i. fundamentals, Proceedings of the IEEE 85, (jan 1997). [10] Tenney, R. R. and Sandell, N. R., Detection with distributed sensors, IEEE Trans. on Aerospace and Electronic Systems 17, (1981). [11] Tsitsiklis, J. N., [Decentralized detection], Massachusetts Institute of Technology, Laboratory for Information and Decision Systems (1989). [12] Tsitsiklis, J. N., Decentralized detection by a large number of sensors, Mathematics of Control, Signals, and Systems (MCSS) 1, (1988) /BF [13] Poor, H. V., [An Introduction to Signal Detection and Estimation], Springer, New York, 2 ed. (1994). [14] Cover, T. and Thomas, J., [Elements of information theory], Wiley Series in Telecommunications and Signal Processing, Wiley-Interscience, 2 ed. (2006).

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