Detection Performance and Energy Efficiency of Sequential Detection in a Sensor Network

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1 Detection Performance and Energy Efficiency of Sequential Detection in a Sensor Network Lige Yu, Student Member, IEEE and Anthony Ephremides, Fellow, IEEE Department of Electrical and Computer Engineering University of Maryland College Park, MD 2742 {lige, tony}@eng.umd.edu Abstract Wireless sensor networks with event detection mechanisms are considered. Based on a simplified sensor network model, a distributed sequential detection scheme is proposed and investigated. Optimal sequential decision rule is derived and its detection performance is compared with that of the non-sequential detection. Our results show that on average the sequential detection requires fewer measurements than the nonsequential detection to achieve the same detection accuracy. Furthermore, energy efficiency of the two schemes is compared, and distinctive performance for different values of system parameters is observed. I. INTRODUCTION The rapid development in small, low-power, low-cost microelectromechanical (MEMs) sensor technology has led to the emergence of wireless sensor networks as a new class of system with wide variety of purposes. In particular, with the ability of sensors to observe, process, and transmit data, they are well suited to perform event detection, which has been studied previously in [1] [8]. In an event detection scenario, a number of sensor nodes are deployed into a hostile environment to collect measurement data from their surroundings. The measured data can be processed at the sensor nodes if necessary, then sent to a fusion center where a final decision is made. In the previous work of [5], we have investigated three detection schemes: the centralized, distributed and quantized scheme. They have different strategies of local processing and data transmission, and consequently result in different detection performance. It has been shown that the centralized scheme always has the best detection accuracy of all, given the same values of the system parameters. However, it consumes much more energy for data transmission than the other two schemes, especially for the case of long distance from sensors to the fusion center, or a large network with many sensor nodes. On the other hand, it has also been demonstrated that the detection accuracy of the distributed scheme can be improved significantly with the increase of the number of sensors and the amount of measured data at each sensor. In that case the difference of the detection accuracy between the distributed and the centralized scheme is negligible. Additionally, the distributed scheme has been proven to outperform the other two schemes in energy This material is based upon work supported by the U.S. Army Research Laboratory and the U.S. Army Research Office, DOD, under contract No. DAAD efficiency for the case of large networks of powerful sensor nodes and long distance between sensors and the fusion center. Therefore the distributed detection scheme should be preferred in scenarios where a large sensor network is needed. In this paper, we focus on the distributed detection schemes. In particular, we consider a sequential detection strategy, which is different from the non-sequential detection that has been studied in [5]. The problem of sequential detection in a decentralized environment has drawn a lot of attentions [9] [12]. Most of the previous work on this issue has been focused on the case where the sequential test is carried out only at the fusion center, and sensors are therefore only responsible of collecting and transmitting data. Similar to the centralized scheme, this method suffers from the large amount of data that has to be transmitted from sensor nodes to the fusion center. It also requires a two-way communication channel to enable the fusion center to send feedbacks to the sensor nodes. To reduce the transmitted data, we consider a distributed sequential detection mechanism that implements the sequential test at each sensor node rather than at the fusion center. Specifically, instead of collecting a fixed number of measurements, each sensor node will collect measurements one by one, until by some sequential decision rule the collection stops, and a local decision is made. The local decisions will be sent to the fusion center, where a final decision is made using the same decision rule of the distributed scheme in [5]. Compared with the centralized sequential detection mechanism of the previous work, our scheme reduces the transmitted data by sending only the local decisions from sensor nodes, and it does not need feedbacks from the fusion center. However, it requires more powerful sensor nodes to perform sequential test locally. Within such a framework, we are interested in the performance of the sequential detection, especially when compared with the non-sequential detection where the number of measurements collected by each sensor node is fixed. First of all, we derive the optimal sequential decision rule at the sensor nodes and the corresponding final decision rule at the fusion center. An iterative algorithm is proposed to compute the approximate values of the optimal thresholds. With the optimal decision rules determined, we are able to assess the detection accuracy of the sequential detection in terms of the probability of incorrect detection. We then compare the sequential detection with the nonsequential detection by two criteria. The first one is detection /6/$2. (C) 26 IEEE 1

