Landmarks: A New Model for Similarity-Based Pattern Querying in Time Series Databases

Transcription

1 Lmarks: A New Model for Similariy-Based Paern Querying in Time Series Daabases Chang-Shing Perng Haixun Wang Sylvia R. Zhang D. So Parker perng@cs.ucla.edu hxwang@cs.ucla.edu Sylvia Zhang@cle.com so@cs.ucla.edu Universiy of California Los Angeles, CA Absrac In his paper we presen he Lmark Model, a model for ime series ha yields new echniques for similariy-based ime series paern querying. The Lmark Model does no follow radiional similariy models ha rely on poinwise Euclidean disance. Insead, i leads o Lmark Similariy, a general model of similariy ha is consisen wih human inuiion episodic memory. By racking differen specific subses of feaures of lmarks, we can efficienly compue differen Lmark Similariy measures ha are invarian under corresponding subses of six ransformaions; namely, Shifing, Uniform Ampliude Scaling, Uniform Time Scaling, Uniform Bi-scaling, Time Warping Non-uniform Ampliude Scaling. A mehod of idenifying feaures ha are invarian under hese ransformaions is proposed. We also discuss a generalized approach for removing noise from raw ime series wihou smoohing ou he peaks booms. Beside hese new capabiliies, our experimens show ha Lmark Indexing is considerably fas.. Inroducion Time series daa is ubiquious in science, engineering business. Recenly here has been a surge of ineres in managing his kind of daa, in processing similariybased queries in ime series daabases. Daa mining knowledge discovery in ime series daabases[] have also enjoyed his ineres. Research in similariy-based paern querying can be classified by hree crieria: he similariy model, he daa represenaion, he index srucure. The similariy model defines he semanics of paern queries. Alhough he similariy of wo ime series is direcly compuable, for Currenly wih Cle Corp., El Segundo, CA 95 mos similariy models his is oo expensive in pracice. Insead, feaures wih good properies are exraced from he raw daa o form feaure ses, which hen can be compared for similariy. Each feaure se is used o represen a porion of he original ime series. Then feaure ses are indexed sored based on muli-dimensional indexing srucures. For example, he pioneering work by Agrawal e al[] Falousos e al[] uses Euclidean disance as he similariy model, he coefficiens of he moving-window Discree Fourier Transform (DFT) as he daa represenaion, an -ree as he index srucure. The similariy model has been exended in many differen direcions: aking ime warping ino accoun [, 5,, 7]; allowing ampliude shifing [9, 5, 7]; allowing ime series segmens of differen ampliude scales o be similar [9,, 8, 7]. Some work also akes smoohing or noise removal ino accoun. Rafiel e al[] proposed a similariy measuremen based on moving averages. Agrawal e al[] suggesed eliminaing gaps before ime series segmens are compared. Even he simples similariy measures are ofen oo expensive o apply on raw daa. The siuaion grows worse as he similariy model is made invarian under ransformaions o he daa (see Secion.3). Assuming he oal lengh of he ime series in a daabase is, he search space is for fixed-lengh paern querying for variable-lengh paern querying. Wih a linear ime comparison algorihm, he overall ime complexiy can be respecively. For example, [] uses an algorihm wih ime complexiy o hle ime warping. Real ime series daabases are no queryable wihou a sub-linear ime algorihm. So various feaure exracion mehods have been proposed in order o provide an indexable search space. The majoriy of hese [,, 9,, 7] use a few DFT coefficiens for each ime window. Wavele coefficiens are used in [5]. Shakay [5] suggesed breaking sequences ino meaningful subsequences represening hem using real-valued funcions.

