robust distributed detection, localization, and estimation of a diffusive ...

ROBUST DISTRIBUTED DETECTION, LOCALIZATION, AND ESTIMATION OF A DIFFUSIVE TARGET IN CLUSTERED WIRELESS SENSOR NETWORKS Sami Aldalahmeh1 and Mounir Ghogho1,2 1

2

University of Leeds, United Kingdom International University of Rabat, Morocco {elsaa, m.ghogho}@leeds.ac.uk

ABSTRACT Robust operation of wireless sensor networks deployed in harsh environment is important in many application. In this paper, we develop a robust technique for distributed detection, localization, and estimation of a diffusive target. The algorithm shows superior performance to the conventional non-linear least square method under low measurement SNR and small number of sensor nodes. 1. INTRODUCTION Wireless sensor networks (WSNs) are increasingly used in mission-critical applications for surveillance and monitoring due to their low cost, scalability, and robustness [1]. This class of applications requires a fast and reliable response to sudden events, e.g. enemy intrusion, natural disasters, or monitoring critical civilian structures. In contrast to the simple time-and-space invariant signal models, which are widely used in the WSN literature, the data collected by sensor nodes (SNs) in the above-mentioned application (e.g. heat or chemical concentration) is better characterized by diffusion laws. The problem of detecting and localizing a diffusive source has been addressed in [2] using generalized likelihood ratio test (GLRT) and maximum likelihood estimator (MLE) or nonlinear least square estimation (NLSE) in the case of Gaussian noise. This approach is deemed unfit for low SNR and harsh environment due to two reasons. First, the nonlinear nature of the problem magnifies the effects of outliers. Second, the least squares (LS) method performs poorly in the presence of large errors. This renders the above-mentioned estimation method unsuitable for harsh environments which usually experience low SNR measurements or/and faulty SNs. The approach in [3] uses minimum mean square error (MMSE) estimation in the context of belief propagation, but its performance too suffers from the effects mentioned above. To our knowledge, the problem of robust estimation for the diffusive data model has not been addressed before. However, several robust processing algorithms have been proposed in the context of acoustic source localization. In [4], the bi-square ρfunction was used instead of LS to provide robustness. A

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major drawback of this method is that the use of the earlier loss function leads to a non-convex optimization problem. Semidefinite programing relaxation was used in [5] to provide a convex robust localization of acoustic source through using min-max loss function instead of LS. The min-max approach behaves well in the presence of outliers but when the errors are small, its performance deviates from the optimal performance provided by the LS method. In our previous work, we proposed a fast, low complexity, energy and bandwidth efficient algorithm for distributed detection, localization, and estimation (DDLE) in time-critical WSN with a diffusive target [6]. However, due to its dependency on the LS method, it is not suited for harsh environments. In this paper, we propose a robust distributed detection, localization, and estimation algorithm by first replacing the nonlinear dependence on the variables with a linear function, and then using the robust Huber loss function, instead of the LS, to solve the linear system. In the next section, the measurement model and the DDLE algorithm are presented. In Section 4 we propose the robust distributed detection, localization, and estimation algorithm. The results are presented in Section 5 with discussions. Finally the paper is concluded in Section 6 and future work is suggested.

2. MEASUREMENT MODEL We model the target as a diffusion point source, e.g, heat, with a constant heat generation rate of μ (Watts/m3 ) located at rt = (xt , yt , zt )T , starting at time instant τ . The sensing field here is an open environment described by the closed halfspace z ≥ 0, with the xy-plane representing the flat ground. It is assumed to be homogeneous, isotropic, and source free. In other words, it consists only of one substance and energy flows equally in all directions. Furthermore, we assume that diffusion is the only means of energy dissipation. κ∇2 e(r, t) =

∂e(r, t) ∂t

(1)

ICASSP 2011

where ∇2 is the Laplacian operator. The solution of (1) gives the instantaneous energy   ⎧ r−rt  μ ⎨2 × √ t>τ erfc 4πκr−rt  2 κ(t−τ ) e(r, t) = (2) ⎩ 0 t≤τ 2

where κ is the diffusivity in m /s, it quantifies how fast does the energy spread throughout the designated area. The factor ’2’ in (2) is due to having an insulating ground which acts as a mirror that creates an identical source image at (xt , yt , −zt ), hence the resulting energy is the sum of both sources. Each SN samples the field with sampling frequency 1/Ts and acquires N measurements corrupted by zero mean additive Gaussian noise with known variance which is temporally and spatially independent. The measured data by the ith SN is fi [n] = ei [n] + wi [n]

(3)

where ei [n] = e(ri , nTs ), and wi [n] = w(ri , nTs ) is the noise at the SNs. If Ts is sufficiently small, then we can assume that τ  nt Ts with nt being a positive integer.

Ii is one, i.e. if

N −1 

fi2 [n] ≥ γ

(4)

i=0

The local threshold, γ, is readily computed from the local constant false alarm rate Pf a by γ = σ 2 Q−1 (Pf a ), where χ2 N

(·) is the inverse of the tail probability of the central Q−1 χ2 N

Chi squared distribution, χ2N with N degrees of freedom. Upon positive decision, the SN proceeds to the local estimation phase, where intermediate parameters are estimated to simplify the problem and decouple it in order to facilitate local processing. Let   bi ei [n] = ai erfc √ = ai hi [n; ϕi ] n − nt √ where ai = μ/2πκri − rt , bi = ri − rt /2 κTs , and ϕi = (bi , nt )T . Now the SN finds the MLE of the intermediate parameters ai and bi as well as the original parameter nt {ˆ ai , ϕˆi } = arg min

ai ,ϕi

3. DISTRIBUTED PROCESSING

N −1 

2

(fi [n] − ai hi [n; ϕi ])

(5)

n=0

The intermediate parameters are estimated by solving We instrument a WSN to detect the diffusive target and estimate its unknown parameters (μ, rt , τ, κ). Although diffusivity, κ, is of little interest to us, it has to be estimated in order to estimate the target’s amplitude, μ, as shown later. Consider a WSN which is deployed randomly in the sensing field with a total of K SNs with the coordinates of the ith sensor being denotes by ri = (xi , yi , 0)T . Conventionally, SNs collect measurements and send them to a sink or a FC regularly. However, these measurements may be of low quality or even contain just noise. This leads to waste of the valuable resources of energy and bandwidth. In our previous work [6] we suggested the DDLE algorithm which is resource efficient and able to respond promptly to sudden events in the sensing field and meet time-critical missions. The main idea there is to process the measured data locally and then send compressed information to the FC for final treatment. We now describe the DDLE briefly to provide sufficient background to proceed to the robust processing in Section 4. 3.1. Local processing

ˆ i = arg max fiT Pi (ϕ)fi ϕ ϕi

a ˆi

=

(hTi hi )−1 fiT hi

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(7)

where Pi (ϕ) = (hTi hi )−1 hi hTi is a projection matrix, fi = (fi [0], · · · , fi [N − 1])T , and hi = (hi [0; ϕi ], · · · , hi [N − 1; ϕi ])T . Since (6) is nonconvex but resembles quadratic programing, we choose the efficient sequential quadratic programming (SQP) algorithm to solve the optimization problem. Once this is done, the intermediate parameters, only three values (ˆ ai , ˆbi , τˆi ), are sent to the FC in lieu of the N measurement sampls; this greatly preserve resources. 3.2. Global processing After receiving information from the active SNs, the FC performs global detection using the counting rule K 

Each SN performs local detection after collecting the measurements samples. If the SN decides that the measurements come solely from noise, it returns to sleep. We propose to use the energy detector (ED) as the local detection algorithm. Unlike sophisticated detectors, such as the generalized likelihood ratio test (GLRT), the low complexity of the ED needs relatively less hardware (or software) contributes in saving energy. The ith SN fully wakes up if the local binary decision

(6)

Ii ≷ Γ

(8)

i=1

The global threshold Γ, is determined as in [6]. This provides another layer of robustness against defective SNs that may have a large false alarm rate. Upon positive decision, the FC uses the intermediate parameters to perform global estimation of the target’s parameters. For the purpose of localization, either a ˆi or ˆbi can be used. Choosing ˆbi is preferable

though since it depends on only two parameters, rt and κ. So we proceed by solving the following {ˆrt , κ ˆ} κ ˆ

