IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 6, JUNE 2011
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Maximin Distributed Detection in the Presence of Impulsive Alpha-Stable Noise Jintae Park, Student Member, IEEE, Georgy Shevlyakov, Member, IEEE, and Kiseon Kim, Senior Member, IEEE
Abstract—The distributed detection problem in wireless sensor networks is studied under the impulsive 𝛼-stable noise assumption. Since symmetric 𝛼-stable density does not have a closed form, its approximation, the bi-parameter Cauchy Gaussian mixture model, is used to describe the impulsive behavior of 𝛼-stable noises. With this model, we propose a low-complexity robust fusion rule by taking the maximin setting with respect to the detection probability. An explicit formula for the detection probability is derived. Robustness of the proposed maximin fusion rule is justified by numerical and simulation results for 𝛼-stable noises. Index Terms—Distributed detection, maximin, 𝛼-stable density, wireless sensor networks.
D
I. I NTRODUCTION
UE to advances in wireless technologies, wireless sensor networks (WSNs) have received much research interest, and consequently the wireless channel layer becomes an important issue in the design of the distributed detection framework in WSNs [1]. In WSNs, many applications can be regarded as two-hypotheses detection problems corresponding to target-present or target-absent. Recently, in order to improve the detection accuracy, distributed detection algorithms have been developed under the Gaussian noise assumption or with the knowledge of channel noise statistics [2]-[4]. In practical problems of detection, it is frequently observed that many physical phenomena are definitely non-Gaussian with impulsive behavior such as underwater acoustic, low frequency atmospheric, and various man-made noises [5], [6]. Among many non-Gaussian distributions, the symmetric 𝛼stable distribution is widely used to model such impulsive perturbations [6]-[8]. In [8], the snapping shrimp dominated ambient noise in Singapore waters has been accurately described by the 𝛼-stable density with typical values of 𝛼 in the range of 1.6–1.9. Although the symmetric 𝛼-stable distribution has proved to be a good model for impulsive noises, designing of the optimal detector based on the likelihood ratio is not available because of the nonexistence of a closed form for the probability density function (pdf) [6]. The density function of a symmetric 𝛼-stable distribution is represented by the characteristic function, and it does not have
Manuscript received March 7, 2010; revised September 13, 2010 and January 13, 2011; accepted March 20, 2011. The associate editor coordinating the review of this paper and approving it for publication was D. I. Kim. This work was supported by a grant (ADD080601) from the basic research program of the Agency for Defense Development (ADD), Korea. J. Park and K. Kim are with the Department of Information and Communications, Gwangju Institute of Science and Technology, Gwangju, 500-712, Republic of Korea (e-mail: {jtpark, kskim}@gist.ac.kr). G. Shevlyakov is with the Department of Applied Mathematics, St. Petersburg State Polytechnic University, St. Petersburg, 195251, Russian Federation (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2011.040411.100333
