Sensors & Transducers 2015 by IFSA Publishing, S. L.

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1 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp Sensors & Transducers 05 by IFSA Publshng, S. L. SGR: A New Effcent Kernel for Outler Detecton n Sensor Data Mnmzng Mse Seema SHARMA, Dr. C.P. GUPTA, Roht JAIN Department of Computer Scence and Engneerng Rajasthan Techncal Unversty, Kota-300, Inda E-mal: seema_rtu@redffmal.com, guptacp@redffmal.com, rohtjan0@gmal.com Receved: June 05 /Accepted: 6 May 05 /Publshed: 30 June 05 Abstract: In sensor network, collected data s error prone due to errors durng sensors and transmsson. Sometmes, the sensed data may appear to be erroneous due to large devaton from normal data dstrbuton. Such data ponts termed as outlers may contan some mportant pattern. Outlers, f neglected as erroneous data, may result n falure to detect mportant phenomenon. Hence, t s necessary to not only detect such data ponts but analyze them further to establsh the reason behnd such data values. The presence of outlers may dstort contaned nformaton. To ensure that the nformaton s correctly extracted, t s necessary to dentfy the outlers and solate them durng knowledge extracton phase. In ths paper, we propose a novel unsupervsed algorthm for detectng outlers based on densty by couplng two prncples: frst, kernel densty estmaton and second assgnng an outler score to each object. A new kernel functon buldng a smoother verson of densty estmate s proposed. An outler score s assgned to each object by comparng local densty estmate of each object to ts neghbors. The two steps provde a framework for outler detecton that can be easly appled to dscover new or unusual types of outlers. Experments performed on synthetc and real datasets suggest that the proposed approach can detect outlers precsely and acheve hgh recall rates whch n turn demonstrate the potency of the proposed approach. Copyrght 05 IFSA Publshng, S. L. Keywords: Kernel, Kernel densty estmaton, Mean ntegrated squared error, Outler detecton.. Introducton Tremendous growths n wreless sensor network technology have enabled the decentralzed processng of enormous data generated n network, scentfc and envronmental sensng applcatons at low communcaton and computatonal cost. Generated sensor data may pertan to physcal phenomenon (lke temperature, humdty, and ambent lght), network traffc, spatotemporal data about weather pattern, clmate change or land cover pattern etc. Avalablty of vast amount of sensor data and mmnent need for transformng such data nto true knowledge or nto useful nformaton requre contnuous montorng and analyss as they are hghly senstve to varous error sources. True knowledge provdes useful applcaton-specfc nsght and gves access to nterestng patterns n data; the dscovered pattern can be used for applcatons such as fraud detecton, ntruson detecton, earth scence etc. Sudden changes n the underlyng pattern may represent rare events of nterest or may be because of errors n the data. Outler detecton refers to detectng such abnormal patterns n the data. Several defntons have been proposed, but none of them s unversally accepted because, the measures and defnton of outlers vary wdely. Barnett, et al. [] defned outlers as an observaton or subset of observatons whch appears to be nconsstent wth the remander of that set of data. 97

2 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp Outlers may arse due to fraudulent behavor, human error, malfunctonng or njecton n sensng devces, faults n computng system and uncontrolled envronment. Outler shows devaton from normal behavor. Declaraton of outler based on observed devaton n the values s a subjectve judgement and may vary dependng upon applcaton. Several approaches for detectng outlers have been proposed [-]. Technques for outler detecton can be classfed as ether statstcal approach [5], dstance based approaches, densty based approaches, proflng methods, or model based approaches. In statstcal approach, data ponts are frst modeled usng stochastc dstrbuton, and then are labeled as outlers based on ther ftness wth the dstrbuton model. An outler score s assgned to each object based on ther nearest neghbor dstances by dstance based outler detecton technque. In densty based approach, an outler score s computed by