2 performance. That is, we attempt to assess how many measurements are needed to achieve a certain detection accuracy by each scheme. The numerical results show that the sequential detection performs more efficiently than the non-sequential detection in that it always requires fewer measurements on average to obtain the same probability of incorrect detection. This is consistent with our expectation since the sequential detection is more adaptive to the measurement data. We conclude that the sequential detection effectively reduces the number of measurements, which is an improvement over the non-sequential detection. The second criterion is energy efficiency, which is represented by the amount of energy that is needed to achieve a certain detection accuracy. Energy efficiency has always been a key issue for wireless sensor networks as sensor nodes must rely on small, non-renewable batteries. A lot of work has been done to improve the energy efficiency of sensor networks. They have focused on clustering mechanism [13], [14], routing algorithms [16], energy dissipation schemes [14], [17] and sleeping schedules [15]. In their work energy is usually traded with detection latency [15], [16], network density [15], [16], or computation complexity [14], [17]. Meanwhile, the energy concern associated with the specific mission of wireless sensor networks has also been investigated. For example, the tradeoff between energy consumption and detection accuracy is studied in [5], where different detection schemes have been demonstrated to have distinctive performance regarding the accuracy-energy tradeoff. Therefore driven by the concern of the energy constraint, we compare the two schemes in terms of energy efficiency. Specifically, we adopt an energy consumption model where energy is consumed for data observing, processing and transmission. We then compute and compare the energy efficiency of the two schemes for different values of the system parameters by simulations. It shows that the sequential detection does not always outperform the non-sequential detection in energy efficiency. In stead the result mainly depends on the values of the system parameters. Specifically, the relative value of energy for processing versus energy for observing determines the performance of the two schemes regarding energy efficiency. To summarize, the major contributions of this paper are: 1) A distributed sequential detection scheme is proposed that performs the sequential test at each sensor node. The optimal sequential decision rule is derived. 2) The sequential detection is compared with the nonsequential detection. It is shown that the sequential detection effectively improves the detection performance by reducing the number of measurements, while the effect on the energy efficiency mainly depends on the values of the system parameters. The remainder of the paper is organized as follows: the wireless sensor network model is described in Section II along with the distributed sequential detection scheme. In Section III we derive the optimal decision rule for the sequential detection. Section IV analyzes the detection performance of the sequential detection and compares it with that of the non-sequential detection. An energy consumption model is introduced in Section V, along with the comparison of the two schemes in terms of energy efficiency. Finally the paper is concluded in Section VI. II. MODEL DESCRIPTION A. A typical sensor network A typical sensor network consists of a number of sensor nodes and a fusion center. To perform an event detection function, each sensor node is responsible of collecting measurement data from surrounding environment, processing the data if needed, then routing the processed data to the fusion center. The fusion center makes a final decision based on all the data it receives from the sensor nodes. Figure 1 shows the structure of such a sensor network. Fig. 1. fusion center A Typical Sensor Network for Detection B. Simplified Sensor Network Model For a sensor network to perform a detection function, usually routing is needed to transmit data in a multi-hop environment [13], [14], [16]; spatial and temporal correlations exist among measurements across or at sensor nodes [3]; interference is always a problem for the wireless channel [3], [8]. However to focus on the detection function, we start our study from the simplified sensor network model introduced in [5], where the above issues are ignored. The model assumes that each sensor node independently observes, processes and transmits data; given the hypothesis, measurements are independently and identically distributed (i.i.d.) across sensor nodes and at each single node; each sensor node sends data to the fusion center via a single hop; and there is no noise or any other interference. The simplified sensor network model is illustrated in Figure 2. Fig. 2. fusion center Simplified Sensor Network Model In this study a binary hypothesis testing is considered. H indicates whether an event happens (H 1 ) or not (H ), with the 2

3 prior probabilities P [H 1 ]=pand P [H ]=1 p( <p<1). Decision errors are penalized through a decision cost function of C ij, which denotes the cost of choosing Ĥ = H i when H j is true. For simplicity we assume uniform costs as C ij = for i = j and C ij =1for i j. We have K sensor nodes, each collecting a number of binary measurements. Measurements at each node and across nodes areassumedtobei.i.d. random variables conditioned on a certain H, with the conditional pmf of P [1 H ]=p and P [1 H 1 ]=p 1 ( <p <p 1 < 1). The measurement data can be processed locally at each sensor node if needed. The processed data is transmitted to the fusion center, where a final decision Ĥ is made. The Bayesian optimization problem is to minimize the overall Bayesian cost over all admissible decision rules at the fusion center and each sensor node. C. Non-sequential detection For the distributed non-sequential detection, each sensor node collects a fixed T measurements before a local decision is made (denoted by b i for the i th node). Then the binary quantities of {b 1,b 2,,b K } are sent to the fusion center, where Ĥ is decided. A local decision rule is