2 (a) MOT (b) BLL (c) DG (d) MIR Figure. Insances of Double Boom paern. These chars are found by our prooype. Given a good daa represenaion, he final issue is how o choose an indexing srucure wih good performance. The -ree, -ree, -ree simple invered files are common choices. Alhough previous work has generalized he similariy model in differen direcions, here is no apparen way o unify all his work under a generalized similariy model. The above argumens can be illusraed by he following brief example: Figure show some insances of he double boom paern. Humans can spo he resemblance beween hese chars almos immediaely, which means hese chars are similar o some degree alhough hey are noisy have differen levels, scales, ime spans. To our bes knowledge, currenly here is no echnique ha can efficienly suppor paern querying using he similariy model implici in hese chars. We also quesion he adequacy of using Euclidean disance as a similariy measuremen. From previous research i has become clear ha ordinary Euclidean disance is a poor similariy measuremen. Is inadequacies are hard o enumerae, bu for example:. Euclidean disance works only on same-lengh segmens. Even a small difference in lengh requires operaions like inerpolaion in order o align ime series segmens. Rafiei Mendelzon [] have also addressed his issue.. Euclidean disance can be srongly influenced by scale (ampliude): similariy in a lower range can be overwhelmed by mild subsequen dissimilariy in a higher range. By conras, similariy among volaile ime series someimes can be relaively insensiive o scale. This is exemplified by Figure, paricularly by recen sock marke rends: since he second half of 997, many Inerne-relaed US socks have followed similar wild growh paerns. Beside hese drawbacks, he presence of noise also affecs he similariy significanly. Noise accompanies almos every real measuremen. Humans usually perceive similariy of paerns wih an implici smoohing procedure. Mos char readers have long known ha every paern is only recognizable on cerain ime scales. In chars wih long ime scales, small flucuaions are reaed as noise. Smoohing is an essenial issue in defining paerns. Mos previous work does no ake smoohing as an inegral par of he process of paern definiion, index consrucion, query processing. Insead, his work ends o apply smoohing echniques firs, hen build an index on he resul. Bu commonly-used smoohing echniques, such as various kinds of moving averages, eiher lag or miss he imporan peaks booms. Peaks booms are generally very significan, have meaning. Smoohing or removing hem can lead o a considerable loss of informaion. Also, he parameers used in curren smoohing echniques ofen lack clear meaning. In his paper, we propose a new echnique called he Lmark Model. Is underlying similariy model, Lmark Similariy, is consisen wih human inuiion episodic memory. Lmark Similariy is defined in a way ha a variey of similariy measuremens each invarian under (i.e., insensiive o, oblivious of) a subse of six basic ransformaions on ime series can be seleced by users. To accomplish his efficienly we also propose a new daa represenaion mehod, a procedure o find a minimal feaure se for any non-degenerae subse of hese ransformaions. A smoohing echnique ha can be parameerized inuiively is also inroduced. Then we reduce he indexing problem o a sring indexing problem.. Similariy model daa represenaion In mos previous work, similariy models daa models are differen. I is hen imporan o esablish a connecion beween he wo. For example, he Parseval heorem relaes poin-wise Euclidean similariy wih a Fourier se- The smoohness of a curve is measured by he frequency of direcion changes. So removing major peaks booms is no necessary when smoohing a curve.

3 ries model. This separaion also makes compleeness (no false dismissals) soundness (no false alarms) wo serious issues in paern querying. Soundness can be guaraneed by checking he original daa. Compleeness is ofen more difficul, because when a search hrough indices fails, here may be no way o avoid scanning he whole daabase. A common sraegy is o relax error olerance allow more false alarms in order o reduce or eliminae false dismissals. Evenually, boh compleeness soundness grow ino performance problems. This separaion beween daa model similariy model is no necessary. In his secion, we inroduce he concep of he Lmark Model, which is boh a similariy a daa model... Lmark concep Researchers in Psychology Cogniive Science have amassed considerable evidence ha human animals depend on lmarks in organizing heir spaial memory [6]. Research ino episodic memory has also produced resuls for organizing memory around lmark evens [, 3]. This all conforms o our daily experience. If one is asked o look a Figure (a) for a shor period hen duplicae he char, a relaively successful sraegy is o memorize he posiions of he urning poins reconnec hem. These urning poins serve as he lmarks in heir chars. The success of his sraegy also implies ha humans, o some exen, consider wo chars similar if heir urning poins are similar he res of he chars are curves ha connec he urning poins. Exreme poins also are significan o char readers. Taking sock prices as an example, every rader would wish he/she had bough (covered) a every local minimum, sold (shored) a every local maximum, oherwise did lile. The curves beween he exreme poins are indifferen o he maximal poenial profi or he opimal rading sraegy. Based