a convenient analytic form. Therefore, the implementation of the optimal detector in an 𝛼-stable context is difficult. In order to solve this problem, several approximation models for the 𝛼-stable density have been proposed [9]-[11]. In [9], a scale mixture of the Gaussian pdf has been proposed by using a finite mixture of the Gaussians. To capture the algebraic tails of the 𝛼-stable density, this model requires the large number of Gaussian components. In [10], the Cauchy Gaussian mixture (CGM) model has been introduced based on a triple parameter. Recently in [11], to reduce the computational burden of parameters, the bi-parameter CGM (BCGM) model has been proposed by simplifying the CGM model. In this paper, we use the BCGM model to represent the impulsive 𝛼-stable noise density. Based on the BCGM, we adapt Huber’s minimax approach to the distributed detection problem considered in [4]. Although the central part of the BCGM model fits the 𝛼-stable density well, there is a little discrepancy in pdf tails [11]. Moreover, the implementation of the asymptotic fusion rule using the maximum likelihood score function is not easy due to its complex and nonlinear functional form. Therefore, we propose a low-complexity robust fusion rule by using the maximin approach. The remainder of this paper is organized as follows. In Section II, we present the summary of asymptotic fusion results in [4], and then describe the symmetric 𝛼-stable noise density and its approximation model. In Section III, we propose a robust fusion rule within the maximin approach. In Section IV, the performance of the proposed fusion rule is evaluated numerically and experimentally. Finally, we conclude in Section V. II. P RELIMINARIES In this section, we first introduce our results on the asymptotic fusion rule in distributed detection problems [4] obtained for the case of given channel noise pdfs. Then, the symmetric 𝛼-stable noise and its approximation model is described. A. Asymptotic Fusion Results 1) Fusion Model: The parallel fusion model incorporated with a communication channel layer is shown in Fig. 1, where a number of sensor nodes observe data generated according to either 𝐻0 (target-absent) or 𝐻1 (target-present), which are the two hypotheses under testing. In this paper, we assume that the observations are independent across sensors conditioned on any hypothesis. After processing its observations, each sensor node makes a preliminary decision about the hypothesis based on the its local decision rule. The 𝑘th sensor makes a binary decision: 𝑢𝑘 = 𝜃 if 𝐻1 is chosen, and 𝑢𝑘 = 0 otherwise. The detection performance of each sensor node is characterized by the false alarm and detection probabilities given by