needed at each sensor node as well as a final decision rule at the fusion center. D. Sequential detection For the distributed sequential detection, the number of measurements collected by each node (denoted by N i for the i th node) is a random variable. Each node will collect measurements one by one, until by some sequential decision rule the collection stops, and a local decision is made (denoted by b i for the i th node). Same as the non-sequential detection, the local decisions are sent to the fusion center and a final decision is made there. III. ANALYSIS In this section we first study the single-node case to derive the optimal local decision rule for the non-sequential detection, and the optimal sequential decision rule for the sequential detection. Then we consider the multi-node case where a final decision rule at the fusion center is needed. Finally we will determine the probability of incorrect detection for these two detection schemes. A. Local decision rule In the distributed non-sequential detection each node applies a local decision rule to make a binary decision based on the T measurements. A question arises here: whether we should have an identical local decision rule for all the nodes or not. Generally, an identical local decision rule cannot guarantee the optimal performance from the global point of view. However, it is still a suboptimal scheme if not the optimal one, which has been observed by some previous work. Irving and Tsitsiklis [7] showed that for the binary hypothesis detection, no optimality is lost with identical local detectors in a two-sensor system; Chen and Papamarcou [4] showed that identical local detectors are asymptotically optimum when the number of sensors tends to infinity. Furthermore the identical local decision rule reduces the computation complexity dramatically. Therefore we will simply apply the identical local decision rule to our approach. We derive the optimal local decision rule for the single-node case and then apply it as the identical rule to all the sensor nodes in the multi-node case. Therefore the local decision rule depends only on the parameters of {T, p, p,p 1 },whilethe number of total nodes (K) is not available to the decisionmaking at sensor nodes. It has been shown in [5] that the number of 1 s out of the T measurements (denoted by x i for the i th node) is a sufficient statistic for the detection, and the optimal local decision rule for the i th node is given by { 1 if xi γ; b i = (1) if x i <γ. where the identical threshold for all the sensor nodes is γ = ln 1 p p B. Sequential decision rule + Tln 1 p 1 p 1 ln p1(1 p) p (1 p 1). (2) Similar to the non-sequential detection, we derive the optimal sequential decision rule from the study of the single-node case, and all the sensor nodes of the multi-node case will simply adopt it. For the single-node case we denote the local decision as b. A sequential decision rule consists of a stopping rule (represented by φ) and a terminal decision rule (represented by δ). They operate as follows: For an observation sequence {Y k ; k =1, 2, },therule (φ,δ) makes the decision δ N (Y 1,Y 2,,Y N ), where N is the stopping time defined by N = min{n φ n (Y 1,Y 2,,Y n )=1}. (3) That is, φ decides when to stop taking measurements by the mechanism that when φ n (Y 1,,Y n ) =, the node takes another measurement and when φ n (Y 1,,Y n )=1, the node stops and makes a decision. Hence the number of measurements (N) is a random variable since it depends on the sequence of input measurements. The terminal decision rule δ makes the decision when the node stops taking measurements, which is the local decision sent to the fusion center, i.e., b = δ N (Y 1,Y 2,,Y N ). The optimal sequential decision rule is defined to minimize the overall Bayesian cost for a given system. In addition to the uniform costs of C ij, a per-measurement-cost of C is assigned to each measurement that is taken, hence the cost of taking n measurements is nc. We will first derive the formulas of the optimal sequential decision rule, then propose an iterative algorithm to compute the approximate values of the optimal thresholds. 3

4 1) Formulas: From [18] we know the optimal sequential decision rule has the following form: { if π <λn (Y φ n (Y 1,,Y n )= 1,,Y n ) < π; (4) 1 otherwise. and { 1 if λn (Y δ n (Y 1,,Y n )= 1,,Y n ) π; (5) if λ n (Y 1,,Y n ) π. where λ n (Y 1,,Y n ) is the Likelihood Ratio of the n measurements, which is given by λ n (Y 1,,Y n )=[ p 1(1 p ) p (1 p 1 ) ]x ( 1 p 1 ) n (6) 1 p for the conditional i.i.d. measurements, where x indicates the number of 1 s out of the n measurements. On the other hand, the thresholds can be computed as π = (1 p)π L (7) p(1 π L ) and π = (1 p)π U p(1 π U ), (8) where π L, π U are determined by the following functions of and V (π L )=π L, (9) V (π U )=1 π U. (1) Furthermore, we have the function of V (p) as V (p) =minr(φ,δ), ( p 1) (11) φ,δ where r(φ,δ) represents the overall Bayesian cost, which is given by r(φ,δ)=(1 p)r (φ,δ)+pr 1 (φ,δ). (12) R α (φ,δ) represents the conditional risk for a given sequential decision rule, when H α is true (α =, 1). They can be computed as R (φ,δ) = P [b =1 H ]C 1 + P [b = H ]C + CE {N} = E {δ N } + CE {N}; (13) R 1 (φ,δ) = P [b =1 H 1 ]C 11 + P [b = H 1 ]C 1 + CE 1 {N} as = 1 E 1 {δ N } + CE 1 {N}. (14) Note that N is the random variable representing the stopping time; E α {z} denotes the expected value of z when H α is true; E α {δ N } is the abbreviation of E α {δ N (Y 1,,Y N )}; and clearly P [b =1 H α ]=1 P[b = H α ]=E α {δ N }. From Eqs. (9) (14) we obtain the formulas for π L and π U as π L = π U = E {δ N } + CE {N} E 1 {δ N } CE 1 {N} + E {δ N } + CE {N} ; (15) E {δ N } + CE {N} 1 E 1 {δ N } CE 1 {N} + E {δ N } + CE {N} 2. (16) 2) Iterative algorithm: Based on the formulas above, we can determine the approximate values of the optimal thresholds by an iterative