Index-based Most Similar Trajectory Search

Transcription

1 Index-based Ms Similar rajecry Search Elias Frenzs, Ksas Grasias, Yannis hedridis Labrary f Infrmain Sysems eparmen f Infrmaics Universiy f Piraeus Hellas echnical Repr Series UNIPI-ISL-R-6- Nvember 6

2 Index-based Ms Similar rajecry Search Elias Frenzs, Ksas Grasias, Yannis hedridis eparmen f Infrmaics, Universiy f Piraeus, Greece {efrenz, grasias, yhed}@unipi.gr Absrac he prblem f rajecry similariy in mving bjec daabases is a relaively new pic in he spaial and spaiempral daabase lieraure. Exising wr fcuses n he spaial nin f similariy ignring he empral dimensin f rajecries and disregarding he presence f a general-purpse spaiempral index. In his wr, we address he issue f spaiempral rajecry similariy search by defining a similariy meric, prpsing an efficien apprximain mehd reduce is calculain cs, and develping nvel merics and heurisics suppr -ms-similar-rajecry search in spaiempral daabases expliing n exising R-reelie srucures ha are already fund here suppr mre radiinal queries. Our experimenal sudy, based n real and synheic daases, verifies ha he prpsed similariy meric efficienly rerieves spaiemprally similar rajecries in cases where relaed wr fails, while a he same ime he prpsed algrihm is shwn be efficien and highly scalable.. Inrducin Wih he rapid grwh f wireless cmmunicains and psiining echnlgies, he cncep f Mving Objec aabases (MO) has been in he cre f he spaial and spaiempral daabase research. An ineresing ype f query ha is useful in MO search is he s-called rajecry similariy prblem, which aims find similar rajecries f mving bjecs. illusrae he prblem, cnsider he fllwing example. Suppse ha he mer newr f a ciy has been recenly exended, iniiaing a new ransprain line, in view f prviding ranspr services a majr par f he residens f he ciy suburbs. his mer newr exensin requires he re-designing f he exising ransprain newr (buses, ram, rlleybuses, ec.). Expers in he field wuld be assised if hey culd pse queries abu he similariy beween he rajecries f he exising ranspr means and he new mer line. As such, hey wuld be able, fr example, change he imeable f a bus line, if i maches in a cerain day wih he imeable f he new mer line, r even abr i. handle such queries efficienly, MO sysems shuld include mehds fr answering he scalled Ms-Similar-rajecry (MS) search als discussed in [4]. rajecry similariy search is a relaively new pic in he lieraure; he majriy f he mehds prpsed s far are based n eiher he cnex f ime series analysis r he Lnges Cmmn SubSequence (LCSS) mdel [] and he recenly prpsed Edi isance n Real Sequence (ER) [5]. Hwever, all hese mehds have he main drawbac ha hey eiher ignre he ime dimensin f he mvemen, herefre calculaing he spaial (and n he spaiempral) similariy beween he rajecries, r assume ha he rajecries are f he same lengh and have he same sampling rae. exemplify he prblem derived when differen sampling raes are presen, cnsider Figure presening w rajecries and Q wih heir psiin being sampled in differen raes. Figure. rajecries wih differen sampling raes Q While Q and sample heir psiin 4 and 3 imes respecively, hey have apprximaely he same lengh raversing hrugh he same area. hugh he w rajecries are bviusly similar, mehds based n he LCSS r he ER mdel cann deec his ind f similariy since hey ry mach rajecry sampled psiins ne by ne, which clearly des n happen in he abve (real wrld) example. Mrever, he majriy f he prpsed appraches expli specialized index srucures in rder prune he search space and rerieve he ms similar a query rajecry. he challenge acceped in his paper, is efficienly suppr he -MS search in MOs sring hisrical rajecry infrmain, expliing exising R-ree-lie srucures which can als be used suppr her ypes f queries as discussed in [] and [6]. Our main cnribuins are ulined as fllws: We define a dissimilariy meric (ISSIM) fr he measuremen f he spaiempral dissimilariy beween w rajecries and we prpse an efficien apprximain mehd vercme is csly calculain. Adping he MINIS calculain beween a rajecry and an index nde prpsed in [6], we

3 presen a se f nvel merics, and prvide several lemmas, be used fr pruning. Using he abve merics, we prpse a bes-firs query prcessing algrihm perfrm -MS search n R-ree-lie srucures. We cnduc a cmprehensive se f experimens ver synheic and real daases demnsraing ha he prpsed similariy meric efficienly rerieves spaiemprally similar rajecries in cases where relaed wr fail, and he prpsed MS search algrihm is highly scalable and efficien, in erms f execuin ime and pruned space. We have pin u ha his algrihm des n require any dedicaed index srucure and can be direcly applied any member f he R-ree family used index rajecries. he bes f ur nwledge, his is he firs wr prviding echniques fr a spaiempral index suppr bh classical range, plgical and similariy based queries. he res f he paper is srucured as fllws. Relaed wr is discussed in Secin, while Secin 3 inrduces he dissimilariy meric, as well as he se f merics ha suppr ur pruning sraegies. Secin 4 describes in deail he bes-firs query prcessing algrihm perfrm MS search ver hisrical rajecry infrmain. Secin 5 presens he resuls f ur experimenal sudy and Secin 6 cncludes he paper giving hins