on his observaion, we define Lmarks in ime series o be hose poins (imes, evens) of greaes imporance. The gis of he Lmark Model is o use lmarks insead of he raw daa for processing. Differen lmarks arise in differen applicaion domains, heir definiion can range from simple predicaes (for example, local maxima, local minima, inflecion poins, ec.) o more sophisicaed consrucs. Since mos imporan poins possess some mahemaical properies, a more generic way is o caegorize hem mahemaically. We call a poin an -h order lmark of a curve if he -h order derivaive is on he poin. So local maxima minima are firs-order lmarks, inflecion poins are second-order lmarks. The decision as o which kinds of poins can be lmarks amouns o a radeoff beween wo exremes. The more differen ypes of lmarks in use, he more accu Original Daa Rebuil from lmarks Rebuil from DFT 6 8 Figure. Cisco sock price from 6//998 o /3/998. The original ime series, he wo ime series reconsruced from firs-order lmarks from DFT coefficiens. raely a ime series will be represened, hence he more deail paerns are defined. However, using fewer lmarks will resul in smaller index rees. The decision abou where o balance his radeoff should be based on he naure of he daa. In our empirical sudy in sock marke daa, his decision was resolved easily. As shown in Table, even for IBM sock (which is supposed o be comparably more sable han oher socks), poins ou of almos half of he records are eiher local minima or maxima. Also, he normalized error (Appendix B) is reasonably small when he curve is reconsruced from he lmarks. So, for he res of his paper, we resric discussion o only firs-order lmarks (alhough in oher applicaions differen lmarks migh be more useful). A somewha surprising fac abou lmarks is ha he more volaile he ime series, he less significan he higher-order lmarks. Only slowly changing ime series, in which he disances beween exrema are long, require higher-order lmarks for accurae reconsrucion. Given a sequence of lmarks, he curve can be reconsruced by segmens of real-valued funcions. In Appendix A, we show how o reconsruc ime series from a sequence of lmarks. Figure shows he ime series reconsruced from lmarks DFT. Noe ha he DFT uses only coefficiens o represen he window of lengh we have chosen. In a ime series of lengh, here are roughly coefficiens o be processed because he DFT has o be performed on every railing window. Our sudy of socks in S&P5 index shows he average number of lmarks is less han, regardless of he ime span. The DFT coefficiens lmarks are no he acual amoun of informaion ha needs o be sored.

4 (x,y) d (x,y) v MDPP(D,P) d<d *v/(y+y)<p Figure 3. Minimal Disance/Percenage Principle days) regards a gain or loss as significan, hen he/she simply uses MDPP(, ) o smooh he daa. This approach ensures ha no price movemen larger han is smoohed ou. In conras, he DFT does no scale as well as he MDPP. Figure 5 shows he error generaed from DFT MDPP. This is a fair comparison because he DFT mus be performed on every railing window (assuming he DFT is performed on all elemens in a sliding fixed-size window). The Lmark Model has anoher desirable propery ha all he peaks (local maxima) booms (local minima) are preserved, while hey are ypically filered ou by boh he DFT DWT (being capured in coefficiens of higher frequencies), as shown in Figure... Smoohing Original Daa MDPP(,.5) MDPP(7,.5) Real world daa are usually noisy. Even for he mos ypical paern like Figure, one canno expec smooh ransiions from each major lmark (for example, he wo booms he local maximum beween hem) o he nex. Low-pass filers like he DFT moving averages are ofen inroduced o eliminae noise in hese ransiions. Moving averages, like he DFT, end o smooh ou peaks booms along wih noise. Moving averages are also known o be lagging indicaors, which have a phase delay comparing o he original daa. While here are infiniely many possible ways o classify lmarks, we inroduce he Minimal Disance/Percenage Principle (MDPP). MDPP is a smoohing process ha can be implemened as a linear ime algorihm. I is defined as follows: Given a sequence of lmarks, a minimal disance a minimal percenage, remove lmarks if We use MDPP(, ) o represen his process. Figure 3 illusraes how MDPP works. Figure shows he effec of MDPP while using differen disances percenages. Table shows how he parameers affec he number of remaining lmarks he normalized error. The real power of he Lmark Model MDPP can be illusraed by he las cell in Table. We can use ( ) of he original poins o represen he whole ime series wih only normalized error. This is no a special case. Our sudies on financial daa shows almos every sock wih sufficienly long hisory gives similar resuls. The parameers of MDPP have inuiive meaning. For example, if a sock rader rades once a week (5 business (a) Varying he MDPP disance parameer Original Daa MDPP(7,.5) MDPP(7,.7) 6 8 (b) Varying he MDPP percenage parameer Figure. Sensiiviy of he Minimal Disance/ Percenage Principle. A difficul decision o make wih he DFT approach is which window size o choose. In conras, MDPP is almos invarian of he window size. In fac, neiher raw lmarks nor MDPP is based on moving windows, so he lengh of ime series has very lile effec on he qualiy of he Lmark Model. The MDPP preserves he offses of each lmark. I is possible o design differen smoohing mehods ha remove he noisy segmens suppor a similariy model similar o he one inroduced by Agrawal e al[].