c 2011 IEEE 1536-1276/11$25.00 ⃝
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Pd1 /Pf1
Sensor node 1
u1
Communication channels
h1 H0 / H1
Pd2 /Pf2
Sensor node 2
u2 h2
. . .
PdK /PfK
Sensor node K
n1
y2 Fusion Center
n2
uK
Generalizing the fusion decision rule, we consider the fusion test statistic based on an arbitrary score function 𝜓
y1
H0 or H1
yK hK
nK
Fig. 1. The parallel fusion model consisting of 𝐾 sensor nodes and the fusion center.
𝑃𝑓 𝑘 = 𝑃 [𝑢𝑘 = 𝜃∣𝐻0 ] and 𝑃𝑑𝑘 = 𝑃 [𝑢𝑘 = 𝜃∣𝐻1 ], respectively. The received decision 𝑦𝑘 from the communication channel is represented as 𝑦𝑘 = ℎ𝑘 𝑢𝑘 + 𝑛𝑘 , where ℎ𝑘 > 0 is the attenuation of fading and 𝑛𝑘 is the additive channel noise with a symmetric pdf 𝑓 (𝑥). Similar to [2], phase coherent reception is assumed for this fusion model. Here, we apply the weak signal approach introduced in [5] to model the sensor output for decision of target-present by replacing the sample size by the number of sensor nodes 𝐾. √ Thus, the sensor decision 𝜃 is represented by 𝜃 = 𝜃𝐾 = 𝜈/ 𝐾 with the finite constant 𝜈 > 0. Hence, we can keep the network energy constant even if a large number of sensors are fused to increase detection performance. 2) Asymptotic Fusion Rule: Since the independence of channel noises 𝑛𝑘 induces the conditional independence of local observations 𝑦𝑘 at each sensor, however not generally their unconditional independence, the pdfs of 𝑦𝑘 under hypotheses 𝐻1 and 𝐻0 denoted by 𝑓 (𝑦𝑘 ∣𝐻1 ) and 𝑓 (𝑦𝑘 ∣𝐻0 ) are given by 𝑓 (𝑦𝑘 ∣𝐻1 ) = 𝑃 [𝑢𝑘 = 𝜃∣𝐻1 ]𝑓 (𝑦𝑘 ∣𝑢𝑘 = 𝜃) + 𝑃 [𝑢𝑘 = 0∣𝐻1 ]𝑓 (𝑦𝑘 ∣𝑢𝑘 = 0) = 𝑃𝑑𝑘 𝑓 (𝑦𝑘 − ℎ𝑘 𝜃) + (1 − 𝑃𝑑𝑘 )𝑓 (𝑦𝑘 )
𝐾 1 ∑ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝜓(𝑦𝑘 ) 𝑇𝐾 (y) = √ 𝐾 𝑘=1
𝐻1
≷
𝐻0
𝜆𝜏 ,
(2)
where 𝜆𝜏 is the threshold value providing the required rate of the false alarm probability 𝑃𝐹 = 𝜏 . The score function 𝜓(𝑥) is following conditions: 𝜓 = ∫ an odd function satisfying ∫ the ′ ′ = 𝜓(𝑥)𝑓 (𝑥)𝑑𝑥 = 0, 𝜓 𝜓 (𝑥)𝑓 (𝑥)𝑑𝑥 < ∞ and 𝜓 2 = ∫ 2 𝜓 (𝑥)𝑓 (𝑥)𝑑𝑥 < ∞. Here we consider the score functions 𝜓 introduced in [12] as a generalization of the ML score function 𝜓𝑀𝐿 implicitly defining an 𝑀 -estimate which in its turn is a generalization of the maximum likelihood estimate. Thus, the proposed generalization allows for further development of suboptimal and robust fusion rules. 3) Asymptotic Performance Results: In [4] under the Neyman-Pearson (NP) criterion, the asymptotic results for the detection probability and the threshold value have been derived from the central limit theorem and from the first order Taylor expansion for the mean and variance of 𝑇𝐾 . These results are summarized as follows. The detection probability is given by ) ( 𝑃𝐷 = 1 − Φ Φ−1 (1 − 𝜏 ) − 𝜈(𝐻/𝑉 (𝜓, 𝑓 ))1/2 , (3) where 𝐻 =
𝐾 1 ∑ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )2 ℎ2𝑘 , and 𝑉 (𝜓, 𝑓 ) = 𝐾→∞ 𝐾
lim
𝑘=1
𝜓 2 /(𝜓 ′ )2 is the asymptotic variance of Huber’s 𝑀 -estimates ∫∞ of location [12] with 𝜓 2 = −∞ 𝜓 2 (𝑥)𝑓 (𝑥) 𝑑𝑥 and 𝜓 ′ = ∫∞ ′ ∫𝑧 2 𝜓 (𝑥)𝑓 (𝑥) 𝑑𝑥; Φ(𝑧) = (2𝜋)−1/2 −∞ 𝑒−𝑡 /2 𝑑𝑡 is the −∞ Gaussian cumulative. The threshold value for 𝑃𝐹 = 𝜏 has the following form 𝜆𝜏 = 𝜈𝜓 ′ 𝐻𝑓 + Φ−1 (1 − 𝜏 )(𝜓 2 𝐻)1/2 ,