algorithm, which is operated as follows: Set arbitrary initial values for π L and π U that satisfy <π L <π U < 1. Compute π and π by Eqs. (7) (8), then we have a fixed sequential decision rule by Eqs. (4) (5). Compute E α {N} and E α {δ N } for α =, 1. This part is difficult because we do not have a closeform expression for E α {N} or E α {δ N }. However we can compute approximate values of them. That is, we simulate the detection process with the fixed sequential decision rule for many times (denoted by m), and each time a stopping time (N) and the corresponding decision (δ N ) are obtained. Then we compute the average values of the m quantities as the expected values that we desire, i.e., E α {N} and E α {δ N }. Compute π L and π U by Eqs. (15) (16). Repeat from the second step until π L and π U converge. Thus the optimal sequential decision rule is determined. Since the emphasis of this study is the performance of the sequential detection in terms of accuracy and energy, we would not elaborate the details of how the detection is operated. Therefore the proof for the convergence of {π L,π U } is not provided here. On the other hand, we assume that all the needed information {C, p, p,p 1 } is known by the fusion center, thus the optimal thresholds {π L,π U } can be computed and transmitted to each sensor node prior to the detection. From the results of the last round of simulation, the expected stopping time (E{N}) and the probability of incorrect detection ( ) of the single-node case can be computed as E{N} =(1 p)e {N} + pe 1 {N}; (17) =(1 p)e {δ N } + p(1 E 1 {δ N }). (18) C. Final decision rule When the multi-node case is considered, a final decision rule is needed at the fusion center. As we have argued, for both the sequential and non-sequential detection, identical decision rule at the sensor nodes will be applied. Assume we have K sensor nodes. The optimal final decision rule for the non-sequential detection has been derived in [5] { H1 if k Γ; Ĥ = if k<γ. H (19) where k = K i=1 b i denotes the total number of 1 s out of the K binary quantities, and the threshold Γ is given by where Γ= ln 1 p p P F = P [b =1 H ]= + Kln 1 PF 1 P D ln PD(1 PF ) P F (1 P D), (2) T ( T i )p i (1 p ) T i, (21) i= γ 4

5 P D = P [b =1 H 1 ]= T ( T i )p i 1(1 p 1 ) T i (22) i= γ are the false alarm and detection probability of the single-node case. On the other hand the sequential detection has the same final decision rule as the non-sequential detection, since for both schemes the final decision will be made based on {b 1,b 2,,b K }, which are the only data available at the fusion center. Whether there are a fixed or a variable number of measurements at each sensor node does not matter to the final decision rule. Therefore Eqs. (19) (2) also apply as the optimal final decision rule for the sequential detection, where the false alarm and detection probability of the single-node case are changed to P F = E {δ N } and P D = E 1 {δ N }. D. Detection accuracy The overall false alarm and detection probability of the nonsequential detection for the multi-node case are given by P f = P [k Γ H ]= P d = P [k Γ H 1 ]= K ( K k )PF k (1 P F ) K k ; (23) k= Γ K ( K k )PD(1 k P D ) K k. (24) k= Γ For the sequential detection the formulas are the same except that P F = E {δ N } and P D = E 1 {δ N }. The overall probability of incorrect detection for both schemes can be computed as = p(1 P d )+(1 p)p f. (25) IV. DETECTION PERFORMANCE ANALYSIS In this section we investigate the detection performance of the sequential detection by simulations. We will first examine some properties of the sequential detection, basically regarding the per-measurement-cost C. Then the detection performance of the sequential detection will be compared with that of the non-sequential detection in terms of the number of measurements that are needed to achieve a certain probability of incorrect detection. Throughout the simulations in this paper we set m = 1 as the number of rounds to obtain the expected values in the iterative algorithm. We also adopt that the iterative algorithm stops when the updated values of the thresholds are within.1 from the previous ones. As to the system parameters, we fix p =.5, and the results are presented for two combinations of {p,p 1 } as {.2,.7} and {.1,.4}. We will see that even though different values of {p,p 1 } have different numerical results, they both display the same properties. A. Properties of sequential detection First we explore some properties of the sequential detection regarding the parameter of per-measurement-cost (C). Specifically we examine how the optimal thresholds of the sequential decision rule ({π L,π U }) and the expected number of measurements (E{N}) are affected by the value of C. Since the sequential decision rule does not depend on the number of sensor nodes, we study the single-node case. We vary C from 1 4 to.5, with a total of 14 discrete values as: {.1,.2,.3,.4,.5,.1,.2,.3,.4,.5,.1,.2,.3,.5}. Fig. 3. π L & π U : optimal thresholds optimal thresholds versus C π U π L C: per measurement cost (log scale) Optimal Thresholds versus C Figure 3 shows the curves of optimal thresholds versus C for the two combinations of {p,p 1 }, where {p =.2,p 1 =.7} is denoted by and solid curve, and {p =.1,p 1 =.4} is denoted by + and dotted curve. C is in log scale. As we can see both thresholds of π L and π U show monotonicity with respect to C all the time, which is exactly what we expect because the increase of C means more cost to take one measurement, therefore sensor nodes prefer to terminate the detection with fewer measurements, which results in a higher π L and a lower π U. It also shows that when C goes towards, π L converges to and π U converges to 1. This is because that if C were, no cost of taking measurements means that the sensor nodes would always prefer to take infinite measurements to make the