5 D/P % % 6% 8% % % % 6% 8% 6/.5% 56/.7% 56/.8% 56/.8% 5/.8% 5/.8% 5/.8% 5/.8% 5/.8% /% /.5% /.6% 6/.7% /.7% /.7% /.7% /.7% /.7% 6 /.% 9/.9% 58/3.% 36/3.% 3/3.% 3/3.% 8/3.% 8/3.% 6/3.% 8 /.% 78/3.% 36/3.5% /3.7% /3.9% 96/% 9/% 9/% 9/% /.% 7/3.3% 6/3.6% /3.9% 88/.3% 8/.3% 8/.3% 8/.3% 8/.3% /.% 7/3.3% 6/3.6% /3.9% 8/.% 76/.% 7/.% 7/.% 7/.% /.% 7/3.3% /3.8% 98/.% 78/.6% 68/.8% 66/.7% 66/.7% 66/.7% 6 /.% 66/3.% 6/% 9/.3% 7/.8% 6/5% 58/.9% 58/.9% 58/.9% 8 /.% 66/3.% /.% 86/.5% 6/5% 5/5.% 5/5.% 5/5.% 5/5.% Table. The number of remaining lmarks he normalized error generaed by MDPP wih differen minimal disances(d) minimal percenages(p). The original daa conains 85 closing prices of IBM. The number of raw lmarks is 38. For example, afer applying MDPP(,%), he number of remaining lmarks is he normalized error is %. Normalized Error(%) Window Size MDPP(3,3%) MDPP(5,5%) MDPP(7,7%) DFT(3) DFT(5) DFT(7) Figure 5. Normalized error generaed by he MDPP DFT. DFT( ) is he ime series reconsruced from coefficiens. The daa used here reflecs differen ime series window lenghs for he daily Dow Jones Indusrial Average ending on /3/ Transformaions A similariy measure is invarian under a family of ransformaions if applying hem o ime series never alers similariy. As previously menioned, he more ransformaions included in a similariy model, he more powerful he similariy model. Mos relaed work has considered wo or hree ransformaions. In his paper, we consider six. Given an univariae ime series, assume is a coninuous funcion obained by inerpolaing beween he poins in. The ransformaions are each defined as a family of funcionals:. Shifing such ha consan.. Uniform Ampliude Scaling a consan. such ha 3. Uniform Time Scaling such ha a posiive consan.. Uniform Bi-scaling such ha is a posiive consan. where is a where is where is where 5. Time Warping(or Non-uniform Time Scaling)! such ha! # % where is posiive monoonically increasing. 6. Non-uniform Ampliude Scaling such ha every, ('!& if only if )'. where for These ransformaions can be composed o form new ransformaions. The composiion order is flexible, in he sense ha for any wo ransformaions *,+ -/., here exis alernaive )' ' such ha *3+5/ *3.:8. The composiion is also idempoen, in he sense ha for any ransformaion * parameers, here exiss a parameer ; such ha *,<=>*3+*?.. Wih hese wo properies, we can use basic ransformaions o represen a composie ransformaion. The purpose of inroducing hese ransformaions is no acually o perform hem, bu insead o exend he semanics of similariy o ignore hem. For example, ime series segmens are similar (acually: idenical) modulo Shifing if here exis a consan such ha for all in he Puing i anoher way,

6 6 8 6 f(x) f()+k f() f()*k f() f(k*) 6 8 (a) Shifing (b) Uniform Ampliude Scaling (c) Uniform Time Scaling f() k*f(/k) f() f(g()) (d) Uniform Bi-scaling (e) Time Warping (f) Non-uniform Ampliude Scaling Figure 6. The six ransformaions in he Lmark Model f() g() he se of funcions ha are similar modulo Shifing is invarian under Shifing ransformaions. There is no need o find a specific value for a consan or funcion in he definiions above. In Secion 3 we will show ha no every composiion is meaningful... Lmark similariy The error olerance in mos similariy models is a single value ha is compued from poinwise differences in ampliude. This simple error measuremen is no longer sufficien when ransformaions like Uniform Time Scaling Uniform Bi-scaling are aken ino accoun. In he Lmark Model, drif on he ime axis also can be significan. Furhermore, he scales on he ampliude-axis imeaxis are incomparable, which means he -dimensional Euclidean disance is meaningless. Hence we mus generalize he dissimilariy measuremen. Definiion Given wo sequences of lmarks '3 ' ' where ' ' ', he disance beween he -h lmarks is defined by ' 3 ' ' where ' '! 8 # 8 8 # 8 &% ) #) 8 ) ) 8 &% if ' oherwise if (' oherwise The disance beween he wo sequences is ' ' ' where is a vecor norm, viewing boh ' ' as -vecors. The max norm +* -,/. ofen works well on financial ime series. Abusing language, we use o denoe he disance beween wo ime series segmens when he parameers 3 are clear from conex. 3 We define ' & ' ' if ' ' '5 '. Lemma The lmark disance funcion saisfies he riangle inequaliy. Tha is, for any lmark sequences, ', ' ', ' ' 6' ' ' ' '. Given fixed MDPP parameers, since each ime series segmen is

7 * * mapped o a unique sequence of lmarks, he inequaliy propery also applies. Wih his dissimilariy measuremen, we now can define he similariy in he Lmark Model. Definiion A lmark similariy relaion is a binary re- where are MDPP parameers, is a se of basic ransformaions, is an error olerance on he ime-axis is an error olerance on he ampliude-axis. Given wo ime series segmens, le be he lmark sequences afer smoohing. Then if only if here exis wo parameerized ransformaions of such ha. laion on ime series segmens defined by a 5-uple Figure 7 illusraes he operaional srucure of lmark similariy. Exrac Raw Lmarks Smooh wih MDPP Apply Transformaion Compare Figure 7. The operaional srucure of Lmark similariy. In comparing wo ime series segmens, we firs exrac lmarks apply MDPP on he raw lmarks. The dissimilariy of he wo ime series segmens is he minimal disance error beween he lmark sequences under he given se of ransformaions. 3. Daa represenaion Up o his poin, we have used only simple coordinaes of lmarks in modeling ime series. Bu a sequence of lmarks denoed by coordinaes represens only a paricular ime series segmen. The similariy we seek is o rea a family of ime series segmens as equivalen under he six ransformaions we inroduced. The soluion we propose is o use various feaures of lmarks ha are invarian under he ransformaions o represen ime series. Given a sequence of lmarks where, we can define as many feaures as possible. In his paper, we use a small feaure se * 3 for demonsraion purposes, defined by:. All hese feaures are generaed from he coordinaes of lmarks, bu each has differen characerisics. In paricular, every feaure is invarian under some ime series ransformaions. Table indicaes which feaures are invarian under each ransformaion: The invarian feaure se of a composie ransformaion is he inersecion of he invarian feaure ses of is componens. By observing he invarian ses, i is easy o see ha no every composiion of hese ransformaions is meaningful. Time series migh be over-ransformed, he similariy relaion become a complee relaion (in which each segmen is similar o all ohers) if he ime series segmens are long enough. This happens when he ransformaion has an empy invarian feaure se. For example, under Time Warping Non-uniform Ampliude Scaling of a ime series, segmens can be ransformed o any shape if hey are sufficienly long ha he inersecion of heir invarian se is empy. On he oher h, one basic ransformaion can be subsumed by anoher ransformaion. For example, Uniform Time Scaling is subsumed by Time Warping. A composie ransformaion ha conains boh UTS TW is idenical o he ransformaion wihou UTS as a componen. A family of ime series can be reconsruced from he values of feaures. Assume * * * is a feaure se. Given a mulivariae sequence where *, we define he quoien funcion such ha ime series segmen he lmarks of have he same feaure value as 3 is used only when a user requires a paern o appear a cerain offse. We found his happened only rarely, so is no included in feaure lis.

8 Shifing (SH) Uniform Ampliude Scaling (UAS) Uniform Time Scaling (UTS) Uniform Bi-scaling (UBS) Time Warping (TW) Non-uniform Ampliude Scaling (NAS) Table. Invarians of ransformaions Abusing language slighly, we le * denoe he family of ime series segmens defined by values in he feaure se * of a sequence of lmarks, where is clear from conex. By observing he dependency relaion, we have he following lemma. * Lemma If is a se of feaures, denoes disjoin union: * * * * * *,, * * * * * * The above lemma should be inerpreed as a se of rewrie rules ha reduces he number of feaures. Having fewer feaures o exrac manipulae leads o more efficien execuion. Example A user chooses o consruc a lmark se under Shifing, Uniform Time Scaling Time Warping. The feaure se is. By Lemma, we can use only as he feaure se. Given a error olerance, he range of he values of an invarian is bounded, as shown in Figure 8. We use o denoe he lower bound upper bound of respecively. Table 3 shows he lower upper bounds of he feaures discussed in his paper. These lower upper bounds can be simplified if he ampliudes of ime series elemens are always posiive.. Querying lmark sequences Unlike oher se-oriened daa represenaions, lmarks are sequenial. Based on his fac, lmark sequences are more like srings han muli-dimensional objecs. Consequenly, sring indexing echniques are more suiable han R-ree-like srucures. Ampliude L i- (-d)h (+d)h h L i (-da)y y (+da)y Time Figure 8. Possible rangeof a lmark wih error olerance (denoed in he figure). Our approach is o adap spaial indexing srucures for query processing. A major difference beween emporal daa sequences srings is ha srings have a welldefined, fixed alphabe. So, we consruc an alphabe o ranslae lmark sequences o srings. Indexing mulidimensional spaial objec sequences (in his case, lmark sequences) is sill a rarely discussed opic. For his purpose, we propose he -Tree [6], an index srucure for subsequence maching of spaial objecs. Due o space limiaions, we canno explain he srucure of -Tree in deail, bu very briefly: he -Tree is a combinaion of wo ree srucures: (i) he X-ree, which provides a clusering mehod of spaial objecs. The -Tree convers he spaial objecs ino binary encodings according o clusering. A parial order in he binary encodings reveals relaionships among he original spaial objecs. (ii) The suffix ree, which implemens subsequence maching on sequences of he binary encodings. A dominan facor in query processing performance is he size of he index. In Figure 9, we show he resuls of some experimens. The daa for experimen is he -year closing price of socks in he Sard & Poor 5 index. We use he Java floa ype for prices, so each occupies

9 Lower Bound Upper Bound,,/. -, -,/., -,/. Table 3. The lower upper bounds of feaures byes. Index Size (Mega byes) MDPP(,%) MDPP(,%) MDPP(6,6%) MDPP(8,8%) MDPP(,%) MDPP(,%) 6 8 # of records in daabase (x) Figure 9. Index size vs. daabase size 5. Conclusion In his paper, we have proposed he Lmark Model, a new model for similariy-based paern querying in ime series daabases. The Lmark Model inegraes similariy measuremen, daa represenaion smoohing echniques in a single framework. Concepually, he model is based on he fac ha people recognize paerns in chars by idenifying imporan poins. The idea of using lmarks also urns ou o have good mahemaical properies. Furhermore, lmarks can represen ime series more accuraely wih less informaion. In conras, DFT-based echniques require compuing low-frequency coefficiens for every sliding window, which can resul in longer processing ime. We have inroduced he Minimal Disance/Percenage Principle (MDPP) as a smoohing mehod for he Lmark Model. The MDPP parameers are inuiive. We have shown ha he MDPP is scalable linear-ime compuable. The Lmark Model suppors a very general similariy model ha permis similariy comparison modulo six very naural ransformaions of ime series. This is done by comparing feaures ha are invarian under hese ransformaions. The flexibiliy of his model ss in conras wih he rigidness of similariy models ha ignore arificial ransformaions /or a limied number of ransformaions. For example, DFT-based echniques permi similariy comparison modulo Shifing (by ignoring he -h coefficien) Uniform Ampliude Scaling (by soring normalized coefficiens insead of heir absolue values). However, i is generally no easy for DFT-based echniques o incoperae he oher four ransformaions discussed in his paper. We have proposed a wo-dimensional dissimilariy measuremen funcion ha considers ime drif ampliude difference separaely. The relaion beween error olerance invarian feaures is also designed so ha users only need o work on seing he value of he error olerance wihou being disraced by he choice of invarians. Summarizing, we feel he Lmark Model is inuiive in several ways. Firs, i is designed so ha every parameer error olerance has an inuiive meaning. The similariy model is defined relaive o ransformaions which capure six naural ways ha people feel wo ime series mach. Finally, he Lmark Model does no require some cerain assumpions, such as ha several Discree Fourier Transform coefficiens is a good model for ime series segmens, or ha similariy based on Euclidean disance is reasonable. References [] R. Agrawal, C. Falousos, A. N. Swami. Efficien similariy search in sequence daabases. In FODO, 993. [] R. Agrawal, K.-I. Lin, H. S. Sawhney, K. Shim. Fas similariy search in he presence of noise, scaling, ranslaion in ime-series daabases. In VLDB, 995. [3] G. A.M. Wha memory is for. Behavioral Brain Sciences, (), 997. [] D. J. Bernd J. Clifford. Finding paerns in ime series: A dynamic programming approach. In Advances in Knowledge Discovery Daa Mining, pages 9 8. MIT Press, 996. [5] K.-P. Chan A.-C. Fu. Efficien ime series maching by waveles. In ICDE, 999.