(4)
𝐾 1 ∑ 𝑃𝑓 𝑘 (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ2𝑘 . 𝐾→∞ 𝐾
where 𝐻𝑓 = lim
𝑘=1
and 𝑓 (𝑦𝑘 ∣𝐻0 ) = 𝑃𝑓 𝑘 𝑓 (𝑦𝑘 − ℎ𝑘 𝜃) + (1 − 𝑃𝑓 𝑘 )𝑓 (𝑦𝑘 ),
B. Symmetric 𝛼-Stable Noise Density
The symmetric 𝛼-stable density is best described by the characteristic function as [13] ∫ ∞ 1 𝐾 ∏ exp(𝑖𝛿𝜔 − 𝛾∣𝜔∣𝛼 )𝑒−𝑖𝜔𝑥 𝑑𝜔 (5) 𝑓𝛼 (𝑥; 𝛿, 𝛾) = 𝑃𝑑𝑘 𝑓 (𝑦𝑘 − ℎ𝑘 𝜃) + (1 − 𝑃𝑑𝑘 )𝑓 (𝑦𝑘 ) 𝑓 (y∣𝐻1 ) 2𝜋 −∞ = Λ𝐾 (y) = 𝑓 (y∣𝐻0 ) 𝑃𝑓 𝑘 𝑓 (𝑦𝑘 − ℎ𝑘 𝜃) + (1 − 𝑃𝑓 𝑘 )𝑓 (𝑦𝑘 ) 𝑘=1 where 𝛾 is the pdf dispersion, 𝛿 is the location parameter, 𝑇 and 𝛼 is the characteristic exponent. Here, we assume that the where y = [𝑦1 , . . . , 𝑦𝐾 ] is the vector of observations. The center of symmetry is zero, i.e., 𝛿 = 0. The two well-known fusion rule utilizing this LR statistic is asymptotically optimal 𝛼-stable pdfs are the Gaussian with zero mean and variance 2𝛾 for a given noise pdf 𝑓 . when 𝛼 = 2, and the Cauchy centered at zero with dispersion Based on the asymptotic weak signal approach when 𝜃 → 0 𝛾 corresponding to 𝛼 = 1. For the other cases of 𝛼 with 𝛿 = 0, as 𝐾 → ∞ [5], the asymptotic optimal fusion statistic is the closed form expressions do not exist, but the asymptotic derived from the Taylor expansion of log Λ𝐾 having the expansions are known, valid for either small or large argument following form [4] 𝑥 [6]. Since the 𝛼-stable density does not have a closed form, 𝐾 one has difficulties in design and implementation, and also 1 ∑ 𝑀𝐿 (y) = √ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝜓𝑀𝐿 (𝑦𝑘 ), (1) performance analysis of the optimal detector. Thus, the design 𝑇𝐾 𝐾 𝑘=1 of suboptimal detectors is the main issue in this 𝛼-stable where 𝜓𝑀𝐿 (𝑥) = −𝑓 ′ (𝑥)/𝑓 (𝑥) is the maximum likelihood context. In [7], a parametric suboptimal detector has been (ML) score function. proposed based on the Cauchy score function introducing two thus forming the optimal likelihood ratio (LR) fusion rule [2]
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scaling parameters. In this work, we develop a score function by approaching the pdf itself with the following noise model. Recently, several approximation mixture models have been proposed to represent the impulsive behavior of the 𝛼-stable density [9]-[11]. Among them we are interested in the biparameter Cauchy-Gaussian mixture model introduced in [11]. The Cauchy-Gaussian mixture model with the bi-parameter (𝜀, 𝜎) is defined as
dispersion parameter 𝜎. To consider the maximin setting, we assume that the pdf 𝑓 is not exactly known belonging to the class ℱ𝜀 . From (3) it directly follows that the maximin problem with respect to the detection power 𝑃𝐷 (𝜓, 𝑓 ) is equivalent to the Huber minimax problem with respect to the asymptotic variance 𝑉 (𝜓, 𝑓 ) of 𝑀 -estimators:
𝑓𝐵 (𝑥) = (1 − 𝜀)𝑓𝐺 (𝑥) + 𝜀𝑓𝐶 (𝑥) ( ) 𝑥2 1 𝜀𝜎 (6) = (1 − 𝜀) √ exp − 2 + 2 2 𝜋𝜎 4𝜎 𝜋(𝑥 + 𝜎 2 )
