Research ArticleBathtub-Shaped Failure Rate of Sensors for DistributedDetection and Fusion

Junhai Luo and Tao Li

School of Electronic Engineering University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Junhai Luo junhai luouestceducn

Received 5 December 2013 Revised 28 February 2014 Accepted 3 March 2014 Published 2 April 2014

Academic Editor Xin Wang

Copyright copy 2014 J Luo and T Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study distributed detection and fusion in sensor networkswith bathtub-shaped failure (BSF) rate of the sensorswhichmay or notsend data to the Fusion Center (FC)The reliability of semiconductor devices is usually represented by the failure rate curve (calledthe ldquobathtub curverdquo) which can be divided into the three following regions initial failure period random failure period and wear-out failure period Considering the possibility of the failed sensors which still work but in a bad situation it is unreasonable to trustthe data from these sensors Based on the above situation we bring in new characteristics to failed sensors Each sensor quantizesits local observation into one bit of information which is sent to the FC for overall fusion because of power communication andbandwidth constraints Under this sensor failure model the Extension Log-likelihood Ratio Test (ELRT) rule is derived Finallythe ROC curve for this model is presentedThe simulation results show that the ELRT rule improves the robust performance of thesystem compared with the traditional fusion rule without considering sensor failures

1 Introduction

Distributed detection and decision fusion using multiplesensors have attracted significant attention because of theirwide applications such as security traffic battlefield surveil-lance and environmental monitoring Considering a wirelesssensor network consisting of 119872 distribution sensors and aFusion Center (FC) each sensor makes a binary hypothesisdecision then their decisions are sent to the FC to make aglobal decision which decides whether the target is present[1ndash8] At each local sensor the local likelihood ratio testis conducted and the local decision then is sent to theFC to make a global log-likelihood ratio test which is theoptimal test statistic [4 5] The methods of decision fusionsuch as Chair-Varshney Test (CVT) rule Counting Rule testGeneralized Likelihood Ratio Test and Bayesian View havebeen compared in [6] Considering the imperfect communi-cation channel between the sensors and the FC the authorsin [7] extended the classical parallel fusion structure byincorporating the fading channel layer that is omnipresent insensor networks Then the likelihood ratio based fusion rulegiven fixed local decision devices is derived In [1] the authorshave derived the optimal power allocation between training

and data at each sensor over orthogonal channels that aresubject to pathloss Rayleigh fading and Gaussian noise In[9] the authors have proposed a new adaptive decentralizedsoft decision combining rule for multiple-sensor distributeddetection systems with data fusion which does not requirethe knowledge of the false alarm and detection probabilitiesof the distributed sensors Based on multiple decisions fromeach individual sensor assuming that the probability distri-butions are not known the fusion rule which is insensitiveto instabilities of the sensors probability distributions wasderived in [10] In [11] the authors studied an efficient designof a sensor network the graph model as the formal tool todescribe the interaction among the nodes the distributedestimation problem and the network topologyDecentralizeddetection through distributed sensors that perform level-triggered sampling and communicate with a FC via noisychannels was studied in [12] The problem of decentralizeddetection in wireless sensor networks in the presence of oneor more classes of misbehaving nodes was considered in [13]In [14] the authors considered the problemof sensor resourcemanagement for multitarget tracking in decentralized track-ing systems In [15] the decentralized estimation of unknownrandom vectors by using wireless sensors and a FC was

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 202950 8 pageshttpdxdoiorg1011552014202950

2 Mathematical Problems in Engineering

studied In [16] the authors proposed a unified framework forsensor management in multimodal sensor networks whichis inspired by the trading behavior of economic agents incommercial markets The computationally efficient fusionrulewhich involves injecting a deliberate randomdisturbanceto the local sensor decisions before fusionwas designed in [8]

However the failed sensors which may or not sendtheir untrusted data to the FC are not taken into accountMost sensor fusion methods rely on the strong assumptionthat the sensors generate the data operation according to apredetermined characteristic at all times But when a sensorfails or wears out detection systems demonstrate behaviorsdifferent from the expected In [17] it has been shownthat Bayes Risk Error is a Bregman divergence betweenthe actual and estimated prior distribution probabilities In[18] the authors have studied a different approach and haveconstructed a fusion rule which is sensor-failure-robust byincluding a sensor failure model and minimizing expectedBayesian risk

In this paper we raise a new approach which expandsthe traditional log-likelihood ratio test by including a sensorfailure model We assume that the failed sensors which stillsend their data to the FC will follow new characteristics Ofcourse we do not pay attention to the failed sensors whichwill not send their data to the FC In order to study thesensor failures the analysis of reliability is introduced Formanymechanical and electronic components the failure ratefunction (or the hazard rate function) which determines thenumber of failures occurring per unit time has a bathtubshape The bathtub-shaped failure rate life distribution hasbeen derived in [19 20]Thenwe can calculate the probabilityof sensor failures by using the initial number of all sensorsthe number of sensors received by the FC and the number offailed sensors through the failure rate Ourmethodmakes thelog-likelihood ratio test robust by expanding the traditionallog-likelihood ratio test including BSF

The remainder of this paper is organized as followsSensor mode is presented in Section 2 In Section 3 weintroduce the analysis of reliability and propose the conceptof failure of sensors In Section 4 the ELRT rule whichconsiders the failure of sensors is derived and the numericalresults based on the Monte Carlo are given At last wesummarize in Section 5

2 Sensor Mode

We consider119872 sensor nodes which are deployed in a RegionOf Interest (ROI) with area 119860

2 Noises at local sensors areindependent and identically distributed (iid) and follow thestandard Gaussian distribution with zero mean and variance1 which can be written as

119899119894sim 119873 (0 1) 119894 = 1 119872 (1)

We assume that sensors make their local decisions inde-pendently without collaborating with other sensors Eachsensor makes a binary local decision choosing from

1198671 119903119894= 119904119894+ 119899119894

1198670 119903119894= 119899119894

(2)

where 1198671is the hypothesis of target presence and 119867

0is the

hypothesis of target absence 119903119894is the signal received by the

sensor 119894 and 119899119894denotes the noise observed by the sensor 119894

The signal strength 119904119894decays as the sensor moves away from

the target and follows the isotropic attenuation power modelas defined in [3]

1199042

119894=1198750119889120574

0

119889120574

119894

(3)

where 1198750is the original signal power from the target at a

reference distance1198890and119889119894represents the Euclidean distance

between the target and the sensor 119894 The signal attenuationexponent 120574 ranges from 2 to 3

Assuming that the local sensor 119894 uses the threshold 120591119894to

make the local binary decisions 119868119894in which 1 represents the

target is present and 0 represents the target is absent thenall decisions are sent to the FC denoted by I = (119868

1 119868

119872)

to make a final decision 119885 where 1 represents the target ispresent and 0 represents the target is absent According tothe Neyman-Pearson lemma [6] the local sensor-level falsealarm rate and probability of detection are given by

119875119891119894= 119875 (decision of sensor 119894 = 1 | 119867

0) = 119876 (120591

119894)

119875119889119894= 119875 (decision of sensor 119894 = 1 | 119867

1) = 119876 (120591

119894minus 119904119894)

(4)

where 119876(119909) = intinfin

119909(1radic2120587)119890

minus(11991022)119889119910 is the complementary

distribution function of the standard Gaussian

3 Analysis of Reliability

31 Failure of Sensors

Definition 1 Sensor failures are that their lifetimes havereached or been reaching to the end They are divided intotwo classes Little Bad (LB) sensor which still sends itsuntrusted data to the FC and Fully Bad (FB) sensor whichhas not worked

We define119877119894as the event that the sensor 119894 fails Obviously

the event of sensor failures is a Bernoulli event with 119875119894=

119875(119877119894= 0) denoting the probability of the event of failure of

the sensor 119894

A sensor deployment example is shown in Figure 1 How-ever in this paper for simplicity we assume that the sensorsare located uniformly in the ROI The circles represent thegood sensors The squares represent the FB sensors Thetriangles represent the LB sensors And the star represents thetarget Our attention of course is not paid to the state of FBsensors but the state of LB sensors is our interest Accordingto Definition 1 the data sent by the LB sensors is untrustedthat is their characteristics of 119875

119891119894and 119875

119889119894have changed We

assume that the LB sensors characteristics of 119875119891119894and 119875

119889119894are

denoted by 1198751015840119891119894and 1198751015840

119889119894 respectively

32 Analysis of Reliability Analysis of reliability life datarefers to the study and modeling of observed product livesLife data can be lifetimes of products in the marketplace

Mathematical Problems in Engineering 3

0 50 100 150 2000

20

40

60

80

100

120

140

160

180

200

X-coordinate

Y-c

oord

inat

e

Figure 1 A sensor deployment example In sensor networks fordistribution detection the local observations have to be quantizedbefore being transmitted to the FC which is demanded to make thefinal decision about the targetrsquos presence Area of the square region1198602 circle the good sensors square the failed sensors which are bad

and do not send their data to the FC triangle the failed sensorswhich still send their untrusted data to the FC and star the target

Time (hours miles cycles etc)

Randomfailure period

Wear-outperiod

Failu

re ra

te (f

ailu

res p

er u

nit t

ime)