fr fuure wr.. Relaed Wr Similariy search has been well sudied in he ime series analysis dmain. As a measure f apprximae maching, Agrawal e al. [] prpsed he uilizain f he iscree Furier ransfrmain (F). An alernaive ime series maching echnique hrugh dimensin reducin was prpsed by Chan and Fu [3], using he iscree Wavele ransfrmain (W). In rder cmpare sequences wih differen lenghs, Bernd and Cliffrd [] used he ynamic ime Warping (W) echnique ha allwed sequences be sreched alng he ime axis minimize he disance beween sequences. Alhugh W incurred a heavy cmpuain cs, i was mre rbus agains nise. In [] an indexing mehd fr prcessing shapebased similariy queries fr rajecry daabases was presened. he prpsed mehd was based n Euclidean isance. Hwever i culd be applied nly n rajecries wih same lenghs being valid during he same ime inerval. Cai and Ng [4] prpsed he uilizain f Chebyshev plynmials fr apprximaing and indexing rajecries fr similariy maching purpses. Sill, his mehd suffered frm he requiremen ha he rajecries shuld be f he same lengh (in erms f he number f spaiempral pins ha are cmpsed f). Vlachs e al. [9] presened a disance measure ha allwed find similar rajecries under ranslain, scaling and rainal ransfrmains. he firs sep f heir mehd was he mapping f each rajecry a rajecry in a rain invarian space. Fr he calculain f he disance beween w rajecries in he new rain invarian space, he W echnique was uilized. Saurai e al. [3] prpsed an imprved versin f W, he Fas search mehd fr ynamic ime Warping (FW), based n a new lwer bunding measure fr he apprximain f he ime warping disance. hey prved ha FW culd prune a significan prin f he search space, leading a significan reducin f he search cs. Recenly, Lin and Su [] have sudied he ime independen similariy search prblem f mving bjec rajecries. he ne way disance (OW) funcin is inrduced fr cmparing he spaial shapes f rajecries alng wih apprpriae algrihms fr cmpuing OW. heir experimenal sudy shws ha he adpin f OW funcin uperfrms W algrihm in erms f precisin and perfrmance. Several appraches are based n he Lnges Cmmn Sub Sequence (LCSS) similariy measure. LCSS measure maches w sequences by allwing hem srech, wihu rearranging, he sequence f he elemens, bu allwing sme elemens be unmached (which is he main advanage f he LCSS measure cmpared wih Euclidean isance and W). herefre, LCSS can efficienly handle uliers and differen scaling facrs. Vlachs e al. [] adped he uilizain f he LCSS mehd. Inrducing w similariy measures allwing ime sreching and ranslains respecively, he auhrs prpsed nn-meric similariy funcins, which were very rbus he presence f nise and prvided an inuiive nin f similariy beween rajecries by giving mre weigh he similar prins f he rajecries. Mrever, an efficien index srucure (based n hierarchical clusering) fr similariy queries was presened. Hwever, as will be shwn in he experimenal sudy, he prpsed mehd suffers when rajecries have differen sampling raes. In [5] a disance funcin, called Edi isance n Real Sequences (ER), was inrduced. his disance funcin, based n edi disance, was shwn be mre rbus han W and LCSS ver rajecries wih nise. he efficiency f his disance funcin was imprved by he applicain f hree pruning sraegies, which reduced he respecive cmpuainal cs in erms f cmpuains beween he query and daa rajecries wihu inrducing false dismissals. On he her hand, same as LCSS, ER deermines spaial similariy nly, ignring ime, while rajecries wih differen sampling raes cann be handled efficienly, as i will be shwn in he experimenal sudy. Mrever, bh [] and [5] prpse he emplymen f dedicaed indexes prune he search space s as efficienly suppr -MS search. Recenly, Kegh e al. [9] presened an algrihm (based n he LB_Kegh funcin inrduced in [8]), which dramaically reduced he ime cmplexiy f he 3

4 calculain f he Euclidean isance measure. his speed up was furher achieved by allwing indexing. Hwever, he abve algrihm, which was generalized her disance measures, such as W and LCSS, culd be applied nly shapes. Acnwledging he cnribuins f he abve prpsals, in he sequel we prpse nvel merics and algrihms fr rajecry similariy search n R-ree-lie srucures. Nain able. able f nains escripin, Q an indexed and a query rajecry, Q he h line segmen f r Q a imesamp Q,() funcin f disance in ime beween Q and a, b, c E Q, facrs f Q,() rinmial calculain errr f he dissimilariy beween rajecries disance beween rajecries V relaive speed beween mving bjecs N R-ree nde MINIS(Q,N) minimum disance beween Q and N he sum f he maximum speed f indexed rajecries V max plus he maximum speed f he query rajecry he se f line segmens already rerieved frm he S R index he se f rajecries wih line segmens already S C rerieved frm he index bu n ye fully cmpleed inside he given ime perid. 