where 𝜓 is an arbitrary score function within a class Ψ. In Huber’s minimax approach [12], it is known that the least favorable density 𝑓 ∗ minimizes Fisher information for location. In addition, the ML score function 𝜓 ∗ for 𝑓 ∗ maximizes the detection probability 𝑃𝐷 (𝜓 ∗ , 𝑓 ∗ ):
where 𝜀 is the mixture ratio 0 ≤ 𝜀 ≤ 1, and 𝜎 is the dispersion parameter of the 𝛼-stable distribution. Note that this model is a particular case of the 𝜀-contaminated Gaussian noise with the Cauchy contamination. For 𝜀 = 0, the BCGM corresponds to the Gaussian, and it yields the Cauchy when 𝜀 = 1. The dependence between 𝛼 and 𝜀 is given by 𝜀=
2Γ(−𝑝/𝛼) − 𝛼Γ(−𝑝/2) , 2𝛼Γ(−𝑝) − 𝛼Γ(−𝑝/2)
(7)
where the parameter 𝑝 represents the fractional lower order moment 𝑝 < 𝛼 [11]. The effectiveness of the above equation depends on 𝑝, and it should be chosen to minimize the difference between the ML estimate value used as the benchmark and 𝜀 given by (7). The test results are shown in [11] with the selected parameter value 𝑝 = −0.25. III. ROBUST F USION RULE In this section, we propose a robust fusion rule based on the BCGM density and introduced asymptotic fusion results. For the BCGM, the asymptotic fusion statistic with the ML score function 𝜓𝐵 (𝑥) is given by 𝐾 1 ∑ 𝐵 (y) = √ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝜓𝐵 (𝑦𝑘 ) 𝑇𝐾 𝐾 𝑘=1
where
(8)
) ( √ 4 𝑥2 𝑥 + (𝑥𝜀8𝜎 𝜋(1 − 𝜀)𝑥 exp − 4𝜎 2 2 +𝜎 2 )2 𝜓𝐵 (𝑥) = ( 𝑥2 ) √ 4𝜀 . + 𝑥4𝜎 2𝜎 2 𝜋(1 − 𝜀) exp − 4𝜎 2 2 +𝜎 2
In the NP setting, the maximum detection probability 𝑃𝐷 under the bounded false alarm probability 𝑃𝐹 ≤ 𝜏 is achieved by choosing the ML score function 𝜓𝑀𝐿 (𝑥) = −𝑓 ′ (𝑥)/𝑓 (𝑥). Given 𝑓 with the ML score function, we have 𝑉 (𝜓𝑀𝐿 , 𝑓 ) = 1/𝐼(𝑓 ) where 𝐼(𝑓 ) is Fisher information, and the corresponding detection probability is given by ( ) 𝑃𝐷 (𝜓𝑀𝐿 , 𝑓 ) = 1 − Φ Φ−1 (1 − 𝜏 ) − 𝜈(𝐻𝐼(𝑓 ))1/2 . (9) Although the central part of the BCGM model well fits the 𝛼-stable density, there is a little discrepancy in pdf tails [11]. Therefore, we adapt Huber’s minimax approach to propose a low-complexity robust fusion statistic. In [12], Huber introduced the class of 𝜀-contaminated Gaussian pdfs as follows ℱ𝜀 = {𝑓 : 𝑓 (𝑥) = (1 − 𝜀)𝑁 (𝑥; 0, 𝜎) + 𝜀ℎ(𝑥)} , where ℎ(𝑥) is an arbitrary pdf. Thus, the BCGM belongs to this class with ℎ(𝑥) given by the Cauchy density with
max min 𝑃𝐷 (𝜓, 𝑓 ) 𝜓∈Ψ 𝑓 ∈ℱ
𝑓 ∗ = arg min 𝐼(𝑓 ), 𝑓 ∈ℱ
⇐⇒
min max 𝑉 (𝜓, 𝑓 ),
𝜓∈Ψ 𝑓 ∈ℱ
𝜓 ∗ (𝑥) = −𝑓 ∗ ′ (𝑥)/𝑓 ∗ (𝑥).
The least favorable density in this class consists of two parts: the Gaussian in the center and the exponential tails [12], 𝑓 ∗ (𝑥) = (1 − 𝜀)(2𝜋)−1/2 exp(−𝑥2 /2) for ∣𝑥∣ ≤ 𝜅 and 𝑓 ∗ (𝑥) = (1 − 𝜀)(2𝜋)−1/2 exp(𝜅2 /2 − 𝜅∣𝑥∣) for ∣𝑥∣ > 𝜅, respectively. Thus, the ML score function 𝜓𝜀∗ for least favorable density 𝑓 ∗ is given by { 𝑥/𝜎 2 , for ∣𝑥∣ ≤ 𝜅 𝜎 ∗ 𝜓𝜀 (𝑥) = 𝜅 sign(𝑥)/𝜎, for ∣𝑥∣ > 𝜅 𝜎, where the connection between 𝜅 and 𝜀 is given by 2𝜑(𝜅)/𝜅 − 2Φ(−𝜅) = 𝜀/(1 − 𝜀) with the standard normal density 𝜑(𝑥) = Φ′ (𝑥) [12]. Finally, we propose the maximin fusion rule given by 𝐾 1 ∑ √ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝜓𝜀∗ (𝑦𝑘 ) 𝐾 𝑘=1
𝐻1
≷
𝐻0
𝜆∗𝜏 .