best decision, which has the zero probability of incorrect detection. Thus {π L,π U } have to be {, 1} to ensure the process of taking measurements will never stop. However this is not realistic since it would take forever. Therefore C must be positive. Figure 4 presents the curves of expected number of measurements versus C for the two combinations of {p,p 1 }, where C is in log scale. Here C is also varied from 1 4 to.5 and each certain value of C results in a certain value of E{N}. E{N} decreases monotonically with the increase of C, and the same explanation can be applied here as above for the thresholds. It is also observed that E{N} has different rate of change for different {p,p 1 }. Specifically, E{N} drops faster for {.1,.4} than for {.2,.7} with the increase of C. This is consistent with the result of the optimal thresholds, where the rate of the change on {π L,π U } due to the change on C is larger for {.1,.4} than for {.2,.7}. 5

6 Fig. 4. E{N}: expected number of measurements E{N} versus C p =.2,p 1 =.7 p =.1,p 1 = C: per measurement cost (log scale) Expected Number of Measurements versus C values of E{N} and the corresponding. For the nonsequential detection, T is set to be varying in the same range as E{N} of the sequential detection, which also generates different values of. In both figures is presented in log scale. (log scale) vs. E{N} or T (K=1) sequential: {.2,.7} non sequential: {.2,.7} sequential: {.1,.4} non sequential: {.1,.4} Though it is not shown on the figure, from the analysis of the thresholds we expect E{N} to go to infinity when C approaches. For{p =.2,p 1 =.7}, it is observed that at the other end of the curve E{N} =1when C =.5. It means that the sensor node always makes the decision right after the first measurement, that δ 1 =1if it is 1 and δ 1 =if it is. This is the worst detection accuracy that a sequential detection can achieve. For C beyond this point, i.e., C >.5, it is always the worst case as E{N} remains as 1, since it cannot be less than 1. Therefore there is a certain point of C that can be considered as a threshold, that for all the values of C beyond this point, the sensor node will always make a decision after the first measurement, and the detection accuracy will always be the worst. Furthermore it is observed that this certain point of C depends on the values of {p,p 1 }. For example, the worst performance does not occur in the case of {p =.1,p 1 =.4} when C =.5. To summarize, given other parameters fixed, the optimal thresholds ({π L,π U }) and the expected number of measurements (E{N}) show monotonicity with respect to the permeasurement-cost (C). When C approaches, π L will converge to, π U will converge to 1, E{N} will go to infinity, which means the detection will never stop. Therefore C should always be positive. On the other hand when C increases to a certain point E{N} will decrease to 1, which corresponds to the worst detection accuracy. This certain point of C varies for different values of {p,p 1 }. B. Detection performance comparison Next we compare the sequential detection and the nonsequential detection in terms of detection performance. That is, we compare the number of measurements (E{N} for the sequential detection and T for the non-sequential detection) that are needed to achieve the same probability of incorrect detection ( ). Figure 5 and 6 present the results of the single-node case and the multi-node case where K =1. Same as before {p,p 1 } are set as {.2,.7} and {.1,.4}. The different values of are obtained by varying the number of measurements. For the sequential detection, C is varied from 1 4 to.5 as before, which generates different E{N} or T Fig. 5. Probability of Incorrect Detection versus Number of Measurements for K=1 (log scale) vs. E{N} or T (K=1) E{N} or T sequential: {.2,.7} non sequential: {.2,.7} sequential: {.1,.4} non sequential: {.1,.4} Fig. 6. Probability of Incorrect Detection versus Number of Measurements for K=1 From the figures it is observed that in both schemes mostly decreases with the increase of E{N} or T, which is consistent with our expectation since more measurements produce more accurate results. The exceptions (variations from monotonicity) are due to the integer fluctuations of the parameters, which has been explained for the distributed scheme in [5]. Obviously the sequential detection outperforms the nonsequential detection everywhere in the sense that it needs fewer measurements on average to achieve the same probability of incorrect detection. The intuition behind is that the sequential detection is more adaptive to the measurement data by making a decision after each new measurement is collected. In another word, it takes advantage of the varying number of measurements. Therefore it is supposed to perform more efficiently than the non-sequential detection, which fixes the number of measurements at sensor nodes. Another observation is about the number of extra measurements that the non-sequential detection needs to achieve the 6

