Sensors 014, 14, 7655-7683; do:10.3390/s140507655 Artcle OPEN ACCESS sensors ISSN 144-80 www.mdp.com/journal/sensors An Uncertanty-Based Dstrbuted Fault Detecton Mechansm for Wreless Sensor Networks Yang Yang *, Zhpeng Gao, Hang Zhou and uesong Qu State Key Laboratory of Networkng and Swtchng Technology, Bejng Unversty of Posts and Telecommuncatons, No.10 tucheng Road, Hadan Dstrct, Bejng 100876, Chna; E-Mals: gaozhpeng@bupt.edu.cn (Z.G.); zhouhangcauc@163.com (H.Z.); xsqu@bupt.edu.cn (.Q.) * Author to whom correspondence should be addressed; E-Mal: yyang@bupt.edu.cn; Tel./Fax: +86-10-6119-8090. Receved: 14 February 014; n revsed form: 14 Aprl 014 / Accepted: 17 Aprl 014 / Publshed: 5 Aprl 014 Abstract: Exchangng too many messages for fault detecton wll cause not only a degradaton of the network qualty of servce, but also represents a huge burden on the lmted energy of sensors. Therefore, we propose an uncertanty-based dstrbuted fault detecton through aded judgment of neghbors for wreless sensor networks. The algorthm consders the serous nfluence of sensng measurement loss and therefore uses Markov decson processes for fllng n mssng data. Most mportant of all, fault msjudgments caused by uncertanty condtons are the man drawbacks of tradtonal dstrbuted fault detecton mechansms. We draw on the experence of evdence fuson rules based on nformaton entropy theory and the degree of dsagreement functon to ncrease the accuracy of fault detecton. Smulaton results demonstrate our algorthm can effectvely reduce communcaton energy overhead due to message exchanges and provde a hgher detecton accuracy rato. Keywords: fault detecton; uncertanty; evdence fuson; data mssng; nformaton entropy 1. Introducton Sensors can be rapdly deployed nto large areas and perform montorng tasks by autonomous wreless communcaton methods. In dsaster preventon applcatons, for example, nodes detect and estmate envronmental nformaton, and then forecast when and where a natural calamty may occur.
Sensors 014, 14 7656 Although users always hope that the network wll provde excellent montorng and gatherng functons, t seems nevtable that sensors to experence faults caused by some extrnsc and ntrnsc factors. Generally, a fault s an unexpected change n the network, whch leads to measurement errors, system breakdown or communcaton falure. Faults are generally classfed as crash, tmng, omsson, ncorrect computaton, fal breakdown, authentcated Byzantne, etc. [1]. From another pont of vew, crash faults are classfed as communcaton faults, snce under those condtons, a sensor can t communcate wth others because t has a falure n ts communcaton module or the lnk s down. Contrarly, all other faults are vewed as data faults, whch means the faulty sensors can communcate wth each other, but the sensed or transmtted data s not correct [,3]. To avod erroneous judgments due to faults, broken-down nodes should be detected and solated from other functonng nodes. Fault detecton should make an unambguous decson about whether the behavor of a sensor devates from other common measurements. Sensors always form a local vew of the fault state of sensors by collectng measurements from ther one-hop neghbors. Neghbor cooperaton s one approach to fault detecton, whereby a sensor uses neghbor measurements to decde ts own fault state collaboratvely [4 6]. Ths s demonstrated to be effcacous for fault nformaton collecton and dagnoss because t allevates the overheads of snk nodes or base statons n order to avod network bottlenecks. Accordngly, a novel challenge for fault detecton n wreless sensor networks (WSNs) s how to reduce the energy consumpton when exchangng messages s the man means of fault detecton n the dstrbuted envronment. Moreover, the dynamc network topology and sgnal loss caused by long propagaton delays and sgnal fadng nfluence the effcency of fault detecton n some advanced medcal care or battlefeld response applcatons. In the majorty votng algorthm based on neghbor cooperaton detecton, the normal measurements of sensors that are located close to each other are assumed to be spatally correlated, whle the fault data are uncorrelated. The tendency state of a sensor s determned as possbly faulty (LF) or possbly good (LG) by comparng ts own readngs wth those of ts one-hop neghbors n a votng process. If the number of LG neghbors that have correlated readngs s greater than or equal to half, then t s fault-free, otherwse t s deemed faulty. The weghted votng approach uses geographc dstance or degree of trust as the decdng factor when calculatng the sensor state, but these methods perform n WSNs better based on the hypothess of hgher average connectvty degree. Actually, sensors are usually deployed n a lower connectvty envronment, n whch exchanged readngs are too few to make an approprate comparson and decson (e.g., n Fgure 1a fronter node only has one neghbor). Then the detecton accuracy rate decreases as the fault rate ncreases. Moreover, the faults caused by attacks are unevenly dstrbuted n the case of ntruson montorng because the hostle sgnals wthout a fxed routng wll randomly affect or tamper wth readngs. In ths paper, we manly focus on sensng faults other than communcaton faults. After analyzng the defects of tradtonal algorthms, we present an Uncertanty-based Dstrbuted Fault Detecton (udfd) mechansm for wreless sensor networks. The man contrbutons of udfd are as follows: (1) Propose the uncertanty-based dstrbuted fault detecton algorthm, whch can avod decreasng fault detecton accuracy when the falure rato becomes hgher. In addton, the accuracy of fault detecton remans at a hgh level regardless of a lower connectvty scene;
Sensors 014, 14 7657 () Data loss wll nfluence the fault judgment because each sensor determnes ts own state step by step accordng to ts neghbors measurements. The paper represents a data forecast model based on a Markov decson processes for fllng n lost data to provde reference data for others state determnatons; (3) We classfy two types of sensors tendency states: Possble Good (LG) and Undetermned (Un). LG nodes contrbute to judge nodes ultmate state. The Un nodes are both n an uncertanty status, so we must determne the ultmate status of an Un node. Here we desgn belef probablty assgnment (BPA) functons for dfferent evdences that reflect the states of Un nodes. What s more, an evdence fuson rule based on nformaton entropy theory s used to avod evdence conflcts. Fgure 1. Fault detecton llustraton. F 1 4 F 3 F F 6 10 5 11 G G G G F 6 F F 3 G 1 5 G G 7 8 G 7 8 9 G G G 4 F (a) (b) The rest of the paper s organzed as follows: Secton descrbes some related works n the area of fault detecton n WSNs. Secton 3 ntroduces our Uncertanty-based Dstrbuted Fault Detecton algorthm (udfd) and the concrete mechansms nvolved. Secton 4 depcts the smulaton results wth respect to typcal fault detecton algorthms lke DFD and IDFD, and demonstrates our algorthm s effcency and superorty. In Secton 5, we conclude the paper.. Related Works In ths secton, we brefly revew related works n the area of dstrbuted and centralzed fault detecton n WSNs. The authors n [4] proposed and evaluated a localzed fault detecton scheme (DFD) to dentfy faulty sensors. An mproved DFD scheme was proposed by Jang n [5]. Neghbors always exchange sensng measurements perodcally, therefore a sensor judges ts own state (as good or faulty) accordng to neghbors values. A faulty dentfcaton algorthm reported n [7] s completely localzed and requres lower computatonal overhead, and t can easly be scaled to large sensor networks. In the algorthm, the readng of a sensor s compared wth ts neghbors medan readngs. If the dfference s large or large but negatve, the sensor s deemed as faulty. If half of neghbors are faulty and the number of neghbors s even, the algorthm cannot detect faults. Krshnamachar and co-workers proposed n [8] a dstrbuted soluton for the canoncal task of bnary detecton of nterestng envronmental events. They explctly take nto account the possblty of
