Distributed Fault-Tolerant Detection via Sensor Fault Detection in ...

4 downloads 866 Views 116KB Size Report
Email: [email protected]. †. Dept. of Computer ... the local sensors sequentially send their decisions to a fusion center. A collaborative ... fusion rule, assuming identical local decision rules and fault-free environments, an upper bound ...
Distributed Fault-Tolerant Detection via Sensor Fault Detection in Sensor Networks Tsang-Yi Wang∗ , Li-Yuan Chang† , Dyi-Rong Duh‡ , and Jeng-Yang Wu§ ∗ Institute

of Communications Eng., National Sun Yat-sen Univ., Taiwan. Email: [email protected] of Computer Science and Information Eng., National Cheng Kung Univ., Taiwan. Email: [email protected] ‡ Dept. of Computer Science and Information Eng., National Chi Nan Univ., Taiwan. Email: [email protected] § Dept. of Computer Science and Information Eng., National Chi Nan Univ., Taiwan. Email: [email protected]

† Dept.

Abstract— This work addresses the design of a distributed fault-tolerant decision fusion in the presence of sensor faults when the local sensors sequentially send their decisions to a fusion center. A collaborative sensor fault detection (CSFD) scheme is proposed here to eliminate unreliable local decisions when performing distributed decision fusion. Based on the pre-designed fusion rule, assuming identical local decision rules and fault-free environments, an upper bound is established on the fusion error probability. According to this error bound, a criterion is proposed to search the faulty nodes. Once the fusion center identifies the faulty nodes, all corresponding local decisions are removed from the computation of the likelihood ratios that are adopted to make the final decision. Performance evaluation results indicate a significant improvement in fault-tolerance capability from the proposed approach over the conventional decision fusion without utilizing CSFD.

Keywords: Wireless sensor networks, fault-tolerance, sensor fault detection, decision fusion, distributed detection I. I NTRODUCTION The problem of distributed decision fusion in wireless sensor networks (WSN) has received considerable attention recently because of many important applications [1]–[10]. In WSN, sensor nodes are prone to damage. Therefore, the decision fusion rules employed in WSN need to be faulttolerant. A possible approach to provide fault-tolerant decision fusion for diversified sensor faults is to design the robust fusion rules [3], [11]. However, all those approaches consider the fusion performance in one snapshot. This paper addresses the problem of fault-tolerant sequential decision fusion by isolating the faulty nodes and eliminating all incorrect local decisions resulting from sensor faults during the fusion process. Therefore, the proposed on-line collaborative sensor fault detection (CSFD) scheme is central to the proposed distributed decision fusion. The problem formulation in this paper differs from the canonical sequential detection [12], [13] in that the fusion center makes global decisions at every time interval. The considered scenario has many applications, such as forest fire monitoring and military aircrafts surveillance. In these applications, the deployed WSN may have to report their decision in every time interval, enabling the selection of appropriate strategies at the earliest or most time. To perform decision fusion at anytime with the fault-tolerance capability, the proposed on-line CSFD has to be conducted at each time

interval as well. The work in [14] very recently addressed faulty sensor node detection in WSN adopting sensor fusion techniques and nonparametric statistical methods to detect faulty nodes behaving arbitrarily. However, unlike in [14], the eventual objective of the method proposed herein is to improve the performance of final decision-making at the fusion center, instead of aiming at the performance of the misbehavior sensor identification. The following example illustrates the essential difference between them. Suppose that the deployed WSN monitors whether an event of target-present happens. A malfunctioning node with stuck-at faults, which always transmits its decision sequentially in favor of target-present to the fusion center irrespective of the true presence of the target, should be identified as a faulty node in terms of the identification of misbehavior sensors. However, such a node should not be seen as a faulty node in terms of the performance of making final decisions, despite its physical defects, if the target appears. Definition 1: Physical sensor faults represent true sensor malfunctioning, where the distributions of local decisions are not the same as those of normally operating sensor nodes. Definition 2: Pseudo-sensor faults are faults in which the distributions of local decisions cause the distributed decision fusion performance to degrade. Notably, pseudo-faults are often associated with a given hypothesis in this paper. A sensor node could be pseudo-faulty for one or both of the hypotheses. In this paper, the distributed decision fusion employing the proposed is designed as follows. The optimal fusion rule for each pair of the numbers of operating sensor nodes and observation samples at each node is first obtained under the assumption of fault-free situations. The fusion performance is then characterized by establishing an upper bound on the fusion error probability. A pseudo-faulty node search criterion is proposed based on this performance bound. A set of pseudofaulty nodes is determined at each time according to this proposed criterion and the estimated distributions of local decisions. Once the fusion center identifies the faulty nodes, all their local decisions are eliminated from the computation of the likelihood ratio, which are utilized to make final decisions regarding which hypothesis is true. Performance evaluation of the proposed approach was conducted in both faulty and fault-free cases. Simulated results

show that the proposed approach has a significant improvement in fault-tolerance capability as compared with the conventional decision fusion without employing CSFD. The remainder of this paper is organized as follows. System models and problem formulation are given in Section II. Section III describes the proposed fault-tolerant distributed decision fusion approach. Section IV presents the performance evaluation of the proposed approach based on simulation. Finally, conclusions are drawn in Section V. II. S YSTEM MODELS AND PROBLEM FORMULATION The problem to be solved including the network operation and sensor fault models is first described formally. A. Network operation Fig. 1 depicts a typical structure of parallel fusion networks. Let S = {s1 , . . . , sN } be a finite set corresponding to the N sensor nodes observing sensor measurement sequences generated from a common phenomenon according to either H0 or H1 , the two hypotheses under test. The prior probabilities of H0 and H1 are assumed to be known, and are denoted by π0 and π1 , respectively. The observation sequences taken by sensor sn are denoted by {xnt }∞ t=1 , where t represents the time index and n is the node index. Let Xtn denote a random variable corresponding to xnt . The following assumptions are made. Phenomenon x1t

?

Sensor node s1 ×

u ˇ1t u1t

?

x2t

···

Sensor node s2

···

u ˇ2t

···

u2t

···

?

×

?

xN t

?

Sensor node sN ×

u ˇN t uN t

?

Assumption 2: The sensor node makes a binary decision based on its own observation, independent of all other nodes. Furthermore, no local memory at all nodes is considered here. That is, the local decision u ˇnt at node sn is constrained to n n ˇt = φnt (Xtn ) . depend only on Xt , i.e., u Assumption 3: An identical local decision rule is employed at each node and in each time, i.e., φ = φnt for all t and n. Employing the identical local decision rule at each node can scale well with the network size. Moreover, for the parallel fusion network, employing identical local decision rules at each node is asymptotically optimal based on the error exponent when the number of sensors approaches very large, i.e., the error exponents of the identical local decision rules are equivalent to that of the nonidentical local decision rules obtained by system-wide optimization [15]. Therefore, this approach is quite suitable for large-scale sensor networks. After each node makes its decision locally at time t, sensor ˇnt to the fusion center. A ‘0’ is sn sends its binary decision u sent if the node makes a decision in favor of H0 , and a ‘1’ is sent otherwise. Due to channel transmission errors, the fusion ˇnt . center receives unt , where unt may or may not equal u Assumption 4: The event of communication error, i.e, ˇnt ], is independent in time and from sensor to [unt = u sensor, and is also independent of the observations 2 ∞ N ∞ {Xt1 }∞ t=1 , {Xt }t=1 , · · · , {Xt }t=1 as well as the true hypothesis H . We denote Pr [unt = u ˇnt ] by nt . Throughout this paper, we n will set  = t for all t and n without loss of generality. By Assumptions 1 2, 3, and 4, unt , n = 1, . . . , N, t = 1, . . . , ∞ are mutually independent and identically distributed in time as well as from sensor to sensor conditioned on each hypothesis. We denote pi = Pr [un1 = i|H ] when node sn operates in a fault-free situation. Given the local decision rule φ, we can obtain p1 = (1 − )E [φ|H ] +  (1 − E [φ|H ]) ,  = 0, 1

× = Channel error

Fusion Center

u0t

?

Fig. 1. System model for distributed decision fusion using collaborative on-line sensor fault detection.

Assumption 1: The sensor observations Xtn , n = 1, . . . , N, t = 1, . . . , ∞ are conditionally independent and identically distributed in time as well as across sensors given each hypothesis. Let p (x11 |H ),  = 0, 1 be the probability measure on the continuous space X1 describing the conditional distribution of X11 given H . The joint conditional distribution N ∞ of {x1t }∞ t=1 , · · · , {xt }t=1 is then well defined according to Assumption 1 if p (x11 |H ),  = 0, 1 are assumed to be known.

(1)

and p0 = 1 − p1 . Additionally, we denote pn1 = Pr [un1 = 1|H ] for  = 0, 1 at node sn in the general case, i.e., in either sensor fault present or fault-free situation. Considering the fusion center is processing its information at time T . All  local decisions up to timeT T from all the nodes, uT = {u1t }Tt=1 , {u2t }Tt=1 , · · · , {uN t }t=1 are available at the fusion center. At time T , the fusion center begins its fusion process by employing the proposed collaborative sensor fault detection scheme to identify the pseudo-faulty nodes {sn }n∈FT , where FT is the set of pseudo-faulty nodes at time T . This mapping is denoted by FT = γ(uT ). The number of faulty nodes occurs at time T in the distributed decision fusion network is also denoted by |FT |. The fusion center then eliminates the local decision information of faulty nodes {sn }n∈FT from consideration. The final decision u0T at time T is then made at the fusion center based on the local decision information {{unt }Tt=1 }n∈S\FT where n ∈ S \ FT denotes n ∈ S and n ∈ FT . Therefore, at time T , the fusion rule φ0 can be expressed as   u0T = φ0 γ(uT ), {{unt }Tt=1 }n∈S\FT .

Also, if (N T + K)/2N T < p¯11 < 1, then PM satisfies that:

B. Sensor fault models The wireless sensor network considered herein is very likely to contain faulty nodes because of random deployment in a harsh environment. As mentioned earlier, sensor faults considered in this paper may include all misbehavior. In one fault model, the local decisions are independent of the true hypothesis. For instance, the faulty node sn may be frozen to a fixed decision ’1’, Pr{ˇ unt = 1|H } = 1 for  = 0, 1, and is independent of the local detection accuracy. This type of sensor fault is called a stuck-at-one fault. Similarly, Pr{ˇ unt = 1|H } = 0,  = 0, 1 for a stuckat-zero fault. A random fault may occur, in which Pr{ˇ unt = 1|H } equals a particular value regardless of the hypothesis present. Another sensor fault model may depend on the present hypothesis. For example, the sensor offset bias transforms the sensor measurement uniformly to a certain value, and turns out to alter the value of Pr{ˇ unt = 1|H } depending on which hypothesis is true. Additionally, a sensor with the compromised fault may always send out the opposite decision to its real one. Hence, when sensor faults occur, pn1 is no longer given by (1) but could become any value ranging from min{, 1 − } to max{, 1 − }.

N T +K

PM < A(N, T )¯ p11 2

A. The design of collaborative sensor fault detection Since the ultimate aim of this work is to improve the performance of distributed hypothesis testing, a faulty node was removed from the detection system by investigating whether the faulty node increases the probability of detection error. By Assumption 3, the optimal fusion rule in our case is reduced to the K-out-of-N T fusion rule, where T is the current time index. The optimal value of K(N, T ) can be obtained as  K ∗ , if K ∗ ≥ 0; (2) K(N, T ) = 0, otherwise, where · denotes the standard ceiling function, and   N T  1−p10 π0 log π1 1−p11 . K∗  log {p11 (1 − p10 )/p10 (1 − p11 )} N Let p¯1 = (1/N ) n=1 pn1 ,  = 0, 1. Define PF  Pr(u0T = 1|H0 ) and PM  Pr(u0T = 0|H1 ). We can upperbound PF and PM of the distributed detection system by the following theorem. Theorem 1: Assume −N T < K < N T , where K = 2K(N, T ) − N T . If 0 < p¯10 < (N T + K)/2N T , then PF satisfies that: N T +K

(1 − p¯10 )

N T −K 2

.

(3)

,

(4)

A(N, T ) N T −K

K

2N T NT + K NT − K + = (5) NT + K NT − K NT + K Theorem 1 demonstrates that the bounds of PF and PM can be adopted as the criterion of removing unwanted local decisions {{unt }Tt=1 }n∈FT causing the detection performance to degrade1 . Let N  = N − |FT |, K = 2K(N  , T ) − N  T , and T 1 n u . (6) pˆn = T t=1 t Notably, pˆn is the maximum likelihood estimate (MLE) for both pn11 and pn10 . Based on (6) and inspired by (3) and (4), the CSFD rule can reasonably be defined by choosing FˆT , if ˆ (S \ FT , T ) , FˆT = arg min B

(7)

FT ∈2S

where N ˆ (S \ FT , T ) = A(N  , T )˜ B p

This section presents the collaborative sensor fault detection scheme, which is the main technique of the proposed faulttolerant decision fusion. The CSFD based on the derived error probability bound is first designed.

N T −K 2

where,

III. C OLLABORATIVE SENSOR FAULT DETECTION

p10 2 PF < A(N, T )¯

(1 − p¯11 )

and p˜ = (1/N  )

N 

 T +K 2

(1 − p˜)

N  T −K 2

,

(8)

1 {sn ∈ S \ FT } pˆn ,

n=1

where 1{·} is the indicator function. ˆ (S \ FT , T ) in criterion (7) However, we do not know B corresponds to which bound in Theorem 1, i.e., the bound of PF or PM , because we do not know which hypothesis is true. Notably, if we knew, we do not need to perform the hypothesis test, and this contradicts the problem considered in this paper. ˆ (S \ FT , T ) in criterion (7) is Moreover, from Theorem 1, B valid only if p˜ satifies either the condition 0 < p˜ < (N  T + K )/2N  T or the condition (N  T + K )/2N  T < p˜ < 1, but not both. Unfortunately, which condition should be satisfied can not be judged by the received local decisions. The above argument implies that the search space of FT in (7) contains the sets that are associated with the hypothesis different from the true one under test, and we are not able to know which hypothesis a FT is associated with. Hence, when the combined effect of the number of faulty nodes and the sensor fault types is great, (7) could have more possibility to result in a FT that is associated with a wrong hypothesis. To reduce the possibility that the rule (7) results in a set of pseudo-faulty nodes associating with a wrong hypothesis, another constraint of the maximum |FT | on utilizing the sensor faults search policy should be set. This is reasonable, since if |FT | is too large, then the search criterion confuses the “faulty nodes” 1 Although the best criterion here should be the minimization of the probability of error, it is almost impossible to evaluate the minimum probability of error in our on-line search requirement because of the involvement of evaluating the discontinuous function min(·).

with “non-faulty nodes” more easily. Hence, the following CSFD rule is adopted in this paper. FˆT = arg

min

FT ∈2S ,|FT |≤NF

ˆ (S \ FT , T ) , B

(9)

where NF is chosen depending on the requirement of faulttolerance capability, the size of networks, and the sensor fault types frequently appearing in sensor networks. IV. S IMULATION RESULTS This section investigates the performance of the suggested fault-tolerant distributed decision fusion approach in the presence of sensor faults, for which the types of faults and the number of faulty nodes are not known in advance. The simulations consider the additive noise model for the sensor measurement, where the sensor observation is given by H0 H1

: :

xnt = gtn xnt = γo + gtn , for all t and n = 1, . . . , N (10)

where γo is the signal strength when hypothesis H1 is present and gtn is the sensor measurement noise at node n and time t, which is assumed to be independent across sensors and times, and Gaussain distributed with zero mean and unit variance. Correspondingly, the observation signal-to-noise power ratio (OSNR) can be defined as 20 log10 γo . Additionally, each sensor makes the local decision based on the bisection threshold  γo /2. Accordingly, E [φ|H ] = 1 − Φ (−1) γo /2 , where Φ(·) is the standard normal cumulative distribution function. Additionally, the communication channel between each local sensor to the fusion center is considered to be over either an AWGN channel or a flat fading channel when only the average channel signal-to-noise power ratio (CSNR) is known at the fusion center. Additionally, binary antipodal signaling is assumed to be employed for data transmission. Hence, both channels result in a binary symmetric channel with a known . In this example,  = 0.02 is set. A system with N = 10 is considered in this example. In the following, the performance of the proposed approach employing CSFD rule in (9) is evaluated, and is compared with that of the conventional approach. The conventional approach refers to the distributed detection employing the likelihood ratio fusion rule without removing unreliable local decisions from the likelihood ratio computation. We choose NF = 4 for CSFD rule in this evaluation. Throughout this section, Monte Carlo methods are used to obtain the simulated performance. Several fault types are investigated here as well as different number of faulty nodes. Fig. 2 shows the results of evaluating two schemes at 0 dB OSNR in a fault-free situation. The two schemes were found to have almost indistinguishable performance, implying that the proposed decision fusion scheme has exceedingly small possibility to remove the normally operating nodes, resulting in almost no degradation of performance. Next, the fault-tolerance capability of the proposed approach was demonstrated when two of deployed nodes send the decision “1” to the fusion center with probability 0.79, and two of deployed nodes send out the decision “1” with probability

0.41. OSNR is still set to 0 dB. These faulty nodes were uniformly drawn from the ten deployed nodes in the network. Notably, these two probabilities are picked up in an ad hoc manner. The purpose of this evaluation was to show that the proposed approach can tolerate any fault. Fig. 3 illustrates that CSFD outperformed the conventional approach in this situation. Figs. 4 and 5 summarize the performance of the two approaches at 0 dB OSNR when two and three stuck-at-zero faults occur in the network, respectively. As can be seen, CSFD performed better than the conventional approach. V. C ONCLUSIONS The problem of fault-tolerant distributed decision fusion in the presence of sensor faults when the fusion center sequentially receives local decisions from local sensors is considered in this paper. First, the fusion performance is analyzed when some sensor nodes fail to operate normally, but the fusion center is unaware of these sensor faults and still employing the pre-designed fusion rule based on the identical local decision rules and fault-free situation assumptions. Based on this performance analysis, the collaborative sensor fault detection scheme is proposed to remove unreliable local decisions when performing the likelihood ratio fusion to make global decisions in favor of one of the hypotheses. Simulation results also indicate that the distributed decision fusion incorporating the proposed CSFD significantly improves the performance of the conventional approach. ACKNOWLEDGMENT This work was supported in part by Research Center of Wireless Network and Multimedia Communication under “Aim for Top University Plan” project of NSYSU and Ministry of Education, Taiwan and by the NSC of Taiwan, R.O.C., under grant NSC 95-2221-E-110-142. R EFERENCES [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A survey on sensor networks,” IEEE Communications Magazine, pp. 102–114, August 2002. [2] L. Dan, K. D. Wong, H. H. Yu, and A. M. Sayeed, “Detection, classification, and tracking of targets,” IEEE Signal Processing Magazine, vol. 19, pp. 17–29, March 2002. [3] B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, “Channel Aware Decision Fusion in Wireless Sensor Networks,” IEEE Trans. Signal Processing, vol. 52, no. 12, pp. 3454–3458, December 2004. [4] Y. Yuan and M. Kam, “Distributed decision fusion with a randomaccess channel for sensor network applications,” IEEE Transactions on Instrumentation and Measurement, vol. 53, no. 4, pp. 1339–1344, August 2004. [5] J.-F. Chamberland and V. V. Veeravalli, “Asymptotic results for decentralized detection in power constrained wireless sensor networks,” IEEE Journal of Selected Areas in Communications, vol. 22, no. 6, pp. 1007– 1015, August 2004. [6] R. Niu, B. Chen, and P. K. Varshney, “Decision fusion rules in wireless sensor networks using fading channel statistics,” in 2003 Conference on Information Sciences and Systems, The Johns Hopkins University, March 2003. [7] H. Wang, J. Elson, L. Girod, D. Estrin, and K. Yao, “Target classification and localization in habitat monitoring,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2003), Hong Kong, China, April 2003.

