Research Article Time Series Outlier Detection Based on Sliding Window Prediction

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1 Mathematcal Problems n Engneerng, Artcle ID , 14 pages Research Artcle Tme Seres Outler Detecton Based on Sldng Wndow Predcton Yufeng Yu, Yuelong Zhu, Shjn L, and Dngsheng Wan CollegeofComputer&Informaton,HohaUnversty,Nanjng210098,Chna Correspondence should be addressed to Yufeng Yu; hhuheyun@126.com Receved 18 July 2014; Accepted 15 September 2014; Publshed 30 October 2014 Academc Edtor: Jun Jang Copyrght 2014 Yufeng Yu et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. In order to detect outlers n hydrologcal tme seres data for mprovng data qualty and decson-makng qualty related to desgn, operaton, and management of water resources, ths research develops a tme seres outler detecton method for hydrologc data that can be used to dentfy data that devate from hstorcal patterns. The method frst bult a forecastng model on the hstory data and then used t to predct future values. Anomales are assumed to take place f the observed values fall outsde a gven predcton confdence nterval (PCI), whch can be calculated by the predcted value and confdence coeffcent. The use of PCI as threshold s manly on the fact that t consders the uncertanty n the data seres parameters n the forecastng model to address the sutable threshold selecton problem. The method performs fast, ncremental evaluaton of data as t becomes avalable, scales to large quanttes of data, and requres no preclassfcaton of anomales. Experments wth dfferent hydrologc real-world tme seres showed that the proposed methods are fast and correctly dentfy abnormal data and can be used for hydrologc tme seres analyss. 1. Introducton As the fundamental resources for water resources management and plannng, long-term hydrologcal data are sets of dscrete record values of hydrologcal elements that are collected wth tme and have been frequently analyzed n the feld such as flood and drought control, water resources management, and water envronment protecton. Wth the development of data acquston technology and data transmsson technology, hydrologcal departments collected everncreasng amounts of tme seres data from automatc montorng systems va loggers and telemetry systems. Wthn these datasets, hydrologc tme seres analyss becomes workable and credble for buldng mathematcal model to generate synthetc hydrologc records, to forecast hydrologc events, to detect trends and shfts n hydrologc records, and to fll n mssng data and extend records [1]. However, a hydrologc tme seres s generally composed of a stochastc component supermposed on a determnstc component and usually shows stochastc, fuzzy, nonlnear, nonstatonary, and multtemporal scale characterstcs [2]. So, t s a challengng task to process and nterpret the orgnal hydrologcal tme seres due to the followng: () the large volumes of data, () the parameter pattern beng specfc and changng to dfferent hydrology acquston system due to multtemporal scale characterstc, () abnormal events or dsturbances that create spurous effects n the data seres and result n unexpected patterns, (v) naccuraces n hydrologcal models due to mprecse and outdated nformaton, logger and communcatons falures, poor calbraton, and lack of system feedback. Consequences of such stuatons n hydrologcal nformaton systems may result n the DRQP (data rch, but qualty poor) phenomenon. Consequently, the orgnal montorng data (.e., precptaton, dscharge, and water levels) should undergo a preprocessng step to elmnate the negatve nfluence caused by ncorrect or abnormal data due to

2 2 Mathematcal Problems n Engneerng nstrumentaton faults, data nherent change, operaton error, or other possble nfluencng factors [3]. Therefore, outler detecton usually becomes a vtal step for hydrologc tme seres analyss based on the montorng data. Regardng small montorng datasets, data managers can detect and deal wth outlers drectly wth a smple graphcal or manual process. However, for massve datasets or data stream, an automatc and objectve technque for effectvely detectng and treatng outlers s necessary. Ths study develops a real-tme outler detecton method that employs a wndow-based forecastng model for hydrologc tme seres collected from automatc montorng systems.themethodbuldsaforecastngmodelfromasequence of hstorcal pont values wth a gven wndow to predct future values. If the observed value dffers from the predcted valuebeyondacertanthreshold,anoutlerwouldbendcated. The method uses predcton confdence nterval (PCI) as threshold n consderaton of uncertanty n the data seres parameters n the forecastng model. Data are classfed as anomalous/nonanomalous based on whether or not they fall outsde a gven PCI. Thus, the method provdes a prncpled framework for selectng a threshold. Ths method does not requre any preclassfed examples of data, scales well to large volumes of data, and allows for fast ncremental evaluaton of data as t becomes avalable. In order to evaluate the proposed method, t was appled to two dfferent hydrologcal varables, water level and daly flow, from Huayuankou (hereafter HYK, N, E) and Lanzhou (hereafter LZ, N, E) statons obtaned from natonal hydrology database of MWR, Chna. The results show that the proposed method can exactly detect the outlers n the hydrologcal tme seres wth near neglgble false postve rate. Furthermore, the algorthm s effcency sanalyzedbasedonthedetectonresults. The rest of the paper s organzed as follows. In the next secton (Secton 2) we present the related work to ths area of research. In Secton 3 we present detals about the proposed algorthm for outler detecton n tme seres based on predcton confdence nterval. A number of experments wth the proposed method usng real-world hydrologcal tme seres are reported n Secton 4.Fnally,Secton 5 gves conclusons and suggestons for further research. 2. Related Work 2.1. Tme Seres Analyss. A tme seres (TS) X={x(t) 1 t m}s a sequence of d-dmensonal observatons vector x(t) = (x 1 (t), x 2 (t),...x d (t)) ordered n tme. Mostly these observatons are collected at equally spaced, dscrete tme ntervals. It s called a unvarate (or sngle) tme seres when d s equal to 1 and a multvarate tme seres when d s equal to or greater than 2 [4]. Generally, a tme seres can be regarded as a sample realzaton from an nfnte populaton of such tme seres generated by a stochastc process, whch can be statonary or nonstatonary. In addton, the understandng of the structure and dependence of tme seres s acheved through tme seres analyss. Tme seres analyss s the nvestgaton of a temporally dstrbutedsequenceofdataorthesynthessofamodel for predcton wheren tme s an ndependent varable; as a consequence, the nformaton obtaned from tme seres analyss can be appled to forecastng, process control, outler detecton, and other applcatons [5]. A basc assumpton n tme seres analyss s that some aspects of the past pattern wll contnue to reman n the future. Also under ths setup, often the tme seres process s assumed to be based on past values of the man varable but not on explanatory varables whch may affect the varable/system. In hydrology, tme seres analyss s one of fronter scentfc ssues because t can detect and descrbe quanttatvely each of the hydrologc processes underlyng a gven sequence of observatons. Moreover, hydrologc tme seres analyss can also be used for buldng mathematcal models to generate synthetchydrologcrecords,toforecasthydrologcevents,to detect trends and shfts n hydrologc records, and to fll n mssng data and extend records. Consequently, tme seres analyss has become a vtal tool n hydrologcal scences and ts mportance has been dramatcally enhanced n the recent past due to ever-ncreasng nterest n the scentfc understandng of clmate change [6]. Inthetmeseresanalyss,tsassumedthatthedata (observatons) consst of a systematc pattern and stochastc component; the former s determnstc n nature, whereas the latter accounts for the random error and usually makes the pattern dffcult to be dentfed. Prevous research usually equates stochastc component to system error and then smply dscards t so as to not complcate the statstcal analyses. However, the stochastc component potentally ncludes nterestng and meanngful nformaton; t must be treated wth cauton. It s for ths reason that outler detecton becomes a hotspot research ssue n recent years Outler Detecton. An outler can be defned as observaton that devates so much from other observatons as to arouse suspcon that t was generated by a dfferent mechansm [7], or patternsndatathatdonotconformtoa well-defned noton of normal behavor [8]. Generally, an outler may be ncorrect and/or the crcumstances around themeasurementmayhavechangedovertme.thus,the dentfcaton and treatment of outlers consttute an mportant component of the tme seres analyss before modelng because outlers can have negatve mpacts on the selecton of the approprate model as well as on the estmaton of the assocated parameters. Outler detecton, also known as anomaly detecton n some lteratures, s an mportant long-standng research problem n the domans of data mnng and statstcs. The major objectve of outler detecton s to dentfy data objects that are markedly dfferent from, or nconsstent wth, the remanng set of data [9, 10]. In recent decades, ths research problem s attractng sgnfcant research attentons n the felds of statstcal analyss, machne learnng, and artfcal ntellgence due to ts mportant applcatons n a wde range of areas n busness, securty, nsurance, health care, and engneerng, to name a few. Chandola et al. [8]provdeacomprehensve classfcaton of the outler detecton technques

3 Mathematcal Problems n Engneerng 3 and classfy the technques nto sx categores: classfcatonbased [11], nearest-neghbor-based [12], clusterng-based [13], statstcal [14], nformaton theory-based [15], and spectral theory-based [16]. In addton to the abovementoned algorthms, other approaches have been adopted to solve the outler detecton problem. For nstance, sgnal processng technques such as wavelet transform [17] and Fourer transform [18] have been used to detect outler regons n meteorologcal data. Outler detecton s a very broad feld and has been studed n the context of a large number of applcaton domans where many detecton methods have been proposed accordng to the dfferent data characterstcs. Recently, there has been sgnfcant nterest n detectng outlers n tme seres. Generally, methods for tme seres outler detecton should consder the sequence nature of data and operate ether on a sngle tme seres or on a tme seres database. The goal of outler detecton on a sngle tme seres s to fnd an anomalous subregon, whle the goal of the latter s to dentfy a few sequences as outlers or to dentfy a subsequence n a test sequence as an outler. In some cases, a sngle tme seres s converted to a tme seres database through the use of a sldng wndow [19]. Gven a sngle tme seres, one can fnd partcular elements (or tme ponts) wthn the tme seres as outlers or fnd subsequence outlers. Fox defnes two types of outlers (type I/addtve and type II/nnovatve) based on the data assocated wth an ndvdual object across tme, gnorng the communty aspect completely [20]. Snce then, a large number of t predcton models and profle based models have been proposed to fnd pont outler wthn a tme seres. A straghtforward method for outler detecton n tme seres s based on forecastng [21 23]. Ths approach frst bulds a predcton model from the hstorcal values andsthenusedtopredctfuturevalues.ifapredcted value dffers from the observed value beyond a certan threshold, an outler would be ndcated. The defnton of the value of the threshold to be used for detectng outlers s the man problem of outler detecton based on predcton. The dffcultes of forecastng-based outler detecton have motvated the proposal of anomaly subsequences detecton technques, based on outler score calculated wth respect to smlar measure between subsequences [24 27]. Keogh et al. [24] proposed usng one nearest-neghbor approach to detect maxmal dfferent subsequences wthn a longer sequence (called dscords). Subsequence comparsons can be smartly ordered for effectve prunng usng varous methods lke heurstc reorderng of canddate subsequences [25], localty senstve hashng [26], Haar wavelet [27], and SAX wth augmented tres [28]. The problem of outler detecton n tme seres database manly focuses on how to fnd all anomalous tme seres. It sassumedthatmostofthetmeseresnthedatabaseare normal whle a few are anomalous. Smlar to the tradtonal outler detecton, the usual recpe of solvng such problems s to frst learn a model based on all the tme seres sequences nthedatabaseandthencomputeanoutlerscoreforeach sequence wth respect to the model [29]. Statstcs-based methods are stll conducted to detect outlers n tme seres database because of ther effcent structure. In 2012, Zhang developedanaverage-basedmethodologybasedontme seres analyss and geostatstcs, whch acheved satsfactory detecton results n short snapshots [30]. However, t wll fal f the outlers gather together closely n the same short tme slot. Hence, recent researches have manly focused on nonparametrc outler detecton methods such as Bayesan method and dscrete wavelet transform (DWT). Freda proposed a Bayesan approach to outler modelng, approxmatng the posteror dstrbuton of the model parameters by applcaton of a componentwse Metropols-Hastngs algorthm [31]. Apart from the Bayesan method, Grané and Vega [32] dentfed the outlers as those observatons n the orgnal seres whose detal coeffcents are greater than a certanthresholdbasedonwavelettransform.theyterated the process of DWT and outler correcton untl all detal coeffcents are lower than the threshold. Based on real-world fnancal tme seres, ther method acheved a lower average number of false outlers than Blen and Huzurbazar s [33]. However,ntheseworks,thresholdsaremanlysetsubjectvely, whch makes these methods neffcent and nsenstve when there are several dfferent knds of outlers appearng n thesametmeseres Outler Detecton n Hydrologcal Tme Seres. Hydrologc systems nvolvng outlers nvarably represent complex dynamcal systems. The current state and future evolutons of such dynamcal systems depend on countless propertes and nteractons nvolvng numerous hghly varable physcal elements. The representaton of such dynamcal systems n ther correspondng models s complcated because certan relatonshps can only be developed through analyses. Outler detecton n hydrologc data s a common problem whch has receved consderable attenton n the unvarate framework. In the multvarate settng, the problem s well establshed n statstcs. However, n the hydrologc feld, the concepts are much less establshed. A poneerng work n ths drecton was recently presented by Chebana and Ouarda [34]. Moreover, many outler detecton technques, such as Chauvenet s method, Dxon-Thompson outler test, and Rosner s test [35], are statstcs based on the prncple of hypothess testng wth the underlyng assumpton of log-pearson type III (LP3) dstrbuton [36], whch may not always be readly avalable for hydrologcal tme seres. Hyndman and Shang s [37] methods, on the bass of real data, are graphcal and consst frst n vsualzng functonal data through the ranbow plot and then n dentfyng functonal hydrologcal outlers usng the functonal bag-plot and the functonal hghest-densty regon box-plot. The methods can detect outler curves that may le outsde the range of the majorty of the data or may be wthn the range of the data but have a very dfferent shape. However, as ndcated by Chebana and Ouarda [34], the ponts outsde the fence of the bag-plot or box-plot are consdered as extremes rather than outlers. On ths bass, Chebana et al. proposed a nongraphcal outler detectonmethodbasedonthebvaratescoreponts,whch were obtaned from the frst two prncpal components score vectors generated by functonal prncpal component analyss, to detect outlers n flood frequency analyss [38].

4 4 Mathematcal Problems n Engneerng Ng et al. [39] found that chaotc approach can determne the level of complexty of a system after they appled the chaotc analytcal technques to daly hydrologc seres comprsng of outler. It adopts box-plot [40] as the outler detecton toolsbasedonts(1)statstcalpopularty,(2)abltyto handle and detect multple outlers smultaneously wthout precedng to an teratve detecton, and (3) ablty to detect outlers by block detecton where no maskng effects are nvolved before conductng the chaotc analyss. The method can effectvely detect numerous outlers n the orgnal daly dscharge dataset but may understate the actual number of outlers because the data s usually clustered around the zero mean and consequently results n less skewness n the data. Although many outler detecton methods exst n the lterature, there s a lack of dscusson on the selecton of a proper detecton method for hydrologcal outlers. It s manly because of the fact that most of outler detecton methods belong to statstcal approaches and demanded that thedatamustfollowsomedstrbutons,andtheselectonof a sutable outler detecton method s crtcally determned bythententofanalystandthentendeduseoftheresults. Analysts have to consder several techncal aspects n ts decsons-makng such as the tradeoff between accurate and effcent, the evaluaton of consequences subject (.e., maskng and swampng), the desgn assumptons and the lmtaton of dfferent methods, and the preference on parametrc or nonparametrc approach. Wthout a thorough understandng of outler phenomena, t s dffcult to determne a sutable outler detecton method. Faced wth such challenges, ths work proposes a new method to detect outlers that splts gven hstorcal hydrologcal tme seres nto subsequences by a sldng-wndow andthenanautoregressve(ar) predctonmodeloftme seres and predcton confdence nterval (PCI) calculated from nearest-neghbor hstorcal data to dentfy tme seres anomales. The method used an autoregressve predcton model, whch belongs to data-drven tme seres model essentally, rather than a physcs-based tme seres model, due to thefactthattssmplertodevelopandcanrapdlyproduce accurate short forecast horzon predctons. Data are classfed as anomalous/nonanomalous based on whether or not they fall outsde a gven PCI.ThePCI whch s employed n ths study not only accounts for the fact that the correlatons between adjacent data ponts n the tme seres are hgher than those farther away but also avods fallng nto neffcent and nsenstve dlemma caused by a subjectve-settng threshold. Moreover, the PCI canbecalculateddynamcally accordng to dfferent nearest-neghbor wndows sze and confdence coeffcent of dfference users, whch make t sutable for dfferent varables of hydrologc tme seres outler detecton for dfferent user s demand. In concluson, ths method does not requre any preclassfed examples of data, scales well to large volumes of data, and allows for fast ncremental evaluaton of data as t becomes avalable. 3. Wndow-Based Outler Detecton In ths secton, we formulate the outler detecton problem and gve a formal defnton of some concepts whch are used n the proposed algorthm. And then we wll ntroduce thealgorthmdetectngoutlersntmeseresbasedonthe sldng-wndow predcton model. In addton, we also menton effcent strateges to choose the optmal parameters to meet the users requrements. Next s the formal formulaton of the contextual outler detecton algorthm Problem Defnton. Hydrology s a tme-varyng phenomenon, the change of whch s referred to hydrologcal processes. As mportant scentfc data resources, hydrologcal data are the dscrete records of hydrologcal processes and couldbedvdedntoflow,waterlevel,ranfall,evaporaton, and other hydrologc tme seres accordng to the physcal quanttes of ts representaton. Defnton 1 (hydrologcal tme seres). A hydrologcal tme seres T s a set of real-valued data n successve order, occurrng unform tme nterval. In ths work, T = d 1 =(V 1,t 1 ), d 2 =(V 2,t 2 ),...d m =(V m,t m ),wherems the length of the tme seres and pont d = (V,t ) stands for the observaton V at the moment t ;moreover,t s strctly ncreased. For outler detecton purposes, we are typcally not nterestednanyoftheglobalpropertesofatmeseres;rather,we are nterested n local subsectons of the tme seres, whch are called subsequences. Defnton 2 (subsequence). Gven a tme seres T of length m, asubsequencec of T s a samplng of length n mof contguous poston from p; thats,c= d p =(V p,t p ), d p+1 = (V p+1,t p+1 ),...d p+n 1 = (V p+n 1,t p+n 1 ) for 1 p m n+1.partcularly,subsequencec may contan only one data pont on the condton that n=m. Snce all subsequences may potentally be abnormal, any algorthm wll eventually have to extract all of them; ths can be acheved by use of a sldng wndow. Defnton 3 (sldng wndow). Gven a tme seres T of length m and a user-defned subsequence length of n, all possble subsequences can be extracted by sldng wndow of sze n across T and consderng each subsequence C p. Generally, the frst problem of tme seres outler detecton s to defne what knd of data n a gven dataset s abnormal. That s, the defnton of outler determnes the outler detectons goals. In hydrologc tme seres, tme sequences whch are composed of dfferent physcal quanttes show great dfference anomaly characterstcs; therefore, t s dffcult to gve a unform defnton of abnormalty. In ths paper, we dentfy a subsequence to be outler based on ts nearestneghbor. Defnton 4 (k-nearest-neghbor). Gven a tme seres T of length m and a data pont d ( < m),thek-nearest-neghbor of d s a samplng of length 2k of contguous poston η (k)

5 Mathematcal Problems n Engneerng 5 from kto +kand does not contan ; thats,η (k) = {d k,...d 1,d +1,...d +k } for +1 k m. Defnton 5 (tme seres outler). Gven a tme seres T of length m and the k-nearest-neghbor η (k) of d, d wll be dentfed as an outler f the observed value of d falls outsde a PCI calculated by confdence coeffcent p and predcted value accordng to η (k). From the above defnton, one can see that the nearestneghbors wndow sze k and confdence coeffcent p become key parameters of outler detecton. Therefore, t can dynamcally adjust k and p for dfferent users and dfferent hydrologcal elements to acheve the optmal detecton result Algorthm Descrpton. After studyng the current stuaton and challenge of hydrologcal tme seres and ts outlers, ths study proposes a new outler detecton method that uses a sldng wndow of hydrologcal tme seres T m = d 1 =(V 1,t 1 ), d 2 =(V 2,t 2 ),...d m =(V m,t m ), where pont d = (V,t ) stands for the measurement V at the moment t, to classfy a partcular data pont as anomalous ornot.adatapontd s classfed as anomalous f ts measurement devates sgnfcantly from ts k-nearest-neghbor (KNN) predcton value calculated usng ts neghborng pont set η (k) as nput. Upon ntalzaton, the method flls the wndow wth the most recent measurements and commences classfcaton wth the next measurement taken by the tme seres. Inbref,themethodconsstsofthefollowngstepsbegnnng at tme wthn a gven hydrologcal tme seres T. Step 1. Defne k-nearest-neghbor wndow η (k) for the pont d, based on the data source, from where outlers are to be detected (hstory data ncludng complete sequence or only past data to forecast new ncomngs); the η (k) can be dvded nto one-sded-wndows and two-sded-wndows types. Step 2. Buld a nearest-neghbor-wndow predcton model that takes η (k) as nput to predct V +1,theexpectedvalueofthe pont d +1 ; n addton, calculate the upper and lower bounds of the range wthn whch the measurement should le (.e., the predcton confdence nterval, PCI). Step 3. Compare the actual measurement at tme +1wth the range calculated n Step 2 and classfy t as anomalous f t falls outsde the range; otherwse, classfy t as nonanomalous. Step 4. Modfy T by removng d k+1 from the back of the wndow and addng d +1 whch holds the value V +1 to the front of the wndow to create T +1 ; slde one step and modfy the η (k) to create η (k) +1 f the measurement s classfed as anomalous; else, modfy T by removng d k+1 from the back of the wndow and addng d +1 whch takes ts orgnal value to the front of the wndow to create T +1 ;sldeonestepand modfy the η (k) to create η (k) +1. Step 5. Repeat Steps 1 4, untl all sequence has been detected. Tme seres dataset T m ={ d 1,d 2,d 3,...d k+1...d,...d m } Defne d s k-nearest-neghbor wndow η (k) ={d 2k, d 2k+1,...d 1 } Predct Calculate PCI Compare wth PCI extends PCI? Yes Mark d as outlers and correct t End Fgure 1: Dagram of the proposed outler detecton method. The detecton process of data pont d s llustrated wth a flow chart n Fgure 1. The remander of ths secton descrbes these steps n detal Wndow Defnton. The frst step of ths outler detecton process, the KNN wndow of the test pont d n tme seres data, s defned to llustrate the relatons between the data pont and ts nearest-neghbor. And then, the predcton model can use only the test pont s KNN wndow to predct the measurement of d forthepurposeofsmplfyngthe computatonal complexty. Generally, there are two types of hydrologcal data from where outlers are to be detected: hstory tme seres data or real-tme data. The dfference between them s prmarly based on the fact that the former uses the prevous and subsequent neghbor wndow as nput parameters to detect the outler whle the latter only uses the prevous neghbor wndow as nput parameters. Then, the neghbor wndow can be dvded nto one-sded and two-sded types. (1) Two-Sded Neghbor Wndows. The two-sded-wndows outler detecton method uses a data ponts prevous (left) and subsequent (rght) neghbor s data pont to determne whether ths data pont s outler or not. Gven a water No

6 6 Mathematcal Problems n Engneerng level tme seres T m = d 1 =(V 1,t 1 ), d 2 =(V 2,t 2 ),...d m = (V m,t m ), the two-sded-wndows neghborng pont sets η (k) for the pont d can be defned as follows: η (k) ={d k,...d 1,d +1,...d +k }. (1) Note that 2k s the sze of the neghborhood wndow, startng at kand endng at +k(not ncludng ). (2) One-Sded Neghbor Wndows. As the rght neghbors may contan outlers that had not been detected, therefore, n the real-tme detecton applcaton, only left neghbor can be avalable. So, one-sded-wndows outler detecton method only chooses left neghbors whch removed (or revsed) the outlers had been detected wll make the detecton results more meanngful. Other than the two-sded-wndows method, one-sded-wndows method outler detecton algorthms defne neghborhood pont set η (k) for the pont d,as follows: η (k) ={d 2k,d 2k+1,...d 1 }. (2) Here, 2k s the sze of the neghborhood wndow for the pont d,startngat 2kand endng at Model Selecton. In ths step, an autoregressve (AR) predcton model s bult to forecast the partcular pont s valueusngtsneghborhoodpontsetsasnput.thear model forecasts future measurements n tme seres datasets usng only a specfed set of observatons, that s, η (k),from thesamedschargeste;theyareusedbecausetheyavod complcatons caused by dfferent samplng frequences that can arse f a heterogeneous set of tme seres data was used; moreover, the use of AR model reduces the number of predctons that cannot be made due to nsuffcent data caused by the embedded telemetry equpment that went offlne. The neghborhood pont sets η (k) areusedasnputto the AR model of the tme seres to predct the followng observaton: d =M(η (k) ), (3) where M( ) s the model. Ths method assumes that the behavor of the processes at tme t+1canbedescrbedby a fnte set of k prevous measurements; thus t mplctly assumes that the tme seres s an order k Markov process. Lterature [22] compares natve, nearest cluster (NC), and sngle-layer lnear network (LN) and multlayer perceptron (MLP) on dfferent dataset and concludes that LN and MLP would obtan better detecton result than other models. Based on ths work, we use LN as predcton model and assume that observaton V s a lnear combnaton of two-sded neghbors wndows data pont: V = ( k j=1 w jv j + k j=1 w +jv +j ) ( k j=1 w j + k j=1 w, (4) +j) where w k,...,w 1,w +1,w +k stands for the weght of neghborhood, whch defnes the relatonshp between the two-sded-wndows neghborhood {d k,...d 1,d +1,... d +k } and the expected value of d. Generally, the weght of the neghborng pont d j s nversely proportonal to the dstance between the pont d and d j ;thats,thelargerthe dstance s, the small the weght w s. For smplcty, t assgns the weght vector w k,...,w 1,w +1,w +k wth the values 1,2,...k,k,...2,1. Generally, two-sded neghbor wndows need ponts prevous and subsequent neghbors; however, the rght neghbors may contan outlers that had not been detected, whch may affect detecton results subsequently. So, n some applcaton felds, only prevous (left) neghbor-wndow data can be used to predct and dentfy forthcomng outlers. Therefore, t often uses a smple modfcaton of the twosded-wndows model to predct the measurement at tme : j=1 w jv j V = 2k 2k j=1 w j. (5) Smlar to two-sded neghbors wndows, the weght vector w 2k,w 2k+1,...w 1 stands for the weght of neghborhood and s assgned wth the values 1,2,...2k Outlers Identfcaton. Gven the model predcton from Secton based on test pont s neghbor,the data pont can then be classfed as anomalous usng a confdence boundary PCI calculated va predcted value and confdence coeffcent. The PCI gves the range of plausble values that the test measurement can take; the confdence coeffcent (p = 100(1 α)) ndcates the expected frequency wth whch measurements wll actually fall n ths range. If t s assumed that the model resduals have a zero-mean Gaussan dstrbuton, the p% PCI canbecalculatedasfollows[22]: PCI = V t+1 ±t α/2,2k 1 s k, (6) where V t+1 s the predcton value of the test pont, t α/2,n 1 s the pth percentle of a Student s t-dstrbuton wth 2k 1 degrees of freedom, s s the standard devaton of the model resdual, and k s the wndow sze used to calculate s. Ths type of PCI s a type of t-nterval because t reles on Student s t-dstrbuton. If the test pont s actual measurement falls wthn the bounds of the PCI, then the pont s classfed as nonanomalous; otherwse, t s classfed as anomalous. Thus, the PCI represents a threshold for acceptance or rejecton of a data pont. The beneft of usng the PCI nstead of an arbtrary threshold s that the predcton level gudes the selecton of the nterval wdth. The two-sded-wndows approach for outler detectng s llustrated wth a smple example n Fgure 2 wth a neghborhood wndow wdth of k=4. It should be noted that the area between the two green lnes n Fgure 2 coverng the neghborhood of data ponts ndcates the confdences bound (PCI). In ths example, d 7 s an outler n the current neghborhood of ponts and proposes to be replaced by the

7 Mathematcal Problems n Engneerng Experments and Analyss Varable 2 1 η 7 (4) ={d 3,d 4,d 5,d 6,d 8,d 9,d 10,d 11 } Predcted value Tme Confdence bound Outler To demonstrate the effcacy of the outler detecton methods developed n ths study for data QA/QC, t wll be appled to hydrologcal data seres from natonal hydrology database of MWR, Chna. In the followng, the real data are descrbed and are functonal and results are presented anddscussed.moreprecsely,tfrstusestheprevously presented approaches to dentfy outlers; then, some performanceevaluatonwllbedscussedandnterpretedonthe bass of hydrologcal data; and some results usng multvarate approaches for comparson purposes wll be provded at last. Fgure 2: Outler detecton example, where the raw data, predcted value, confdence bound, and outler are dentfed by dfferent symbols. In ths example, d 7 s an outler n the current neghborhood of ponts and has been detected correctly. predcted value V 7 for analyss and modelng of the tme seres Parameter Optmzaton. In order to detect the outlers n the tme seres, the proposed methods should calculate the plausble values range PCI va predcted value and confdence coeffcent based on the test pont s k-nearestneghbor. Hence, the values of the two parameters, k and p, become the key ssues for mprovng the outler detecton methods mentoned n the prevous secton. In ths case, we can use the followng experments to choose a proper value of those two parameters. (1) The wndow wdth (k) of the methods controls how many neghborng ponts are ncluded n the calculaton of thepredctonvalue.thelargerthewndowwdths,themore ponts are ncluded to compute the predcton value. It vares k from 3 to 15 n ncrements of one; that s, k = {3, 4,..., 15}. (2) The confdence coeffcent p calculates the plausble values range PCI toclassfyadatapontasanoutler.the larger the confdence coeffcent s, the more plausble the predcton value s. Therefore, t vares p from 85% to 99% n ncrements of one percent. To tune the best combnaton of algorthm parameters that maxmzes the rato of detecton, the cross-valdaton scheme s appled. The complete sample s splt nto two segments: a tranng dataset and a testng dataset. Ths s a way of cross-valdatng whether the parameters found durng the frst perod, the tranng phase, are consstent and stll vald n a dfferent perod, the testng phase. The prncple of cross-valdaton s a generc resource to valdate statstcal procedures and t has been appled n dfferent contexts. Furthermore, n tranng phase, the tranng set s dvded nto 10 nonntersectng subsets of equal sze, chosen by random samplng. The model s then traned 10 tmes, each tme reservng one of the subsets as a valdaton set on whch the model error s evaluated whle fttng the model parameters usng the remanng nne subsets. The model parameters wth the lowest mean squared error among the 10 tranng models arethenselectedforthefnalmodel[41] Study Area and Data. In ths subsecton we report on a number of experments usng two dfferent hydrologc elements, water level (m) and daly flow (m 3 s 1 ), from LZ and HYK statons wth reference numbers and , respectvely, obtaned from natonal hydrology database of MWR, Chna. The LZ and HYK statons are the mportant flood preventon and control statons and are generally known as typcal hydrologcal statons of upper and mddle reaches of the Yellow Rver. They play an mportant role n downstream of Yellow Rver n the felds such as hydrologcal data collecton and forecastng, flood control schedulng, rver control experments, and water resources development. Fgure 3 ndcates the geographcal locaton of LZ and HYKstatons. The data downloaded from the natonal hydrology database of MWR, Chna, were avalable as raw data. In addton, we followed the procedures descrbed n the prevous secton. Fgures 4 and 5 llustrate the whole dataset wthn a gven tme seres; t shows that the dataset s nearly perodc and has some suspcous data pont obvously Experment Result. Snce the data used n ths study were subjectedtomanualqualtycontrolbeforebengarchved to the natonal hydrology database, t was expected that the detectors would not dentfy many data outlers n the archve. However, we can easly see that some data ponts devate from ther neghbor. And then, we apply our methods to detect the outlers n the gven hydrologcal tme seres wth the wndow sze k=6and probablty p=95%. And the detecton results fortheproposedmethodsareshownnfgures6, 7, 8,and9. Fgures 6 9 depct real and predcted values wth the proposed outler detected methods n the daly flow and water level tme seres of LZ and HYKstatons, respectvely. In each of these graphcs the PCI for the predctons s also depcted. These graphcs show that most of the real values are very near the respectve predcted value, whle a spot of them les outsde the PCI boundares for the predctons. An outler detecton method based on PCI wouldndcateanoutlerf the real value les outsde the confdence bounds, as descrbed n Secton 3.2. The experments reported here showed that these bounds, bult from cross-valdaton scheme on tranng and valdaton sets, can correctly bnd the regon of normalty. Furthermore, these ntervals could be used to ndcate the level of suspcon assocated wth a gven pont n the future.

8 8 Mathematcal Problems n Engneerng N Staton Fgure 3: Geographcal locaton of LZ and HYK statonsn Yellow Rver. Water level (m) Staton ( ) raw water level data: from 1998/1/1 to 1999/12/31 Daly flow (m 3 /s) Staton ( ) raw daly flow data: from 1998/1/1 to 1999/12/31 Tme (day) Tme (day) (a) Water level (b) Daly flow Fgure 4: Raw hydrologcal tme seres of LZ staton Experment Evaluaton Parameters of Qualty. The detecton results for the proposed methods wth the two dfferent hydrologc elements from LZ and HYK statons llustrate that the outler detecton methods are promsng and can successfully detect a consderable amount of outlers from the hydrologc dataset. However, we also note that there are some vald data ponts whch had been detected as outlers and mputed n error. The objectve of the evaluaton analyss descrbed n ths secton s to assess the effectveness of method; we can classfy the results from our experment nto four categores (see Table 1). The categores n Table 1 correspond to the four possble outcomes of one expermental run, whch conssts of usng both methods wth a partcular combnaton of wndow wdth (k) and probablty (p). CategoresA and D are deal stuatons n whch a pont can be detected correctly, whle categores B and C are undesred, because the methods are not able to dstngush between outlers and exactness. Accordng to these defntons, the Senstvty s the probablty that the proposed methods dscovered a real outler.itsformulasdefnedasfollows: Senstvty = TP (TP + FN). (7) Truth Outler Not an outler Table 1: Assessment of both methods. Outler TruepostvesorTP(A) data ponts that are outlers and dentfy as outlers False postves or FP (B) data ponts that are normal but dentfy as outlers Detecton Not an outler False negatves or FN (C) data ponts that are outlers but dentfy as normal True negatves or TN (D) data ponts that are normal and dentfy as normal Another relevant parameter s the Specfcty,the proporton of negatve test results among the normal; the mathematcal expresson s as follows: Specfcty = TN (TN + FP). (8) The postve predctve value (PPV)stheprobabltythat a detected outler s ndeed a real one. Its formula s as follows: PPV = TP (TP + FP). (9)

9 Mathematcal Problems n Engneerng 9 Staton ( ) raw water level data: from 2006/1/1 to 2007/12/31 Staton ( ) raw daly flow data: from 2006/1/1 to 2007/12/31 Water level (m) Daly flow (m 3 /s) Tme (day) Tme (day) (a) Water level (b) Daly flow Fgure 5: Raw hydrologcal tme seres of HYK staton. Staton ( ) raw water level data: from 1998/1/1 to 1999/12/31 Staton ( ) raw water level data: from 1998/1/1 to 1999/12/31 Water level (m) Tme (day) Water level (m) Tme (day) Predcted value Confdence bound Outler Predcted value Confdence bound Outler (a) Two-sded wndow detecton results (b) One-sded wndow detecton results Fgure6: Detecton resultsoverlz water level. Staton ( ) detecton result: from 1998/1/1 to 1999/12/31 Staton ( ) detecton result: from 1998/1/1 to 1999/12/ Daly flow (m 3 /s) Daly flow (m 3 /s) Tme (day) 0 Tme (day) Predcted value Confdence bound Outler Predcted value Confdence bound Outler (a) Two-sded-wndow results (b) One-sded-wndow detecton results Fgure 7: Detecton results over LZ daly flow.