3. Merics fr -MS Search In his secin we will define he nin f spaiempral dissimilariy used in he res f he paper fllwed by a series f merics and heurisics used in ur algrihms fr MS Search (able presens he nains used in he res f he secin). As already menined, exising wr in he dmain f rajecry similariy search, eiher ignres he ime dimensin f he mvemen, as such calculaing he spaial similariy beween rajecries r assumes ha rajecries have he same lenghs (in erms f he number f spaiempral pins ha are cmpsed f) and he same sampling rae. Frm a differen perspecive, exending he well nwn Euclidean isance meric als used in [] and [5], we define he nin f spaiempral dissimilariy beween w rajecries and Q bh being valid during a definie ime inerval [, n ], by inegraing heir Euclidean disance in ime. efiniin : he issimilariy ISSIM(Q,) beween rajecries Q and being valid during he perid [, n ] is defined as he definie inegral f he funcin f ime f he Euclidean disance beween he w rajecries during he same perid: n ISSIM ( Q, ) ( ) d =, Q, where Q, () is he funcin f he Euclidean disance beween rajecries Q and wih ime. Hwever, since each rajecry is represened by a cllecin f discree pins where linear inerplain is applied in beween, he definiin f dissimilariy is ransfrmed : n + =, ISSIM ( Q, ) ( ) d = where are he imesamps ha bjecs and Q recrded heir psiin. Obviusly, in real wrld applicains, he sampling raes f rajecries may vary, resuling in rajecries wih psiins sampled a differen imesamps; hwever, cnsidering w rajecries wih his characerisic, he psiin f he firs bjec a he ime insance when he secnd recrded is psiin can be apprximaed by applying linear inerplain. he Euclidean disance beween w pins mving wih linear funcins f ime beween cnsecuive imesamps, was defined in [6]: Q, a b c Q, ( ) = + +, where a, b, c are he facrs f his rinmial (real numbers, a ). In rder calculae he inegral f Q, (), we disinguish beween he fllwing w cases fr he value f he nn-negaive facr a : a=. As shwn in [6], i implies ha b=. Hence, + Q, ( ) d= + c a>. Accrding Merania and By []: Q, d a b c + a+ b b 4ac a+ b ( ) = + + arcsinh 4a 8a a 4ac b In rder avid such a cmpuainally heavy perain, we adp he uilizain f he rapezid Rule fr he cmpuain f he inegral, resuling in he fllwing Lemma. Lemma : he dissimilariy value beween w pins mving linearly wih ime can be apprximaed by he fllwing expressin: n ISSIM ( Q, ) (( Q, ( ) + Q, ( + )) ( + )) = wih he errr f he apprximain, which depends n, + values, being bunded by: 3 ( + ) ( b ), if b a a ( ) E if Q, + n 3 + Q,, ( ), b Q + < + < a = 3 ( + ) Q, ( ), if b < < + Prf: in Appendix. Figure demnsraes he rapezid apprximain illusraing he apprximain errr E in he hree abve cases: he value f b a is he flex f ; E is a Q, calculaed based n he value f ( + ) (case b), E + Q, i+ i 4

5 is calculaed based n he value f and E + is calculaed based n Figure. rapezid apprximain, ( b a) (case a) Q ( + ) (case c). Q, S far we have defined he dissimilariy beween w rajecries (efiniin ) and have apprximaed his measure wih a less expensive cmpuain and a bunded errr. As already menined, he lcain f nn-recrded imesamps is apprximaed by linear inerplain beween cnsecuive recrder pins. (Suppr f nnlinear e.g. arc, mvemen is lef as a as fr fuure wr.) In he sequel, we will prvide a series f merics ha will be used in ur MS search algrihm. n L +3 + b a + (a) Figure 3. L definiin 3. Speed-ependen Merics V< isance E + In his secin we define w merics, namely OPISSIM and PESISSIM, and prvide several lemmas be used fr pruning purpses during MS Search. Befre prceeding in he cre f he secin, we define he Linearly epended issimilariy (L) which is used in he definiin f ur merics: efiniin : he Linearly epended issimilariy (L) beween w mving bjecs wih iniial disance mving cllinearly wih relaive speed V during he perid = [, n ], is given by he fllwing expressin: ( + V ), if + V L(, V, ) = ( V ), herwise he relaive speed V is a negaive (psiive) number when he disance beween he w bjecs decreases (increases, respecively). illusrae his definiin, cnsider Figure 3 where L is described as he shaded area encmpassed by he inclined line represening a disance funcin beween w bjecs mving wards each her wih relaive speed V, wih he hriznal lines and n defining. he w cases f L definiin are illusraed in Figure 3(a) and Figure 3(b), respecively. n Q, E E + isance L (b) V< isance Any algrihm used fr MS search will have calculae he dissimilariy beween a query rajecry and several (indexed r n) rajecries; bviusly, a any ime insance such an algrihm will have rerieved several pars f candidae MSs. n= OPISSIM Figure 4. OPISSIM definiin Alhugh we cann calculae he exac ISSIM f hese parially rerieved rajecries frm he query rajecry, we can safely esimae a lwer bund fr i, called OPISSIM. Cnsider, fr example, Figure 4 ha illusraes OPISSIM f a parially rerieved candidae rajecry frm he query rajecry Q. OPISSIM parially cnsiss f he dissimilariy f he enries already rerieved frm he index (he shaded area during he ime inervals [, ] and [ 3, 4 ]). Regarding he perid [ 4, 5 ], he smalles pssible dissimilariy is given assuming ha he mving bjec sared frm is psiin a 4 appraching he query bjec wih he maximum pssible speed (he inclined line beween 4 and 5 ). Finally, when dealing wih inermediae ime inervals such as [, 3 ], ne has calculae he ime insance in which he bjec spped is mvemen wards he query rajecry (he inclined line beween and ) and hen reurned is nwn psiin a he ime insance 3 (he inclined line beween and 3 ). Nw we can prceed wih he frmal definiin f OPISSIM: efiniin 3: he ms pimisic ISSIM (OPISSIM) beween a query rajecry Q and an indexed rajecry wih line segmens parially rerieved frm he index, during a perid [, n ], is defined as: OPISSIM ( Q,,, n ) = ISSIM ( Q, ), if S R; L( Q, ( + ), Vmax,( + )), if SR, = ; n L( Q, ( ), Vmax,( + )), if SR, = n ; = L( Q, ( ), Vmax,( )) + L( Q, ( + ), Vmax,( + )), herwise where Q, is he funcin f disance wih ime beween rajecries Q and, S R is he se f line segmens already rerieved frm he index, V max is he sum f he maximum speed f indexed rajecries plus he maximum speed f he query rajecry, and is given by he fllwing expressin: Q, ( + (, +, ) max) = + + ( ) ( ) V Q Q V max isance 5

6 Recalling Figure 4, he value f is sraighfrward uilizing he fac ha he slpe f he w inclined lines beween [, ] and [, 3] is he same and equal V max. Having defined OPISSIM, we can prvide he fllwing lemma, which will als urn u be useful fr pruning purpses: Lemma : A rajecry indexed by an R-ree-lie srucure wih line segmens parially rerieved frm he index cann have smaller ISSIM frm a query rajecry Q during a perid [, n ] han is respecive OPISSIM. Prf: Accrding efiniin 3, OPISSIM is he sum f he ISSIM f he rajecry enries already rerieved frm he index (belnging he se S R ), a value which is fixed, plus he ISSIM f an bjec which apprached he query rajecry wih he maximum pssible speed (V max ) during he ime inervals n already rerieved frm he index, wih he cnsrain ha he bjec has be fund a given psiins a he sar and/r he end f he inerval. herefre, since he w bjecs apprach each her wih he maximum pssible speed during hse perids, he disance beween hem is minimized; hence minimizing he crrespnding inegral and cnsequenly heir dissimilariy. n= p PESISSIM Q, Figure 5. PESISSIM definiin Liewise, by adping he same scenari where an MS algrihm has nly parially rerieved rajecries, ne can esimae an upper bund, fr he ISSIM beween he query and a parially rerieved rajecry, named PESISSIM. As illusraed in Figure 5, PESISSIM wrs in a fashin similar OPISSIM wih he difference ha during ime inervals where he mvemen f he bjec is n nwn, he bjec is assumed diverge (and n apprach) he query rajecry wih he maximum pssible speed V max. In he same way, we frmally define PESISSIM: efiniin 4: he ms pessimisic ISSIM (PESISSIM) beween a query rajecry Q and an indexed rajecry wih line segmens parially rerieved frm he index, during a perid [, n ], is defined as: PESISSIM ( Q,,, n ) = ISSIM ( Q, ), if S R; L( Q, ( + ), Vmax,( + )), if SR, = ; n L( Q, ( ), Vmax,( + )), if SR, = n ; = p L( Q, ( ), Vmax,( )) + p L( Q, ( + ), Vmax,( + )), herwise V max isance where Q,, S R and V max are as defined in previus definiins, and is given by he fllwing expressin: p ( + (,, + ) max) p = + + ( ) ( ) V Q Q he fllwing lemma is direcly derived by he definiin f PESISSIM. Lemma 3: A rajecry indexed by an R-ree-lie srucure wih line segmens parially rerieved frm he index cann have ISSIM frm a query rajecry Q during a perid [, n ] greaer han is respecive PESISSIM. Prf: Accrding efiniin 4, PESISSIM is he sum f he ISSIM f he rajecry enries already rerieved frm he index (belnging he se S R ), a value which is fixed, plus he ISSIM f an bjec which diverged he query rajecry wih he maximum pssible speed (V max ) during he ime inervals n already rerieved frm he index, wih he cnsrain ha he bjec has be fund in given psiins a he sar and/r he end f he inerval. herefre, he disance beween he w rajecries during hse perids is maximized, hence maximizing heir dissimilariy. 3. Speed-Independen Merics he uilizain f he previusly defined merics in an MS search algrihm can significanly enhance is perfrmance by pruning several candidae rajecries. Hwever, hese merics are relaively lse, since hey are based n he maximum speed V max which, hereically speaing, culd be rders f magniude higher han he mean bjec speed. herefre, we need define her merics n influenced by V max, suppring ur speedindependen MS search algrihms. hese merics can be develped when an MS algrihm reprs index ndes in incremenal rder f heir MINIS frm he query rajecry. Obviusly, his is a reasnable assumpin cnsidering R-ree lie srucures where a bes-firs sraegy lie he ne prpsed in [7] can be uilized. n= OPISSIMINC MINIS(Q,N) Q, Figure 6. OPISSIM INC definiin isance Cnsider, fr example, Figure 6 ha describes he ISSIM f a parially rerieved candidae rajecry frm he query rajecry Q; Accrding ur previus discussin, he ISSIM beween [, ] and [ 3, 4 ] is accuraely defined. In his case hwever, we can uilize he fac ha index ndes are accessed in incremenal rder 6

7 f heir MINIS frm he query rajecry. Cnsequenly, any line segmen n ye rerieved frm he index, cann be clser Q han MINIS(Q,N) where N is he nex index nde in he queue, and he lwer bund f ISSIM urns in he shaded area f Figure 6. Mre frmally, we define OPISSIM INC as fllws: efiniin 5: Assuming ha index ndes are repred in incremenal rder f heir MINIS frm he query rajecry, he ms pimisic ISSIM beween a query rajecry Q and an indexed rajecry during a perid [, n ] having a line segmen inside a ree nde N, is given by he fllwing expressin: OPISSIM INC ( Q,, N,, n) = n ISSIM ( Q, ), if S R; = MINIS ( N, ) ( + ), herwise where S R is he se f line segmens already rerieved frm he index. Using he abve definiin f OPISSIM INC, we can als define he minimum ISSIM f an index nde N: efiniin 6: Assuming ha index ndes are repred in incremenal rder f heir MINIS frm he query rajecry, he minimum ISSIM beween a rajecry, indexed by an R-ree-lie srucure having a line segmen inside a nde N, and a query rajecry Q during a perid [, n ], is defined as: MINISSIM INC ( Q, N,, n) = MINIS ( Q, N) ( n ) min OPISSIM INC ( Q,, N,, n), SC where S C, is he se f he rajecries wih line segmens already rerieved frm he index bu n ye fully cmpleed inside he perid [, n ]. Lemma 4: Assuming ha index ndes are repred in incremenal rder f heir MINIS frm a query rajecry Q, a rajecry ha is parially sred inside a ree nde N cann have smaller ISSIM frm Q during he ime perid [, n ] han he nde s respecive MINISSIM INC. Prf: Any line segmen inside N resides in a rajecry ha eiher belngs S c r n. In he frmer case, cnsidering ha ndes are repred in incremenal rder, rajecry enries n ye rerieved cann be clser he query bjec han he MINIS f he nde in which hey belng. S, he minimum dissimilariy f an bjec f S c is he sum f he dissimilariy f is enries already rerieved frm he index, plus he dissimilariy f an bjec being as clse as MINIS he query rajecry during he res f he query ime perid - a sum which crrespnds OPISSIM INC definiin. In he laer case, where he rajecry des n belng S c, he line segmen cann belng an bjec fully rerieved frm he index because his wuld lead duplicae line segmens in he index. Hence he line segmen belngs a mving bjec wih n segmens rerieved frm he previusly accessed ndes and i cann be clser he query rajecry han MINIS. hus, in he bes case, is disance frm he query bjec during he query perid is equal MINIS and is ISSIM is equal MINIS ( Q, N). 3.3 Heurisics he lemmas prvided in previus secins suppr he fllwing heurisics direcly used in he MS Search algrihm ha will be presened in Secin 4. Heurisic : Every rajecry wih OPISSIM greaer han he curren ms similar (i.e. he ne wih he smalles calculaed ISSIM - r PESISSIM if here is n a fully calculaed ISSIM) cann be mre similar he query rajecry han he curren ms similar; as such, i can be pruned frm he candidaes lis. Heurisic : When leaf and inernal ndes are repred in incremenal rder f heir MINIS frm he query rajecry, every rajecry line segmen cnained in a nde wih MINISSIM INC greaer han he curren ms similar belngs a mving bjec, which cann be mre similar he query rajecry, hence, he nde can be pruned frm he candidaes lis. Mrever, since any nde repred afer he ne prcessed will have MINIS greaer r equal MINIS f he curren nde, accrding efiniin 6 he same will hld fr he respecive values f MINISSIM INC. As a resul, all hese ndes will have MINISSIM INC greaer han he curren ms similar, and he algrihm can be erminaed since all he remaining ndes can be pruned. 4. A -MS Search Algrihm he prpsed BFMSSearch algrihm (illusraed in Figure 7) accesses he ree srucure in a bes-firs mde, calculaing he apprpriae MINISs beween he query rajecry and he ree ndes, hus repring leaf and inernal ree ndes in incremenal rder f heir MINIS frm he query rajecry. A leaf level, he algrihm uses hree hashed in-memry srucures: One wih he cmpleed rajecries (Cmpleed), ne wih he parially cmpleed rajecries (Valid) and ne wih he parially cmpleed neverheless already rejeced (Rejeced) rajecries. Bh Cmpleed and Valid in-memry srucures sre liss. Each lis cnains he mving bjec s ime inervals alng wih heir saring and ending disances, is (parial) ISSIM he respecive calculain errr and he OPISSIM and PESISSIM values. he Rejeced in-memry srucure cnains nly rajecry ids. When an inernal nde is prcessed (lines 3-37) he algrihm calculaes he MINIS beween he nde and he par f he query rajecry Q being inside he empral exend f he nde and hen is enqued. When a leaf enry is prcessed (lines 9-3), he algrihm checs wheher i belngs a Rejeced mving bjec (by simply using is id) and rejecs i if i des (line ). In he sequel 7

8 i checs wheher he enry belngs a Valid mving bjec and if s rerieves is lis L; herwise i creaes a new lis and adds i Valid (line 3). he algrihm uses a plane sweep mehd which scans leaf enries and rajecry segmens in heir empral dimensin in a single pass. his requires ha he leaf enries are previusly sred accrding heir empral rder (line ), unless he underlying ree srucure (such as he Bree) sres hem in empral rganizain anyway. When a leaf enry and a query rajecry segmen verlap in he empral dimensin, he algrihm adds he perid he lis L (line 7), calculaing ISSIM, OPISSIM and PESISSIM, geher wih he respecive calculain errr (line 8). If he lis L is cmpleed, i is remved frm he Valid and added he Cmpleed, while is ISSIM is checed agains he curren ms similar; if smaller, aes is psiin in MSim (lines -). In he case where L is n ye cmpleed, is PESISSIM is checed agains he curren ms similar and, if smaller, aes is psiin in MSim (lines 4-5); is OPISSIM is als cmpared wih he curren ms similar and, if greaer, he lis is mved frm Valid Rejeced applying heurisic (lines 6-7) Algrihm BFMSSearch (R-ree R, rajecry Q, ime perid Q per) EnQueue Queue, R.RNde,, Q O WHILE Queue.Cun > Elemen = equeue(queue) N=Elemen.Nde:Q=Elemen.Queryrajecry IF Cmpleed.Cun> IF MINISSIM INC(Q,N)>MSim.ISSIM Reurn MSim ELSE IF N is leaf nde Sr(N, S) FOR EACH leaf enry E in leaf nde N IF Rejeced n cnains E.Id IF Valid cnains E.Id rerieve lis L ELSE creae lis L: Add L in Valid FIN nex query enry QS in Q wih QS. e<n. S: QE=QS O UNIL QE. S > E. e Inerplae prduce ne, nqe in perid (, ):Add (, ) in L Calc ISSIM,PESISSIM, OPISSIM,ERR) IF L is cmpleed Mve L frm Valid Cmpleed IF ISSIM<MSim.ISSIM Updae Msim wih ne,issim ELSE IF PESISSIM<MSim.ISSIM Updae MSim wih ne,pesissim IF OPISSIM>MSim.ISSIM Mve L frm Valid Rejeced NEX query enry QE Reurn QE in he query enry QS NEX ELSE FOR EACH enry E in he nde Elemen IF (Q. S,Q. E) Overlaps (E. S,E. E) Inerplae prduce nqe in perid (, ) is = Minis(nQ, E) EnQueue Queue, E, is, nq NEX LOOP Figure 7. BFMS Search pseud-cde In bh cases where a nde (leaf r inernal) is prcessed, he algrihm firs checs wheher is MINISSIM INC is greaer han he curren ms similar and if s, he algrihm erminaes applying heurisic, and reurns he curren ms similar as he query reply (lines 5-7). Ne ha in rder avid calculaing all he OPISSIM INC values invlving in he MINISSIM INC definiin (e.g. SC in definiin 6), we firs chec wheher he MINIS ( Q, N) ( n ) value f he nde is less han he curren ms similar. In such a case, he calculain f he OPISSIM INC values is mied, since he value f MINISSIM INC will be less han he curren ms similar regardless f he OPISSIM INC values. 4.3 Exending -MS algrihms In he same fashin as in [6], we generalize he abve algrihm suppr he -ms similar rajecry search by cnsidering he fllwing: using a buffer f a ms (curren) ms similar rajecries sred by heir acual dissimilariy frm he query rajecry; erminaing he algrihm execuin when prcessing a nde wih MINISSIM INC greaer han he dissimilariy f he mre dissimilar bjec in he buffer, when exending he BFMSSearch algrihm. 4.4 Errr Managemen he abve MS algrihm calculaes dissimilariy beween query and indexed rajecries using he apprximain inrduced in Lemma, cmpuing a he same ime he apprpriae apprximain errr (dened as ERR in Figure 7). Hwever, apar frm is cmpuain, he usage f he errr is fundamenal in rder cmpue exac and crrec resuls, a as n explicily discussed in he descripin f he BFMS algrihm fr sae f clariy. Acually, hree mdificains mus be inrduced in he algrihm s as incrprae he rle f he apprximain errr: A candidae ms similar rajecry, n already cmpleed, is cmpared agains he curren h ms similar by using he value f PESISSIM-ERR. A cmpleed candidae ms similar rajecry is cmpared agains he curren h ms similar using he value ISSIM-ERR. Insead f using ne h ms similar, i is required uilize a buffer f he candidae h ms similar rajecries. hese will be all he rajecries wih ISSIM greaer han he h ms similar and ISSIM- ERR less han i. Finally, a ps prcessing sep is required afer he execuin f he MS algrihm in rder deermine he definie MSs by calculaing he acual dissimilariy f each candidae rajecry agains he query rajecry. Alhugh, his is a cmpuainal heavy perain, i nly 8

9 happens when he errr buffer cnains mre han ne rajecry, r when he rder in which he rajecries are repred frm he -buffer can be affeced by he calculain errr f each rajecry s similariy. As an indicain, during he enire experimenal sudy, here was n experimen ha his case appeared. 5. Experimenal Sudy he abve illusraed algrihm can be implemened in any R-ree-lie srucure sring hisrical mving bjec infrmain such as he 3 R-ree [8], he SR-ree [] and he B-ree []. Amng hem, we have chsen he 3 R-ree and he B-ree ha have excellen perfrmance in specific radiinal rajecry queries []. We used a page size f 4KB and a (variable size) buffer fiing he % f he index size, wih a maximum capaciy f pages. he experimens were perfrmed in a PC running Micrsf Windws XP wih AM Ahln 64 3GHz prcessr, 5 MB RAM and several GB f dis space. 5. aases Alhugh exising wr n rajecry similariy [], [5] uilized real daa, hese daases are n suiable fr ur bjecives due he fac ha hey are cmpsed by prjecins f rajecries wihu any infrmain abu he sampled imesamps; a reasnable fac, bearing in mind ha he similariy measured in hse papers nly depends n he spaial and n he spaiempral rajecry similariy. On he her hand, several real daases recenly became available fr experimenain purpses [5]; hese daases (represening he mvemen f a flee f rucs) were used in ur experimens evaluae he qualiy f he prpsed similariy measure (secin 5.). Hwever, since hey are relaively small (73 rajecries and 3 line segmens), hey culd n expse he acual perfrmance f he algrihms; herefre, he perfrmance sudy (secin 5.3) was cnduced using synheic daases generaed by a cusm generar based n he GS daa generar [6]. aase able. Summary daase infrmain # Objecs # Enries (xk) Speed isribuin ype σ Index Size (MB) 3 R- ree Bree rucs 73 Real daa 3..8 S Lgnrmal S Lgnrmal S 5 5 Lgnrmal S Lgnrmal In rder achieve scalabiliy in he vlumes f he daases, we generaed synheic rajecries f, 5, 5 and mving bjecs resuling in daases f K, 5K, K, and K enries, respecively (he psiin f each bjec was sampled apprximaely imes), hus building indices f up MB size. Regarding he res parameers f he generar, he iniial disribuin and he heading f bjecs in all cases was randm, while heir speed was ruled by a nrmal r lgnrmal disribuin. able illusraes summary infrmain abu he real and he generaed daases and he crrespnding indexes. Ne ha each synheic daase is dened by is cardinaliy (e.g. he S cnsiues frm rajecries). 5. Experimens n he qualiy In rder evaluae he qualiy f he prpsed similariy measure we cnduced an exensive se f experimens using he real rucs daase. All rajecries f he daase were cmpressed using he -R algrihm described in [] prducing hus arificial rajecries, which were similar (bu n idenical) he nes f he riginal daase. hen, we used each cmpressed rajecry query he riginal daase, expecing he algrihm reurn he crrespnding riginal rajecry as ms similar. We run ne se f queries seing = and we cuned he number f imes he query failed reurn he riginal rajecry as he ms similar. We als scaled he value f he -R parameer p frm.% % f he lengh f each rajecry, in rder achieve differen values f similariy since an increasing -R parameer prduces a cmpressed rajecry wih fewer sampled pins and greaer dissimilariy regarding he riginal rajecry. As an example, Figure 8 illusraes (a) an riginal rajecry and he rajecries prduced using he -R algrihm wih (b, c, d) differen values f p. A majr bservain derived frm Figure 8 is ha while he general sech f he rajecry remains unaffeced wih he evluin f p, he number f verices ulining he rajecry decreases and he lcal deails are vanished. p= (68 Verices) p=. % (65 Verices) p= % (9 Verices) p= % ( Verices) (a) (b) (c) (d) Figure 8. ifferen degree f cmpressin n a rajecry Amng he relaed wr we have chsen run he same experimens using he LCSS [] and ER [5] similariy measures. We did n include W [] in ur experimenal sudy, since bh LCSS and ER were shwn uperfrm i [], [5]. We se he value f he parameer ε fr hese w measures be a quarer f he maximum sandard deviain f rajecries, which leads he bes clusering resuls, accrding [5]. We als nrmalized he rajecry daase as suggesed in he same paper. Furhermre, fr a fair cmparisn, we made an 9

10 bvius imprvemen ver LCSS and ER, by manually adding samples in he under-sampled (query) rajecry wih linear inerplain a he imesamps he checed daase rajecry was sampled. We called hese imprved versins LCSS-I and ER-I respecively. he resuls f he experimens evaluaing he qualiy f he prpsed similariy meric are illusraed in Figure 9. Clearly, he prpsed dissimilariy measure (ISSIM) uperfrms bh is cmpeirs in all seings, regarding als heir imprved versins. Acually, in he larges par f he experimens, ISSIM crrecly idenifies he riginal rajecry frm which he query ne has been prduced. On he her hand, i prduces false respnses nly when he value f p exceeds 5%, verifying ha i is a very rbus similariy meric. LCSS (and LCSS-I) als achieves gd qualiy classifying crrecly he query rajecry in he majriy f he experimenal seings; neverheless, i is always less accurae han ISSIM. Regarding ER and ER-I, i urns u ha fr p values greaer han % hey cmpleely fail describe he similariy beween rajecries, since he false respnses exceed 6%. False Resuls (%) %.%.% 5.%.% -R Parameer Figure 9. False resuls increasing he value he -R parameer he reasn f he pr perfrmance f ER similariy measure demnsraed in hese experimens can be explained cnsidering is definiin: ER is he number f inser, delee, r replace perains ha are needed cnver rajecry A in B [5]. hus, suppsing ha n is he number f verices in A and m is he number f verices in (he cmpressed) A c, ER( A, Ac ) n m since a leas n-m verices are needed be added in A c s as cnver i A. Fr an arbirary daase rajecry wih verices being spaially away frm A, i can be easily shwn ha ER beween and A c is a ms max(m, ). herefre, if a daase cnains a rajecry wih verices and max( m, ) n m, e.g. a rajecry cmpsed by a small number f verices, hen i als hlds ha ER(, A ) ER( A, A ). 5.3 Experimens n he perfrmance c he prpsed algrihm was evaluaed wih hree ses f 5 queries accrding he seings presened in able 3. As such he effecs f cardinaliy (Q), query lengh (Q) and (Q3) were evaluaed using bh 3 R- and B-rees. (Here, we have ne ha alhugh relaed wr als uses index srucures prune he search space and c ISSIM LCSS LCSS-I ER ER-I suppr efficien -MS search, hey uilize dedicaed indices n designed suppr her ypes f queries. ue his fac, hey are n cmparable wih ur prpsal, hence hey are n included in ur perfrmance sudy.) able 3. Query Seings Query Query rajecry (as par f a aases Se randm daa rajecry) Q S S 5% Q S 5 % % Q3 S 5 5%.. Figure illusraes he execuin ime and he achieved pruned space fr he query ses Q (scaling wih he daase cardinaliy), Q (scaling wih he query lengh) and Q3 (scaling wih he number f ) evaluaing he BFMS search algrihm. Clearly, he implemenain f he prpsed algrihm in bh indices demnsraes high pruning pwer, pruning ver 9% in all he experimenal seings. Mrever, as als demnsraed in he same figures, he pruning pwer remains alms cnsan r decreases a a lw rae - regardless f he scaling facr. Execuin ime (ms) Execuin ime (ms Execuin ime (ms R - BFMS B - BFMS 5 Mving Objecs Pruning pwer R - BFMS B - BFMS 5 Mving Objecs (a Q) (b Q) 3 R - BFMS B - BFMS.5 Query Lengh Pruning pwer R - BFMS B - BFMS.5 Query Lengh (a Q) (b Q) 3 R - BFMS B - BFMS 5 Pruning pwer R - BFMS B - BFMS 5 (a Q3) (b Q3) Figure. Scaling wih he daase cardinaliy (Q) he query lengh (Q) and he number f (Q3) Regarding execuin ime, bh rees achieve gd execuin imes, due he fac ha he algrihm prunes mainly by he MINISSIM INC heurisic, which direcly rejecs all ree ndes n ye prcessed by he ime i realizes. he execuin ime appears be linear wih he

11 number f mving bjecs, quadraic wih he query lengh and sub-linear wih. Mrever, he B-ree uperfrms he 3 R-ree as he query lengh increases, while in he res f he experimenal seings, i is he ppsie ha is repred. 6. Cnclusins and Fuure Wr Relaed wr n similariy query prcessing eiher ignres ime dimensin f rajecries r cnsiders rajecries wih he same sampling rae. In his wr, we relaxed hese assumpins by defining a nvel meric, called ISSIM, and hen we presened a cmplee reamen f hisrical MS queries ver mving bjec rajecries sred n R-ree lie srucures aviding he drawbacs f he exising mehds. Using ur prpsed merics and heurisics fr rdering and pruning purpses, we presened a bes-firs MS algrihm. Under varius synheic and real rajecry daases, we illusraed he supeririy f he prpsed ISSIM meric agains relaed wr [], [5], in erms f qualiy, while ur algrihm was shwn have high pruning abiliy when prcessing MS queries, als verified in he case f -MS queries. Fuure wr includes he develpmen f algrihms suppr ime-relaxed MS queries ver rajecries indexed by R-ree lie srucures using he prpsed ISSIM meric. his ype f query calculaes he minimum dissimilariy beween rajecries regardless f he ime insance in which he query bjec sars. A secnd research direcin includes he develpmen f seleciviy esimain frmulae fr query pimizain purpses invesing n he wr presened in [7] fr predicive spaiempral queries. Acnwledgemens Research parially suppred by FP6/IS Prgramme f he Eurpean Unin under he GePK prjec (5-8). References [] Agrawal, R., Faluss, C., and Swami, A., Efficien Similariy Search in Sequence aabases, Prceedings f FOO, 993. [] Bernd, J. and Cliffrd, J., Finding paerns in ime series: A dynamic prgramming apprach, Advances in Knwledge iscvery and aa Mining. AAAI/MI Press. [3] Chan, K.P., and Fu, A.W-C., Efficien ime series maching by Waveles, Prceedings f ICE, 999. [4] Cai, Y., and Ng, R., Indexing spai-empral rajecries wih Chebyshev plynmials, Prceedings f ACM SIGMO, 4 [5] Chen, L., amer Özsu, M., and Oria, V., Rbus and Fas Similariy Search fr Mving Objec rajecries, Prceedings f ACM SIGMO, 5. [6] Frenzs, E., Grasias, K., Peleis, N., and hedridis, Y., Algrihms fr Neares Neighbr Search n Mving Objec rajecries. Geinfrmaica, appear [7] Hjalasn, G., and Same, H., isance Brwsing in Spaial aabases, ACM ransacins in aabase Sysems, vl. 4, pp , 999. [8] Kegh, E., Exac indexing f dynamic ime warping, Prceedings f VLB, [9] Kegh, E., Wei, L., Xi, X., Lee, S.H., and Vlachs, M., LB_Kegh Supprs Exac Indexing f Shapes under Rain Invariance wih Arbirary Represenains and isance Measures, Prceedings f VLB, 6. [] Lin, B., and Su, J., Shapes Based rajecry Queries fr Mving Objecs, Prceedings f ACM GIS, 5. [] Merania, N., and By, R., Spaiempral Cmpressin echniques fr Mving Pin Objecs, Prceedings f EB, 4 [] Pfser., Jensen C. S., and hedridis, Y., Nvel Appraches he Indexing f Mving Objec rajecries, Prceedings f VLB,. [3] Saurai, Y., Yshiawa, M., and Faluss, C., FW: Fas Similariy Search under he ime Warping isance, Prceedings f POS, 5. [4] hedridis, Y., en Benchmar Queries fr Lcainbased Services, he Cmpuer Jurnal, vl. 46(6), pp , 3 [5] hedridis, Y., he R-ree Pral. URL: (accessed 5 March 6). [6] hedridis, Y., Silva, J.R.O., and Nascimen, M. A., On he Generain f Spai-empral aases, Prceedings f SS, 999. [7] a, Y., Sun, J., and Papadias,., Analysis f predicive spai-empral queries, ACM ransacins n aabase Sysems vl. 8(4), pp , ecember 3. [8] hedridis, Y., Vazirgiannis, M., and Sellis,., Spaiempral Indexing fr Large Mulimedia Applicains, Prceedings f ICMCS, 996. [9] Vlachs, M., Gunpuls,., and as, G., Rain Invarian isance Measures fr rajecries, Prceedings f SIGK, 4. [] Vlachs, M., Kllis, G., and Gunpuls,., iscvering Similar Mulidimensinal rajecries, Prceedings f ICE,. [] Yanagisawa, Y., Aahani, J., and Sah,., Shape-Based Similariy Query fr rajecry f mbile Objecs, Prceedings f MM, 3. Appendix (Prf f Lemma ) he rapezid apprximain n ( f ) f x n x f ( x) dx assciaed wih he pariin x < x <... < xn is given by: n ( f ) = ( xn x)[ f ( x) + f ( x) f ( xn ) + f ( xn)] If f ( x ) is cninuus in [ x, x n ], hen he errr En ( f ) in he rapezid rule is bunded as fllws: 3 ( xn x) En( f ) f ( M ) f ( M ) f ( x) x x, xn n In ur case, by seing n=, we finally calculae:,where [ ]

12 + Q, ( ) d ( Q, ( ) + Q, ( + )) ( + ) wih he errr f ur apprximain being bunded by: 3 ( + ) Q, Q, E ( M ) herefre, we deermine he maximum value f ( ) ( ) 3 Q, ( ) = 4ac b 4( a + b+ c) in [, + ].Since he firs derivaive f = b a and 3 ( 4(4 ac b ) ) (4) Q Q, ( ),, ( b a) (3) Q, ( ) zeres a 5 ( ) = 3 a ( 4a) (since a ), he larges value f Q, ( ) in R is Q, ( b a). Finally, we disinguish beween hree cases: (a) b a +.In his case, Q, ( M ) = Q, ( b a) 3 and he errr is E ( ) ( b a) ; Q, + Q, (b) < < b a. In his case, Q, ( M ) = Q, ( + ) + and he errr is E ( ) ( ) ; 3 Q, + Q, + (c) b a< < +. In his case, and he errr is Q, ( M ) = 3 Q, + Q,, ( ) Q E ( ) ( ). Summing he n- equains f he dissimilariy errr calculain by sides, i implies ha he apprximain errr E Q, is cmpued as presened in Lemma.