Initial failureperiod

Figure 2 Bathtub-shaped failure with three stages which are initialfailure period random failure period and wear-out failure periodrespectively

such as the time the product operated successfully or thetime the product operated before it failed These lifetimescan be measured in hours miles cycles-to-failure stresscycles or any other metrics with which the life or exposureof a product can be measured All such data of productlifetimes can be encompassed in the term of life data ormore specifically product life data Otherwise in reliabilityone often characterizes a lifetime distribution through threefunctions [21] reliability function failure rate function andmean residual life

Furthermore for many mechanical and electronic com-ponents the failure rate function has a bathtub shape As isshown in Figure 2 the failure rate has three stages initial

failure period which is considered to occur when a latent isformed random failure period which occurs once deviceshaving latent defects have already failed and been removedand wear-out failure period which occurs due to the agingof devices from wear and fatigue In the random failureperiod the remaining high-quality devices operate stablyThefailures that occur during this period can usually be attributedto randomly occurring excessive stress such as power surgesand software errors

In [19] the authors have studied a simplemodel by addingtwo Weibull survival functions The failure rate function isgiven by

119903 (119905) = 119886119887(119886119905)119887minus1

+ 119888119889(119888119905)119889minus1

119905 119886 119888 ge 0 119887 gt 1 119889 lt 1

(5)

where 119886 119887 119888 and 119889 are related with life data of the productsOtherwise Bebbington et al [20] argue that it is worth addinga constant say ℎ

0ge 0 to the failure rate function This yields

the reduced additive Weibull failure rate function

119903 (119905) = 119886119887(119886119905)119887minus1

+ (119886

119887) (119886119905)

1119887minus1+ ℎ0 ℎ0ge 0 (6)

In (6) let 119889 = 1119887 and 119888 = 119886 to simplify (5)

33 Probability of Random Variable 119877119894 Let 119872 be the initial

number of the sensors and let119873 be the number of the sensorsreceived by the FC at time 119905 Let 119898 be the failure numberincluding all the FB and LB sensors at time 119905

From (6) we know

119898 = lceilint

119905

0

119903 (119905) 119889119905rceil (7)

where lceil119909rceil denotes the smallest integer which is not less than119909

The number of good sensors 119899 at time 119905 can be easilycalculated as follows

119899 = 119872 minus 119898 (8)

If 119873 gt 119899 it denotes that the 119873 sensors include the LBsensors which is

119904 = 119873 minus 119899 (9)

where 119904 denotes the LB sensors received by the FC at time 119905In the119873 sensors there is equal possibility for each sensor

to fail So the probability of 119877119894= 0 can be calculated as

follows

119875119894=

119904

119873119868119873+ (119873 minus 119899) (10)

where 119868119873+(119873 minus 119899) is the indicator function and 119873

+ is thepositive integer set 119868

119873+(119873 minus 119899) = 1 if 119873 minus 119899 gt 0 otherwise

119868119873+(119873 minus 119899) = 0 if119873 minus 119899 le 0

4 Mathematical Problems in Engineering

4 Fusion Rule

41 CVT Rule Based on the local sensors decision set thetraditional log-likelihood ratio test rule is given by [4]

Λopt0

= log119875 (I | 119867

1)

119875 (I | 1198670)

=

119873

sum

119894=0

(119868119894log

119875119889119894

119875119891119894

+ (1 minus 119868119894) log

1 minus 119875119889119894

1 minus 119875119891119894

)

(11)

42 ELRT Rule At the local sensor 119894 the likelihood functionunder the hypothesis of119867

1can be expressed as

119875 (119868119894| 1198671) = sum

119877119894

119875 (119868119894| 1198671 119877119894) 119875 (119877

119894| 1198671)

= 119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1)

(12)

With the local decisions which aremutually independentthe likelihood function at the FC in the hypothesis of119867

1is

119875 (I | 1198671) =

119873

prod

119894=1

119875 (119868119894| 1198671) (13)

From (12) and (13) we obtain

119875 (I | 1198671) =

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1))

=

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) (1 minus 119875

119894)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875

119894)

= prod

1198781

(119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894)

sdotprod

1198780

((1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894)

(14)

where the second equality of (14) comes from the fact that119877119894and 119867

1are uncorrelated The third equality of (14) is the

application of (4) 1198781(or 1198780) denotes the set of 119868

119894= 1 (or 119868

119894=

0)

Similarly under the hypothesis of1198670 we obtain

119875 (119868119894| 1198670) = sum

119877119894

119875 (119868119894| 1198670 119877119894) 119875 (119877

119894| 1198670)

= 119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0)

(15)

119875 (I | 1198670) =

119873

prod

119894=1

(119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0))

= prod

1198781

(119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894)

sdotprod

1198780

((1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894) 119875119894)

(16)

Finally combining (14) and (16) into (11) the ELRT ruleis therefore

Λopt1

= log119875 (I | 119867

1)

119875 (I | 1198670)

= log

(prod

1198781

[119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894]

timesprod

1198780

[(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894])

times (prod

1198781

[119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894]

timesprod

1198780

[(1 minus 119875119891119894)(1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894])

minus1

= log[

[

prod

1198781

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

timesprod

1198780

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

=

119873

sum

119894=1

[

[

119868119894log

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

+ (1 minus 119868119894) log

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

(17)

The advantage of such a formulation is that it allows usto consider all states of the sensors Such a fusion rule obeys

Mathematical Problems in Engineering 5

Chair-VarshneyChair-Varshney with ELRT with t1 = 1

t1 = 1

Chair-Varshney with t2 = 2ELRT with t2 = 2

Pd

Pfa

10minus210minus3 10minus110minus1

100

100

Figure 3 The ROC curves for different fusion rules and differenttimes 120574 = 2 119860 = 200119898 119889

0= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 171198751015840

119889119894= 11987511988911989451198751015840119891119894= 05 119905

1= 1 1199052= 2 119886 = 025

119887 = 8 and ℎ0= 4

our intuitions at the extreme cases When 119875119894= 0 the case

that 119904 lt 0 in (10) indicates no sensors fail then the ELRTrule is simplified to the traditional fusion rule defined in (11)With the increase of 119875

119894 the performance of ELRT rule will

be more better than the traditional rule without consideringsensor failures

43 Location of the Target From (3) and (4) we know119875119889119894

is the function of 119889119894 if the 120591

119894is given So the CVT

rule and the ELRT rule both require the knowledge of thecoordinates of the targetwhich are denoted by (119909

119905 119910119905) In [22]

the authors proposed to use Generalized Likelihood RatioTest (GLRT) statistic which uses the Maximum LikelihoodEstimate (MLE) of (119909

119905 119910119905) assuming 119867

1is true The authors

showed that the GLRT fusion rulersquos performance was close tothat of the state which the targetrsquos location was fully knownto the FC In [6] the uniform prior for the target is chosenwhen nothing else is available However in this paper thecoordinates of the target are given by (119909

119905 119910119905) = (50 45) to

simplify the process because ourmajor interest is the problemof failure

44 Numerical Results Figure 3 shows the ROC curves forthe two different test statistics CVT rule and ELRT rule Inthis simulation 104 Monte Carlo runs have been used Wechoose 119905

1= 1 and 119905

2= 2 for simulating At 119905

1(or 1199052)

the characteristics of 119875119889119894

and 119875119891119894

have changed for the LBsensors We choose 119875

1015840

119889119894= 1198751198891198945 and 119875

1015840

119891119894= 05 for the LB

sensors due to their poor performances The worst case thatthe failed sensors are all the LB sensors and they make wrongdecisions under the hypothesis of 119867

1(or 119867

0) is only taken

05

055

06

065

07

075

08

085

09

095

1

Pfa

Pd

10minus2 10minus1 100

Figure 4The ROC curves obtained by CVT at different times 120574 =

2 119860 = 200119898 1198890= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue

CVT 119905 = 1 red + circle CVT 119905 = 2 black CVT 119905 = 45 yellowCVT 119905 = 5 green CVT 119905 = 6 cyan CVT 119905 = 63 red + pentagramCVT 119905 = 67

into consideration Note that the performance of the CVTrule without failed sensors serves as an upper bound for otherfusion methods But the performance of the CVT rule withfailed sensors at 119905

1= 1 (or 119905

2= 2) rapidly decreases Their

performances are outperformed by the ELRT rule So theELRT rule improves the robust performances of the system

In order to reflect the influence of failed sensors weperform more simulations In Figure 4 the ROC curvescorresponding to the CVT at different times are plotted Ineach time we assume that all the failed sensors send either1 or 0 to the FC that is when the sensor fails it sends 1(or 0) to the FC all the time without considering the trueenvironment The failed sensors are randomly selected fromall the sensors In this simulation we use 119905 = 1 2 45 5 6 63and 67 to conduct the experiment In Figure 4 we see that theperformance of the system decreases over time Especially attime 63 and 67 the ROC curves sharply decrease Anotherobservation is that when at low time the ROC curves areclose to each other The reason of these phenomena is thatthe number of failed sensors changes over time At low timea small number of failed sensors have a negative but not bigeffect on the performance of the system At high time a largenumber of failed sensors have a major negative effect on theperformance of the system

To show the number of failed sensors how to changeover time the bar chart of the number of failed sensors isplotted in Figure 5 The major parameters are set as before119886 = 025 119887 = 8 and ℎ

0= 4 One observation is that the

number of failed sensors begins to increase slowly at low timeWhen at time 6 the number of failed sensors increases at highspeed Another observation is that the cutoff point betweenthe random failure period and the wear-out period is withinthe range of 5 to 6 The curve of the number of failed sensorsover time also reflects the features of bathtub-shaped failure

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 65 70

20

40

60

80

100

120

Time

Num

ber o

f fa

iled

sens

ors

Number of failed sensors

Figure 5 The number of failed sensors obtained by (7) at different times 119886 = 025 119887 = 8 and ℎ0= 4

05

055

06

065

07

075

08

085

09

095

1

Pd

Pfa

10minus2 10minus1 100

Figure 6 The ROC curves obtained by ELRT mode at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue ELRT 119905 = 1 red + upward-pointing triangle ELRT 119905 = 2 black ELRT

119905 = 45 yellow ELRT 119905 = 5 green ELRT 119905 = 6 cyan ELRT 119905 = 63 red + asterisk ELRT 119905 = 67

In Figure 6 the ROC curves corresponding to ELRT overtime are plotted In this simulation we also assume that eachsensor sends 1 (or 0) to the FC all the time when it becomesfailure The time 119905 = 1 2 45 5 6 63 and 67 are alsochosen to conduct the experiment The failed sensors arerandomly selected from all the sensors From the figure weshow that the ROC curves decrease over time but become asharp decline at high time because of a large number of failedsensors These features of ELRT are similar to the CVTTheirperformances of the system are both affected by the failedsensors

To compare the ROC curves between the ELRT and theCVT their comparisons at each time are plotted in Figure 7

All the basic parameters are set as before In this simulationwe also assume that each sensor sends 1 (or 0) to the FCall the time when it becomes failure The failed sensors arerandomly selected from all the sensors From Figure 7(a) toFigure 7(i) all the ROC curves of ELRT are above the ROCcurves of the CVT that is the performances of the CVTare outperformed by the ELRT rule with failed sensors Attime 67 the performance of the system has a sharp declinebecause of a large number of failed sensors

In summary the performances of the ELRT and CVTare both affected by the bathtub-shaped failure Because theELRT considers the failed sensors their performances arebetter than those of the CVT without considering the failed

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

studied In [16] the authors proposed a unified framework forsensor management in multimodal sensor networks whichis inspired by the trading behavior of economic agents incommercial markets The computationally efficient fusionrulewhich involves injecting a deliberate randomdisturbanceto the local sensor decisions before fusionwas designed in [8]

However the failed sensors which may or not sendtheir untrusted data to the FC are not taken into accountMost sensor fusion methods rely on the strong assumptionthat the sensors generate the data operation according to apredetermined characteristic at all times But when a sensorfails or wears out detection systems demonstrate behaviorsdifferent from the expected In [17] it has been shownthat Bayes Risk Error is a Bregman divergence betweenthe actual and estimated prior distribution probabilities In[18] the authors have studied a different approach and haveconstructed a fusion rule which is sensor-failure-robust byincluding a sensor failure model and minimizing expectedBayesian risk

In this paper we raise a new approach which expandsthe traditional log-likelihood ratio test by including a sensorfailure model We assume that the failed sensors which stillsend their data to the FC will follow new characteristics Ofcourse we do not pay attention to the failed sensors whichwill not send their data to the FC In order to study thesensor failures the analysis of reliability is introduced Formanymechanical and electronic components the failure ratefunction (or the hazard rate function) which determines thenumber of failures occurring per unit time has a bathtubshape The bathtub-shaped failure rate life distribution hasbeen derived in [19 20]Thenwe can calculate the probabilityof sensor failures by using the initial number of all sensorsthe number of sensors received by the FC and the number offailed sensors through the failure rate Ourmethodmakes thelog-likelihood ratio test robust by expanding the traditionallog-likelihood ratio test including BSF

The remainder of this paper is organized as followsSensor mode is presented in Section 2 In Section 3 weintroduce the analysis of reliability and propose the conceptof failure of sensors In Section 4 the ELRT rule whichconsiders the failure of sensors is derived and the numericalresults based on the Monte Carlo are given At last wesummarize in Section 5

2 Sensor Mode

We consider119872 sensor nodes which are deployed in a RegionOf Interest (ROI) with area 119860

2 Noises at local sensors areindependent and identically distributed (iid) and follow thestandard Gaussian distribution with zero mean and variance1 which can be written as

119899119894sim 119873 (0 1) 119894 = 1 119872 (1)

We assume that sensors make their local decisions inde-pendently without collaborating with other sensors Eachsensor makes a binary local decision choosing from

1198671 119903119894= 119904119894+ 119899119894

1198670 119903119894= 119899119894

(2)

where 1198671is the hypothesis of target presence and 119867

0is the

hypothesis of target absence 119903119894is the signal received by the

sensor 119894 and 119899119894denotes the noise observed by the sensor 119894

The signal strength 119904119894decays as the sensor moves away from

the target and follows the isotropic attenuation power modelas defined in [3]

1199042

119894=1198750119889120574

0

119889120574

119894

(3)

where 1198750is the original signal power from the target at a

reference distance1198890and119889119894represents the Euclidean distance

between the target and the sensor 119894 The signal attenuationexponent 120574 ranges from 2 to 3

Assuming that the local sensor 119894 uses the threshold 120591119894to

make the local binary decisions 119868119894in which 1 represents the

target is present and 0 represents the target is absent thenall decisions are sent to the FC denoted by I = (119868

1 119868

119872)

to make a final decision 119885 where 1 represents the target ispresent and 0 represents the target is absent According tothe Neyman-Pearson lemma [6] the local sensor-level falsealarm rate and probability of detection are given by

119875119891119894= 119875 (decision of sensor 119894 = 1 | 119867

0) = 119876 (120591

119894)

119875119889119894= 119875 (decision of sensor 119894 = 1 | 119867

1) = 119876 (120591

119894minus 119904119894)

(4)

where 119876(119909) = intinfin

119909(1radic2120587)119890

minus(11991022)119889119910 is the complementary

distribution function of the standard Gaussian

3 Analysis of Reliability

31 Failure of Sensors

Definition 1 Sensor failures are that their lifetimes havereached or been reaching to the end They are divided intotwo classes Little Bad (LB) sensor which still sends itsuntrusted data to the FC and Fully Bad (FB) sensor whichhas not worked

We define119877119894as the event that the sensor 119894 fails Obviously

the event of sensor failures is a Bernoulli event with 119875119894=

119875(119877119894= 0) denoting the probability of the event of failure of

the sensor 119894

A sensor deployment example is shown in Figure 1 How-ever in this paper for simplicity we assume that the sensorsare located uniformly in the ROI The circles represent thegood sensors The squares represent the FB sensors Thetriangles represent the LB sensors And the star represents thetarget Our attention of course is not paid to the state of FBsensors but the state of LB sensors is our interest Accordingto Definition 1 the data sent by the LB sensors is untrustedthat is their characteristics of 119875

119891119894and 119875

119889119894have changed We

assume that the LB sensors characteristics of 119875119891119894and 119875

119889119894are

denoted by 1198751015840119891119894and 1198751015840

119889119894 respectively

32 Analysis of Reliability Analysis of reliability life datarefers to the study and modeling of observed product livesLife data can be lifetimes of products in the marketplace

Mathematical Problems in Engineering 3

0 50 100 150 2000

20

40

60

80

100

120

140

160

180

200

X-coordinate

Y-c

oord

inat

e

Figure 1 A sensor deployment example In sensor networks fordistribution detection the local observations have to be quantizedbefore being transmitted to the FC which is demanded to make thefinal decision about the targetrsquos presence Area of the square region1198602 circle the good sensors square the failed sensors which are bad

and do not send their data to the FC triangle the failed sensorswhich still send their untrusted data to the FC and star the target

Time (hours miles cycles etc)

Randomfailure period

Wear-outperiod

Failu

re ra

te (f

ailu

res p

er u

nit t

ime)

Initial failureperiod

Figure 2 Bathtub-shaped failure with three stages which are initialfailure period random failure period and wear-out failure periodrespectively

such as the time the product operated successfully or thetime the product operated before it failed These lifetimescan be measured in hours miles cycles-to-failure stresscycles or any other metrics with which the life or exposureof a product can be measured All such data of productlifetimes can be encompassed in the term of life data ormore specifically product life data Otherwise in reliabilityone often characterizes a lifetime distribution through threefunctions [21] reliability function failure rate function andmean residual life

Furthermore for many mechanical and electronic com-ponents the failure rate function has a bathtub shape As isshown in Figure 2 the failure rate has three stages initial

failure period which is considered to occur when a latent isformed random failure period which occurs once deviceshaving latent defects have already failed and been removedand wear-out failure period which occurs due to the agingof devices from wear and fatigue In the random failureperiod the remaining high-quality devices operate stablyThefailures that occur during this period can usually be attributedto randomly occurring excessive stress such as power surgesand software errors

In [19] the authors have studied a simplemodel by addingtwo Weibull survival functions The failure rate function isgiven by

119903 (119905) = 119886119887(119886119905)119887minus1

+ 119888119889(119888119905)119889minus1

119905 119886 119888 ge 0 119887 gt 1 119889 lt 1

(5)

where 119886 119887 119888 and 119889 are related with life data of the productsOtherwise Bebbington et al [20] argue that it is worth addinga constant say ℎ

0ge 0 to the failure rate function This yields

the reduced additive Weibull failure rate function

119903 (119905) = 119886119887(119886119905)119887minus1

+ (119886

119887) (119886119905)