10 10 Mathematcal Problems n Engneerng Water level (m) Staton ( ) detecton result: from 2006/1/1 to 2007/12/31 Tme (day) Water level (m) Staton ( ) detecton result: from 2006/1/1 to 2007/12/31 Tme (day) Predcted value Confdence bound Outler Predcted value Confdence bound Outler (a) Two-sded-wndow detecton results (b) One-sded-wndow detecton results Fgure8: DetectonresultsoverHYK water level. Staton ( ) detecton result: from 2006/1/1 to 2007/12/31 Staton ( ) detecton result: from 2006/1/1 to 2007/12/ Daly flow (m 3 /s) Daly flow (m 3 /s) Tme (day) 0 Tme (day) Predcted value Confdence bound Outler Predcted value Confdence bound Outler (a) Two-sded-wndow detecton results (b) One-sded-wndow detecton results Fgure 9: Detecton results over HYK daly flow. Fnally, the negatve predctve value (NPV) s the proporton of nonoutlers among subjects wth a negatve test result. Its formula s as follows: NPV = TN (TN + FN). (10) Quantfyng the Outler Occurrences. Astatstcalmeasureoftheaccuracysprovdednthssecton.Theparameters used are the ones descrbed n Secton and summarzedntables2 and 3. Note that all parameters refer to the whole year from 2006 to 2007 of HYK staton and 1998 to 1999 of LZ staton. And just for smplcty, only the detecton result wth three pars of parameters and the explanaton for the water level where (k, p) takes of the value (6, 0.95) s provded, snce the reasonng for the remanng hydrologc elements and parameters s analogous. Note that the optmal values for the parameters (k, p) would be(6,0.95)on water level tme seres of HYK staton accordng to parameter optmzaton method descrbed n Secton 3.3. As t s shown n Table 2, durng the one-sded methods detectng process on water level tme seres of HYK staton wth (k, p) takng the value of (6, 0.95), there were 14 outlers properly detected; that s, TP = 14. On the other hand, there were two occasons whch orgnally represent normal event but are to be consdered an outler by the methods; therefore, FP = 2. Furthermore, there was one day whch was outler and was not detected by the methods; that s, FN=1.And the remanng 713 days were correctly judged as normal event. Thus, TN = 713 ( =730forecasteddaysfrom 2006 to 2007). The results show great accuracy for both features. Partcularly remarkable are the values reached by the specfcty. In partcular, t exceeds 99% n all stuatons. These values mean that when the approach classfes the day to be predcted as normal,tdoestwthhghrelablty.

11 Mathematcal Problems n Engneerng 11 Table 2: Statstcal analyss of one-sded methods wth dfferent parameters of HYK staton. Parameters Water level Daly flow (5, 0.95) (6, 0.95) (6, 0.96) (7, 0.95) (5, 0.95) (5, 0.96) (6, 0.96) (5, 0.97) TP TN FP FN Senstvty 80.00% 93.33% 80.00% 73.33% 83.33% 88.89% 77.78% 72.22% Specfcty 99.72% 99.72% 99.44% 99.58% 99.30% 99.72% 99.44% 99.58% PPV 85.71% 87.50% 75.00% 78.57% 75.00% 88.89% 77.78% 81.25% NPV 99.58% 99.86% 99.58% 99.44% 99.58% 99.72% 99.44% 99.30% Table 3: Statstcal analyss of both methods wth optmal parameters of gven dataset. LZ staton HYK staton Parameters Water level Daly flow Water level Daly flow One-sded (6, 0.96) Two-sded (6, 0.95) One-sded (7, 0.96) Two-sded (6, 0.95) One-sded (6, 0.95) Two-sded (6, 0.96) One-sded (5, 0.96) Two-sded (6, 0.95) TP TN FP FN Senstvty 90.91% 81.82% 85.71% 90.48% 93.33% 86.67% 94.44% 94.44% Specfcty 99.44% 99.44% 99.58% 99.29% 99.72% 99.30% 99.72% 99.44% PPV 83.33% 81.82% 85.71% 79.17% 87.50% 72.22% 89.47% 80.95% NPV 99.72% 99.44% 99.58% 99.72% 99.86% 99.72% 99.86% 99.86% As for the senstvty, all the results reached values greater than 73% (except for daly flow wth the stuaton (k, p) takng thevalueof(5,0.97),nwhchtreaches72.22%)andobtans 81.6% on average. The PPV reached values smlar to those of the senstvty, n partcular, slghtly greater (82.8% on average). Therefore, t can be stated that when the proposed method determnes that there s an upcomng outler, t s hghly relable. Fnally, NPV provdes smlar values for all the stuatons, reachng 99.58% on average; that s, the rate of real outlers not found by the approach cannot be consdered sgnfcant. As for Table 3, t can be easy to draw a concluson that dfferent hydrologc elements of the same staton and the same hydrologc elements of dfferent staton may take dfferent optmal parameters values for (k, p) from experments descrbed n Secton 3.3. Moreover, we can see that the detecton accuracy of one-sded method s better than twosded one as a whole. For example, there were 15 outlers n water level dataset of HYK staton; one-sded method can correctly detect 14 abnormal and 713 normal data ponts; smultaneously, t masks one outler as normal and swamps two normal samples as outlers. As a comparson, two-sded methods can correctly detect 13 abnormal and 710 normal data ponts; correspondngly, the maskng and swampng effects reach to 2 and 5, respectvely. The reason for ths dfference may le n that the latter method needs both sdes neghbors whch may contan outlers, whch may lead to the fact that maskng and swampng event occurs Analyss and Dscusson. We compared our methods wth other methods such as SVM (support vector machne) [42], box-plot technques [40], and medan method[23] on thesametestdatasets.thecomparsonresultswlldsplay n the recever operatng characterstc (ROC) [43] curves, whch wll be shown later. By conventon, the ROC curve dsplays senstvty (TPR) on the vertcal axs aganst the complement of specfcty (1-specfcty or FPR) on the horzontal axs. The ROC curve then demonstrates the characterstc recprocal relatonshp between senstvty and specfcty, expressed as a tradeoff between the TPR and FPR. Ths confguraton of the curve also facltates calculaton of the area beneath t as a summary ndex of overall test performance. Therefore, the larger the area under the ROC curve, the better the performance of the technque. Fgure 10 reveals the ROC curves obtaned by proposed methods and other technques. For these datasets, the performances of proposed methods are satsfactory and stable. For the water level dataset of LZ, our methods obtan smlar results; ther ROC curves show better performance than those of SVM and box-plot methods, whle medan method shows second-better ROC curves. For the daly flow dataset of LZ, our methods and SVM show better performance than medan method and box-plot method; moreover, the performance of one-sded-wndow method s slghtly better than that of two-sded-wndow. For the water level of HYK,ourmethods show preferable results, whle SVM method obtans relatvely better results than box-plot and medan methods. For the

12 12 Mathematcal Problems n Engneerng Table 4: Comparsons of area under ROC curves. AUCs Medan Box-plot SVM Two-sded One-sded LZ water level LZ daly flow HYK water level HYK daly flow Medan Box-plot SVM Two-sded One-sded Medan Box-plot SVM Two-sded One-sded (a) ROC of LZ water level (b) ROC of LZ daly flow Medan Box-plot SVM Two-sded One-sded (c) ROC of HYK water level Medan Box-plot SVM Two-sded One-sded (d) ROC of HYK daly flow Fgure 10: ROC curves for dfferent datasets. Fve methods are compared: medan method, box-plot method, SVM method, and our method, whch are descrbed usng dfferent colored lnes. daly flow ofhyk,the ROC curves nterlace wth each other, so t s dffcult to evaluate the performance only accordng totheobservaton.wehavealsocalculatedtheaucs for the ROC curves to compare the results (see Table 4). Note that there s a remarkable mprovement n the detecton performance. The average AUCs of our method are and 0.942, whch are the hghest two among these fve methods. The average AUCs of medan, box-plot, and SVM methods are 0.892, 0.878, and 0.897, respectvely. For our method, t can be nferred that the anomales canbeeffectvelydetectedbythewndow-basedforecastng model, whch s constructed usng AR predcton model and dynamcally boundary movement strateges. The results suggest that our methods mprove the robustness of the overall decson, whle the other methods show dfferent performances on dfferent datasets. The other three technques have the smlar characterstc; they acheve better or worse results on dfferent datasets. Results ndcate that the proposed framework can fnd robust PCI as a means for defnng the thresholds for detectng noveltes n tme seres and can therefore mprove performance of forecastng-based tme seres novelty detecton. Moreover, our method s robust

13 Mathematcal Problems n Engneerng 13 to detect anomales n dfferent datasets, whch means that t s more data ndependent. It s mportant to menton that we have also valdated our method on dfferent datasets. The results were smlar to those reported n ths secton. Furthermore, ths paper used PCI rather than an arbtrary, user-defned threshold, whch s manly because of the fact that PCI can provde gudance (n the form of the confdence coeffcent) for the calculaton of the normal boundary regon wthout requrng any knowledge of the process varables beng measured. The p% PCI ndcates the regon lkely to contan at least p% of the possble sensor measurements, and we expect that approxmately (1 p)% of the nonanomalous data wll be msclassfed as anomalous. Ths ndcates that the normal assumpton mplct n the PCI calculaton s reasonable for the hydrologcal tme seres data. One lmtaton of our methods s that we dentfy a data pont to be an outler or not by comparng new observed value to PCI calculated dynamcally accordng to dfferent nearest-neghbor-wndows sze and confdence coeffcent of dfference users, whch may cost a certan amount of tme complexty to calculate optmzaton parameter for best detecton results. Notwthstandng that lmtaton, ths study does suggest that the proposed method can mprove the performance of outler detecton. Results confrm that our methods can acheve a more robust performance than other outler detecton technques. 5. Conclusons Outler detecton, one of the classcal topcs of data mnng, has generated a great deal of research n recent years owng to the new challenges posed by large hgh-dmensonal data. In the meantme, outlers n hydrologcal tme seres have many practcal applcatons, such as data QA/QC, adaptve samplng, and anomalous event detecton. Ths research developed a tme seres outler detecton method that employs a wndow-based forecastng mode n conjuncton wth PCI to detect noveltes n hydrologcal tme seres. The method frst splts gven hstorcal hydrologcal tme seres nto subsequences by a sldng-wndow, and then an autoregressve predcton model of tme seres was bult from ts nearestneghbor-wndow to predct future values. Anomales are assumed to take place f the observed values fall outsde a gven PCI, whchcanbecalculateddynamcallyaccordng to dfferent nearest-neghbor-wndows sze and confdence coeffcent of dfferent user. Moreover, experments wth two dfferent hydrologcal varables, water level and daly flow, from LZ and HYK statons obtaned from natonal hydrology database of MWR, Chna, were performed to examne the effects of the method. In the experments, sngle-layer lnear networks were used for forecastng model; confdence coeffcent and wndow sze are assgned to 95% and 6 respectvely n the ntalzaton phase. Moreover, the best combnaton of parameters confdence coeffcent and wndow sze can be tuned accordng to dfferent user s requrements and tme seres features. The case study results suggest that the proposed outler detecton methods developed n ths study are useful tools for dentfyng anomales n hydrologcal tme seres. Snce these methods only requre a tme-seres model of the tme seres, they can be easly appled to many real-tme hydrologcal tme seres. However, t should be noted that, whle the LN produced the best model for the water level and daly flow tme seres consdered n the case study, ths model may not be the most approprate choce for other types of hydrologcal data. Furthermore, the optmzaton combnaton of parameters confdence coeffcent and wndow sze may cost a certan amount of tme complexty for dfferent hydrologcal element and dfferent user s requrements. In spte of ths, these methods do not requre any preclassfed examples of data, scale well to large volumes of data, and allow for fast ncremental evaluaton of data, whch make them an deal choce for correctly and effcently hydrologcal tme seres outler detecton, especally for cases n whch there s lttle nformaton about how to set the threshold value. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. Acknowledgments ThsworkssupportedbytheNaturalScenceFoundaton of Chna (nos , , and ) and the Natonal Scence and Technology Infrastructure of Chna (no. 2005DKA32000). References [1] J. D. Salas, Analyss and modelng of hydrologc tme seres, n Handbook of Hydrology,vol.19,pp.1 72,1993. [2] W. Gujer, Systems Analyss for Water Technology, Sprnger, Berln, Germany, [3] N. Lauzon, Water resources data qualty assessment and descrpton of natural processes usng artfcal ntellgence technques [Ph.D. thess],unverstyofbrtshcolumba,2003. [4] K. 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15 Advances n Operatons Research Advances n Decson Scences Appled Mathematcs Algebra Probablty and Statstcs The Scentfc World Journal Internatonal Dfferental Equatons Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Complex Analyss Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Dscrete Mathematcs Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Internatonal Stochastc Analyss Optmzaton