7 same detection accuracy as the sequential detection. Basically the gap between T and E{N} for the same increases with the increase of T or E{N}. That is, more extra measurements are needed with the increase of the number of measurements. This is not a surprising result too, since a decrease of means an increase of E{N} or T, and to achieve the same the non-sequential detection always needs more measurements than the sequential detection, which makes the increment of T even larger than that of E{N}. A combined examination of the two figures also shows that the above results are not affected by the parameters of K or {p,p 1 }. It indicates that the increase of the number of sensor nodes improves the detection accuracy for both schemes, while it does not change the advantage of the sequential detection over the non-sequential detection in detection performance. For {p,p 1 }, although different values of them yield distinctive performance for the two schemes, the conclusion about detection performance remains the same. To summarize, in spite of a few variations, decreases monotonically with the increase of E{N} or T. The sequential detection always needs fewer measurements on average to obtain the same probability of incorrect detection as the nonsequential detection, and this is not affected by the number of sensor nodes or the values of {p,p 1 }. Meanwhile, the gap between E{N} and T for the same increases with the increase of E{N} or T. V. ENERGY EFFICIENCY ANALYSIS Although the sequential detection performs better than the non-sequential detection in terms of detection performance, it might not be the case when the energy consumption is considered. The sequential detection achieves a better detection accuracy by making a decision after each measurement comes, thus more energy has to be consumed on data processing for each sensor node due to the more complicated detection mechanism. This can be considered as a tradeoff between detection accuracy and energy consumption, as demonstrated in [5]. Therefore we do not expect the sequential detection to perform as well in energy efficiency as it does in detection performance, when compared with the non-sequential detection. In this section we investigate the energy efficiency of the sequential detection, and compare it with that of the nonsequential detection. We will first introduce a simple energy consumption model, where energy is assumed to be consumed for three distinct components of data observing, processing, and transmission. To compute the energy consumption, we consider some basic operations, assume a unit-energy for each of them, and count the number of operations that have been executed. The total energy consumption of each operation will then be the product of its unit-energy and the number of operations. Based on that we will compare the energy efficiency of the two schemes by examining energy consumption versus detection accuracy for different values of the system parameters. An analysis of the numerical results is also provided. A. Energy consumption model In our study we only consider the energy consumed at sensor nodes, and we do not take into account the energy consumption for the fusion center, which is assumed to have less stringent energy constraints. For the simplified sensor network model in [5], energy is considered to be consumed for data processing and transmission, while the energy consumed for data observing is ignored since the same amount of measurements are collected at each sensor node for all the three detection schemes that have been studied. However we have to consider the energy for data observing for our problem, because the number of measurements collected at sensor nodes is a random variable for the sequential detection, which probably will be different from that of the non-sequential detection. 1) Energy for data observing This part of energy consumption is considered as the cost of taking measurements. We assume the unit-energy to take one measurement is a constant, denoted as E o, then the energy consumed for taking n measurements at a sensor node is E o n. Note that C is also considered as the cost of taking one measurement, thus we assume that there is a proportional relation between E o and C, i.e., E o = C e, where e is a constant parameter. Therefore the expected value of energy consumed per node for data observing is given by E O = E{N} E o = E{N} C e (26) for the sequential detection, or E O = T E o = T C e (27) for the non-sequential detection. 2) Energy for data processing Energy consumed for data processing depends on the quantity of processed data and the complexity of the processing operations. Here as argued before, we assume that the system parameters of {T, C, p, p,p 1 } are all known to the fusion center, thus the needed information, such as local thresholds and {p,p 1 }, can be computed if necessary, then transmitted to each sensor node in advance. Hence, even though the Likelihood Ratio has to be computed every time when a new measurement comes, we assume that the values can be computed and saved in the sensor node s database prior to the detection, as {p,p 1 } are already available to each node. Thus after taking a new measurement the corresponding value of Likelihood Ratio can be found by looking up the database, without any extra computation. Energy involved in this procedure is neglected. Therefore we consider that energy for data processing is only consumed for the comparisons of the Likelihood Ratio and the thresholds. Hence we adopt comparison as the basic operation for data processing, and we have a simple model to calculate the energy consumption as follows: 7

8 Each time a new measurement comes, except for the last one, exactly 2 comparisons are needed, i.e., λ n is compared with π L and π U respectively to draw the conclusion of π L <λ n <π U. In total there will be 2(E{N} 1) comparisons. For the last measurement, suppose λ N has the equal probability to be first compared with π L or π U. In the case it is compared with π L first, if λ N π L, then it will stop right here, otherwise it will need another comparison with π U to draw the conclusion of λ N π U. Thus the expected number of comparisons for this case can be computed as P [λ N π L ]+2P [λ N π U ]=1+E{δ N }. While for the other case, when λ N is first compared with π U, similarly the expected number of comparisons can be calculated as 2P [λ N π L ]+P [λ N π U ]=2 E{δ N }. Hence the overall expected number of comparisons needed for the last measurement is given by 1 2 [1 + E{δ N }]+ 1 2 [2 E{δ N }] = 3 2. (28) Thus the expected energy consumption per node of data processing for the sequential detection is given by E P = E c [2(E{N} 1) ]=E c[2e{n} 1 ], (29) 2 where E c denotes the unit-energy for one comparison operation. 3) Energy for data transmission For both of the sequential and non-sequential detection, each node sends exactly one-bit data of the local decision to the fusion center. Therefore the energy for data transmission will be the same for both schemes, given the same number of sensor nodes. For simplicity we ignore this part of energy consumption as it does not matter to the comparison of the two schemes. Finally, the total energy consumption per node for the sequential detection is given by the summation of Eqs. (26) and (29) as E seq = E{N}Ce + E c [2E{N} 1 ]. (3) 2 On the other hand for the non-sequential detection, at each sensor node T measurements are collected and one comparison of the Likelihood Ratio with the local threshold is made to determine the local decision. Thus the total energy consumption per node is computed as E non = TE o + E c = TCe+ E c. (31) In Eq. (31) E o represents the energy to take one measurement, which is a constant and it does not change with T. Since we have assumed E o = Ce, here for the nonsequential detection we consider it as C takes a fixed value and T can be varied independently with C or E o. Therefore we fix C and E o, and vary T to obtain different performance for the non-sequential detection. On the other hand for the sequential detection we have shown that the expected number of measurements (E{N}) varies with C given all the other parameters are fixed. Thus we cannot assume a constant C and a varying E{N}. Therefore for the sequential detection we consider it as C takes different values, which result in different values of E{N}. Meanwhile E o is varying too since it is a function of C for a given e. In one word, we vary C to generate different values of E{N} and E o, and consequently different performance for the sequential detection. B. Energy efficiency comparison As we have argued, the sequential detection is not expected to perform as well in energy efficiency as it does in detection performance, compared with the non-sequential detection. That is, although the sequential detection needs fewer measurements on average than the non-sequential detection to obtain the same detection accuracy, it might consume more energy simply because it involves a more complicated data processing at sensor nodes. Furthermore this is illustrated by comparing Eqs. of (3) and (31): E seq E non = E o [E{N} T]+E c [2E{N} 3 ]. (32) 2 Here we assume E o is fixed for both schemes. From the results of the detection performance, for the same detection accuracy that is obtained by the two schemes, we have E{N} <T and 2E{N} > 3 2. Therefore the key factor is E o versus E c. To be specific, if the energy of taking one measurement (E o ) is much larger than the energy of performing one comparison (E c ), then the energy consumption of the non-sequential detection (E non ) is likely to exceed that of the sequential detection (E seq ), and vice versa. Figure 7 1 present the results of energy consumption versus detection accuracy for different values of the system parameters, where E c = 1(nJ/bit),p =.5 are fixed, and K, e, p,p 1 are varied. For the sequential detection, C is varied as before from 1 4 to.5 and the 14 discrete values are sampled. For the nonsequential detection, two cases of C being fixed as.1 and.5 have been studied, and T is varied adaptively to ensure that the resulted probabilities of incorrect detection fall into the same range as the sequential detection. We consider the combination of parameters in Figure 7 as a basic case, and change the value of parameters one at a time in the following figures to inspect the impact of each individual parameter on the performance of the two schemes. Specifically the values of parameters in the four figures are set as 1) Figure 7 (basic case): K =1,e= 2(nJ/bit),p =.2,p 1 =.7; 2) Figure 8 (varying e): K =1,e= 1(nJ/bit),p =.2,p 1 =.7; 3) Figure 9 (varying K): K =2,e= 2(nJ/bit),p =.2,p 1 =.7; 4) Figure 1 (varying p,p 1 ): K =1,e= 2(nJ/bit),p =.1,p 1 =.4. First of all, a tradeoff between energy consumption and detection accuracy is observed almost everywhere through 8

9 the curves, as the energy consumption generally increases with the decrease of the probability of incorrect detection. Several exceptions are due to the integer fluctuations of the parameters that occur to the two schemes. However with a careful comparison of the curves, it is found that different parameters have distinctive impacts on the performance of the two schemes. E: energy consumption per node (nj) E vs. (K=1,e=2,p =.2,p 1 =.7) sequential:c=1 4.5 non sequential:c=.5 non sequential:c= : probability of incorrect detection (log scale) Fig. 7. Energy Consumption vs. Detection Accuracy (K = 1,e = 2,p =.2,p 1 =.7) In Figure 7, for the sequential detection E o = Ce is varied from.2(nj/bit) to 1(nJ/bit). For the non-sequential detection of C =.1, E o is computed as 2(nJ/bit). Both of the E o s of these two schemes are relatively small compared with the value of E c, i.e., 1(nJ/bit). When these two schemes are compared, the sequential detection is not as efficient as the non-sequential detection, as it always consumes more energy to achieve the same detection accuracy. Meanwhile the gap of energy consumption of the two schemes increases with the decrease of the probability of incorrect detection, which is due to the increase of the number of measurements that are involved. On the other hand, the non-sequential detection of C =.5 yields E o = 1(nJ/bit), which equals the value of E c. When it is compared with the sequential detection, the performance of these two schemes is much closer. This is due to the increase of energy consumption for the non-sequential detection, which is caused by the increase of E o. However it is also observed that in this case, the sequential detection performs more efficiently for relatively small value of, or large number