1119887minus1+ ℎ0 ℎ0ge 0 (6)

In (6) let 119889 = 1119887 and 119888 = 119886 to simplify (5)

33 Probability of Random Variable 119877119894 Let 119872 be the initial

number of the sensors and let119873 be the number of the sensorsreceived by the FC at time 119905 Let 119898 be the failure numberincluding all the FB and LB sensors at time 119905

From (6) we know

119898 = lceilint

119905

0

119903 (119905) 119889119905rceil (7)

where lceil119909rceil denotes the smallest integer which is not less than119909

The number of good sensors 119899 at time 119905 can be easilycalculated as follows

119899 = 119872 minus 119898 (8)

If 119873 gt 119899 it denotes that the 119873 sensors include the LBsensors which is

119904 = 119873 minus 119899 (9)

where 119904 denotes the LB sensors received by the FC at time 119905In the119873 sensors there is equal possibility for each sensor

to fail So the probability of 119877119894= 0 can be calculated as

follows

119875119894=

119904

119873119868119873+ (119873 minus 119899) (10)

where 119868119873+(119873 minus 119899) is the indicator function and 119873

+ is thepositive integer set 119868

119873+(119873 minus 119899) = 1 if 119873 minus 119899 gt 0 otherwise

119868119873+(119873 minus 119899) = 0 if119873 minus 119899 le 0

4 Mathematical Problems in Engineering

4 Fusion Rule

41 CVT Rule Based on the local sensors decision set thetraditional log-likelihood ratio test rule is given by [4]

Λopt0

= log119875 (I | 119867

1)

119875 (I | 1198670)

=

119873

sum

119894=0

(119868119894log

119875119889119894

119875119891119894

+ (1 minus 119868119894) log

1 minus 119875119889119894

1 minus 119875119891119894

)

(11)

42 ELRT Rule At the local sensor 119894 the likelihood functionunder the hypothesis of119867

1can be expressed as

119875 (119868119894| 1198671) = sum

119877119894

119875 (119868119894| 1198671 119877119894) 119875 (119877

119894| 1198671)

= 119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1)

(12)

With the local decisions which aremutually independentthe likelihood function at the FC in the hypothesis of119867

1is

119875 (I | 1198671) =

119873

prod

119894=1

119875 (119868119894| 1198671) (13)

From (12) and (13) we obtain

119875 (I | 1198671) =

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1))

=

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) (1 minus 119875

119894)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875

119894)

= prod

1198781

(119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894)

sdotprod

1198780

((1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894)

(14)

where the second equality of (14) comes from the fact that119877119894and 119867

1are uncorrelated The third equality of (14) is the

application of (4) 1198781(or 1198780) denotes the set of 119868

119894= 1 (or 119868

119894=

0)

Similarly under the hypothesis of1198670 we obtain

119875 (119868119894| 1198670) = sum

119877119894

119875 (119868119894| 1198670 119877119894) 119875 (119877

119894| 1198670)

= 119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0)

(15)

119875 (I | 1198670) =

119873

prod

119894=1

(119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0))

= prod

1198781

(119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894)

sdotprod

1198780

((1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894) 119875119894)

(16)

Finally combining (14) and (16) into (11) the ELRT ruleis therefore

Λopt1

= log119875 (I | 119867

1)

119875 (I | 1198670)

= log

(prod

1198781

[119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894]

timesprod

1198780

[(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894])

times (prod

1198781

[119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894]

timesprod

1198780

[(1 minus 119875119891119894)(1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894])

minus1

= log[

[

prod

1198781

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

timesprod

1198780

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

=

119873

sum

119894=1

[

[

119868119894log

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

+ (1 minus 119868119894) log

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

(17)

The advantage of such a formulation is that it allows usto consider all states of the sensors Such a fusion rule obeys

Mathematical Problems in Engineering 5

Chair-VarshneyChair-Varshney with ELRT with t1 = 1

t1 = 1

Chair-Varshney with t2 = 2ELRT with t2 = 2

Pd

Pfa

10minus210minus3 10minus110minus1

100

100

Figure 3 The ROC curves for different fusion rules and differenttimes 120574 = 2 119860 = 200119898 119889

0= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 171198751015840

119889119894= 11987511988911989451198751015840119891119894= 05 119905

1= 1 1199052= 2 119886 = 025

119887 = 8 and ℎ0= 4

our intuitions at the extreme cases When 119875119894= 0 the case

that 119904 lt 0 in (10) indicates no sensors fail then the ELRTrule is simplified to the traditional fusion rule defined in (11)With the increase of 119875

119894 the performance of ELRT rule will

be more better than the traditional rule without consideringsensor failures

43 Location of the Target From (3) and (4) we know119875119889119894

is the function of 119889119894 if the 120591

119894is given So the CVT

rule and the ELRT rule both require the knowledge of thecoordinates of the targetwhich are denoted by (119909

119905 119910119905) In [22]

the authors proposed to use Generalized Likelihood RatioTest (GLRT) statistic which uses the Maximum LikelihoodEstimate (MLE) of (119909

119905 119910119905) assuming 119867

1is true The authors

showed that the GLRT fusion rulersquos performance was close tothat of the state which the targetrsquos location was fully knownto the FC In [6] the uniform prior for the target is chosenwhen nothing else is available However in this paper thecoordinates of the target are given by (119909

119905 119910119905) = (50 45) to

simplify the process because ourmajor interest is the problemof failure

44 Numerical Results Figure 3 shows the ROC curves forthe two different test statistics CVT rule and ELRT rule Inthis simulation 104 Monte Carlo runs have been used Wechoose 119905

1= 1 and 119905

2= 2 for simulating At 119905

1(or 1199052)

the characteristics of 119875119889119894

and 119875119891119894

have changed for the LBsensors We choose 119875

1015840

119889119894= 1198751198891198945 and 119875

1015840

119891119894= 05 for the LB

sensors due to their poor performances The worst case thatthe failed sensors are all the LB sensors and they make wrongdecisions under the hypothesis of 119867

1(or 119867

0) is only taken

05

055

06

065

07

075

08

085

09

095

1

Pfa

Pd

10minus2 10minus1 100

Figure 4The ROC curves obtained by CVT at different times 120574 =

2 119860 = 200119898 1198890= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue

CVT 119905 = 1 red + circle CVT 119905 = 2 black CVT 119905 = 45 yellowCVT 119905 = 5 green CVT 119905 = 6 cyan CVT 119905 = 63 red + pentagramCVT 119905 = 67

into consideration Note that the performance of the CVTrule without failed sensors serves as an upper bound for otherfusion methods But the performance of the CVT rule withfailed sensors at 119905

1= 1 (or 119905

2= 2) rapidly decreases Their

performances are outperformed by the ELRT rule So theELRT rule improves the robust performances of the system

In order to reflect the influence of failed sensors weperform more simulations In Figure 4 the ROC curvescorresponding to the CVT at different times are plotted Ineach time we assume that all the failed sensors send either1 or 0 to the FC that is when the sensor fails it sends 1(or 0) to the FC all the time without considering the trueenvironment The failed sensors are randomly selected fromall the sensors In this simulation we use 119905 = 1 2 45 5 6 63and 67 to conduct the experiment In Figure 4 we see that theperformance of the system decreases over time Especially attime 63 and 67 the ROC curves sharply decrease Anotherobservation is that when at low time the ROC curves areclose to each other The reason of these phenomena is thatthe number of failed sensors changes over time At low timea small number of failed sensors have a negative but not bigeffect on the performance of the system At high time a largenumber of failed sensors have a major negative effect on theperformance of the system

To show the number of failed sensors how to changeover time the bar chart of the number of failed sensors isplotted in Figure 5 The major parameters are set as before119886 = 025 119887 = 8 and ℎ

0= 4 One observation is that the

number of failed sensors begins to increase slowly at low timeWhen at time 6 the number of failed sensors increases at highspeed Another observation is that the cutoff point betweenthe random failure period and the wear-out period is withinthe range of 5 to 6 The curve of the number of failed sensorsover time also reflects the features of bathtub-shaped failure

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 65 70

20

40

60

80

100

120

Time

Num

ber o

f fa

iled

sens

ors

Number of failed sensors

Figure 5 The number of failed sensors obtained by (7) at different times 119886 = 025 119887 = 8 and ℎ0= 4

05

055

06

065

07

075

08

085

09

095

1

Pd

Pfa

10minus2 10minus1 100

Figure 6 The ROC curves obtained by ELRT mode at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue ELRT 119905 = 1 red + upward-pointing triangle ELRT 119905 = 2 black ELRT

119905 = 45 yellow ELRT 119905 = 5 green ELRT 119905 = 6 cyan ELRT 119905 = 63 red + asterisk ELRT 119905 = 67

In Figure 6 the ROC curves corresponding to ELRT overtime are plotted In this simulation we also assume that eachsensor sends 1 (or 0) to the FC all the time when it becomesfailure The time 119905 = 1 2 45 5 6 63 and 67 are alsochosen to conduct the experiment The failed sensors arerandomly selected from all the sensors From the figure weshow that the ROC curves decrease over time but become asharp decline at high time because of a large number of failedsensors These features of ELRT are similar to the CVTTheirperformances of the system are both affected by the failedsensors

To compare the ROC curves between the ELRT and theCVT their comparisons at each time are plotted in Figure 7

All the basic parameters are set as before In this simulationwe also assume that each sensor sends 1 (or 0) to the FCall the time when it becomes failure The failed sensors arerandomly selected from all the sensors From Figure 7(a) toFigure 7(i) all the ROC curves of ELRT are above the ROCcurves of the CVT that is the performances of the CVTare outperformed by the ELRT rule with failed sensors Attime 67 the performance of the system has a sharp declinebecause of a large number of failed sensors

In summary the performances of the ELRT and CVTare both affected by the bathtub-shaped failure Because theELRT considers the failed sensors their performances arebetter than those of the CVT without considering the failed

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 50 100 150 2000

20

40

60

80

100

120

140

160

180

200

X-coordinate

Y-c

oord

inat

e

Figure 1 A sensor deployment example In sensor networks fordistribution detection the local observations have to be quantizedbefore being transmitted to the FC which is demanded to make thefinal decision about the targetrsquos presence Area of the square region1198602 circle the good sensors square the failed sensors which are bad

and do not send their data to the FC triangle the failed sensorswhich still send their untrusted data to the FC and star the target

Time (hours miles cycles etc)

Randomfailure period

Wear-outperiod

Failu

re ra

te (f

ailu

res p

er u

nit t

ime)

Initial failureperiod

Figure 2 Bathtub-shaped failure with three stages which are initialfailure period random failure period and wear-out failure periodrespectively

such as the time the product operated successfully or thetime the product operated before it failed These lifetimescan be measured in hours miles cycles-to-failure stresscycles or any other metrics with which the life or exposureof a product can be measured All such data of productlifetimes can be encompassed in the term of life data ormore specifically product life data Otherwise in reliabilityone often characterizes a lifetime distribution through threefunctions [21] reliability function failure rate function andmean residual life

Furthermore for many mechanical and electronic com-ponents the failure rate function has a bathtub shape As isshown in Figure 2 the failure rate has three stages initial

failure period which is considered to occur when a latent isformed random failure period which occurs once deviceshaving latent defects have already failed and been removedand wear-out failure period which occurs due to the agingof devices from wear and fatigue In the random failureperiod the remaining high-quality devices operate stablyThefailures that occur during this period can usually be attributedto randomly occurring excessive stress such as power surgesand software errors

In [19] the authors have studied a simplemodel by addingtwo Weibull survival functions The failure rate function isgiven by

119903 (119905) = 119886119887(119886119905)119887minus1

+ 119888119889(119888119905)119889minus1

119905 119886 119888 ge 0 119887 gt 1 119889 lt 1

(5)

where 119886 119887 119888 and 119889 are related with life data of the productsOtherwise Bebbington et al [20] argue that it is worth addinga constant say ℎ

0ge 0 to the failure rate function This yields

the reduced additive Weibull failure rate function

119903 (119905) = 119886119887(119886119905)119887minus1

+ (119886

119887) (119886119905)

1119887minus1+ ℎ0 ℎ0ge 0 (6)

In (6) let 119889 = 1119887 and 119888 = 119886 to simplify (5)

33 Probability of Random Variable 119877119894 Let 119872 be the initial

number of the sensors and let119873 be the number of the sensorsreceived by the FC at time 119905 Let 119898 be the failure numberincluding all the FB and LB sensors at time 119905

From (6) we know

119898 = lceilint

119905

0

119903 (119905) 119889119905rceil (7)

where lceil119909rceil denotes the smallest integer which is not less than119909

The number of good sensors 119899 at time 119905 can be easilycalculated as follows

119899 = 119872 minus 119898 (8)

If 119873 gt 119899 it denotes that the 119873 sensors include the LBsensors which is

119904 = 119873 minus 119899 (9)

where 119904 denotes the LB sensors received by the FC at time 119905In the119873 sensors there is equal possibility for each sensor

to fail So the probability of 119877119894= 0 can be calculated as

follows

119875119894=

119904

119873119868119873+ (119873 minus 119899) (10)

where 119868119873+(119873 minus 119899) is the indicator function and 119873

+ is thepositive integer set 119868

119873+(119873 minus 119899) = 1 if 119873 minus 119899 gt 0 otherwise

119868119873+(119873 minus 119899) = 0 if119873 minus 119899 le 0

4 Mathematical Problems in Engineering

4 Fusion Rule

41 CVT Rule Based on the local sensors decision set thetraditional log-likelihood ratio test rule is given by [4]

Λopt0

= log119875 (I | 119867

1)

119875 (I | 1198670)

=

119873

sum

119894=0

(119868119894log

119875119889119894

119875119891119894

+ (1 minus 119868119894) log

1 minus 119875119889119894

1 minus 119875119891119894

)

(11)

42 ELRT Rule At the local sensor 119894 the likelihood functionunder the hypothesis of119867

1can be expressed as

119875 (119868119894| 1198671) = sum

119877119894

119875 (119868119894| 1198671 119877119894) 119875 (119877

119894| 1198671)

= 119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1)

(12)

With the local decisions which aremutually independentthe likelihood function at the FC in the hypothesis of119867

1is

119875 (I | 1198671) =

119873

prod

119894=1

119875 (119868119894| 1198671) (13)

From (12) and (13) we obtain

119875 (I | 1198671) =

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1))

=

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) (1 minus 119875

119894)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875

119894)

= prod

1198781

(119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894)

sdotprod

1198780

((1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894)

(14)

where the second equality of (14) comes from the fact that119877119894and 119867

1are uncorrelated The third equality of (14) is the

application of (4) 1198781(or 1198780) denotes the set of 119868

119894= 1 (or 119868

119894=

0)

Similarly under the hypothesis of1198670 we obtain

119875 (119868119894| 1198670) = sum

119877119894

119875 (119868119894| 1198670 119877119894) 119875 (119877

119894| 1198670)

= 119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0)

(15)

119875 (I | 1198670) =

119873

prod

119894=1

(119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0))

= prod

1198781

(119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894)

sdotprod

1198780

((1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894) 119875119894)

(16)

Finally combining (14) and (16) into (11) the ELRT ruleis therefore

Λopt1

= log119875 (I | 119867

1)

119875 (I | 1198670)

= log

(prod

1198781

[119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894]

timesprod

1198780

[(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894])

times (prod

1198781

[119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894]

timesprod

1198780

[(1 minus 119875119891119894)(1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894])

minus1

= log[

[

prod

1198781

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

timesprod

1198780

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

=

119873

sum

119894=1

[

[

119868119894log

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

+ (1 minus 119868119894) log

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

(17)

The advantage of such a formulation is that it allows usto consider all states of the sensors Such a fusion rule obeys

Mathematical Problems in Engineering 5

Chair-VarshneyChair-Varshney with ELRT with t1 = 1

t1 = 1

Chair-Varshney with t2 = 2ELRT with t2 = 2

Pd

Pfa

10minus210minus3 10minus110minus1

100

100

Figure 3 The ROC curves for different fusion rules and differenttimes 120574 = 2 119860 = 200119898 119889

0= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 171198751015840

119889119894= 11987511988911989451198751015840119891119894= 05 119905

1= 1 1199052= 2 119886 = 025

119887 = 8 and ℎ0= 4

our intuitions at the extreme cases When 119875119894= 0 the case

that 119904 lt 0 in (10) indicates no sensors fail then the ELRTrule is simplified to the traditional fusion rule defined in (11)With the increase of 119875

119894 the performance of ELRT rule will

be more better than the traditional rule without consideringsensor failures

43 Location of the Target From (3) and (4) we know119875119889119894

is the function of 119889119894 if the 120591

119894is given So the CVT

rule and the ELRT rule both require the knowledge of thecoordinates of the targetwhich are denoted by (119909

119905 119910119905) In [22]

the authors proposed to use Generalized Likelihood RatioTest (GLRT) statistic which uses the Maximum LikelihoodEstimate (MLE) of (119909

119905 119910119905) assuming 119867

1is true The authors

showed that the GLRT fusion rulersquos performance was close tothat of the state which the targetrsquos location was fully knownto the FC In [6] the uniform prior for the target is chosenwhen nothing else is available However in this paper thecoordinates of the target are given by (119909

119905 119910119905) = (50 45) to

simplify the process because ourmajor interest is the problemof failure

44 Numerical Results Figure 3 shows the ROC curves forthe two different test statistics CVT rule and ELRT rule Inthis simulation 104 Monte Carlo runs have been used Wechoose 119905

1= 1 and 119905

2= 2 for simulating At 119905

1(or 1199052)

the characteristics of 119875119889119894

and 119875119891119894

have changed for the LBsensors We choose 119875

1015840

119889119894= 1198751198891198945 and 119875

1015840

119891119894= 05 for the LB

sensors due to their poor performances The worst case thatthe failed sensors are all the LB sensors and they make wrongdecisions under the hypothesis of 119867

1(or 119867

0) is only taken

05

055

06

065

07

075

08

085

09

095

1

Pfa

Pd

10minus2 10minus1 100

Figure 4The ROC curves obtained by CVT at different times 120574 =

2 119860 = 200119898 1198890= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue

CVT 119905 = 1 red + circle CVT 119905 = 2 black CVT 119905 = 45 yellowCVT 119905 = 5 green CVT 119905 = 6 cyan CVT 119905 = 63 red + pentagramCVT 119905 = 67

into consideration Note that the performance of the CVTrule without failed sensors serves as an upper bound for otherfusion methods But the performance of the CVT rule withfailed sensors at 119905

1= 1 (or 119905

2= 2) rapidly decreases Their

performances are outperformed by the ELRT rule So theELRT rule improves the robust performances of the system

In order to reflect the influence of failed sensors weperform more simulations In Figure 4 the ROC curvescorresponding to the CVT at different times are plotted Ineach time we assume that all the failed sensors send either1 or 0 to the FC that is when the sensor fails it sends 1(or 0) to the FC all the time without considering the trueenvironment The failed sensors are randomly selected fromall the sensors In this simulation we use 119905 = 1 2 45 5 6 63and 67 to conduct the experiment In Figure 4 we see that theperformance of the system decreases over time Especially attime 63 and 67 the ROC curves sharply decrease Anotherobservation is that when at low time the ROC curves areclose to each other The reason of these phenomena is thatthe number of failed sensors changes over time At low timea small number of failed sensors have a negative but not bigeffect on the performance of the system At high time a largenumber of failed sensors have a major negative effect on theperformance of the system

To show the number of failed sensors how to changeover time the bar chart of the number of failed sensors isplotted in Figure 5 The major parameters are set as before119886 = 025 119887 = 8 and ℎ

0= 4 One observation is that the

number of failed sensors begins to increase slowly at low timeWhen at time 6 the number of failed sensors increases at highspeed Another observation is that the cutoff point betweenthe random failure period and the wear-out period is withinthe range of 5 to 6 The curve of the number of failed sensorsover time also reflects the features of bathtub-shaped failure

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 65 70

20

40

60

80

100

120

Time

Num

ber o

f fa

iled

sens

ors

Number of failed sensors

Figure 5 The number of failed sensors obtained by (7) at different times 119886 = 025 119887 = 8 and ℎ0= 4

05

055

06

065

07

075

08

085

09

095

1

Pd

Pfa

10minus2 10minus1 100

Figure 6 The ROC curves obtained by ELRT mode at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue ELRT 119905 = 1 red + upward-pointing triangle ELRT 119905 = 2 black ELRT

119905 = 45 yellow ELRT 119905 = 5 green ELRT 119905 = 6 cyan ELRT 119905 = 63 red + asterisk ELRT 119905 = 67

In Figure 6 the ROC curves corresponding to ELRT overtime are plotted In this simulation we also assume that eachsensor sends 1 (or 0) to the FC all the time when it becomesfailure The time 119905 = 1 2 45 5 6 63 and 67 are alsochosen to conduct the experiment The failed sensors arerandomly selected from all the sensors From the figure weshow that the ROC curves decrease over time but become asharp decline at high time because of a large number of failedsensors These features of ELRT are similar to the CVTTheirperformances of the system are both affected by the failedsensors

To compare the ROC curves between the ELRT and theCVT their comparisons at each time are plotted in Figure 7

All the basic parameters are set as before In this simulationwe also assume that each sensor sends 1 (or 0) to the FCall the time when it becomes failure The failed sensors arerandomly selected from all the sensors From Figure 7(a) toFigure 7(i) all the ROC curves of ELRT are above the ROCcurves of the CVT that is the performances of the CVTare outperformed by the ELRT rule with failed sensors Attime 67 the performance of the system has a sharp declinebecause of a large number of failed sensors

In summary the performances of the ELRT and CVTare both affected by the bathtub-shaped failure Because theELRT considers the failed sensors their performances arebetter than those of the CVT without considering the failed

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

4 Fusion Rule

41 CVT Rule Based on the local sensors decision set thetraditional log-likelihood ratio test rule is given by [4]

Λopt0

= log119875 (I | 119867

1)

119875 (I | 1198670)

=

119873

sum

119894=0

(119868119894log

119875119889119894

119875119891119894

+ (1 minus 119868119894) log

1 minus 119875119889119894

1 minus 119875119891119894

)

(11)

42 ELRT Rule At the local sensor 119894 the likelihood functionunder the hypothesis of119867

1can be expressed as

119875 (119868119894| 1198671) = sum

119877119894

119875 (119868119894| 1198671 119877119894) 119875 (119877

119894| 1198671)

= 119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1)

(12)

With the local decisions which aremutually independentthe likelihood function at the FC in the hypothesis of119867

1is

119875 (I | 1198671) =

119873

prod

119894=1

119875 (119868119894| 1198671) (13)

From (12) and (13) we obtain

119875 (I | 1198671) =

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) 119875 (119877

119894= 1 | 119867

1)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875 (119877

119894= 0 | 119867

1))

=

119873

prod

119894=1

(119875 (119868119894| 1198671 119877119894= 1) (1 minus 119875

119894)

+ 119875 (119868119894| 1198671 119877119894= 0) 119875

119894)

= prod

1198781

(119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894)

sdotprod

1198780

((1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894)

(14)

where the second equality of (14) comes from the fact that119877119894and 119867

1are uncorrelated The third equality of (14) is the

application of (4) 1198781(or 1198780) denotes the set of 119868

119894= 1 (or 119868

119894=

0)

Similarly under the hypothesis of1198670 we obtain

119875 (119868119894| 1198670) = sum

119877119894

119875 (119868119894| 1198670 119877119894) 119875 (119877

119894| 1198670)

= 119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0)

(15)

119875 (I | 1198670) =

119873

prod

119894=1

(119875 (119868119894| 1198670 119877119894= 1) 119875 (119877

119894= 1 | 119867

0)

+ 119875 (119868119894| 1198670 119877119894= 0) 119875 (119877

119894= 0 | 119867

0))

= prod

1198781

(119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894)

sdotprod

1198780

((1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894) 119875119894)

(16)

Finally combining (14) and (16) into (11) the ELRT ruleis therefore

Λopt1

= log119875 (I | 119867

1)

119875 (I | 1198670)

= log

(prod

1198781

[119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894]

timesprod

1198780

[(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894])

times (prod

1198781

[119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894]

timesprod

1198780

[(1 minus 119875119891119894)(1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894])

minus1

= log[

[

prod

1198781

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

timesprod

1198780

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

=

119873

sum

119894=1

[

[

119868119894log

119875119889119894(1 minus 119875

119894) + 1198751015840

119889119894119875119894

119875119891119894(1 minus 119875

119894) + 1198751015840

119891119894119875119894

+ (1 minus 119868119894) log

(1 minus 119875119889119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119889119894) 119875119894

(1 minus 119875119891119894) (1 minus 119875

119894) + (1 minus 119875

1015840

119891119894)119875119894

]

]

(17)

The advantage of such a formulation is that it allows usto consider all states of the sensors Such a fusion rule obeys

Mathematical Problems in Engineering 5

Chair-VarshneyChair-Varshney with ELRT with t1 = 1

t1 = 1

Chair-Varshney with t2 = 2ELRT with t2 = 2

Pd

Pfa

10minus210minus3 10minus110minus1

100

100

Figure 3 The ROC curves for different fusion rules and differenttimes 120574 = 2 119860 = 200119898 119889

0= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 171198751015840

119889119894= 11987511988911989451198751015840119891119894= 05 119905

1= 1 1199052= 2 119886 = 025

119887 = 8 and ℎ0= 4

our intuitions at the extreme cases When 119875119894= 0 the case

that 119904 lt 0 in (10) indicates no sensors fail then the ELRTrule is simplified to the traditional fusion rule defined in (11)With the increase of 119875

119894 the performance of ELRT rule will

be more better than the traditional rule without consideringsensor failures

43 Location of the Target From (3) and (4) we know119875119889119894

is the function of 119889119894 if the 120591

119894is given So the CVT

rule and the ELRT rule both require the knowledge of thecoordinates of the targetwhich are denoted by (119909

119905 119910119905) In [22]

the authors proposed to use Generalized Likelihood RatioTest (GLRT) statistic which uses the Maximum LikelihoodEstimate (MLE) of (119909

119905 119910119905) assuming 119867

1is true The authors

showed that the GLRT fusion rulersquos performance was close tothat of the state which the targetrsquos location was fully knownto the FC In [6] the uniform prior for the target is chosenwhen nothing else is available However in this paper thecoordinates of the target are given by (119909

119905 119910119905) = (50 45) to

simplify the process because ourmajor interest is the problemof failure

44 Numerical Results Figure 3 shows the ROC curves forthe two different test statistics CVT rule and ELRT rule Inthis simulation 104 Monte Carlo runs have been used Wechoose 119905

1= 1 and 119905

2= 2 for simulating At 119905

1(or 1199052)

the characteristics of 119875119889119894

and 119875119891119894

have changed for the LBsensors We choose 119875

1015840

119889119894= 1198751198891198945 and 119875

1015840

119891119894= 05 for the LB

sensors due to their poor performances The worst case thatthe failed sensors are all the LB sensors and they make wrongdecisions under the hypothesis of 119867

1(or 119867

0) is only taken

05

055

06

065

07

075

08

085

09

095

1

Pfa

Pd

10minus2 10minus1 100

Figure 4The ROC curves obtained by CVT at different times 120574 =

2 119860 = 200119898 1198890= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue

CVT 119905 = 1 red + circle CVT 119905 = 2 black CVT 119905 = 45 yellowCVT 119905 = 5 green CVT 119905 = 6 cyan CVT 119905 = 63 red + pentagramCVT 119905 = 67

into consideration Note that the performance of the CVTrule without failed sensors serves as an upper bound for otherfusion methods But the performance of the CVT rule withfailed sensors at 119905