2 M   ri − rt  ˆ bi − √ = arg min rt ,κ 2 κTs i=1 = (4Ts bT ˜ r)−1 ˜rT ˜r

μ ˆ = τˆ =

2πˆ κ M

M −1 

a ˆi ri − ˆrt 

(9) (10) (11)

i=0

M −1 Ts  n ˆt M i=0 i

(12)

4. ROBUST DISTRIBUTED DETECTION LOCALIZATION AND ESTIMATION In this section, we address the outlier sensitivity issue in two steps. First, we discuss replacing the nonlinear structure in the global estimation (9) by a linear form. Second, we use the Huber loss function instead of the LS metric to solve the optimization problem. Note that the latter step is a complement to the first one, and will be shown to improve performance. We advocate clustering in the WSN for two reasons. First, the transmission distance to the cluster head (CH) is considerably lower than the one to the FC, so communication energy is saved. Second, due to the symmetry between local and global processing, (6) and (9) respectively, any active SN can serve as a CH. Furthermore, the reduced complexity algorithm suggested next for global processing also helps in the same way. Final results are relayed to the sink which is usually located outside the sensing field. As mentioned previously, performing global detection in the CH provides detection robustness. As for localization, the nonlinearity in √ the global optimization problem (9) comes from having κ in the denominator of ˆbi . To circumvent this problem, we eliminate dependency on κ by computing the ratios of several parameter estimates ˆbi from different SNs, similar to [7]. Thus, every active SN sends its ˆbi value amongst the other intermediate parameters to the CH where the ratios are computed as (13)

If the parameters bi are perfectly estimated, we have that Kij =

ri − rt  rj − rt 

rt − cij 2 = ρ2ij ri −K 2 rj

(14)

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(15)

K 2 ri −rj 

where cij = 1−Kij2 and ρ = ij1−K 2 . It is well known ij ij that the nonlinear least square problem (15) can be converted into a linear least square by taking an arbitrary SN as a reference node indexed 0, e.g. the CH, and forming pairs of hypersphere equations to acquire the linear equations 2(ci0 − cj0 )T rt = (c2i0 − ρ2i0 ) − (c2j0 − ρ2j0 )



 uT l

where ˜r = (r0 − ˆrt , · · · , rM −1 − ˆrt )T and M is the active number of SNs. Again (9) is nonlinear and solved using SQP. Due to the nonlinear structure of the global estimation problem (9), any errors in the local estimation of ˆbi will cause significant errors in the final results. Moreover, using the LS metric in (9) may further degrade performance.

ˆ  ij  bi K ˆbj

To determine rt , the CH solves the hypersphere equations

(16)

ζl

for every active pair of SNs i and j. Compiling all the values in matrix form we have U = [uT1 · · · uTL ]T and ζ = [ζ1 · · · ζL ]T , where L is the number of equations. Since the estimates of the bi ’s are not error-free, the estimate of rt can be obtained after replacing Kij by their estimates. The LS solution is given in closed form. Several other methods can be used though, to provide flexibility such as introducing constraints on the unknown parameters or using different norms, all at the expense of more computational complexity. Here, we propose solving the above linear system by minimizing the following convex optimization problem

  t − ζ ˆ min H Ur (17) rt

s.t. − rmax  rt  rmax  and ζˆ are evaluated using the estimated values of where U  Kij . The inequality  is element wise, x is the L2 norm of vector x, and H(u) is the Huber loss function which is defined as [8]  u2 | u |≤  H(u) = (18) 2 | u | −1 | u |>  For small errors, the Huber function behaves like the least square, whereas for large errors it behaves as the l1 norm which is robust against outliers. Note that (18) reduces to LS when  → ∞. Here, we choose  = 1. Estimating rt by solving the above problem will be shown to provide better performance than the LS-based solution when the SNR is low and/or the number of activated nodes is low. Once the estimate of the target’s location, ˆ rt , is obtained. The other target’s parameters are readily obtained from (10), (11), and (12). 5. SIMULATIONS We consider a randomly deployed WSN having 50 nodes. Every SN is assumed to have a sampling period of Ts = 1 msec and acquires N = 200 samples before processing. The target is located at the origin for convenience. It starts diffusing heat at time instant τ = 50 msec with amplitude μ = 20 (Watts/m3 ) and diffusivity κ = 80 m2 /s. We assume that, according to the DDLE algorithm in [6], there is a

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Fig. 1: Estimation and localization for different total number of sensor nodes. positive global detection with 0.01 local probability of false alarm. We run 100 Monte Carlo iterations. We compare three DDLE algorithms: the nonlinear least square (NLSE) in (9), the ratio-based algorithm with LS solution of (16) (RDDLE-LS), and the ratio-based algorithm with Huber solution of (16) (R-DDLE-Hub). In Fig. 1, the performance is displayed against different WSN sizes for SNR=20dB1 . Significant improvement is evident by the suggested algorithms for low number of SNs, with best performance obtained with R-DDLE-Hub, as expected. However, the NLSE is better in larger networks since the other algorithms solve a large system of linear equation causing relative degradation in performance. Note that in Fig. (1d) NLSE is only shown since all algorithms employ the same method and hence get the same results. Fig. 2 shows the effect of measurements SNR on the localization and estimation performance. Both of the suggested algorithms show superior performance at low SNR compared with the NLSE algorithm, which becomes better at higher SNR but not significantly compared to the other suggested methods. 6. CONCLUSIONS AND FUTURE WORK In summary, we have proposed a new robust distributed detection, localization, and estimation algorithm for a diffusive tar1 The

measurement SNR is defined to be the following average SNR =

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(b) RMSE of μ ˆ vs SNR.

Fig. 2: Effect of SNR on localization and estimation.

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(b) RMSE of amplitude, μ ˆ.

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get in clustered WSNs. In addition to being robust to outliers, the proposed algorithm is energy and bandwidth efficient. It has been shown that the proposed algorithms performs well compared to the previously proposed nonlinear least squares estimation method, in the cases of low measurements SNR and/or small number of sensor nodes. An interesting direction for future work would be using the local estimates as a means of weighing to use in the global estimation stage. Another avenue would be fully decentralized localization and estimation of the target’s parameters using robust average consensus. 7. REFERENCES [1] Songhwai Oh, Phoebus Chen, Michael Manzo, and Shankar Sastry, “Instrumenting wireless sensor networks for real-time surveillance,” in ICRA, 2006, pp. 3128–3133. [2] A. Nehorai, B. Porat, and E. Paldi, “Detection and localization of vapor-emitting sources,” Signal Processing, IEEE Transactions on, vol. 43, no. 1, pp. 243–253, Jan 1995. [3] Tong Zhao and A. Nehorai, “Distributed sequential bayesian estimation of a diffusive source in wireless sensor networks,” Signal Processing, IEEE Transactions on, vol. 55, no. 4, pp. 1511–1524, April 2007. [4] Yong Liu, Yu Hen Hu, and Quan Pan, “Robust maximum likelihood acoustic source localization in wireless sensor networks,” in Global Telecommunications Conference, 2009. GLOBECOM 2009. IEEE, nov. 2009, pp. 1 –6. [5] Enyang Xu, Zhi Ding, and S. Dasgupta, “Robust and low complexity source localization in wireless sensor networks using time difference of arrival measurement,” in Wireless Communications and Networking Conference (WCNC), 2010 IEEE, apr. 2010, pp. 1 –5. [6] S. Aldalahmeh, M. Ghogho, and A. Swami, “Fast distributed detection, localization, and estimation of a diffusive target in wireless sensor networks,” in Wireless Communication Systems, 2010. ISWCS2010, Seventh International Symposium on, York, September 2010. [7] Yu Hen Hu and Dan Li, “Energy based collaborative source localization using acoustic micro-sensor array,” Multimedia Signal Processing, 2002 IEEE Workshop on, pp. 371 – 375, dec. 2002. [8] Peter Huber, Robust Statistics, Wiley, New York, 2009.

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