(10)
For the 𝛼-stable pdf 𝑓𝛼 (𝑥) in (5), the detection probability 𝑃𝐷 (𝜓𝜀∗ , 𝑓𝛼 ) and the threshold value 𝜆∗𝜏 (𝜓𝜀∗ , 𝑓𝛼 ) with 𝜓𝜀∗ (𝑥) can be numerically calculated by using equation (3) and (4), respectively. By using 𝑓𝐵 (𝑥), in addition, we can obtain approximate results for 𝜆∗𝜏 (𝜓𝜀∗ , 𝑓𝐵 ) and 𝑃𝐷 (𝜓𝜀∗ , 𝑓𝐵 ) in a closed form as follows 𝜆∗𝜏 (𝜓𝜀∗ , 𝑓𝐵 ) = 𝜈𝜓 ∗ ′𝜀𝐵 𝐻𝑓 + Φ−1 (1 − 𝜏 )(𝜓𝜀∗2𝐵 𝐻)1/2 , where 𝜓 ∗ ′𝜀𝐵 = = and
∫
∫
𝜓𝜀∗ ′ (𝑥)𝑓𝐵 (𝑥)𝑑𝑥
2𝜀 (1 − 𝜀) erf(𝜅/2) + 2 tan−1 (𝜅) 𝜎2 𝜎 𝜋
𝜓𝜀∗ 2 (𝑥)𝑓𝐵 (𝑥)𝑑𝑥 ] √ (1 − 𝜀) [ −𝜅2 /4 2 2 −𝑒 2𝜅/ 𝜋 + erf(𝜅/2)(2 − 𝜅 ) + 𝜅 = 𝜎2 ] 2𝜀 [ + 2 𝜅(1 + 𝜅 tan−1 (1/𝜅)) − tan−1 (𝜅) 𝜎 𝜋 ∫𝑥 2 with the Gauss error function erf(𝑥) = 2 𝜋 −1/2 0 𝑒−𝑡 𝑑𝑡. ∗ The detection probability 𝑃𝐷 (𝜓𝜀 , 𝑓𝐵 ) is given by ) ( 𝑃𝐷 (𝜓𝜀∗ , 𝑓𝐵 ) = 1 − Φ Φ−1 (1 − 𝜏 ) − 𝜈(𝐻/𝑉 (𝜓𝜀∗ , 𝑓𝐵 ))1/2 , / where the asymptotic variance 𝑉 (𝜓𝜀∗ , 𝑓𝐵 ) = 𝜓𝜀∗2𝐵 (𝜓 ∗ ′𝜀𝐵 )2 . 𝜓𝜀∗2𝐵 =
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4 Numerical
0.9
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3
0.8 Detection Probability
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κ* 2 1 0
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Fig. 2.
1.1
1.2
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1.5
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Parameter dependence between 𝛼 and 𝜅.
0 10
IV. P ERFORMANCE E VALUATION In this section, we evaluate the proposed fusion rule under 𝛼-stable channel noises. Throughout this paper, we assume that the sensor level false alarm rate and the detection probability are identical: 𝑃𝑑 = 0.5 and 𝑃𝑓 = 0.05, respectively. Also, the attenuation of the fading is assumed to have the Rayleigh distribution with unit power, i.e., ℎ2 = 1, as used in [2]. In addition, the formal weak signal amplitude parameter 𝜈 can be expressed through the network energy ℰ 𝐾 𝐾 ∑ 1 ∑ as follows: ℰ = 𝑃𝑑𝑘 𝜃2 = 𝜈 2 𝑃𝑑𝑘 = 𝑃𝑑 𝜈 2 . Hence, 𝐾 𝑘=1
𝑘=1
𝜈 = (ℰ/𝑃𝑑 )1/2 . For the given 𝜓𝜀 , the link between 𝜅 and 𝜀 known in [12] is proposed for the least favorable density 𝑓 ∗ , therefore it is not valuable for 𝑓𝛼 and 𝑓𝐵 . Thus, we set 𝜅 in order to maximize the detection probability 𝑃𝐷 . Given 𝜓𝜀∗ , 𝑓𝛼 and 𝜈, the 𝑃𝐷 is a concave function of 𝜅. Here, thus, we sought for 𝜅∗ maximizing 𝑃𝐷 𝜅∗ = arg max 𝑃𝐷 (𝜅; 𝜓𝜀∗ , 𝑓𝛼 ). 𝜅
(11)
Since 𝑓𝛼 does not have a closed-form expression, 𝜅∗ should be computed numerically. For given 𝛼 and chosen 𝑝, we can also obtain 𝜅∗ by using the approximation method based on the 𝑃𝐷 (𝜅; 𝜓𝜀∗ , 𝑓𝐵 ) as follows: 𝜅∗ = arg max 𝑃𝐷 (𝜅; 𝜓𝜀∗ , 𝑓𝐵 ) = 𝜅 arg min 𝑉 (𝜅; 𝜓𝜀∗ , 𝑓𝐵 ). All further results are based on 𝜅∗ 𝜅 obtained by this approximation method. The parameter dependencies between 𝜅∗ and 𝛼 shown in Fig. 2 are obtained by a numerical method with 𝑓𝛼 , by an approximation method using 𝑓𝐵 with 𝑝 = −0.25 used in [11], and by a simplified method using equation 𝜅∗ = 2 𝛼 + 𝑐 with 𝑐 = −1.7. The simplified method is derived by fitting linear equation to numerical results at almost linear region, 1 ≤ 𝛼 ≤ 1.7. Obviously, the asymptotic detection probability is achieved with 𝜅∗ for given 𝛼. In practice, however, the estimation of 𝛼 requires additional costs. Thus, we also evaluate the maximin fusion rule with 𝜅 = 1 obtained by taking the sample mean of 𝜅∗ values corresponding to the linear approximation interval for 𝛼. Fig. 3 shows receiver operating characteristic (ROC) curves for the 𝛼-stable noise density at 𝛼 = 1.8. For simulations, we set ℰ = 20 dB, 𝛾 = 1, and 𝐾 = 100. The random samples of the 𝛼-stable density are generated by
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Fig. 3. ROC curves for the 𝛼-stable noise density at 𝛼 = 1.8. Total number of sensor nodes 𝐾 = 100.
the computer generation method introduced in [13], [14]. In Fig. 3, the asymptotic formula is obtained through numerical computation of 𝑃𝐷 (𝜓𝜀∗ , 𝑓𝐵 ) and the others are obtained by simulation. The likelihood ratio is based on Λ𝐾 (y) and 𝑓𝛼 , the Gaussian score is with score function 𝑥/𝜎 2 for 𝜎 2 = 1, the Cauchy score is with 2𝑥/(𝑥2 + 1), the maximin score (mean 𝜅∗ ) is with 𝜓𝜀∗ for 𝜅 = 1, the maximin score is with 𝜓𝜀∗ for 𝜅∗ = 2.16 calculated at 𝛼 = 1.8, and the BCGM score is with 𝜓𝐵 (𝑥). Although the proposed maximin fusion rule provides slightly lower detection probability than the fusion rule utilizing the BCGM score function, it outperforms both fusion rules utilizing Gaussian score and Cauchy score functions. The asymptotic behavior of the proposed fusion rule for 𝛼 = 1.8 is evaluated in Fig. 4, where detection probability is obtained for the fixed false alarm probability 𝑃𝐹 = 0.01. To consider weak signals, the network energy is set to be ℰ = 15 dB. The legends are the same as in Fig 3. From Fig. 4 it can be seen that with increasing the number of sensor nodes 𝐾, the difference between the optimal likelihood ratio and the maximin fusion rules becomes asymptotically small. The Gaussian score can not mitigate impulsive noise effect, therefore its performance decreases with increasing nodes which result in higher possibility in received samples corrupted by impulsive noise. The ROC curves for 𝛼 = 1.2 are shown in Fig. 5. In this example, ℰ = 20 dB, 𝛾 = 1, and 𝐾 = 100 are used. Since, the asymptotic efficiency is not fully achieved here, we can observe the difference between the likelihood ratio and the asymptotic formula. The maximin fusion rule with the mean value of 𝜅∗ still provides good detection probability while the asymptotic fusion rule with the Gaussian score function suffers serious losses in detection probability. The detection probability as a function of the network energy for 𝛼 = 1.8 is shown in Fig. 6 at 𝑃𝐹 = 0.01, 𝛾 = 1,
PARK et al.: MAXIMIN DISTRIBUTED DETECTION IN THE PRESENCE OF IMPULSIVE ALPHA-STABLE NOISE
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Fig. 6. Detection probability as a function of the network energy for 𝛼 = 1.8 at a fixed 𝑃𝐹 = 0.01. Total number of sensor nodes 𝐾 = 100.