Sensors 014, 14 7658 measurement faults and develop a dstrbuted Bayesan scheme for detectng and correctng faults. Each sensor node dentfes ts own status based on local comparsons of sensed data wth some thresholds and dssemnaton of the test results [9]. Tme redundancy s used to tolerate transent sensng and communcaton faults. To elmnate the delay nvolved n z tme redundancy scheme, a sldng wndow s employed wth some data storage for comparson wth prevous results. The MANNA scheme [10] creates a manager located externally to the WSN. It has a global vson of the network and can perform complex tasks that would not be possble nsde the network. Management actvtes take place when sensor nodes are collectng and sendng temperature data. Every node wll check ts energy level and send a message to the manager/agent whenever there s a state change. The manager can then obtan the coverage map and energy level of all sensors based upon the collected nformaton. To detect node falures, the manager sends GET operatons to retreve the node state. Wthout hearng from the nodes, the manager wll consult the energy map to check ts resdual energy. In ths way, MANNA archtecture s able to locate faulty sensor nodes. However, ths approach requres an external manager to perform the centralzed dagnoss and the communcaton between nodes and the manager s too expensve for WSNs. Tsang-Y et al. [11] proposed a dstrbuted fault-tolerant decson fuson n the presence of sensor faults. The collaboratve sensor fault detecton (CSFD) scheme s proposed to elmnate unrelable local decsons. In ths approach, the local sensors send ther decsons sequentally to a fuson center. Ths scheme establshes an upper bound on the fuson error probablty based on a pre-desgned fuson rule. Ths upper bound assumes dentcal local decson rules and fault-free envronments. They proposed a crteron to search the faulty sensor nodes whch s based on ths error boundary. Once the fuson center dentfes the faulty sensor nodes, all correspondng local decsons are removed from the computaton of the lkelhood ratos that are adopted to make the fnal decson. Ths approach consders crash and ncorrect computaton faults. In [1], a taxonomy for classfcaton of faults n sensor networks and the frst on-lne model-based testng technque are ntroduced. The technque consders the mpact of readngs of a partcular sensor on the consstency of mult-sensor fuson. A sensor s most lkely to be faulty f ts elmnaton sgnfcantly mproves the consstency of the results. A way to dstngush random nose s to run a maxmum lkelhood or Bayesan approach on the mult-sensor fuson measurements. If the accuracy of fnal results of multsensory fuson mproves after runnng these procedures, random nose should exst. To get a consstent mappng of the sensed phenomena, dfferent sensors measurements need to be combned n a model. Ths cross-valdaton-based technque can be appled to a broad set of fault models. It s generc and can be appled to an arbtrary system of sensors that use an arbtrary type of data fuson. However, ths technque s centralzed. Sensor node nformaton must be collected and sent to the base staton to conduct the on-lne fault detecton. Mao et al. [13] presented an onlne lghtweght falure detecton scheme named Agnostc Dagnoss (AD). Ths approach s motvated by the fact that the system metrcs of sensors (e.g., rado-on tme, number of packets transmtted) usually exhbt certan correlaton patterns. Ths approach collects types of metrcs that are classfed nto four categores: (1) tmng metrcs (e.g., RadoOnTmeCounter). They denote the accumulatve rado-on tme; () traffc metrcs (e.g., TransmtCounter). They record the accumulatve number of packets transmtted by a sensor node; (3) task metrcs (e.g., TaskExecCounter). Ths s the accumulatve number of tasks executed; (4) other metrcs
Sensors 014, 14 7659 such as Parent Change Counter, whch counts the number of parent changes. AD explots the correlatons between the metrcs of each sensor usng a correlaton graph that descrbes the status of the sensor node. By mnng through the perodcally updated correlaton graphs, abnormal correlatons are detected n tme. Specfcally, n addton to predefned faults (.e., wth known types and symptoms), slent falures caused by Byzantne faults are consdered. Exchangng too many messages for fault detecton wll cause not only a degradaton of the network qualty of servce, but also a huge burden on the lmted energy of sensors. Hence, we desgn an uncertanty-based dstrbuted fault detecton based on neghbor cooperaton n WSNs. It adopts auto-correlated test results to descrbe dfferent sensng states from day to day, and the nformaton entropy-based D-S evdence theory wll be ntroduced to deduce actual states for undetermned nodes. 3. Uncertanty-Based Fault Detecton Mechansm 3.1. The DFD and IDFD Schemes and Ther Drawbacks Ths secton presents the DFD algorthm proposed by Chen [4] and IDFD algorthm descrbed by Jang [5] to gve an overvew of dstrbuted fault detecton, and then analyzes these algorthms drawbacks. Chen [4] ntroduced a localzed fault detecton method by exchangng measures n WSNs. It s assumed that x s the measurement of node. We defne between node and j at tme t, whle When tl d j d to represent the measured dfference t j s measurement dfference from tme t l to t l+1 : d x x (1) t t t j j d d d ( x x ) ( x x ) () tl tl 1 tl tl 1 tl 1 tl tl j j j j j t d j s less than or equal to a predefned threshold 1, we wll consder a test result c j s set to l 0, or else t contnuously calculates d t l j. If d t j > ( s also a predefned threshold), then c j = 1, otherwse c j = 0. Here the expresson c j = 1 means node and node j are possbly n dfferent states. Next, the tendency status (possbly a faulty LF or possbly a good LG) s determned accordng to followng formula [14]: where N LF f cj N / T jn LG otherwse s the number of one-hop neghbors of node. The formula states that a sensor s deemed to be possbly good only f there are less than N / (3) neghbors whose test results are 1. In order to process the second round test, each node needs to send ts tendency state to ts one-hop neghbors. In the DFD algorthm, n the end state the node Z s decded to be fault-free only f a dfference s greater than or equal to N /, otherwse s undetermned. Here (1 cj ) cj (1 cj ) ( j N, T LG ). In order to promote dentfcaton effcency for undetermned sensors, these nodes j repeatedly check whether ther neghbor s state s fault-free or not. If such a neghbor exsts, then the sensor s faulty (fault-free) accordng to the test result 1(0) between them. A sensor may not decde ts
Sensors 014, 14 7660 own state because the states of neghbors are n conflct, e.g., Z j = Z k = GOOD. At the same tme, c j c k. Then Z s GOOD f T = LG, or else Z s FAULT. Jang [5] consders the determnant condton (1 c ) / j N & T LG j N n the DFD algorthm s too harsh and ths wll lead some normal nodes to be msdagnosed as faulty, so the determnant condton for a normal node s amended as: jn& T j JLG Tj LG J c N / (4) If there s no tendency status of a neghbor as LG, then the fnal determnant status s set as normal (faulty) based on T = LG (T = LF). Although ths mechansm promotes the fault detecton accuracy to a certan extent through smulaton demonstraton, t doesn t have a clear way to resolve conflcts or erroneous judgments as llustrated n Fgure 1. In Fgure 1a, t calculates c 1 = 0, c 13 = 0, and c 14 = 0 for node 1. Then T 1 s set as LG accordng to Equaton (3). In the same way, we get T = LF, T 3 = LF, T 4 = LF. Node 1 has no neghbor whose tendency status s LG, and then the fnal determnant status s set as normal based on the rule of T = LG. Ths s an obvous erroneous judgment. The tendency states n Fgure 1b are calculated as follows: T 1 = LF, T = LG, T 3 = LF, T 4 = LG. For node 1, c 1 0 1 & j, and / / 1 jn T LG T3LG T5LG. The node 1 s decded as J N Tj LG faulty accordng to Equaton (4). Actually, node 1 s a normal sensor. Node 1 wll make a mstake when the number of normal neghbors equals the number of faulty neghbors. The premse s that ther ntal detecton tendency states are LG. By analyzng msjudgment condtons of tradtonal algorthms, a defect s that an ndetermnacy occurs on the condton = n Equaton (4), and thus the node s not reducble to good or faulty. Another s that these algorthms gnore the effect of sensors own measurements whch are approxmate at the same tme on adjacent days (e.g., 8 June and 9 June). The analogous and hstorcal readngs of the same node contrbute to determne the faulty state under vague condtons. Moreover, most dstrbuted fault detecton mechansms assume that sensors have the ablty to acqure every measurement and cooperatvely judge the state of each other. When the sensor s communcaton module has a falure, but the acquston module s actve, the readngs can t be perceved by the sensor. In a dstrbuted collaboratve process, nodes dagnose data faults based prmarly on neghbors data. Once a neghbor s data s mssng, t wll affect the accuracy of fault dagnoss, e.g., n Fgure 1b, node 4 can t determne ts own status when node 1 has no data. 3.. Uncertanty-Based Dstrbuted Fault Detecton Algorthm In the paper, we manly resolve the followng problems: (1) data mssng before exchangng readngs; () msjudgments caused by ndetermnacy condtons. The problem of mssng data due to communcaton faults wll affect the determnaton accuracy when comparng neghbors measurements. To solve the data loss, a faulty sensng node should fll n the mssng measurements to provde the reference. Secondly, the represented algorthm adopts the auto-correlated test results to descrbe the status of dfferences between dfferent days. Fnally, those undetermned appearances may occur n the above-mentoned secton. The nformaton entropy and the degree of dsagreement functon combned
Sensors 014, 14 7661 n evdence fuson theory are mproved accordngly to help to deduce ther actual states. In addton, usng nformaton entropy n the evdence fuson can reduce evdence conflcts and ncrease detecton accuracy. 3..1. Defntons We lst the notatons n the udfd algorthm as follows: p: Probablty of fault of a sensor; N : A set of neghbors of node ;, x Dt : Measurement value of node at tme t on day D; N : Number of one-hop neghbors of node ; t d j : Measurement dfference between node and j at tme t on the same day accordng to Formula (1); tl d : Measurement dfference between node and j from tme t l to t l+1 on the same day j accordng to Formula ();, Δd Dt : Measurement dfference of node at the same tme t on dfferent day; c j : Test result between node and j, cj {0,1} ; T : Tendency value of a sensor, T { LG, Un} ; Z : Determned detecton status of a sensor, Z {GOOD,FAULT} ; t Δ, θ 1, θ, θ 3 : Predefned threshold values about, Δ t D t d d l, Δd ; Num ({G}): Number of good neghbors of node ; Num ({F}): Number of faulty neghbors of node. 3... Fault Detecton j j The man processes of the udfd algorthm based on neghbor cooperaton are summarzed as follows. The key technology for solvng the two problems s descrbed n Sectons 3..3 and 3..4. Stage 1: Each sensor acqures the readngs from ts own sensng module. If no data s acqured, then t flls up the mssng data. After that, t exchanges the measurement at tme t on day D wth ts neghbors and calculates the test result C j (It s assumed that C j = 0 at the ntal tme): 1: If d, then set C j = 0; Dt, : else C j = 1; 3: end f 4: t If dj 1, then C j = 1 3 t t 5: else f dj 1 && d l j, then C j = 1; 6: else C j = 0; 7: end f 8: Repeat the above steps untl all of test results about neghbors are obtaned. Stage : Node generates the tendency value based on c ( j) : j
Sensors 014, 14 766 9: If j c j c N 1, then T = LG; 10: else T = Un; 11: end f 1: Broadcast the tendency status f T = LG. Stage 3: Calculate the determned status of LG nodes: 13: If T = LG && ( j) j { LG} ; Num( Ns ) 1 14: If Cj, then Z = Good; jlg jlg 15: else Z = Fault; 16: end f 17: else f T = LG && no any neghbor s LG, then T = Un; 18: end f 19: A LG node can determne ts own status (good or faulty), and only good sensors broadcast ther states n order to save transmsson overheads. Stage 4: A node whose tendency status s Un determnes the actual state by usng entropy-based evdence combnaton mechansm: 0: Node ( { LF, Un}) receves the evdence of good neghbors. 1: Combne the evdences generated by measurements by adoptng nformaton entropy-based evdence fuson, and acqure the combned BPA functons m * ({ G }), m * ({ F }), and m * ( ); mn mj ( G m ({ G})) ; : Node fnds the node j whch matches the * 3: f c j = 1, then Z = FAULT, else Z = GOOD; 4: end f 5: Determned node broadcasts ts status f t s a good sensor. Broadcastng not only uses up nodes energy but also occupes the channel bandwdth, so the man method of savng energy consumpton n our algorthm s that only partcular states n dfferent stages (LG and GOOD) are broadcast. In Step 1, only the node whose tendency status s equal to LG broadcasts the value. The reason s that only LG neghbors partcpate n fnal state determnaton n Step 14. Smlarly, only good sensors broadcast ther states n order to save energy transmsson overhead. 3..3. Mssng Data Preprocessng Mechansm In the paper, we manly focus on sensng faults rather than communcaton faults. When mssng data, occurs because of a sensng fault, t wll affect the accuracy of fault dagnoss. Ths means Dt has been lost because the communcaton module has faled, whch subsequently nfluences the reference data for other sensors faulty state determnaton. It s necessary for node to fll n the mssng data and send t to neghbors. In ths secton, we use a Markov decson processes based on neghbors hstorcal data to predct the current mssng measurement values of node. Relyng the features of Markov theory whch can reflect the nfluence of random factors and extenson to the stochastc process whch s dynamc and fluctuatng s consdered and we combne the hstorcal data of node wth ts neghbors
Sensors 014, 14 7663 hstorcal data, and then form a fuson hstorcal data vector, whch can be adaptvely adjusted accordng to the sgnfcance of neghbors measurements. Therefore, the state transton matrx of Markov s adopted to predct the value and sgn of the readng dfference between two days. The steps for data mssng preprocess preprocessng are as follows: Steps: (1) For each node j N, where N s the set of all the neghbors of node, fetch the prevous m hstorcal measurements of node j, and these hstorcal measurements correspond to an m dmensonal Dm, t Dm1, t D1, t vector V j, that s V (,,..., ) ; j j j j () Calculate the reputaton value C j for each neghbor of node, that s for each node j N, we 1 m D k, t D k, t k1 j have Cj e, where. Note that for a dfferent node, node j has m dfferent reputaton values and a smaller value for wll ncrease the reputaton value of node j; (3) Here we ntroduce Mahalanobs dstance to evaluate the smlarty dstance between node and ts neghbors. Then the predcton results should keep Mahalanobs dstance changes wthn a predefned threshold. For each node j N V, V between vectors V, calculate the Mahalanobs s dstance D j and V j, n order to evaluate the smlarty of node and all ts neghbors. That s D V V V V V V, where Σ s the covarance matrx of V and V j ; T 1 (, j ) ( j ) ( j ) (4) Assume that * V s a fuson of the hstorcal measurements of node and all ts neghbors, whch s used n the Markov decson processes to predct the current measurement of node. It s also an m dmensonal vector and can be calculated as follows: N C * j V V V j1 N j (5) C k1 k In ths data-fuson formula, the hstorcal measurement vector V j s weghted by the reputaton value of node j, and the factors α and β ( 1) ndcate to what extent a node trusts tself and neghbors. Here 0.5. (5) Accordng to the result of fuson n Step 4, use Markov decson processes to predct the current, measurement of node, then we can get Dt ; (6) For each node j N, recalculate the Mahalanobs s dstance D( V, V ) between vectors V and V j. That s vectors, and covarance matrx of (7) If j N D VV VV V V, here V and T 1 (, j ) ( j ) Σ ( j ) V, Dm, t Dm1, t D1, t D, t (,,,, ) V and V j;, (, j) VV, j adjust the fuson factor Dt, j V j are (m + 1) dmensonal V Dm, t Dm1, t D1, t D, t j ( j, j,, j, j ), Σ s the D VV D, where θ s a predefned threshold, then there s no need to and the predcted value can be adopted. Otherwse, the predcted value ncreases the dfferences between node and node j, so the fuson factor needs to be reduced approprately, n order to decrease the proporton of neghbors measurements n the calculaton of (8) If the fuson factor α has been adjusted n Step 7, then return to Step 4. Otherwse, ths algorthm ends. Dt, * V ;
Sensors 014, 14 7664 In order to predct the mssng data n Step 5 of the above algorthm, we draw on the experence Dt, of Markov decson processes [15,16]. Frstly, accordng to above algorthm, we can get the correspondng vector * V whch s calculated n Step 4 of the and t s, 1,, ( Dm t, D m t,..., D t ) an (m 1) dmensonal vector and can be consdered as an ndependent and dentcal dstrbuted Markov chan. Then, we classfy the state of each component n vector by an average-standard devaton E mn, max, where mn s and classfcaton method. Assume that state s can be expressed as max s ndcate the lower bound and upper bound of state s. Then the sample average s: The standard devaton s: j s s s m 1 Dj, t (6) m 1 S 1 m Dj, t ( ) (7) m j Accordng to central-lmt theorem [17], we dvde the sldng nterval of hstorcal fault data E1 3 S, S, E [ S, 0.5 S), E3 [ 0.5 S, 0.5 S), E4 [ 0.5 S, S), and E5 [ S, 3 S). The state of each component n the dfference vector nto fve states, that s dependng on whch sldng nterval t belongs to. The transton probablty matrx P (1) can be calculated as follows. Assume that (1) M st ndcates the sample numbers that state E S transfers to state E t n one step, and M S ndcates the sample numbers of (1) (1) M st (1) state E S before transfer. Then we get pst, where p st means the transton probablty of M shftng from state E S to state E t by one step. Therefore the 5 5 transton probablty matrx s: For any component Assume that ( D j, t) P (1) s p p p (1) (1) 11 15 p (1) (1) 51 55 (j = 1,, m), the probablty dstrbuton vector s: D j D j D j D j D j π( D j ) (,,,, ) (8) ( D, t) 1 3 4 5 s n state E 3, then the probablty dstrbuton vector of t s πd (0,0,1,0,0). As the probablty dstrbuton vector π D and the transton probablty matrx P (1) are known, then the probablty dstrbuton vector of (1) 1 P ( D 1, t) π D π D, the correspondng state n max{ π ( D 1), s{1,,3, 4,5}} s the state ( D 1, t) as follows: belongs to. If ( D 1, t) s n state s, then the specfc value of s ( D 1, t) s s determned
Sensors 014, 14 7665 D1, t πs 1 D 1 1 1 1 π 1 s s D πs 1 D πs D πs 1 D πs 1 D 1 1 1 1 mns π D π D π D s1 s s 1 mns max 1 1 1 maxs π D π D π D s1 s s1 (9) Contnue to ntroduce Markov decson processes to predct the sgns (postve and negatve) of ( D 1, t) ( Dm, t) ( Dm1, t) D, t. For the vector (,,..., ), we defne that state E 1 corresponds to postve, and state E corresponds to negatve. Then we get the transton probablty p M (1) (1) st st M s (1) (1) (1) p 11 p 1 the transton probablty matrx P (1) (1) whch reflects the probablty of transferences p 1 p ( D j, t) between postve and negatve. Also for any component, (j = 1,,,m), the probablty dstrbuton vector s ( 1 D j, D j the probablty dstrbuton vector of t s πd and the transton probablty matrx vector of the sgn of π D j. Assume that ( D 1, t) max{ ( D 1), s{ 1,}} ndcates the sgn of s ( D, t) and s a postve, then π D (1,0). As the probablty dstrbuton vector s (1) (1) P are known, then the probablty dstrbuton π D 1 π D P, the correspondng state n ( D 1, t) 3..4. Informaton Entropy Based Evdence Confuson. As the Un nodes are both n uncertanty status, we need to fnd a mechansm to determne the status of these nodes. Dempster-Shafer evdence theory s an effectve method for dealng wth uncertanty problems, but the results obtaned are counterntutve when the evdences conflct hghly wth each other [18,19]. In the mproved evdence fuson algorthm we propose, the possble events can be depcted as evdences. Through combnaton rules, evdences are aggregated nto a comprehensve belef probablty assgnment under uncertanty condtons. It s assumed that a set of hypotheses about node status s denoted as frame of dscernment { GF, }. The symbol G represents a good sensor, and F s faulty. The power set Θ ncludes all of subsets of. Here {{ },{ G},{ F},{ }}, each symbol of whch respectvely represents the hypotheses about mpossble, good, faulty, and uncertanty. The belef probablty assgnment (BPA) functons of node are depcted as follows: We defne the BPA functon for good status s: m: [0,1] (10) m Φ 0 (11)
Sensors 014, 14 7666 0.5 1 0 u1 1 1 ( u1 ) 1 1 e m({ G}) u1 1 u1 1 0.5 u1 1 4 1 0 u1 Smlarly, the BPA functon for faulty status s: 0 u1 1 41 ( u1 ) 1 1 e m({ F}) u1 1 u1 1 0.5 u1 1 1 1 u1 (1) (13) The BPA functon for uncertanty status s: u11 41 ( u1 ) 1 m({ }) e u1 1 u1 1 0.5 u1 1 4 1 0 u1 Here we desgn an expectaton devaton functon (14). It s assumed that the measurement value of nodes at tme t on day D s a random varable, whch has the expectaton E and varance. Defne E that means the multple relaton between and the dfference between and E. x ndcates the data offset between node and the average of good neghbors. The larger more probable that the node s faulty. Wth the ncrease of x s, the x, m({g}) reduces, on the contrary, m({f}) rses. In Secton 3.1, we have dscussed that one of the defects of tradtonal algorthms s that an ndetermnacy occurs for the = condton n Equaton (4), and thus the node s not reducble to good or faulty. Therefore, we defne the range ( 1 1, 1 1) wthn whch the status of ths node has hgher uncertanty (the probablty of ths node beng fault s moderate) and m( ) N( 1, 1) when x ( 1 1, 1 1). When x 1, m({ }) 1, whch means the uncertanty reaches the top (depcted n the Fgure ).The defntons of m(g), m(f) and m( ) express ths meanng above and provde a good descrpton of the nfluence of changng x on evdence.
Sensors 014, 14 7667 In Equaton (14), when ( 1 1, 1 1), we can see that: Accordng to the Chebyshev nequalty: x P( ) 1/ x 1 1 e (15) p{ ε} σ / ε E (16) Make ε e then 1/ ε e 1/4 σ. The formula expands as follows: E P{ e } e 1/4 1/ (17) Fgure. The BPA functons. Accordng to Equatons (15) and (16), we get. Here, we defne σ 1 = 0.1, 1/4 1 1 e and then μ 1 = 0.68. After all above, m({g}), m({f}) and m( ) can be calculated by Equatons (1) (14), respectvely. In D-S evdence theory, f there are more than two BPAs that need to be combned, then the combnaton rule s defned as follows: N N m( ) B Z 1 B (18) 1 m Z 1 K where K s the mass that s assgned to the empty set Φ, and K m ( B ). But the N B 1 tradtonal Dempster-Shafer evdence has a very obvous dsadvantage when beng used n our algorthm of fault detecton. For example: m1: m(g) = 0.8, m(f) = 0., m( ) = 0, m: m(g) = 0.8, m(f) = 0., m( ) = 0. The fused result s m(g) = 0.94, m(f) = 0.06, m( ) = 0. However, the result extremely negates F n the tradtonal Dempster-Shafer evdence fuson. Oblteratng conflct roughly and runnng normalzaton processes leads to extreme dfferences between G and F. Ths wll cause errors n the judgment of sensors states when usng udfd. That s because the node wll fnd the node j whch * matches the mn mj ({ G} m ({ G})). Too extreme evdence wll nfluence the effect of N 1
Sensors 014, 14 7668 comprehensve evdence. Based on ths, we propose a new evdence fuson rule combned wth nformaton entropy theory. Accordng to conflcts to the entrety presented by nformaton dvergences, we classfy evdences nto several sets. By fusng the results from dfferent sets, ths prevents extreme extenson of dfferences between G and F. By ths evdence fuson algorthm, we can fnally determne the nodes status. In classcal theores of nformaton, Shannon Entropy measures the amount of nformaton, whle the amount of nformaton reflects the uncertanty n random events. Consderng dfferent evdences should be assgned dfferent fuson weght accordng to ts amount of nformaton, so, n ths secton, the theores of entropy and the degree of dsagreement functon whch measures the nformaton dscrepancy are ntroduced nto combnaton rules for evdence conflcts and ncrease the accuracy of fault determnaton for Un nodes. Frstly, we ntroduce some defntons. The nformaton dvergence D( p q ) between dscrete random varables p and q s defned as below [0]: px ( ) D( p q) p( x) log (19) qx ( ) x It s obvous that D( p q) 0 assume that M l ndcates the lth evdence and Ml ( m1 l, ml,, mnl ), where m l s called a focal element. Here, focal elements n each evdence and s ndcates the amount of evdences. D l s defned as follows: n 1 m 1 1 D D M M m l s s n l l ( l j ) l ln s j1 s j1 1 mj 1, m 0,0 n,1 l s, where n s the number of l m (0) It ndcates the degree of dfferences between M l and the whole evdences. It s determned by the average of the nformaton dvergence between M l and each evdence. After ths, defne δ l as the percentage of the whole dfference degree that M l occupes. It s calculated as follows: D / D (1) l l 1 s Accordng to δ l, evdences are gong to be classfed nto several subsets. Evdences whch have smlar δ l are aggregated n the same subset. Before classfcaton, the demarcaton pont s confrmed as below, whch means the average dfferences between δ l : s s l () ss ( 1) l1 l1 Assume that C r s the resultant subset and P s the collecton of δ l. The pseudo code of the classfcaton algorthm s as follows: 1: r = 0; : Whle P s not empty, do 3: Randomly, choose any one element from P and put t nto C r. Remark ths element as C r1 ; 4: Remove C r1 from P; 5: Loop1, for l = 1 to s 6: Loop, for j = 1 to C r ( C r s the cardnalty of C r )
Sensors 014, 14 7669 7: If l Crj, then contnue loop1; 8: End f; 9: End loop; 10: Put δ l nto C r and remove t from P; 11: End loop1; 1: r++; 13: End whle. After classfcaton, the dfference between δ l of each evdence n the same subset s less than, that s, evdences have a smaller extent of conflct n each subset. Assume that there are m subsets and the weght of each subset s defned as CWr Cr / s, where C r s the number of evdences n each subset. Evdences n each subset wll be fused wth weghts to get the aggregatve center of C r, and the weghtng fuson s based on nformaton entropy. Informaton entropy ndcates the amount of nformaton an evdence has. The larger the nformaton entropy s, the less amount of nformaton the evdence has. Defne that the nformaton entropy of evdence Ml ( m1 l, ml,, mnl ) s calculated as follows [1]: n H m ln m (3) l jl jl j1 If Θ s a focal element n M l, then a larger ml ( ) means M l has less amount of nformaton, so the amount of nformaton of M l can be calculated accordng to the followng formula: Hl Vl 1 ml f Hl 1 ml e (4) A smaller weght wll be assgned to an evdence whch has less amount of nformaton, so the weght allocaton of each evdence s as follows: W l C r V l V, 0 1 l f l where V l l (5) 1 Cr, f l, Vl=0 Then we can get the aggregatve center of C r by usng an mproved D-S formula. For any focal n A p A Kq A, where: element A, the result of fuson s ( ) r p A Cr ml ( A ) (6) C r A 1 1 A l C r q A Wlml( A) (7) l1 K (8) Here, p(a) represents the tradtonal way to fuse evdences and q(a) represents the average support degrees from each evdence to A. When K s large enough, the nfluence from q(a) s ncreased. Assume that there are m subsets and n r s the aggregatve center of C r, then the fnal result of fuson s: m k1 k k (9) A m A ( ( CW n ( A))) ( m( A))
Sensors 014, 14 7670 4. Smulaton Analyss 4.1. Smulaton Settng We use the MATLAB smulaton tool to demonstrate our model. As shown n Fgure 3, a square wth a sde length of 100 m s constructed n our model, n whch sensors are deployed and form the network. Ten temperature sources are deployed n the square as the sensng objects of sensors. The dstance between two sources s no less than L. Every temperature source randomly generates temperature data x whch ranges from 5 to 40 C. These readngs smulate the temperature varaton of four seasons, whch means t has regularty and smoothness. Second, n sensors are deployed n ths square and each of them selects the nearest temperature source whch must be n the sensng range. If no temperature source exsts wthn sensng range, a sensor s set to not work, whch means no sensng from a temperature source and no communcaton wth neghbor nodes. Fgure 3. Topology descrpton. Each workng node establshes ts varaton of sensed data accordng to the dstance to ts temperature source, whch can be descrbed by the formulas below: max x d /10 (30) mn x d /10 (31) max and mn are the upper and lower bounds of the data range, respectvely. x s the temperature generated by the temperature sources and t s unformly dstrbuted n ( mn, max ). d s the dstance between a sensor and ts temperature source. In every sensng moment, a sensor chooses a random value between max and mn as ts sensng data. Each sensor node chooses other nodes whch are wthn ts communcaton range (communcaton radus s represented by R) and have the same temperature source as ts neghbor nodes. After ths, each node creates a set of neghbor nodes and the wreless sensor network s formed. Two cases of unformly dstrbuted fault nodes and ntensvely dstrbuted fault nodes are smulated. The frst case s used n comparson when the number of nodes ranges. In the second case, we set squares
Sensors 014, 14 7671 whch are located at a random coordnate as a fault regon. We compare the detecton effects for dfferent scales of ntensve faults by changng the area of a square. Accordng to the sensng data desgnaton, data rangng from mn to max are treated as good, otherwse, data are treated as faulty. Fault data are set to 5 d / r or 5 d / r. Parameters are ntalzed: L = 30 m, r = 0 m, R = 0 m. In our max mn smulaton, each fnal data result s the average of results from 30 repeats. 4.. Smulaton Result Analyss 4..1. Effect of Data loss At each moment of the data collecton, some nodes are chosen to be unable to sense data to smulate a data loss scenaro. The Data Mssng Preprocess Mechansm proposed n ths paper s compared wth the Data Fllng method based on the k-nearest Neghbor algorthm (df-knn) algorthm. The man dea of df-knn s to select k nodes from neghborhood whch have the shortest dstances, wegh the data of the k nodes accordng to these dstances and fnally sum the data as the nterpolaton result. Here, the data loss rate s set to 5%, 10%, 15%, 0%, 5%, 30% and 35%, respectvely. Fgure 4. Effects of data fllng. As shown n Fgure 4, data loss rate s set as the horzontal ordnate, whch means the rato of the number of the data loss nodes to the sum of workng nodes. The mean resdual s set as the vertcal ordnate, whch means the average of dfferences between nterpolaton data and pre-establshed data and t reflects the fnal accuracy of the algorthms. Mean resdual grows as the loss rate grows. When the loss rate s lower, the mean resdual of udfd s 0.1 lower than that of KNN, and acheves an unremarkable mprovement, but as the loss rate grows hgher, the mprovement turns to be hgher. Approxmately, when the loss rate s hgh enough, the mprovement s 0.5, whch means udfd s more sutable n the large-scale data loss stuaton. Wth the growth of data loss rate, the number of neghbors whch have avalable data reduces, whch means less nformaton could be collected and eventually ths makes the nterpolaton results unrelable. In comparson wth df-knn, udfd adequately nvolves the hstorcal data of neghbors to predct and solve the problem of credt reducton due to less avalable data, whch leads to better results.
Sensors 014, 14 767 4... Evdence Fuson In ths paper, nformaton theory-based evdence reasonng s used to fuse collected evdences before the status judgment of nodes. Orgnal D-S evdence reasonng and an mproved one proposed by Qang Ma et al. [] are used for comparson. The mproved D-S s depcted as below. Defne the dstance between evdence m 1 and m : Defne the smlarty of m 1 and m : d m, m 1/ m m ( m m ) (3) 1 1 1 1 1 T S m, m 1 d( m, m ) (33) Defne the basc credt of m : Here N s the sum of evdences. Defne the weght of m j : Amend all evdences: s ( m 1, 1, m ) (34) jn j / max (35) j 1jN ( ) m A m A (36) m (1 ) m (37) A s the establshed focal element and ψ s the uncertan one. Fuse the amended evdences through the orgnal D-S. The algorthm above measures the degree of conflcts among evdences by nvolvng dstance and amends evdences before fuson. The smulaton results are shown n Table 1. Belef functon and plausblty functon are nvolved to estmate the fuson results. BPA-based belef functon n the frame of dscernment Θ s defned as: Bel A mb ( ) BA (38) BPA-based plausblty functon n the frame of dscernment Θ s defned as: PlA mb ( ) B A (39) Belef nterval s defned as [Bel(A), Pl(A)], whch s shown n Fgure 5. The hypothess that A s true s accepted n [0, Bel(A), s uncertan n [Bel(A), Pl(A)] and s refused n Pl(A), 1]. The length of nterval presents the possblty to make a correspondng concluson to ths hypothess. Fgure 5. Belef nterval.
Sensors 014, 14 7673 Table 1. Results of evdence fuson. Evdences Algorthms Evdence Set 1 m 1 :m(g) = 0.1, m(f) = 0., m(ψ) = 0.7 m :m(g) = 0., m(f) = 0., m(ψ) = 0.6 Evdence Set m 1 :m(g) = 0.1, m(f) = 0., m(ψ) = 0.7 m :m(g) = 0., m(f) = 0., m(ψ) = 0.6 m 3 :m(g) = 0.1, m(f) = 0.1, m(ψ) = 0.8 Evdence Set m 1 :m(g) = 0.1, m(f) = 0., m(ψ) = 0.7 m :m(g) = 0., m(f) = 0., m(ψ) = 0.6 m 3 :m(g) = 0.6, m(f) = 0., m(ψ) = 0. m(g) = 0.34, m(g) = 0.703, m(g) = 0.5978, Orgnal D-S m(f) = 0.3191, m(f) = 0.3514, m(f) = 0.849, m(ψ) = 0.4468 m(ψ) = 0.3784 m(ψ) = 0.1173 Improved D-S by Qang Ma m(g) = 0.34, m(f) = 0.3191, m(ψ) = 0.4468 m(g) = 0.561, m(f) = 0.3388, m(ψ) = 0.7 m(g) = 0.881, m(f) = 0.909, m(ψ) = 0.410 m(g) = 0.71, m(g) = 0.1956, m(g) = 0.3384, udfd m(f) = 0.3088, m(f) = 0.7, m(f) = 0.679, m(ψ) = 0.4641 m(ψ) = 0.577 m(ψ) = 0.397 The belef functons and plausblty functons are shown n Table accordng to the fuson results. Table. Belef functons and plausblty functons of fuson results. Orgnal D-S Improved D-S by Qang Ma udfd Evdence Set 1 Evdence Set Evdence Set 3 Bel(G) = 0.34, Pl(G) = 6808 Bel(G) = 0.703, Pl(G) = 0.6486 Bel(G) = 0.5978, Pl(G) = 0.7151 Bel(F) = 0.3191, Pl(F) = 0.7659 Bel(F) = 0.3514, Pl(F) = 0.791 Bel(F) = 0.849, Pl(F) = 0.40 Bel(ψ) = 0.4468, Pl(ψ) = 1 Bel(ψ) = 0.3784, Pl(ψ) = 1 Bel(ψ) = 0.1173, Pl(ψ) = 1 Bel(G) = 0.34, Pl(G) = 6808 Bel(G) = 0.561, Pl(G) = 0.661 Bel(G) = 0.881, Pl(G) = 0.7091 Bel(F) = 0.3191, Pl(F) = 0.7659 Bel(F) = 0.3388, Pl(F) = 0.7439 Bel(F) = 0.909, Pl(F) = 0.7119 Bel(ψ) = 0.4468, Pl(ψ) = 1 Bel(ψ) = 0.4051, Pl(ψ) = 1 Bel(ψ) = 0.410, Pl(ψ) = 1 Bel(G) = 0.71, Pl(G) = 0.691 Bel(G) = 0.1956, Pl(G) = 0.778 Bel(G) = 0.3348, Pl(G) = 0.731 Bel(F) = 0.3088, Pl(F) = 0.779 Bel(F) = 0.7, Pl(F) = 0.8044 Bel(F) = 0.679, Pl(F) = 0.665 Bel(ψ) = 0.4641, Pl(ψ) = 1 Bel(ψ) = 0.3784, Pl(ψ) = 1 Bel(ψ) = 0.397, Pl(ψ) = 1 The results of these algorthms are smlar to each other when two evdences wth low degree of conflct n evdence set 1 are to be fused, among whch the results of the orgnal D-S and the mproved D-S proposed by Qang Ma [] are the same. A smlar evdence s added to set 1 to form set. Through the analyss of belef functons and plausblty functons of three algorthms, the possblty of orgnal D-S to accept that G s true s 0.703 and the possblty to refuse s 0.3514. The possblty of mproved D-S to accept that G s true s 0.561 and the possblty to refuse s 0.3388. The possblty of udfd to accept that G s true s 0.1956 and the possblty to refuse s 0.7. Accordng to evdence set, the possbltes of the prevous two algorthms to accept and refuse that G s true s too hgh to reflect the actual stuaton (the possbltes to accept and refuse are both lower than 0.) of each evdence. However, the algorthm proposed n ths paper s closer. In udfd, the possbltes to accept and refuse are not rased by reducng the uncertanty, whch makes the fuson result more credble. A completely dfferent evdence s added to set 1 to form set 3 wth hgh degree of conflct. It s obvous that the possblty (0.5978) of orgnal D-S to accept that G s true s so hgh that approaches the one of the last evdence whch s added n set 3 and the possblty (0.881) of mproved D-S s too low to reflect the
Sensors 014, 14 7674 affect caused by hgh degree of conflct. The result of udfd s between the ones of the prevous two algorthms, whch balances the nfluences of all evdences and s more credble. 4..3. Detecton Accuracy DFD, IDFD and udfd are compared based on the constructed wreless sensor network model. Measures to be nvolved are detecton accuracy (the rato of number of correctly detected nodes to the sum of workng nodes), false alarm rate (the rato of number of nodes whch are msjudged from good to false to the sum of workng nodes), mssng alarm rate (the rato of number of nodes whch are msjudged from false to good to the sum of workng nodes). In ths smulaton, each fnal data pont s the average of results from 30 repeats. Frst, we analyze the effects of these three algorthms wth the changng fault rate when nodes are randomly dstrbuted unformly. Consderng that dfferent nfluences are caused by dfferent dstrbuton denstes, cases wth 40, 80 and 10 workng nodes are smulated. Fgure 6 shows the detecton effects of the three algorthms wth 40 workng nodes. In ths case, nodes are dstrbuted sparsely n the smulaton regon. It can be seen from the fgure that wth the ncreasng fault rate, detecton accuracy shows an approxmate lnear downward trend; however, false alarm rate and mssng alarm rate show the opposte trend. Through further calculaton, when the fault rate ranges from 5% to 50%, average detecton accuracy of udfd s 9.33% ponts hgher than that of DFD and 6.5% ponts hgher than that of IDFD; average fault alarm rate of udfd s 6.93% ponts lower than that of DFD and 5.33% ponts lower than that of IDFD; average mssng alarm rate of udfd s.49% ponts lower than that of DFD and 1.0% ponts lower than that of IDFD. The udfd brngs better detecton accuracy under sparse dstrbuton condtons. Fgure 6. Detecton effects under 40 workng node condtons.
Sensors 014, 14 7675 Fgure 7 shows the detecton effects of the three algorthms wth 80 workng nodes. In ths case, nodes are dstrbuted moderately densely n the smulaton regon. As s shown by the fgure, wth the ncreasng fault rate, detecton accuracy shows an approxmately lnear downward trend; however, false alarm rate and mssng alarm rate show the opposte trend. Through further calculaton, when the fault rate ranges from 5% to 50%, average detecton accuracy of udfd s 7.31% ponts hgher than that of DFD and 5.44% ponts hgher than that of IDFD; average fault alarm rate of udfd s 4.1% ponts lower than that of DFD and 3.4% ponts lower than that of IDFD; average mssng alarm rate of udfd s 3.% ponts lower than that of DFD and.04% ponts lower than that of IDFD. The udfd provdes better detecton accuracy n ths condton. Meanwhle, as the fault rate grows, the superorty of the detecton effect of udfd contnues to ncrease. When the fault rate s 50%, the detecton accuracy of udfd s 10.08% ponts hgher than that of DFD and 7.68% ponts hgher than that of IDFD, whch ndcates that udfd adapts better to hgh fault rate condtons. Fgure 7. Detecton effects under 80 workng node condtons. Fgure 8 shows the detecton effects of the three algorthms wth 10 workng nodes. In ths case, nodes are dstrbuted densely n the smulaton regon. It can be seen from the fgure that wth the ncreasng fault rate, detecton accuracy shows an approxmately lnear downward trend; however, false alarm rate and mssng alarm rate show the opposte trend. Through further calculaton, when the fault rate ranges from 5% to 50%, the average detecton accuracy of udfd s 3.75% ponts hgher than that of DFD and 1.74% ponts hgher than that of IDFD; average fault alarm rate of udfd s 1.69% ponts lower than that of DFD and 1.14% ponts lower than that of IDFD; average mssng alarm rate of udfd s.09% ponts lower than that of DFD and 0.6% ponts lower than that of IDFD. The udfd provdes better detecton accuracy under dense dstrbuton condtons. Furthermore, as the fault rate grows, the superorty of the detecton effect of udfd ncreases contnually, whch ndcates that udfd performs better under hgh fault rate condtons.
Sensors 014, 14 7676 Fgure 8. Detecton effects under 10 workng node condtons. Through comprehensve analyss of Fgures 6 8, we can see that detecton accuracy of these algorthms ncreases wth the ncreasng dstrbuton densty of nodes. Ths s due to the ncrease of avalable nformaton when judgng resultng from the growng number of neghbor nodes. By comparson, the advantage of detecton accuracy of udfd ncreased as the dstrbuton densty of nodes decreases, whch ndcates that the detecton accuracy of udfd mproves more than that of DFD and IDFD. Thanks to evdence fuson based on the status of neghbor nodes before judgment, udfd works better under small number of neghbor node condtons. Second, we analyze the detecton accuracy of the three algorthms when fault nodes are ntensvely dstrbuted. The ntensve dstrbuton scheme nvolves settng squares located at random coordnates wth length of 0, 5, 30, 35 and 40 m as the fault regons. In the fault regon, all nodes are set to fault. By changng the sze of the ault regon, we can observe the detecton accuracy for dfferent scales of faulty nodes. Here, the number of workng nodes s 80. Fgure 9 shows detecton effects of the three algorthms wth dfferent fault regon szes. When the fault rate ranges from 5% to 50%, the average detecton accuracy of udfd s.08% ponts hgher than that of DFD and 1.55% ponts hgher than that of IDFD; average fault alarm rate of udfd s 0.84% ponts lower than that of DFD and 0.45% ponts lower than that of IDFD; average mssng alarm rate of udfd s 1.4% ponts lower than that of DFD and 1.09% ponts lower than that of IDFD. It s easy to conclude that udfd can acheve better detecton effects when ntensve faults occur. The udfd takes n more nformaton from the neghborhood to judge when dealng wth ntensve faults stuatons, whch reduces the nfluence from mutual cheatng among faulty nodes. In udfd, θ 1 s the threshold of data from dfferent nodes on same moment, θ s the threshold of data from the same node on dfferent moments and θ 3 s the threshold of data of the same node collected on
Sensors 014, 14 7677 the same moment of dfferent days. In our model, a value of θ rangng wthn the nterval (0, ) has lttle nfluence on detecton effect, so t s set to a fxed value of 1. Based on those values, the best combnaton of θ 1 and θ 3 s gong to be explored. The selecton of θ 1 and θ 3 s drectly related to the judgment results, thus, to explore the best combnaton of θ 1 and θ 3 becomes the key to explore the best detecton effect of udfd. Fgure 10 shows a 3D map of detecton accuracy wth dfferent combnatons of θ 1 and θ 3. Here, the number of workng nodes s 80. It can be seen from the curved surface that the combnaton of θ 1 = 5, θ 3 = 5.4 approaches the peak, whch means the maxmum detecton accuracy s 0.98. Fgure 9. Detecton effects under ntensve fault condtons. Fgure 10. Detecton accuracy wth dfferent combnatons of θ 1 and θ 3.
Sensors 014, 14 7678 4..4. Communcaton Energy Consumpton In our model, communcaton between nodes s smulated by usng the ZgBee protocol. ZgBee, a personal area network protocol based on IEEE80.15.4, supports short-dstance, low-complexty, self-organzng, low-power, hgh-speed and low-cost wreless communcaton technology and apples well n WSNs. The bref frame structure of ZgBee s shown n Fgure 11. The frame head s constructed by the bts from the applcaton layer, network layer, MAC layer and physcal layer. In our model, messages transmtted between nodes nclude prejudged statuses, evdences and fnal judged statuses. The nformaton above can be encapsulated n the payload feld of the applcaton layer frame by analyzng the frame structure of ZgBee. When statuses are to be transmtted, the payload s only 4 bts ( bts present message type and bts present status, namely LG/G) and the total length of a frame s 47 bts (the header length s 43 bts). Fgure 11. Bref frame structure of ZgBee. Rado energy dsspaton model s calculated as ndcated below:,, E l d E l E l d Tx Txelec Txamp leelec l fsd,? d d0 4 leelec l mpd,? d d0 Rx Rxelec elec (40) E l E l le (41) ETx l, d s the transmsson energy, whch ncludes electronc energy ETx elec l energy E l, d and amplfer Tx amp. l s the length of a message. E elec s determned by dgtal codng, modulaton and flterng s fxed as 50 nj/bt. = 10 fs pj/bt/m and = 0.0013 mp pj/bt/m4. d s the transmsson dstance. d 0 s the threshold of d and s set to 0 m. If d s larger than d 0, energy dsspaton s manly caused by free space power loss ( d ). Otherwse, energy dsspaton s manly caused by multpath power loss 4 ( d ). Frst, we analyze the number of messages transmtted n the smulated network under 40 workng node condtons, whch s depcted n Fgure 1. When transmttng messages, nodes exchange statuses by rado broadcastng. An accumulated number of messages s recorded n the process of 30 tests. Wth the growth of test rounds, the number of messages shows an approxmately lnear upward trend. Through further calculaton, approxmate slopes of DFD, IDFD and udfd are 66.7, 180.0 and 103.3, respectvely. Obvously, the rate of ncrease of udfd s the lowest, whch means a mnmum of messages transmtted durng the fault detecton. Ths s because the nodes n DFD and IDFD have to exchange all prejudged states and fnal judged statuses whether they are good or faulty, whch leads to more nteractons, whle udfd only exchanges message f the tendency status s LG or fnal status s good.
Sensors 014, 14 7679 The comparson of average communcaton energy consumpton of each node after 30 tests s shown n Fgure 13. Average energy consumptons of all nodes n DFD, IDFD and udfd are 0.35, 0.176 and 0.08 mj. By countng and comparng the energy consumpton of each node specfcally, we fnd that udfd has the best energy savng performance, whch s caused by the reducton of nteractons. Fgure 1. Number of messages accumulated durng 30 tests under 40 workng node condtons. Fgure 13. Energy consumpton of each node under 40 workng node condtons. Fgures 14 16 show the energy consumpton of each detecton under condtons of 40, 80 and 10 workng nodes. Through further calculaton, under 40 workng node condtons, the average energy consumpton of all detectons n udfd s.87 mj lower than that of IDFD and 5.47 mj lower than that of DFD. Under 80 workng node condtons, the average energy consumpton of all detectons n udfd s 4.77 mj lower than that of IDFD and 10.6 mj lower than that of DFD.