10 [6] K. Cheng M. Spech. Mechanisms of lmark use in mammals birds. In S. Healy, edior, Spaial Represenaion in Animals. Oxford Universiy Press, 998. [7] K. K. W. Chu M. H. Wong. Fas ime-series searching wih scaling shifing. In PODS, 999. [8] G. Das, D. Gunopulos, H. Mannila. Finding similar ime series. In PKDD, 997. [9] D.Q.Goldin P. Kanellakis. On similariy queries for ime-series daa: Consrain specificaion implemenaion. In Inernaional Conference on he Principles Pracice of Consrain Programming, 995. [] C. Falousos, M. Ranganahan, Y. Manolopoulos. Fas subsequence maching in ime-series daabases. In SIG- MOD, 99. [] U. M. Fayyad, G. Piaesky-Shapiro, P. Smyh, R. Uhurusamy, ediors. Advances in Knowledge Discovery Daa Mining. MIT Press, 996. [] M. Humphreys, J. Wiles, S. Dennis. Toward a heory of human memory: Daa srucures access processes. Behavioral Brain Sciences, 7(), 99. [3] D. S. Parker, E. Simon, P. Valduriez. Svp: A model capuring ses, liss, sreams, parallelism. In Very Large Daa Bases (VLDB) Conference, 99. [] D. Rafiei A. O. Mendelzon. Similariy-based queries for ime series daa. In SIGMOD, 997. [5] H. Shakay S. B. Zdonik. Approximae queries represenaions for large daa sequences. In ICDE, 996. [6] H. Wang C.-S. Perng. The : An index srucure for subsequence maching of spaial objecs. Technical Repor 995, Universiy of California, Los Angeles, Compuer Science Deparmen, 999. [7] B.-K. Yi, H. Jagadish, C. Falousos. Efficien rerieval of similar ime sequences under ime warping. In ICDE, 998. Appendix The conceps inroduced in his appendix are no needed in query processing. Insead, hey only serve for demonsraing he qualiy of he lmark model. The similariy model he daa represenaion inroduced in his paper are muually dependen, i.e. he original ime series are idenical o heir lmark represenaions if measured in he lmark similariy model. To avoid his self-reference in showing he accuracy of he lmark model, i is necessary o provide a way o reconsruc a ime series from is lmark represenaion (like invering he DFT from a few coefficiens). However, we again need a good poin-wise similariy measuremen. As remarked in Secion, he Euclidean disance has many undesirable properies, so we propose a new similariy measuremen. A. Reconsrucing ime eries egmens from lmarks Given wo firs-order lmarks we use a cubic funcion, o inerpolae beween wo lmarks. Since lmarks are exreme poins, we have he curve To obain he coefficiens, le. We obain solve % % ' ( % % ' ) %& % %& %&* % %)) B. Normalized error Definiion 3 Given wo sequences of lengh,, he normalized disance funcion is defined by: + -, '! # ' The normalized disance funcion has hree imporan properies:. Symmeric:.. Invariable o ampliude scale: where.,. 3. Non-accumulaive: Assume is he concaenaion op- are lmark sequences where, hen '. eraor, % %&