1= 1 (or 119905

2= 2) rapidly decreases Their

performances are outperformed by the ELRT rule So theELRT rule improves the robust performances of the system

In order to reflect the influence of failed sensors weperform more simulations In Figure 4 the ROC curvescorresponding to the CVT at different times are plotted Ineach time we assume that all the failed sensors send either1 or 0 to the FC that is when the sensor fails it sends 1(or 0) to the FC all the time without considering the trueenvironment The failed sensors are randomly selected fromall the sensors In this simulation we use 119905 = 1 2 45 5 6 63and 67 to conduct the experiment In Figure 4 we see that theperformance of the system decreases over time Especially attime 63 and 67 the ROC curves sharply decrease Anotherobservation is that when at low time the ROC curves areclose to each other The reason of these phenomena is thatthe number of failed sensors changes over time At low timea small number of failed sensors have a negative but not bigeffect on the performance of the system At high time a largenumber of failed sensors have a major negative effect on theperformance of the system

To show the number of failed sensors how to changeover time the bar chart of the number of failed sensors isplotted in Figure 5 The major parameters are set as before119886 = 025 119887 = 8 and ℎ

0= 4 One observation is that the

number of failed sensors begins to increase slowly at low timeWhen at time 6 the number of failed sensors increases at highspeed Another observation is that the cutoff point betweenthe random failure period and the wear-out period is withinthe range of 5 to 6 The curve of the number of failed sensorsover time also reflects the features of bathtub-shaped failure

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 65 70

20

40

60

80

100

120

Time

Num

ber o

f fa

iled

sens

ors

Number of failed sensors

Figure 5 The number of failed sensors obtained by (7) at different times 119886 = 025 119887 = 8 and ℎ0= 4

05

055

06

065

07

075

08

085

09

095

1

Pd

Pfa

10minus2 10minus1 100

Figure 6 The ROC curves obtained by ELRT mode at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue ELRT 119905 = 1 red + upward-pointing triangle ELRT 119905 = 2 black ELRT

119905 = 45 yellow ELRT 119905 = 5 green ELRT 119905 = 6 cyan ELRT 119905 = 63 red + asterisk ELRT 119905 = 67

In Figure 6 the ROC curves corresponding to ELRT overtime are plotted In this simulation we also assume that eachsensor sends 1 (or 0) to the FC all the time when it becomesfailure The time 119905 = 1 2 45 5 6 63 and 67 are alsochosen to conduct the experiment The failed sensors arerandomly selected from all the sensors From the figure weshow that the ROC curves decrease over time but become asharp decline at high time because of a large number of failedsensors These features of ELRT are similar to the CVTTheirperformances of the system are both affected by the failedsensors

To compare the ROC curves between the ELRT and theCVT their comparisons at each time are plotted in Figure 7

All the basic parameters are set as before In this simulationwe also assume that each sensor sends 1 (or 0) to the FCall the time when it becomes failure The failed sensors arerandomly selected from all the sensors From Figure 7(a) toFigure 7(i) all the ROC curves of ELRT are above the ROCcurves of the CVT that is the performances of the CVTare outperformed by the ELRT rule with failed sensors Attime 67 the performance of the system has a sharp declinebecause of a large number of failed sensors

In summary the performances of the ELRT and CVTare both affected by the bathtub-shaped failure Because theELRT considers the failed sensors their performances arebetter than those of the CVT without considering the failed

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Chair-VarshneyChair-Varshney with ELRT with t1 = 1

t1 = 1

Chair-Varshney with t2 = 2ELRT with t2 = 2

Pd

Pfa

10minus210minus3 10minus110minus1

100

100

Figure 3 The ROC curves for different fusion rules and differenttimes 120574 = 2 119860 = 200119898 119889

0= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 171198751015840

119889119894= 11987511988911989451198751015840119891119894= 05 119905

1= 1 1199052= 2 119886 = 025

119887 = 8 and ℎ0= 4

our intuitions at the extreme cases When 119875119894= 0 the case

that 119904 lt 0 in (10) indicates no sensors fail then the ELRTrule is simplified to the traditional fusion rule defined in (11)With the increase of 119875

119894 the performance of ELRT rule will

be more better than the traditional rule without consideringsensor failures

43 Location of the Target From (3) and (4) we know119875119889119894

is the function of 119889119894 if the 120591

119894is given So the CVT

rule and the ELRT rule both require the knowledge of thecoordinates of the targetwhich are denoted by (119909

119905 119910119905) In [22]

the authors proposed to use Generalized Likelihood RatioTest (GLRT) statistic which uses the Maximum LikelihoodEstimate (MLE) of (119909

119905 119910119905) assuming 119867

1is true The authors

showed that the GLRT fusion rulersquos performance was close tothat of the state which the targetrsquos location was fully knownto the FC In [6] the uniform prior for the target is chosenwhen nothing else is available However in this paper thecoordinates of the target are given by (119909

119905 119910119905) = (50 45) to

simplify the process because ourmajor interest is the problemof failure

44 Numerical Results Figure 3 shows the ROC curves forthe two different test statistics CVT rule and ELRT rule Inthis simulation 104 Monte Carlo runs have been used Wechoose 119905

1= 1 and 119905

2= 2 for simulating At 119905

1(or 1199052)

the characteristics of 119875119889119894

and 119875119891119894

have changed for the LBsensors We choose 119875

1015840

119889119894= 1198751198891198945 and 119875

1015840

119891119894= 05 for the LB

sensors due to their poor performances The worst case thatthe failed sensors are all the LB sensors and they make wrongdecisions under the hypothesis of 119867

1(or 119867

0) is only taken

05

055

06

065

07

075

08

085

09

095

1

Pfa

Pd

10minus2 10minus1 100

Figure 4The ROC curves obtained by CVT at different times 120574 =

2 119860 = 200119898 1198890= 1119898 119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue

CVT 119905 = 1 red + circle CVT 119905 = 2 black CVT 119905 = 45 yellowCVT 119905 = 5 green CVT 119905 = 6 cyan CVT 119905 = 63 red + pentagramCVT 119905 = 67

into consideration Note that the performance of the CVTrule without failed sensors serves as an upper bound for otherfusion methods But the performance of the CVT rule withfailed sensors at 119905

1= 1 (or 119905

2= 2) rapidly decreases Their

performances are outperformed by the ELRT rule So theELRT rule improves the robust performances of the system

In order to reflect the influence of failed sensors weperform more simulations In Figure 4 the ROC curvescorresponding to the CVT at different times are plotted Ineach time we assume that all the failed sensors send either1 or 0 to the FC that is when the sensor fails it sends 1(or 0) to the FC all the time without considering the trueenvironment The failed sensors are randomly selected fromall the sensors In this simulation we use 119905 = 1 2 45 5 6 63and 67 to conduct the experiment In Figure 4 we see that theperformance of the system decreases over time Especially attime 63 and 67 the ROC curves sharply decrease Anotherobservation is that when at low time the ROC curves areclose to each other The reason of these phenomena is thatthe number of failed sensors changes over time At low timea small number of failed sensors have a negative but not bigeffect on the performance of the system At high time a largenumber of failed sensors have a major negative effect on theperformance of the system

To show the number of failed sensors how to changeover time the bar chart of the number of failed sensors isplotted in Figure 5 The major parameters are set as before119886 = 025 119887 = 8 and ℎ

0= 4 One observation is that the

number of failed sensors begins to increase slowly at low timeWhen at time 6 the number of failed sensors increases at highspeed Another observation is that the cutoff point betweenthe random failure period and the wear-out period is withinthe range of 5 to 6 The curve of the number of failed sensorsover time also reflects the features of bathtub-shaped failure

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 65 70

20

40

60

80

100

120

Time

Num

ber o

f fa

iled

sens

ors

Number of failed sensors

Figure 5 The number of failed sensors obtained by (7) at different times 119886 = 025 119887 = 8 and ℎ0= 4

05

055

06

065

07

075

08

085

09

095

1

Pd

Pfa

10minus2 10minus1 100

Figure 6 The ROC curves obtained by ELRT mode at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue ELRT 119905 = 1 red + upward-pointing triangle ELRT 119905 = 2 black ELRT

119905 = 45 yellow ELRT 119905 = 5 green ELRT 119905 = 6 cyan ELRT 119905 = 63 red + asterisk ELRT 119905 = 67

In Figure 6 the ROC curves corresponding to ELRT overtime are plotted In this simulation we also assume that eachsensor sends 1 (or 0) to the FC all the time when it becomesfailure The time 119905 = 1 2 45 5 6 63 and 67 are alsochosen to conduct the experiment The failed sensors arerandomly selected from all the sensors From the figure weshow that the ROC curves decrease over time but become asharp decline at high time because of a large number of failedsensors These features of ELRT are similar to the CVTTheirperformances of the system are both affected by the failedsensors

To compare the ROC curves between the ELRT and theCVT their comparisons at each time are plotted in Figure 7

All the basic parameters are set as before In this simulationwe also assume that each sensor sends 1 (or 0) to the FCall the time when it becomes failure The failed sensors arerandomly selected from all the sensors From Figure 7(a) toFigure 7(i) all the ROC curves of ELRT are above the ROCcurves of the CVT that is the performances of the CVTare outperformed by the ELRT rule with failed sensors Attime 67 the performance of the system has a sharp declinebecause of a large number of failed sensors