Fig. 4. Detection probability as a function of the number 𝐾 of sensor nodes for 𝛼 = 1.8 at a fixed 𝑃𝐹 = 0.01.
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setting with respect to detection probability. The dependence among 𝛼 and 𝜀, and also 𝜅 maximizing asymptotic detection probability is evaluated. Robustness of the proposed maximin fusion rule is justified by numerical and simulation results for 𝛼-stable noises.
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R EFERENCES
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[1] J.-F. Chamberland and V. V. Veeravalli, “Wireless sensors in distributed detection applications,” IEEE Signal Process. Mag., vol. 24, pp. 16-25, May 2007. [2] B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, “Channel aware decision fusion in wireless sensor networks,” IEEE Trans. Signal Process., vol. 52, no. 12, pp. 3454-3458, Dec. 2004. [3] R. Niu, B. Chen, and P. K. Varshney, “Fusion of decisions transmitted over Rayleigh fading channels in wireless sensor networks,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 1018-1027, Mar. 2006. [4] J. Park, G. Shevlyakov, and K. Kim, “Fusion of decisions modeled as weak signals in wireless sensor networks,” in Proc. IEEE Globecom 2009, Dec. 2009. [5] S. A. Kassam, Signal Detection in Non-Gaussian Noise. Springer-Verlag, 1988. [6] C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications. Wiley, 1995. [7] S. Zozor, J. Brossier, and P. Amblard “A parametric approach to suboptimal signal detection in 𝛼-stable noise,” IEEE Trans. Signal Process., vol. 54, no. 12, pp. 4497-4509, Dec. 2006. [8] M. A. Chitre, J. R. Potter, and S. Ong, “Optimal and near-optimal signal detection in snapping shrimp dominated ambient noise,” IEEE J. Ocean. Eng., vol. 31, no. 2, pp. 497-503, Apr. 2006. [9] E. E. Kuruoglu, C. Molina, and W. J. Fitzgerald, “Approximation of 𝛼stable probability densities using finite mixtures of Gaussians,” in Proc. EUSIPCO’98, pp. 8-11, Sep. 1998. [10] A. Swami, “Non-Gaussian mixture models for detection and estimation in heavy-tailed noise,” in Proc. IEEE ICASSP 2000, pp. 3802-3805, June 2000. [11] X. T. Li, J. Sun, L. W. Jin, and M. Liu, “Bi-parameter CGM model for approximation of 𝛼-stable PDF,” IET Electron. Lett., vol. 44, no. 18, pp. 1096-1098, Aug. 2008. [12] P. J. Huber, Robust Statistics. Wiley, 1981. [13] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models With Infinite Variance. Chapman & Hall, 1994. [14] R. Weron, “On the Chambers-Mallows-Stuck method for simulating skewed stable random variables,” Statist. Probab. Lett., 28, pp. 165-171, 1996.
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Fig. 5. ROC curves for the 𝛼-stable noise density at 𝛼 = 1.2. Total number of sensor nodes 𝐾 = 100.
and 𝐾 = 100. The proposed maximin fusion rule provides good detection performance for a wide range of the network energy, and it approaches the optimal LR case at the lower level than 15 dB of the network energy. V. C ONCLUSION The distributed detection problem in WSNs in the presence of impulsive 𝛼-stable noise is studied in this paper. The BCGM model is used to describe the impulsive characteristic of 𝛼-stable noises. For BCGM, we first represent the asymptotic fusion rule with the ML score function. Then, we propose a low-complexity robust fusion rule by taking the maximin