0

10

conventional approach CSFD

−1

Probability of error

10

−2

10

−3

10

−4

10

0

10

20

30

40

50 Time

60

70

80

90

100

Fig. 2. Performance comparison of CSFD rule and the conventional approach in a fault-free situation. Probability of error versus time at OSNR=0 dB for symmetric distribution.

0

10

conventional approach CSFD

−1

Probability of error

10

−2

10

−3

10

−4

10

0

10

20

30

40

50 Time

60

70

80

90

100

Fig. 3. Performance comparison of CSFD rule and the conventional approach involving two random faults with the probability of decision “1” equaling 0.79 and two random faults with the probability of decision “1” equaling 0.41. Probability of error versus time at OSNR=0 dB for symmetric distribution.

[8] A. D’Costa and A. M. Sayeed, “Data versus decision fusion in sensor networks,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2003), Hong Kong, China, April 2003. [9] S. A. Aldosari and J. M. F. Moura, “Detection in decentralized sensor networks,” in IEEE International Conference on Accoustics, Speech, and Signal Processing, Montreal, Canada, May 2004. [10] B. Chen and P. K. Willet, “On the optimality of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels,” IEEE Trans. Inform. Theory, vol. 51, no. 2, pp. 693–699, Feb 2005. [11] T.-Y. Wang, Y. S. Han, P. K. Varshney, and P.-N. Chen, “Distributed fault-tolerant classification in wireless sensor networks,” IEEE Journal of Selected Areas in Communications, vol. 23, no. 4, pp. 724–734, April 2005.

[12] V. V. Veeravalli, T. Basar, and H. V. Poor, “Decentralized sequential detection with a fusion center performing the sequential test,” IEEE Trans. Inform. Theory, vol. 39, no. 2, pp. 433–442, March 1993. [13] P. K. Varshney, Distributed Detection and Data Fusion. New York: Springer, 1997. [14] F. koushanfar, M. Potkonjak, and A. Sangiovanni-Vincentelli, “On-line fault detection of sensor measurement,” in Proceedings of IEEE, vol. 2, 2003, pp. 974–979. [15] P.-N. Chen and A. Papamarcou, “New asymtotic results in parallel distributed detection,” IEEE Trans. Inform. Theory, vol. 39, no. 6, pp. 1847–1863, November 1993.

0

10

conventional approach CSFD

−1

Probability of error

10

−2

10

−3

10

−4

10

0

10

20

30

40

50 Time

60

70

80

90

100

Fig. 4. Performance comparison of CSFD rule and the conventional approach involving two stuck-at-zero faults. Probability of error versus time at OSNR=0 dB for symmetric distribution.

0

10

Probability of error

conventional approach CSFD

−1

10

−2

10

0

10

20

30

40

50 Time

60

70

80

90

100

Fig. 5. Performance comparison of CSFD rule and the conventional approach involving three stuck-at-zero faults. Probability of error versus time at OSNR=0 dB for symmetric distribution.