comparng the local densty estmate of each object to the local densty estmate of ts neghbors, and the objects are flagged as outlers based on ther outler score. In proflng methods, dfferent technques of data mnng are used to buld profles of normal behavor, and devatons from these underlyng profles are flagged as outlers. In model-based approaches, frst, by usng some predctve models, the normal behavor s characterzed, and then the devatons from the normal behavor are flagged as outlers. In ths paper, we propose an outler detecton algorthm combnng statstcal and the densty based approaches. Our proposed approach uses kernel densty estmators to approxmate the data dstrbuton and then computes the local densty estmate of each data pont, and thus detects potental outlers. Experments performed on both synthetc and real data sets shows that the proposed approach can detect outlers precsely and acheve hgh recall rates, whch n turn demonstrate the potency of the proposed approach. Rest of the paper s organzed as follows: Secton II descrbe the lterature revew of the work. Secton III explans the kernel densty estmators. Secton IV presents the proposed work. Secton V provdes the dscusson on results. Secton VI concludes the work.. Related Work Several Non-parametrc estmators were presented n [6]. Hstogram was the smplest nonparametrc estmator used for densty estmaton but generated densty estmates were hghly dependent on the startng pont of bns. Sheng, et al. [7] ntroduced Outler Detecton n Wreless Sensor Network based on hstogram method to detect dstance based outlers. In the proposed method, to flter out unnecessary observatons correspond to potental outler the collected hnts about data dstrbuton are modeled as hstogram. The problem assocated wth unsupervsed outler detecton n WSN was addressed by Branch, et al. [8]. The proposed algorthm for outler detecton was generc evdenced by ts sutablty to varous outler detecton heurstcs and t does not requre, for a data source, any pror assumpton about global model. Kernel Densty Estmators [9-0] were used as an alternatve to hstograms. KDE were superor n terms of accuracy and hence, had attracted a great deal of attenton. A smoother verson of densty profle was constructed by kernel densty estmator. Kernel densty estmaton was coupled wth the varous outler detecton methods n order to buld a framework for detectng densty based outlers and the resulted qualty of densty based outler detecton was mproved. Outler score s dependent on the choce of approach used to detect outlers. Breung, et al. [] ntroduced Local Outler Factor (LOF) for detectng outlers n a multdmensonal dataset. In the proposed scheme, local densty estmate of each object were compared wth average densty estmate for MnPts-nearest neghbors. The resulted densty rato was referred as local outler factor. Local outler factor was computed n order to determne the physcal locaton of each object n feature space. An object led deep nsde a cluster when ts local outler factor was approxmately whereas an object that got hgher value of local outler factor corresponds to low neghborhood densty. An object wth hgher local outler factor was flagged as an outler. The proposed method was free from local densty problem but dependent on the choce of MnPts. Local Outler Correlaton Integral (LOCI) [] was based on the concept of mult-granularty devaton factor (MDEF) and dealt wth both local densty and mult-granularty successfully. The scheme had lower senstvty to chosen parameters. The proposed scheme strctly reled on counts and needed to test arbtrary rad. An automatc, data-dedcated cut-off was provded to determne whether a pont s an outler. In the proposed scheme, MDEF was computed for each data pont n feature space. A data pont wth MDEF of 0 sgnfed that t got neghborhood densty equal to average local neghborhood densty whereas a data pont wth large MDEF was flagged as an outler. Palpanas, et al. [3] propose a kernel-based technque for onlne dentfcaton of outlers n streamng sensor data. Ths technque requres not a pror known data dstrbuton and uses kernel densty estmator to approxmate the underlyng dstrbuton of sensor data. Thus, each node can locally dentfy outlers f the values devate sgnfcantly from the model of approxmated data dstrbuton. A value s consdered as an outler f the number of values beng n ts neghborhood s less than a user-specfed threshold. 98