In summary the performances of the ELRT and CVTare both affected by the bathtub-shaped failure Because theELRT considers the failed sensors their performances arebetter than those of the CVT without considering the failed

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 65 70

20

40

60

80

100

120

Time

Num

ber o

f fa

iled

sens

ors

Number of failed sensors

Figure 5 The number of failed sensors obtained by (7) at different times 119886 = 025 119887 = 8 and ℎ0= 4

05

055

06

065

07

075

08

085

09

095

1

Pd

Pfa

10minus2 10minus1 100

Figure 6 The ROC curves obtained by ELRT mode at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898 119910

119905= 45119898

120591119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Blue ELRT 119905 = 1 red + upward-pointing triangle ELRT 119905 = 2 black ELRT

119905 = 45 yellow ELRT 119905 = 5 green ELRT 119905 = 6 cyan ELRT 119905 = 63 red + asterisk ELRT 119905 = 67

In Figure 6 the ROC curves corresponding to ELRT overtime are plotted In this simulation we also assume that eachsensor sends 1 (or 0) to the FC all the time when it becomesfailure The time 119905 = 1 2 45 5 6 63 and 67 are alsochosen to conduct the experiment The failed sensors arerandomly selected from all the sensors From the figure weshow that the ROC curves decrease over time but become asharp decline at high time because of a large number of failedsensors These features of ELRT are similar to the CVTTheirperformances of the system are both affected by the failedsensors

To compare the ROC curves between the ELRT and theCVT their comparisons at each time are plotted in Figure 7

All the basic parameters are set as before In this simulationwe also assume that each sensor sends 1 (or 0) to the FCall the time when it becomes failure The failed sensors arerandomly selected from all the sensors From Figure 7(a) toFigure 7(i) all the ROC curves of ELRT are above the ROCcurves of the CVT that is the performances of the CVTare outperformed by the ELRT rule with failed sensors Attime 67 the performance of the system has a sharp declinebecause of a large number of failed sensors

In summary the performances of the ELRT and CVTare both affected by the bathtub-shaped failure Because theELRT considers the failed sensors their performances arebetter than those of the CVT without considering the failed

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

09091092093094095096097098099

1

ELRT with t = 1Chair-Varshney with t = 1

Pfa

10minus2 10minus1 100

Pd

(a)

09091092093094095096097098099

1

ELRT with t = 2Chair-Varshney with t = 2

Pfa

10minus2 10minus1 100

Pd

(b)

08082084086088

09092094096098

1

ELRT with t = 3Chair-Varshney with t = 3

Pfa

10minus2 10minus1 100

Pd

(c)

082084086088

09092094096098

1

ELRT with t = 45Chai-Varshney with t = 45

08

Pfa

10minus2 10minus1 100

Pd

(d)

08208

084086088

09092094096098

1

ELRT with t = 5Chai-Varshney with t = 5

Pfa

10minus2 10minus1 100

Pd

(e)

07

075

08

085

09

095

1

ELRT with t = 6Chair-Varshney with t = 6

Pfa

10minus2 10minus1 100

Pd

(f)

03040506070809

1

ELRT with t = 63Chair-Varshney with t = 63

Pfa

10minus2 10minus1 100

Pd

(g)

06065

07075

08085

09095

1

ELRT with t = 65Chair-Varshney with t = 65

Pfa

10minus2 10minus1 100

Pd

(h)

0302

040506070809

1

ELRT with t = 67Chair-Varshney with t = 67

Pfa

10minus2 10minus1 100

Pd

(i)

Figure 7 The ROC for comparison between ELRT and CVT at different times 120574 = 2 119860 = 200119898 1198890= 1119898119872 = 196 119875

0= 400 119909

119905= 50119898

119910119905= 45119898 120591

119894= 120591 = 17 1198751015840

119889119894= 1198751198891198945 1198751015840119891119894= 05 119886 = 025 119887 = 8 and ℎ

0= 4 Asterisk ELRT circle CVT (a) 119905 = 1 (b) 119905 = 2 (c) 119905 = 3 (d)

119905 = 45 (e) 119905 = 5 (f) 119905 = 6 (g) 119905 = 63 (h) 119905 = 65 (i) 119905 = 67

sensors when the features of failed sensors meet the bathtub-shaped failure

5 Conclusion

We have derived the ELRT rule based on BSF model fordistributed target detection in sensor networks It is assumedthat each sensor receives a signal that attenuates as a functionof the distance between the target and the local sensor

The BSF model which introduces the analysis of reliabilityclassifies the sensorsrsquo states into three stages initial failureperiod random failure period and wear-out failure periodWe have shown that the ELRT fusion rule outperforms theCVT rule without considering failed sensors The ELRTfusion rule improves the robust performance of the system Inthe future we will investigate the design of 1198751015840

119889119894and 119875

1015840

119891119894when

the sensors become failures and the locationrsquos estimation ofthe target

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no 61001086) and FundamentalResearch Funds for the Central Universities (Grant noZYGX2011X004)

References

[1] H Ahmadi and A Vosoughi ldquoOptimal training and datapower allocation in distributed detection with inhomogeneoussensorsrdquo IEEE Signal Processing Society vol 20 no 4 pp 339ndash342 2013

[2] J Fang Y Liu H Li and S Li ldquoOne-bit quantizer design formultisensor GLRT fusionrdquo IEEE Signal Processing Letters vol20 no 3 pp 257ndash260 2013

[3] R Niu and P K Varshney ldquoPerformance analysis of distributeddetection in a random sensor fieldrdquo IEEE Transactions on SignalProcessing vol 56 no 1 pp 339ndash349 2008

[4] Z Chair and P K Varshney ldquoOptimal data fusion in multiplesensor detection systemsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 22 no 1 pp 98ndash101 1986

[5] I Y Hoballah and P K Varshney ldquoDistributed Bayesian signaldetectionrdquo IEEETransactions on InformationTheory vol 35 no5 pp 995ndash1000 1989

[6] M Guerriero L Svensson and P Willett ldquoBayesian datafusion for distributed target detection in sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 6 pp 3417ndash34212010

[7] B Chen R Jiang T Kasetkasem and P K Varshney ldquoChannelaware decision fusion in wireless sensor networksrdquo IEEE Trans-actions on Signal Processing vol 52 no 12 pp 3454ndash3458 2004

[8] S G Iyengar R Niu and P K Varshney ldquoFusing dependentdecisions for hypothesis testing with heterogeneous sensorsrdquoIEEE Transactions on Signal Processing vol 60 no 9 pp 4888ndash4897 2012

[9] A M Aziz ldquoA new adaptive decentralized soft decision com-bining rule for distributed sensor systems with data fusionrdquoInformation Sciences vol 256 pp 197ndash210 2014

[10] A M Aziz ldquoA new multiple decisions fusion rule for targetsdetection inmultiple sensors distributed detection systemswithdata fusionrdquo Information Fusion vol 18 pp 175ndash186 2014

[11] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo submit-ted httparxivorgabs13071448

[12] Y Yilmaz G V Moustakides and X Wang ldquoChannel-awaredecentralized detection via level-triggered samplingrdquo IEEETransactions on Signal Processing vol 61 no 2 pp 300ndash3152013

[13] E Soltanmohammadi M Orooji and M Naraghi-PourldquoDecentralized hypothesis testing in wireless sensor networksin the presence of misbehaving nodesrdquo IEEE Transactions onInformation Forensics and Security vol 8 no 1 pp 205ndash2152013

[14] R Tharmarasa T Kirubarajan A Sinha and T Lang ldquoDecen-tralized sensor selection for large-scale multisensor-multitarget

trackingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 47 no 2 pp 1307ndash1324 2011

[15] A S Behbahani A M Eltawil and H Jafarkhani ldquoLinearestimation of correlated vector sources for wireless sensornetworks with fusion centerrdquo IEEE Wireless CommunicationsLetters vol 1 no 4 pp 400ndash403 2012

[16] P Chavali and A Nehorai ldquoManaging multi-modal sensornetworks using price theoryrdquo IEEE Transactions on SignalProcessing vol 60 no 9 pp 4874ndash4887 2012

[17] K R Varshney ldquoBayes risk error is a Bregman divergencerdquo IEEETransactions on Signal Processing vol 59 no 9 pp 4470ndash44722011

[18] M Higger M Akcakaya and D Erdogmus ldquoA robust fusionalgorithm for sensor failurerdquo IEEE Signal Processing Society vol20 no 8 pp 755ndash758 2013

[19] M Xie and C D Lai ldquoReliability analysis using an additiveWeibull model with bathtub-shaped failure rate functionrdquoReliability Engineering amp System Safety vol 52 no 1 pp 87ndash931996

[20] M Bebbington C-D Lai and R Zitikis ldquoUseful periods forlifetime distributions with bathtub shaped hazardrate func-tionsrdquo IEEE Transactions on Reliability vol 55 no 2 pp 245ndash251 2006

[21] C D Lai M Xie and D N P Murthy ldquoChapter 3 Bathtub-shaped failure rate life distributionsrdquo in Advances in Reliabilityvol 20 of Handbook of Statistics pp 69ndash104 2001

[22] R Niu and P K Varshney ldquoJoint detection and localization insensor networks based on local decisionsrdquo in Proceedings of the40th Asilomar Conference on Signals Systems and Computers(ACSSC rsquo06) pp 525ndash529 November 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of