of measurements; while the non-sequential detection performs better for the converse case. This is because that for large number of measurements, as we can closely examine from Figure 6, the gap between E{N} and T increases even faster than twice the increase of E{N}, i.e., Δ(T E{N}) > Δ(2E{N}), which results in E seq <E non. In Figure 8, the E o s of the three detection schemes are all increased from the previous case due to the increase of e. For the sequential detection E o is now ranged from.1(nj/bit) to 5(nJ/bit). When it is compared with the non-sequential detection of C =.5, which has E o = 5(nJ/bit), itis clear that the sequential detection performs more efficiently in energy. However, when the sequential detection is compared with the other non-sequential detection of C =.1, where E: energy consumption per node (nj) E vs. (K=1,e=1 3,p =.2,p 1 =.7) sequential:c=1 4.5 non sequential:c=.5 non sequential:c= : probability of incorrect detection (log scale) Fig. 8. Energy Consumption vs. Detection Accuracy (K = 1,e = 1 3,p =.2,p 1 =.7) E o = 1(nJ/bit), they have nearly the same performance, though similar to the case of Figure 7, that the sequential detection consumes less energy for relatively large number of measurements, and vice versa. This is because E o of the first non-sequential detection is much larger than the fixed E c, which is 1(nJ/bit); while the second E o equals E c, and the same explanation applies here as for Figure 7. A combined examination of Figure 7 and Figure 8 validates our analysis on the impact of E o versus E c. That is, when E o is relatively small compared with E c, which occurs in Figure 7, the non-sequential detection is likely to perform better than the sequential detection in terms of energy efficiency; while it would be the opposite for the case of relatively large E o compared with E c, as demonstrated in Figure 8; and for the case that E o approximately equals E c, the sequential detection performs more efficiently for relatively large number of measurements, and vice versa, which has been shown in both figures. The intuitive explanation is that to achieve the same detection accuracy, the sequential detection needs to operate more comparisons in data processing; while the non-sequential detection has to take more measurements. Therefore if it consumes much more energy to operate a comparison than to take a measurement, the sequential detection would suffer more than the non-sequence detection; while if it is the converse case, the sequential detection would be favored. In such a way E o versus E c contributes as a key factor to this issue. It is observed in Figure 9 that the change of K brings only negligible changes. The reason is that the energy consumption per node is not affected by the number of sensor nodes, hence the increase of K only improves the detection accuracy for both schemes, while it does not affect much to the tradeoff between detection accuracy and energy consumption. The curves of Figure 1 are also similar to those of Figure 7, which indicates that the values of {p,p 1 } are not an important factor to the accuracy-energy tradeoff for the two schemes. As a summary, we have examined the impacts of system parameters on the energy efficiency of the sequential and nonsequential detection. It has been shown that E o versus E c is the key factor to affect the performance of the two detection 9

10 E: energy consumption per node (nj) E vs. (K=2,e=2,p =.2,p 1 =.7) sequential:c=1 4.5 non sequential:c=.5 non sequential:c= P : probability of incorrect detection (log scale) e Fig. 9. Energy Consumption vs. Detection Accuracy (K = 2,e = 2,p =.2,p 1 =.7) E: energy consumption per node (nj) E vs. (K=1,e=2,p =.1,p 1 =.4) sequential:c=1 4.5 non sequential:c=.5 non sequential:c= P : probability of incorrect detection (log scale) e Fig. 1. Energy Consumption vs. Detection Accuracy (K = 1,e = 2,p =.1,p 1 =.4) schemes, while other parameters, i.e., {K, p,p 1 }, do not have much impact on this issue. VI. CONCLUSIONS We have studied a simplified sensor network model which performs an event detection mission. We have implemented a distributed sequential detection scheme on the model, derived the optimal sequential decision rule and evaluated the corresponding detection performance. In particular we compared the detection performance of the sequential detection with that of the non-sequential detection, where the number of measurements collected by each sensor is fixed. It has shown that on average the sequential detection needs fewer measurements than the non-sequential detection to achieve the same detection accuracy. Furthermore, we modelled the energy consumption at the sensor nodes. The comparison of the energy efficiency as a function of the system parameters was presented for both detection schemes. A tradeoff between energy consumption and detection accuracy has been observed, yet the two schemes have distinctive performance for different values of the system parameters. The sequential detection performs more efficiently in energy for relatively high values of E o ; while the nonsequential detection performs better for relatively high values of E c ; and for the case that E o and E c are close, the sequential detection outperforms the non-sequential detection for relatively large number of measurements, and vice versa. Though our study has been based on the simplified sensor network model where routing component and measurement correlations are ignored, we believe that our results have presented some inherent properties regarding the comparison of distributed sequential and non-sequential detection schemes. Meanwhile a more realistic model will be an extension of this work, where we will need to consider the spatial and temporal correlations among measurements, as well as routing in a multi-hop environment. 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