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1 Tme Wa Edt Dstance PIERRE-FRNÇOIS MRTEU VLORI UNIVERSITE EUROPEENNE DE RETGNE CMPUS DE TONNIC T. YVES COPPENS P VNNES CEDEX FRNCE FERURY 28 TECNICL REPORT N : VLORI.28.V5 FERURY 28 bstact: Ths techncal eot detals a famly of tme wa dstances on the set of dscete tme sees. Ths famly s constucted as an edtng dstance whose elementay oeatons aly on lnea segments. secfc aamete allows contollng the stffness of the elastc matchng. It s well suted fo the ocessng of event data fo whch each data samle s assocated wth a tmestam not necessaly obtaned accodng to a constant samlng ate. Some oetes vefed by these dstances ae oosed and oved n ths eot. Keywods: Dynamc Tme Wang Elastc Dstances Stffness Contol Tme Sees matchng Tmestamed Data Event Data Lnea Segment Matchng.

2 I. Intoducton t the juncton of symbolc edt dstances [] [3] [5] [7] [] and dynamc tme wang measues [2] [9] [4] [] we oose a famly of Tme Wa Edt Dstance TWED that we denote to efe to the two aametes that chaacteze the famly namely the ga enalty and the stffness aamete. We fst ne wheneve >. Pooston : s a dstance metc. 2. Pooston 2: s ue bounded by twce the L dstance. 3. Pooston 3: s an nceasng functon of and. and then we ove successvely that 4. Pooston 4: Ue-boundng the dstance between a tme sees and ts ecewse constant olygonal aoxmaton Futhe detals and exements on ae descbed n [6].

3 II. Defntons Let U be the set of fnte dscete tme sees: U { / N } { Ω} whee Ω s the emty tme sees wth null length. Let be a tme sees wth dscete ndex vayng between and. Let be the th samle of tme sees. We wll consde that S T whee k S R wth k embeds the multdmensonal sace vaables and T R embeds the tme stam vaable so that we can wte a ' a t whee a S and t T wth the a a condton that t > t wheneve >j tme stam ae stctly nceasng n the seuence of a a j samles. Let us ne as: Γ Λ Mn Γ b' Γ Λ b' wth Γ Γ b' Γ Λ b' Λ d d d b' b' b' d b' whee d s any dstance on R k. In actce we wll choose d b' d ab.d t t whee s a aamete whch chaactezes the LP L a b stffness of the elastc dstance coesonds to a ga enalty. and any ostve constant element n R that

4 The ecuson s ntalzed settng: j j k k d b' d b' k j wth b' by conventon. k k k {..} {.. } III. Pooston : s a dstance on the set U of fnte dscete tme sees Poof: P: non-negatvty Fo all n U U let m. Non-negatvty of s oved by nducton on m. P s tue fo m by nton of and the nducton hyothess holds. Suose P s tue fo all {.. n } m fo some n>. Then fo all n U U such that m n as and ae assumed ostve and as the non-negatvty of dstance d holds s necessay non-negatve showng that P s tue fo all m {.. n}. y nducton P holds fo all m N.

5 P2: dentty of ndscenbles Fo all n U U f by nducton on that f... b'. It s easy to show then and { } then d b' leadng to. Now consde the backwad ooston P 2: nducton on m.. P 2 s oved by P 2 s tue fo m. Suose P 2 s tue fo all {.. n } m fo some n>. Then fo all n U U such that m n and we have necessaly: d b' d b'. We vefy that the cases whee d b' b' o d ae mossble snce d b ' b ' and d a ' a ' ae stctly ostve the eason beng that tme stams ae stctly nceasng. Thus and d b' d b' leadng to and a ' ' b. Fnally and necessaly. P3: Symmety Poof: Snce the dstance d on the samle sace S T s symmetc t s easy to show that s symmetc fo all n U U by nducton on m.

6 P4: Tangle neualty Fo all C n U U U C C. Poof: We wll ove P4 by nducton on m. P4 s tue fo m snce Ω Ω Ω Ω Ω and the nducton hyothess holds. Ω 4: Suose P4 s tue fo all {.. n } m fo some n>. Let Σ C. Then fo all C n U U U such that mn we have bascally 9 dffeent cases to exloe fo the decomoston of and C : st Case: f C C Γ Γ b' b' then Σ C d C d and d vefes the tangula neualty. d b' d b' C d b' d b' snce 4 ales 2 nd Case: f C C Γ Λ b' Γ b' then

7 C d C d d C d b' d b' C b' d b' ales 4 the tangula neualty vefes Σ 3 d Case: f Γ Λ Γ b' C C then C d Σ C d C ales 4 Σ 4 th Case: f Λ Γ Λ Γ C C b' then Σ d C C d C ales 4 Σ 5 th Case: f Λ Γ Γ C C b' then C d C d C ales 4 Σ

8 6 th Case: f Λ Γ Λ Γ C C then C d C d C ales 4 Σ 7 th Case: f Λ Γ Γ b' C C b' then ales. 4 satsfes the tangle neualty C d C d d C b' d b' C b' d b' d Σ 8 th Case: f Λ Γ Λ Γ b' C C then: C d C C d ales 4 Σ 9 th Case: f Λ Γ Λ Γ b' C C b' then: ales. 4 C C Σ