3 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp Ths technque can also be extended to hgh-level nodes for dentfcaton of outler n a more global perspectve. The man problem of ths technque s ts hgh dependency on the defned threshold, whle choce of an approprate threshold s qute dffcult and a sngle threshold may also not be sutable for outler detecton n mult-dmensonal data. Furthermore, the technque does not consder mantanng the model whle sensor data s frequently updated. Several approaches are used for detectng outlers usng kernel densty estmaton. In all these schemes, the densty estmate s constructed wth prevously avalable kernel functons [6]. Most of these works consdered performance measure for outler detecton whle gnorng accuracy of densty estmate. Subramanam, et al. [] further extend the work of Palpanas, et al. [3] and solve the two prevous problems of nsuffcency of a sngle threshold for mult-dmensonal data and mantanng the data model bult by kernel densty estmator. They propose two global outler detecton technques for complex applcatons. One technque allows each node to locally dentfy outlers usng the same technque as Palpanas, et al. [3] and then transmt the outlers to ts correspondng parent to be checked untl the snk eventually determnes all global outlers. In the other technque, each node employs more robust technque called LOCI [] to locally detect global outlers by havng a copy of global estmator model obtaned from the snk. Expermental results show that these technques acheve hgh accuracy n terms of estmatng data dstrbuton and hgh detecton rate whle consumng low memory usage and message transmsson. Problem wth ths technque s ts nablty to detect spatal outlers due to the fact that t does not consder the spatal correlatons among neghborng sensor data. Bettencourt, et al. [5] presented a local outler detecton technque to dentfy errors and detect events n ecologcal applcatons of WSNs. Ths technque can dstngush between erroneous measurements and events by usng the spatotemporal correlatons of sensor data. Each node learns the statstcal dstrbuton of dfference between ts own measurements and each of ts neghborng nodes, as well as between ts current and prevous measurements. A measurement s dentfed as anomalous f ts value n the statstcal sgnfcance test s less than a user-specfed threshold. The scheme reles on the choce of the approprate values of the threshold whch s a major weakness. A varant of LOF was proposed by Lateck, et al. [6] combnng the LOF and kernel densty estmaton n order to utlze the strength of both n densty based outler detecton, whch was referred as outler detecton wth kernel densty functon. In ths approach, frst, a robust local densty estmate was generated wth kernel densty estmator and then by comparng the local densty estmate of each data pont to the local densty estmate of all of ts neghbors, the outlers were detected. Local densty factor (LDF) s computed for each data pont n feature space and the data ponts wth hgher LDF values were flagged as outlers. Kregel, et al. [7] proposed a method for local densty based outler detecton referred as Local Outler Probablty (LoOP) whch was more robust to the choce of MnPts. The proposed method combned the local densty based outler scorng wth probablty and statstcs based methods. An outler probablty n the range of [0, ] was assgned to each data pont as outler score sgnfyng severty of outlerness. More specfcally, hgher the outler score meant more severe a pont to be declared as outler. Most of the densty based outler detecton methods were bounded to detect specfc type of outlers. Schubert, et al. [8] proposed a general framework for densty based outler detecton referred as KDEOS and could be adjusted to detect any specfed types of densty-based outlers. In KDEOS, the densty estmaton and the outler detecton steps were decoupled n order to mantan the strength of both. Quantfcaton of outlerness was frst ntroduced by Breung, et al. []. In the proposed method, user was asked to specfy the value of parameter, MnPts correspond to neghborhood sze of an object x explctly, used for estmatng local densty. Local Outler Factor (LOF) was mplemented by usng three steps descrbed below: Step : Neghborhood Constructon - To construct neghborhood, frst, d border of an object x correspondng to neghborhood border dstance was defned and computed usng:,,, d x MnPts dst x NN x MnPts, () border th NN x MnPts refers to MnPts nearest neghbor of x. Fg. shows the neghborhood of an object x contanng sx objects.e. where,,6,,,,, N x x x x x x x. a b c d e f Neghborhood, comprsed of all objects N x, MnPts N x MnPts of object x x j was constructed usng: x D \ x dst x, x d x, MnPts j tran j border Step : Neghborhood Densty Estmaton - For estmatng neghborhood densty the concept of reachablty dstance was ntroduced, computed usng: 99

4 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp ,, max,,, d x x MnPts d x MnPts dst x x reach j border j j Neghborhood densty of an object x was dependent on two mportant parameters, vz., d and computed usng: N x MnPts, reach the densty estmate quantfed by Mean Integrated Squared Error (MISE), and then ncorporate these densty estmates n the computaton of outler score n order to mprove the effcency of outler detecton method. x, MnPts x Nx, MnPts j N x, MnPts d x, x, MnPts reach j 3. Kernel Densty Estmator Kernel densty estmators belong to nonparametrc [6] class of densty estmators. The nonparametrc estmators ncorporate all data ponts to reach an estmate. Kernel estmator smooths contrbuton of each data pont n densty estmates. Kernel densty estmators place a kernel K on each data pont x n the sample. Let, x, x,,x n be the sample of sze n and dmensonalty dm whch are dentcally and ndependently dstrbuted accordng to some unknown densty f ( x ). Expected densty estmate ˆf x s the convoluton of true unknown densty f ( x) wth kernel K s computed as follows: x x dm, () n fˆ x K n h x h x Fg.. Neghborhood constructon of an object x. Step 3: Neghborhood Densty Comparson - In ths step, local densty estmate of each object were compared wth average densty estmate for MnPtsnearest neghbors. The resulted densty rato was referred as local outler factor (LOF) and computed usng: LOF x, MnPts x Nx, MnPts j x, MnPts x, MnPts. N x, MnPts Local outler factor was computed n order to determne the physcal locaton of each object n feature space. An object led deep nsde a cluster when ts local outler factor was approxmately whereas an object that got hgher value of local outler factor corresponds to low neghborhood densty. An object wth hgher local outler factor was flagged as an outler. It was free from local densty problem but dependent on the choce of MnPts. In ths paper, we propose a kernel functon named SGR that can further mprove the accuracy of j where K s the non-negatve, real-valued kernel functon of order- p (degree of polynomal), and hx ( ) s the bandwdth appled at each data pont x. A unvarate kernel functon K of order- must satsfy the requrements of ) Unt area under the curve; ) Symmetry; 3) Zero odd moments; ) Fnte even moments. Qualty of estmated densty s determned by the choce of both the smoothng parameter and the kernel. A kernel may exhbt ether fnte or nfnte support. A kernel wth fnte support s consdered as optmal. There are varous ways to quantfy the accuracy of a densty estmator. We wll focus here on the mean squared error (MSE) and ts two components, namely bas and standard error (or varance). We note that the MSE of ˆf x s a functon of the argument x: ˆ ˆ MSE f x E f x f x ˆ ˆ ˆ ˆ ˆ ˆ ˆ E f x E f x E f x f x E f x E f x E f x E f x ˆ ˆ ˆ ˆ ˆ E f x f x E E f x f x E f x E f x E f x f x 00

5 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp By usng the defnton of bas and varance, var MSE fˆ x fˆ x Bas fˆ x h K f '' x RK f x, nh where K u Kudu s the second moment and RK K udu roughness respectvely of K u. Above equaton suggests that Mean Squared Error (MSE) s the sum over varance and squared bas. Mean Integrated Squared Error (MISE) s the global measure of accuracy of ˆf x. It s computed by ntegratng MSE wth respect to x Plug-n Bandwdth h plug By puttng RK K /5 n n, Snce, f E f x natural estmator for s, ˆ n Equaton (), n r g fˆ x n r r r r!! () MISE fˆ x MSE fˆ x. dx h K f '' x RK f x dx nh h K f '' xdx R K f xdx nh MISE f x h K f R K nh ˆ, where '' f f x dx. 3.. Bandwdth Selecton 3... Optmal Bandwdth Optmal bandwdth h opt s computed by mnmzng MISE.e. dfferentatng MISE wth respect to h and settng t equal to zero. d 3 MISE f ˆ h 0 K f R K dh nh h 0 RK n K f RK K h h opt n f 5 nh K f R K 5 nh K f R K 5 /5 (3) We note that h opt depends on the sample sze, n, and the kernel, K. However, t also depends on the unknown pdf, f, through the functonal β(f ). Thus as t stands Expresson () s not applcable n practce. However, the plug n estmator of h opt, dscussed next, s smply Expresson () wth β(f) replaced by an estmator. 3.. Optmal MISE Substtutng Equaton () for h n Equaton (), 5 MISE fˆ K R K f /5 n opt In other words, /5 (5) MISE K R K. /5 ( ) ( ), where R(K) K (u) du s the roughness, (K) u K(u) du s second moment of K(u) and f '' s second dervatve of f.. Proposed Work.. Kernel Functon We propose a kernel functon SGR of order- as follows: K 5 SGR, h u for u 5/ (6).37 It satsfes all the requrements of beng a kernel functon whch are descrbed below: ) Area under Curve 5/ 5 uk u du u u du 5/.37 5/ 5 u u / ) Symmetry K u K u 0

6 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp ) Odd Moment 5/. 5 K u du u 5/ / u 5u ) Even Vovent du 5/ 5 u K u du 5) Roughness u u du.37 5/ 5/ u u / / K u du 5 u du.37 5/ 5/ u u u Outler Detecton Outler detecton usng kernel densty estmaton nvolves two prncpled and clear steps, whch are descrbed as follows: Step : Densty Estmaton - In ths step, the densty estmate s constructed wth a nonparametrc estmator whch s superor n terms of accuracy s the kernel. The kernel functon s taken as an nput parameter to the algorthm. We wll use our proposed kernel functon of bandwdth h and dmensonalty d for densty estmaton: K proposed, h u 5 d.37 h h The balloon estmator [8] s: KDE balloon, h ( o) K h( o) ( o p). n p (7) (8) In our approach, we wll use balloon estmator for constructng the densty estmates because t optmzes MISE pontwse [0]. The smoothng parameter appled to the data controls the smoothness of the constructed densty estmate. Algorthm DScore: = -dmensonal matrx of densty estmates, o kmax avelof : = Average value of local outler factor // Kernel Densty Estmaton for each object o Compute: NN : k NearestNeghbors max max for each k n k... k do mn max Compute kernel bandwdth h: mean d p, o p k NearestNeghbor o If h : 0, then: h: compute usng plug-n bandwdth estmator. Else h: h End of If Structure NN ; k max do Compute Eucldean dstance : d o, p for each neghbor p n u end end end :, Dscore o k Dscore o k K u h // Densty Comparson:. For each object o, compute LOF value for each value of k n kmn... k max. Compute avelof for each object o, by takng average of LOF values over the specfed range A nearest-neghbor dstance [] s a classc approach to calculate local kernel bandwdth. Sheather and Jones [] proposed a data-drven procedure for selectng the kernel bandwdths known as plug-n bandwdth estmator. To prevent from dvson by 0 we use ho ( ) mn{ mean p knn d( po, ), }. Selecton of parameter k s non-trval. In our proposed scheme, nstead of choosng a sngle value of k a range of k k mn... kmax s employed that produces a seres of densty estmate, one for each k. The proposed scheme s elegant, computatonally effcent, and produces stable and relable results. Step : Densty Comparson - In ths step, the local densty estmate of each object o s compared to the local densty estmate of all of ts nearest neghbors p knn ( o). Let, be the local densty estmate and Nok (, ) be the number of objects n the k - neghborhood of object o. The resulted densty rato 0

7 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp whch s referred as Local Outler Factor (LOF) s computed usng: LOF ( o) ( p) pknn( o) ( o) N( o, k) (9) decomposton (vz. bas and varance) we reled on overall error that takes nto account both the sources of error.e. error due to bas and varance [6]. The optmal pont drew n Fg. refers to the optmal bandwdth h opt at whch the overall error s mnmzed. An LOF value correspondng to each value of k k... k s computed and then mean of LOFs n mn max s taken over the range n order to produce more stablzed LOF value for each object o. Z score s utlzed to standardze the outler scores and objects wth Z score 3 are declared outlers. 5. Results 5.. Kernel Densty Estmator Table shows statstcs and comparson of our proposed kernel aganst varous prevously avalable kernel functons. Our proposed kernel functon has mnmum MISE. The value of effcency s relatve effcency computed usng ˆ ˆ MISE f usng K MISE f usng K. opt proposed opt For example, the effcency of Gaussan kernel s approxmately 96 %. That s the MISEopt f ˆ obtaned usng proposed kernel functon wth n 96 s approxmately equal to the MISEopt f ˆ obtaned usng a Gaussan kernel functon wth n 00. Fg.. Analyss of statstcal propertes (Roht05). Table. Statstcs and Comparson of Varous Kernel Functons R K Kernels K MISE Effcency SGR Roht 05 [] Epanechnkovv Bweght Trangular Fg. 3. Analyss of statstcal propertes SGR. Fg. reflects the bas effects for a kernel densty estmate. Gaussan Box Fg. and Fg. 3 show the varaton of statstcal propertes of Roht05 and the proposed KDE-SGR wth bandwdth. Bas and varance, the two subcomponents of predcton errors are unable to gve approprate understandng about predcton model behavor as there s always a tradeoff between bas and varance. So, nstead of relyng on specfc Fg.. Vsualzaton of bas effects. 03

8 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp Kernel Densty Estmaton and Outler Detecton Datasets: To evaluate the proposed kernel and outler detecton method, experments were carred out on real as well as synthetc datasets. In our experments, we used two real datasets. The frst dataset contans the data about eruptons of old fathful geyser taken from Wesberg (980) and the second datasets contans the data about temperature, humdty, lght, and voltage collected between February 8 th and Aprl 5 th, 00 from 5 McaDot motes deployed n the Intel Berkeley Research Lab [3]. Evaluaton of Outler Detecton Technque: We have appled the kernel densty estmaton step to approxmate the densty at varous kernel ponts. Fg. 5 and Fg. 6 show the densty estmate constructed from the observatons of eruptons of old fathful geyser and Intel lab data. datasets and have evaluated the mpact of k values on LOF. Fg. 6 shows the mpact of k value on Local Outler Factor (LOF). It demonstrates a smple scenaro where the data objects belong to a Gaussan cluster.e. all the data objects wthn a cluster follows a Gaussan dstrbuton. For each k value rangng from 3 to 00, the mean, mnmum and maxmum LOF values are drawn. It can be observed that, wth ncreasng k value, the LOF nether ncreases nor decreases monotoncally. For example, as shown n Fg. 5, the maxmum LOF value s fluctuatng as k value ncreases contnuously and eventually stablzes to some value showng that a sngle value of k s neffcent to produce a more accurate LOF value. So, mean of LOFs s taken over the range of k k mn... k max n order to produce more stablzed LOF values. These are shown n Fg. 7. Fg. 5. Densty estmate constructed from old fathful geyser data h=0.5. Fg. 7. Fluctuaton of outler factors wthn a Gaussan cluster. In Fg. 8, we present the effect of number of neghbour on the value of LOF. It s clear that value of LOF s dependent upon the number of neghbour. Ths establshes that sngle value of MnPts s napproprate n dentfyng outlers. The fgure also shows comparatve performance of three kernel functons. It s clear that our new SGR s superor n terms of assgnng outlerness scores n comparson to the earler kernels. Fg. 6. Densty estmate constructed from Intel Lab data h=0.5. These densty estmates are ncorporated n outler detecton method to calculate Local Outler Factor (LOF) of each data pont present n the partcular dataset. Computed LOF values wll expose the ndces of potental outlers. We have also appled both of these steps to the synthetc Fg. 8. Varaton of LOF wth dfferent kernels. 0

9 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp Fg. 9 presents the temperature data contanng 30 ponts. In Fg. 0, the values of Z-score are presented usng hstogram. Total 9 values of Z-score have absolute value greater than or equal to 3. The scatter dagram also shows these ponts, whch are clearly dstngushable from rest of the data ponts. Ths proves that our scheme dentfes the outlers correctly. 6. Conclusons Fg. 9. Data plot Intel Lab Data. Fg. 0. Z-Score for Intel Lab Data. In ths paper, a new symmetrc and computatonally effcent kernel named SGR of order- wth lower MISE than the prevously avalable kernels has been proposed. The smulaton results on the datasets produced a more accurate densty estmate. An outler detecton method s also proposed. The proposed algorthm uses our proposed kernel functon to construct densty estmates. The detecton method has two completely decoupled steps. In the frst step, densty estmates are computed usng SGR. The detecton s carred out based on local densty estmates n the second step. Ths preserves the strength of both the methods. Fnally, a pont s declared an outler based on the Z-score. Ths makes the frame adjustable to any applcaton-specfc envronment. Experments performed on both real and synthetc datasets ndcate that the proposed technques can detect outlers effcently. Comparson wth the known kernels suggests that SGR s superor n assgnng outler scores. In future, we propose to estmate the sutablty of SGR for mult-dmensonal and multvarate datasets. References []. Barnett V., Lews T., Outlers n statstcal data, Wley, New York, Vol. 3, 99. []. Chandola V., Banerjee A., Kumar V., Anomaly detecton: A survey, ACM Computng Surveys (CSUR), Vol., No. 3, Artcle 5, 009, pp [3]. Hodge V. J., Austn J., A survey of outler detecton methodologes, Artfcal Intellgence Revew, Vol., No., 00, pp []. Gupta M., Gao J., Aggarwal C. C., Han J., Outler detecton for temporal data: A survey, IEEE Transacton on Knowledge and Data Engneerng, Vol. 5, No., 03. [5]. Knorr E. M., Raymond T. Ng., A Unfed Noton of Outlers: Propertes and Computaton, n Proceedngs of the Internatonal Conference on Knowledge Dscovery and Data Mnng (KDD), [6]. Zucchn W., Berzel A., Nenadc O., Appled smoothng technques, Lecture Notes, 005. [7]. Sheng B., L Q., Mao W., Jn W., Outler Detecton n Sensor Network, n Proceedngs of the Eghth ACM Internatonal Symposum on Moble Ad Hoc Networkng and Computng (MobHoc 07), 007, pp [8]. Branch J. W., Gannella C., Szymansk B., Wolf R., Kargupta H., In-Network Outler Detecton n Wreless Sensor Networks, Knowledge and Informaton System, Vol. 3, No., 03, pp [9]. Aggarwal C. C., Outler analyss, Sprnger, 03. [0]. Slverman B. W., Densty estmaton for statstcs and data analyss, CRC Press, Vol. 6, 986. []. Breung M. M., Kregel H. P., Raymond T. Ng., Sander J., LOF: dentfyng densty-based local outlers, ACM SIGMOD Record, Vol. 9, No., 000, pp []. Subramanam Sharmla, et al., Onlne outler detecton n sensor data usng non-parametrc models, n Proceedngs of the 3 nd Internatonal Conference on Very Large Data Bases (VLDB Endowment), 006, pp [3]. Palpanas Themstokls, et al., Dstrbuted devaton detecton n sensor networks, ACM SIGMOD Record, 3,, 003, pp []. Papadmtrou S., Ktagawa H., Gbbons P. B., Faloutsos C., LOCI: Fast outler detecton usng the local correlaton ntegral, n Proceedngs of the IEEE 9 th Internatonal Conference on Data Engneerng (ICDE 03), Bangalore, Inda, 003, pp [5]. Bettencourt Luís M. A., Arc A. Hagberg, Lev B. Larkey, Separatng the wheat from the chaff: practcal anomaly detecton schemes n ecologcal applcatons of dstrbuted sensor networks, Dstrbuted Computng n Sensor Systems, Sprnger Berln Hedelberg, Berln, 007, pp [6]. Lateck L. J., Lazarevc A., Pokrajac D., Outler detecton wth kernel densty functons, Machne Learnng and Data Mnng n Pattern Recognton, Sprnger Berln Hedelberg, Berln, 007, pp [7]. Kregel H. P., Kröger P., Schubert E., Zmek A., Local outler probabltes (LoOP), n Proceedngs of 05

10 Sensors & Transducers, Vol. 89, Issue 6, June 05, pp the 8 th ACM Conference on Informaton and Knowledge Management (CIKM 09), 009, pp [8]. Schubert E., Zmek A., Kregel H.P., Generalzed Outler Detecton wth Flexble Kernel Densty Estmates, n Proceedngs of the th SIAM Conference on Data Mnng (SDM ), 0, pp SDM0 /KDEOS.pdf [9]. Marron J. S., Wand M. P., Exact mean ntegrated squared error, The Annals of Statstcs, Vol. 0, No., 99, pp [0]. Terrell G. R., Scott D. W., Varable kernel densty estmaton, The Annals of Statstcs, 99, pp []. Loftsgaarden D. O., Quesenberry C. P., A nonparametrc estmate of a multvarate densty functons, The Annals of Mathematcal Statstcs, Vol. 36, No. 3, 965, pp []. Sheather S. J., Jones M. C., A relable data-based bandwdth selecton method for kernel densty estmaton, Journal of the Royal Statstcal Socety, seres B, Vol. 53, No. 3, 99, pp [3]. Intel Lab Data downloaded from []. Jan R., Gupta C., Sharma S., A New Kernel for Outler Detecton n WSNs Mnmzng MISE, n Proceedngs of the th Internatonal Conference on Sensor Networks (SENSORNETS 5), 05, pp Copyrght, Internatonal Frequency Sensor Assocaton (IFSA) Publshng, S. L. All rghts reserved. ( 06