9 So oety P4 holds fo all m n {..n}. y nducton P4 holds fo all m n N and so P4 holds fo all C n U U U. IV. Pooston 2: s ue bounded by twce the L dstance. Pooston 2: 2 > X Y U X Y 2 D X Y wheneve X and Y have the same length. LP Poof: let us consde the seuence of edtng oeatons consstng n m match oeatons whee m s the length of the X and Y. Ths seuence has a cost eual to twce the L-dstance between the two tme sees X and Y. Snce s eual to the cost of the otmal seuence of edt oeatons the esult follows. V. Pooston 3: s an nceasng functon of and Pooston 3: 2 > ' ' X Y U X Y ' ' X Y Poof: Let us consde one of the otmal seuences of edtng oeatons evaluated wth the tule ' ' wth mnmal cost eual to X Y. If we kee ths seuence of edtng ' ' oeaton whle elacng ' ' wth n all the elementay oeaton costs we get a cost fo ths seuence that s lowe than X Y but geate than the cost of the otmal ' ' seuence X evaluated usng. The esult follows. Y

10 VI. Pooston 4: Ue-boundng the dstance between a tme sees and ts ecewse constant olygonal aoxmaton. We ne as a Pece Wse Constant oxmaton PWC of tme sees contanng constant segments and samles. Ths aoxmaton can be obtaned usng any knd of soluton fom heustc to otmal solutons let say the otmal soluton smla to the one oosed n [8]. and have the same numbe of samles namely. Let ~ be the tme sees comosed wth the segment extemtes of. ~ contans samles. Let us smlaly ne ' and fom tme sees ' ~. Pooston 4: X ~ > [ ; [ X U X T 2 whee T s the tme dffeence aveage between two successve samles nsde the ecewse constant segments of the aoxmaton. The oof fo ths ooston s staghtfowad: let us consde the seuence of oeatons consstng n match oeatons fo the end extemtes of the ecewse constant segments and - delete oeatons fo the set of samles n X that ae not end extemtes of the ecewse constant segments. In ths seuence each match oeaton has n aveage the cost / T and each delete oeaton has a fxed enalty and a enalty ootonal to the tme stam dffeence between two successve samles. tmestams tmestams. Then the cost fo ths seuence of edtng oeatons s. T T. Fnally the otmal seuence of edtng oeatons has a cost ~ X X lowe o eual to T 2.

11 VII Concluson We have oosed a famly of tme wa edt dstances fo tme sees matchng. Ths famly nvolves two aametes: the stffness aamete that contols the elastcty of the dstance and the ga enalty that s at of the cost nvolved n the nset o delete oeatons. We have shown that the oosed measue: - s a metc on the set of dscete and fnte tme sees. - s ue bounded by twce the L-dstance. - s an nceasng functon of the ga enalty and the stffness aamete. Futhe moe the dstance between a tme sees and ts ecewse constant aoxmaton s ue bounded by an exesson that only deends on the lengths of the tmes sees the numbe of segments of ts aoxmaton and the two aametes and.

12 Refeences [] L. Chen & R. Ng. On the maage of L-nom and edt dstance. In Poc. 3th Int'l Conf. on Vey Lage Data ases [2] G. Das D. Gunoulos and. Mannla. Fndng Smla Tme Sees. In Poceedngs of the Confeence on Pncles of Knowledge Dscovey and Data Mnng 997. [3] Gotoh O. 982 "n moved algothm fo matchng bologcal seuences." J. Mol. ol [4] E. J. Keogh and M. J. Pazzan smle dmensonalty educton technue fo fast smlaty seach n lage tme sees databases 4th Pacfc-sa Confeence on Knowledge Dscovey and Data Mnng PKDD [5] V. Mäknen. Usng Edt Dstance n Pont-Patten Matchng. In Poc. 8th Symosum on Stng Pocessng and Infomaton Reteval SPIRE 2 IEEE CS Pess [6] Mateau P.F. Tme Wa Edt Dstances wth Stffness djustment fo Tme Sees Matchng IEEE Tansactons on Patten nalyss and Machne Intellgence IEEE Comute Socety Dgtal Lbay. IEEE Comute Socety :-5 <htt://do.eeecomutesocety.og/.9/tpmi.28.76> 3 l 28. [7] Needleman S.. and C.D. Wunsch. 97. geneal method alcable to the seach fo smlates n the amno acd seuences of two otens. Jounal of Molecula ology 48: [8] Peez J.C. Vdal E. Otmum olygonal aoxmaton of dgtzed cuves Patten Recognton Lettes 5: [9]. Sakoe and S. Chba " dynamc ogammng aoach to contnuous seech ecognton" n Poc. 7th Int. Cong. coust. udaest ungay ug [] V. M. Velchko and N. G. Zagouyko "utomatc ecognton of 2 wods" Int. J. Man-Machne Studes vol [] R.. Wagne Mchael J. Fsche The Stng-to-Stng Coecton PoblemJounal of the CM JCM Volume 2 Issue P: