Map-Matching for Low-Sampling-Rate GPS Trajectories

Transcription

1 Map-Maching for Low-Sampling-Rae GPS Trajecorie Yin Lou Microof Reearch Aia Chengyang Zhang Microof Reearch Aia Yu Zheng Microof Reearch Aia Xing Xie Microof Reearch Aia Wei Wang Fudan Univeriy Yan Huang Univeriy of Norh Texa ABSTRACT Map-maching i he proce of aligning a equence of oberved uer poiion wih he road nework on a digial map. I i a fundamenal pre-proceing ep for many applicaion, uch a moving objec managemen, raffic flow analyi, and driving direcion. In pracice here exi huge amoun of low-amplingrae (e.g., one poin every -5 minue) GPS rajecorie. Unforunaely, mo curren map-maching approache only deal wih high-ampling-rae (ypically one poin every 0-0) GPS daa, and become le effecive for low-ampling-rae poin a he uncerainy in daa increae. In hi paper, we propoe a novel global map-maching algorihm called ST-Maching for lowampling-rae GPS rajecorie. ST-Maching conider () he paial geomeric and opological rucure of he road nework and () he emporal/peed conrain of he rajecorie. Baed on paio-emporal analyi, a candidae graph i conruced from which he be maching pah equence i idenified. We compare ST-Maching wih he incremenal algorihm and Average-Fréche-Diance (AFD) baed global map-maching algorihm. The experimen are performed boh on ynheic and real daae. The reul how ha our ST-maching algorihm ignificanly ouperform incremenal algorihm in erm of maching accuracy for low-ampling rajecorie. Meanwhile, when compared wih AFD-baed global algorihm, ST-Maching alo improve accuracy a well a running ime. Caegorie and Subjec Decripor H..8 [Daabae Applicaion]: Spaial Daabae and GIS. General Term Algorihm, Deign Keyword Map-maching, GPS, rajecory, road nework. INTRODUCTION The pa year have een a dramaic increae of handheld or dahboard-mouned ravel guidance yem and GPS-embedded Permiion o make digial or hard copie of all or par of hi work for peronal or claroom ue i graned wihou fee provided ha copie are no made or diribued for profi or commercial advanage and ha copie bear hi noice and he full ciaion on he fir page. To copy oherwie, or republih, o po on erver or o rediribue o li, require prior pecific permiion and/or a fee. ACM GIS '09, November 4-6, 009. Seale, WA, USA (c) 009 ACM ISBN /09/...$0.00 PDA and mar phone. The proliferaion of hee device ha enabled he collecion of huge amoun of GPS rajecorie. More and more applicaion, uch a roue planner [7], ho roue finder [6], raffic flow analyi [5], geographical ocial nework [], have ared o ue informaion from GPS daa o achieve beer qualiy of ervice. Typically a GPS rajecory coni of a equence of poin wih laiude, longiude, and imeamp informaion. However, hi daa i no precie due o meauremen error caued by he limiaion of GPS device and ampling error caued by he ampling rae [7]. Therefore he oberved GPS poiion ofen need o be aligned wih he road nework on a given digial map. Thi proce i called map-maching. Map-maching i a fundamenal pre-proceing ep for many rajecory-baed applicaion, uch a moving objec managemen, raffic flow analyi, and driving direcion. The difficuly of map-maching can grealy differ depending on GPS accuracy and he ampling rae. Thi paper addree he problem of ampling error in paricular. In pracice here exi large amoun of low-ampling-rae (e.g., one poin every minue) GPS rajecorie. They are eiher applicaion-logged daa colleced from ad-hoc locaion-baed querie, or generaed in he cenario where aving of energy co and communicaion co are deired. For example, here are 60,000+ axie in Beijing, among which many are GPSembedded. Since axi driver ravel very frequenly, ampling rae ha o be reduced in order o ave energy conumpion and achieve reaonable repone ime. Unforunaely, curren mapmaching approache only deal wih high-ampling-rae (ypically one poin every 0-0) GPS daa, and become le effecive for low-ampling-rae poin a he uncerainy in daa increae. Mo exiing map-maching approache employ local or incremenal algorihm ha map curren or neighboring poiion ono vecor road egmen on a map. For an approach ha only conider curren poiion, he reul i grealy affeced by meauremen error. The accuracy i generally low becaue he correlaion of neighboring poin i compleely overlooked. The incremenal maching algorihm in [8][7] purue he local maching of a mall porion of he rajecory. When maching a new poiion, i previou poiion and la mached edge are conidered. Alhough fa in compuaion, hi approach performance i eniive o he decreae of ampling frequency. On he oher hand, a global algorihm align enire rajecory wih he road nework. Generally peaking, a global approach achieve beer accuracy a a higher compuaional co. Exiing global

2 maching algorihm are all baed on cerain diance/imilariy meaure (uch a Average Fréche diance propoed in []). The lack of ground ruh (i.e. he rue pah of he moving objec) make i difficul o evaluae he real maching qualiy. Moreover, curren global maching algorihm only employ paial analyi, while neglec he emporal/peed conrain of he rajecorie. Thi make hem vulnerable o he decreae of ampling rae a well. In hi paper we refer o low-ampling-rae a one poin every minue or above. Wih uch ampling rae, he diance beween wo poin may reach over 00m even a vehicle peed i only 40km/h! Thi poe a big challenge for he map-maching problem becaue a he diance beween wo neighboring poin increae, le informaion can be ued o deduce he precie locaion of he objec. The problem i aggravaed when a moving objec i ravelling wih high peed, or here are many inerecion beween wo oberved neighboring poin. To addre he challenge, we provide wo key obervaion ha lead o our approach, a illuraed in he following example. Obervaion : True pah end o be direc, raher han roundabou. Example : Conider he GPS rajecory of a axi viualized in Figure. The axi ravel from norh o ouh. p a, p b and p c are hree conecuive ampling poin. p a p b p c Figure. Illuraion of obervaion Mo map-maching algorihm would mach he circled GPS obervaion (p b ) o i neare road egmen, i.e., he verical one. However, from i previou poiion p a and fuure poiion p c, we may deermine ha p b hould be mached o he horizonal road egmen becaue i i unlikely for hi axi o ake a roundabou rip o he horizonal road fir, and hen reurn o he verical road. Thi implie ha opological informaion of he road nework can be combined wih he poiion conex o provide beer maching reul. Obervaion : True pah end o follow he peed conrain of he road. Figure. Illuraion of obervaion Example : Conider anoher axi GPS rajecory viualized in Figure. Wihou peed informaion, i i nearly impoible o ell wheher hee wo poin belong o he highway or he ervice road. However, if we compue he average peed of hi pah a km/h baed on diance of he wo poin and heir imeamp, we may deermine ha he wo poin are very likely on he highway. Thi implie ha emporal/peed informaion i alo ueful in he maching proce. Baed on above obervaion, we propoe a novel global mapmaching algorihm called ST-Maching for low-ampling-rae GPS rajecorie in hi paper. To he be of our knowledge, our work i he fir map-maching algorihm ha incorporae boh () he paial geomeric and opological rucure of he road nework and () he emporal/peed conrain of he rajecorie. By combing paial and emporal analyi, a candidae graph i conruced from which ST-maching algorihm rie o find a global maching pah wih highe core. In ummary, hi paper make he following conribuion: We propoe a novel global maching algorihm called STmaching for low-ampling-rae GPS rajecorie. I combine boh paial and emporal feaure ha are no explored by previou mehod. We perform exenive experimen on boh ynheic and real daae. The real daae i colleced from he phyical world and labeled by real people. Therefore, i can beer evaluae he performance of map-maching algorihm han ynheic daa ued in mo exiing work. Our algorihm i evaluaed in erm of running ime and maching accuracy. The reul how ha our ST-maching algorihm ignificanly ouperform incremenal algorihm in erm of maching accuracy for low-ampling-rae rajecorie. Meanwhile, when compared wih Average- Fréche-Diance baed global algorihm, ST-Maching alo improve accuracy a well a running ime. The re of he paper i organized a follow. Secion dicue he relaed work. Preliminarie and problem definiion are preened in Secion. Syem overview i given in Secion 4. Then ST-maching algorihm i propoed in Secion 5 wih deailed dicuion and analyi in boh Secion 5 and 6. Secion 7 preen he experimenal evaluaion reul. Finally, we conclude he paper wih Secion 8.. RELATED WORK In hi ecion, we preen he relaed work of he map-maching problem. Since our algorihm relie on hore pah compuaion, we will alo briefly review he hore pah problem.. Map-maching Problem There are a number of udie on maching GPS obervaion on a digial map. Thee approache can be generally claified ino hree clae: local/incremenal mehod [4][8][0], global mehod [][][], and aiical mehod [][8]. The local/incremenal mehod ry o find local mach of geomerie. The incremenal mehod in [8] ue wo imilariy meaure o evaluae he candidae edge, one for diance imilariy and he oher for orienaion imilariy. The combined imilariy meaure i compued a he um of individual core. The ime complexiy i O n once we find adjacen edge for each ample, where n i he number of GPS poin o be mached. The adapive clipping mehod in [0] ue Dijkra algorihm o conruc hore pah on local free pace graph. I run in

3 O mn log m ime, where m and n are number of edge in he road nework and number of GPS poin repecively. On op of incremenal maching, [4] propoe a egmen-baed maching mehod o aign confidence value for differen ampling poin. I mache high-confidence egmen fir, and hen mach lowconfidence egmen uing previouly mached edge. In general, when mach a new poiion, a local/incremenal mehod only conider a mall porion of he rajecory ha i cloe o he poiion. I run fa and perform well when ampling frequency i very high (e.g. -5 econd). However a he ampling rae decreae, he problem of arc-kipping [8] become prominen, cauing ignifican degrade of accuracy. By conra, wih reaonable increae of ime complexiy, our global ST-maching algorihm i more robu o he decreae of ampling rae. The global mehod aim o mach he enire rajecory wih he road nework. Paper [] employ an offline napping mehod ha aim o find a minimum weigh pah baed on edi diance. Oher mehod [][] are baed on Fréche diance or i varian. Fréche diance ake he coninuiy of curve ino accoun and i herefore uiable for comparing rajecorie. In [], he algorihm applie parameric earch over all criical value. Then i olve deciion problem by finding a monoone pah in he free pace from he lower lef corner o he upper righ corner. I run in O(mn log mn ) ime, where m and n are he number of edge and number of node in he road nework repecively. Thi work i exended in [] wih average Fréche diance o reduce he effec of oulier. Paper [] alo ue weak Fréche diance ha run in O(mn log mn ) ime wih imilar maching qualiy. Global mehod aim o minimize he Fréche diance beween he rajecory and he mached road egmen. However, curren mehod have no ried uing rue pah o evaluae he acual maching accuracy. Meanwhile, he emporal/peed informaion in he rajecorie i generally overlooked. A our experimen demonrae, when applied o low-ampling rae real daa wih human labeled rue pah, ST-maching ouperform he mehod baed on Fréche diance boh in erm of maching accuracy and running ime. Saiical model are alo ued for maching GPS obervaion. In [8], a map-maching mehod baed on Bayeian claifier i preened ha incorporae a Hidden Markov Model o model opological conrain of he road nework. An enhanced mehod baed on exended Kalman filer and cubic pline inerpolaion i propoed in []. The aiical approache eem paricularly effecive o handle GPS meauremen error, a demonraed by heir experimen. They may be combined wih our ST-maching mehod o achieve beer reul.. Shore Pah Compuaion Now we briefly review relaed work on hore pah compuaion ince i i ued in our ST-Maching algorihm. Thi i an area ha ha received exenive reearch aenion over decade. A good urvey can be found in [5]. The baic algorihm in hore pah compuaion i Dijkra' algorihm. In pracice, A* algorihm [0] i ofen ued a a more efficien alernaive. A* algorihm ue heuriic funcion o guide he earch oward he deinaion. Oher raegie uch a bidirecional earch [], earch decompoiion [], and hierarchical earch [4] are ofen ued in real applicaion a well. Some pre-proceing ep can be added o peed up he hore pah earch. For example, ALT algorihm in [6] employ he combinaion of landmark and riangular inequaliy o reach a igher lower bound han Euclidean diance ued in A* algorihm. Reach-baed-pruning [9] i anoher mehod o compue lower bound for pruning purpoe.. PROBLEM STATEMENT In hi ecion, we will give he preliminarie and formally define he problem of map-maching for low-ampling-rae GPS rajecorie. Definiion (GPS Log): A GPS log i a collecion of GPS poin L = {p, p,, p n }. Each GPS poin p i L conain laiude p i. la, longiude p i. lng and imeamp p i., a illuraed in he lef par of Figure. Definiion (GPS Trajecory): A GPS Trajecory T i a equence of GPS poin wih he ime inerval beween any conecuive GPS poin no exceeding a cerain hrehold ΔT, i.e. T: p p p n, where p i L, and 0 < p i+. p i. < T ( i < n). Figure how an example of GPS rajecory. ΔT i he ampling inerval. In hi paper, we focu on low ampling rae GPS rajecorie wih T min. Laiude, Longiude, Time p : , 6.9, : p : 9.984, 6.47, :, p n: 9.98, 6.558, :5 Figure. Illuraion of GPS log and GPS rajecory Definiion (Road Segmen): A road egmen e i a direced edge ha i aociaed wih an id e. eid, a ypical ravel peed e. v, a lengh value e. l, a aring poin e. ar, an ending poin e. end and a li of inermediae poin ha decribe he road uing a polyline. Figure 4 how everal real road egmen in Bing Map Search []. Noe ha a road may conain everal road egmen. e e e.ar p p p Figure 4. Illuraion of road egmen Definiion 4 (Road Nework): A road nework i a direced graph G(V, E), where V i a e of verice repreening he inerecion and erminal poin of he road egmen, and E i a e of edge repreening road egmen. Definiion 5 (Pah): Given wo verice V i, V j in a road nework G, a pah P i a e of conneced road egmen ha ar a V i and end a V j, i.e. P: e e e n, where e. ar = V i, e n. end = V j, e k. end = e k+. ar, k < n. Now he problem of map-maching i defined a: p 6 p 7 p 4 p 5 Given a raw GPS rajecory T and a road nework G(V, E), find he pah P from G ha mache T wih i real pah. e e.end e 4

4 4. SYSTEM OVERVIEW The archiecure of our propoed map-maching yem i hown in Figure 5. I i compoed of hree major componen: Candidae Preparaion, Spaial and Temporal Analyi, and Reul Maching. Figure 5. Overview of yem archiecure Candidae Preparaion Thi componen conain a road nework daabae wih indexed edge and verex informaion. I accep given raw GPS rajecory from he uer, and hen rerieve all he poible candidae poin for each ampling poin on he rajecory. Thi ep can be efficienly performed wih he buil-in grid-baed paial index. The oupu of hi componen i a e of candidae poin and he candidae road egmen hey lie on. Spaial and Temporal Analyi Thi componen perform paial analyi followed by emporal analyi on he rerieved candidae e and he rajecory o be mached.. Candidae Preparaion. Spaio-Temporal Analyi. Reul Maching GPS Log Road Nework Candidae Compuaion Candidae Se Spaial Analyi Temporal Analyi Candidae Graph Be Pah Search Maching Reul Uer Inerface Spaial analyi no only conider he diance beween a ingle GPS poin and he candidae road egmen for hi poin, bu alo ake ino accoun he opological informaion of he road nework. To avoid roundabou pah, we employ hore pah o meaure he imilariy beween each candidae pah and he rue pah. Temporal analyi meaure he acual average ravel peed beween any neighboring poin. I hen compare he average peed wih he ypical peed conrain on each candidae pah. The informaion can laer be ued o mach he rajecory o he candidae pah wih mo imilar peed condiion during ha ime inerval. Afer paial and emporal analyi, a candidae graph i conruced a he oupu of hi componen. The node of he graph are he e of candidae poin for each GPS obervaion, and he edge of he graph are e of hore pah beween any wo neighboring candidae poin. The node and edge are all aigned weigh value baed on he reul of paial/emporal analyi. Reul Maching Thi componen evaluae he candidae graph uing he weigh informaion aigned during paial/emporal analyi. I mache given rajecory o he pah wih highe core in he candidae graph. The reul are hen viualized on an inerface ha can be ailored oward differen end-uer device. The reul can alo be ored in a raffic daabae o uppor exernal applicaion uch a raffic managemen or driving direcion. 5. The ST-MATCHING ALGORITHM In hi ecion, we decribe our ST-Maching algorihm in deail. 5. Candidae Preparaion Given rajecory T = p p p n, we fir rerieve a e of candidae road egmen wihin radiu r of each poin p i, i n. Then we compue candidae poin, which are line egmen projecion of p i o hee road egmen defined a follow. Definiion 6 (Line Segmen Projecion): The line egmen projecion of a poin p o a road egmen e i he poin c on e uch ha c = arg min ci e di(c i, p), where di(c i, p) reurn he diance beween p and any poin c i on e. In he re of hi paper, we ue e j j i and c i repecively o denoe he jh candidae edge and candidae poin of p i. A hown in Figure 6, p i candidae poin are c i,c i and c i. Figure 6 Candidae poin for a ampling poin p i To faciliae he earch of candidae poin, he road nework i indexed uing a grid. Once he candidae poin e are rerieved for all he ampling poin on he rajecory T, he problem become how o chooe one candidae from each e o ha P: c j c j c n j n be mache T: p p p n. 5. Spaial Analyi In paial analyi, we ue boh geomeric and opological informaion of he road nework o evaluae he candidae poin found in he previou ep. The geomeric informaion i incorporaed uing obervaion probabiliy, and he opological informaion i expreed uing ranmiion probabiliy. Definiion 7 (Obervaion Probabiliy): The obervaion probabiliy i defined a he likelihood ha a GPS ampling poin p i mache a candidae poin c i j compued baed on he diance beween he wo poin di(c i j, p i ). Generally peaking, he error in a GPS meauremen can be reaonably decribed a a normal diribuion N(μ, σ ) of he diance beween p i and c i j. I indicae how likely a GPS obervaion p i can be mached o a candidae poin c j i on he real road wihou conidering i neighboring poin. Formally, we j define obervaion probabiliy N c i of c j i w.r.. p i a: N c i j e i c i c i = j (x i μ ) πσ e σ () where x i j = di(c i j, p i ) i he diance beween p i and c i j. In hi paper, we ue a zero-mean normal diribuion wih a andard deviaion of 0 meer baed on empirical evaluaion. Obervaion probabiliy doe no ake ino accoun a GPS poin poiion conex. Thi omeime lead o wrong maching reul. Figure 7 how uch an example. The hick line repreen a highway and he hin verical line repreen a local road. Alhough p i i cloer o c i on he local road, we hould c i p i e i e i

5 mach p i o c i on he highway if we already know ha i neighbor p i and p i+ are on he highway. Thi i baed on he obervaion ha a vehicle i unlikely o ake a roundabou pah (Obervaion ). Figure 7. An example ha need ranmiion probabiliy To characerize he above inuiion, we compue hore pah beween wo neighboring candidae poin c i and c i. Then we define ranmiion probabiliy a follow: Definiion 8 (Tranmiion Probabiliy): Given wo candidae poin c i and c i for wo neighboring GPS ampling poin p i and p i repecively, he ranmiion probabiliy from c i o c i i defined a he likelihood ha he rue pah from p i o p i follow he hore pah from c i o c i. We compue ranmiion probabiliy a V c i c i = d i i w i, (i,) () where d i i = di(p i, p i ) i he Euclidean diance beween p i and p i, and w i, (i,) i he lengh of hore pah from c i o c i. Combining Equaion () and (), we define he paial analyi funcion F c i c i a he produc of obervaion probabiliy and ranmiion probabiliy: F c i p i c i p i c i c i = N c i V c i p i+ c i, i n () where c i and c i are any wo candidae poin for wo neighboring GPS poin p i and p i repecively. Equaion () compue he likelihood ha an objec move from c i o c i uing he produc of wo probabiliy funcion, hu geomeric and opological informaion are boh aken ino conideraion. Noe ha in pracice i i unlikely for a moving objec o alway follow he ric hore pah. Therefore he obervaion probabiliy N c i canno be omied from (). Wih paial analyi, for any wo neighboring GPS poin p i and p i, a e of candidae pah c i c i are generaed. Each pah i aigned a paial meauremen value compued from Equaion (). 5. Temporal Analyi Spaial analyi can diinguih he acual pah from he oher candidae pah in mo cae. However, here are iuaion ha paial analyi alone could no handle. Conider he example hown in Figure 8. The hick line i a highway, and he hin line are ervice road cloe o he highway. The paial analyi funcion may produce he ame value wheher wo poin p i and p i are mached o he highway or he ervice road. However, if we calculae he average peed from p i o p i a km/, we would mach hem o he highway conidering he peed limi of he ervice road (Obervaion ). More formally, given wo candidae poin c i and c i for wo neighboring GPS ampling poin p i and p i repecively, he hore pah from c i o c i i denoed a a li of road egmen [e, e,, e k ]. The average peed v i, (i,) of he hore pah i compued a follow:. v i, (i,) = k l u u= i i (4) where l u = e u. l i he lengh of e u, and i i = p i. p i. i he ime inerval beween wo ampling poin p i and p i. Noe ha each road egmen e u i alo aociaed wih a ypical peed value e u. v. We employ coine diance o meaure he imilariy beween he acual average peed from c i o c i and he peed conrain of he pah. Conider he vecor ha conain k elemen of he ame value v i, (i,) and he vecor (e. v, e. v,, e k. v) T. The emporal analyi funcion i defined a follow. F c i c i = k u = A in paial analyi funcion, c i poin for p i and p i repecively. (e u. v v i, (i,)) k u = (e u. v) k u = v i, (i,) (5) and c i are any wo candidae 5.4 Reul Maching Wih he paial and emporal analyi above, we are ready o decribe our ST-Maching algorihm ued in he reul maching componen. In general, afer paial and emporal analyi, we are able o generae a candidae graph G T (V T, E T ) for rajecory T: p p p n. V T i a e of candidae poin for each GPS ampling poin, and E T i a e of edge repreening he hore pah beween any wo neighboring candidae poin, a depiced in Figure 9. Each node in G i aociaed wih N c i. Each edge i aociaed wih V c i c i and F c i c i. P ' candidae P ' candidae P n' candidae c c c c c c c c c c n c n A Highway P i P i- A Service Road Figure 8. An example ha need emporal analyi Figure 9. The candidae graph G T (V T, E T ) Combining Equaion () and (5), we define ST funcion for c i a: c i F c i c i = F c i c i F c i c i, i n (6)

6 A candidae pah equence P c for he enire rajecory T i a pah in he candidae graph, denoed a P c : c c c n n. The overall core for uch a candidae equence i F P c = n F c i i c i i= i. From all he candidae equence we aim o find he one wih he highe overall core a he be maching pah for he rajecory. More formally, he be maching pah P for a rajecory T i eleced a: P = arg max Pc F P c, P c G T (V T, E T ) (7) Algorihm ouline he framework of ST-Maching algorihm. We fir compue he candidae poin e for each GPS ampling poin on T. Then we conruc he candidae graph G T (V T, E T ) baed on paial and emporal analyi. Finally, he algorihm repor he pah equence P wih highe ST-funcion value from G T a he reul. Algorihm ST-Maching Algorihm Inpu: Road nework G, a rajecory T: p p p n. Oupu: The mached equence P: c j c j c n j n in G : Iniialize Li a an empy li; // a li of candidae poin : for i = o n do : = GeCandidae(p i, G, r); // candidae wihin radiu r 4: Li.add(); 5: G T = ConrucGraph(Li); // conruc graph G T 6: reurn FindMachedSequence( G T ); The FindMachedSequence procedure aim o find he longe pah in G T. In general, finding longe pah in a graph i an NPC problem. However, G T i a direced acyclic graph (DAG) and he anwer can be compued efficienly uing he opological order of he graph. I i eay o ee ha uch opological order can be obained from he conrucion of G T. Algorihm how he deail of he procedure FindMachedSequence. Algorihm FindMachedSequence Inpu: Candidae graph G T (V T, E T ) Oupu: The longe equence P: c c c n n in GT : Le f[ ] denoe he highe core compued o far; : Le pre[ ] denoe he paren of curren candidae; : for each c do 4: f c = N c ; 5: for i = o n do 6: for each c i do 7: max = ; 8: for each c i do 9: al = f[c i ] + F c i c i ; 0: if (al > max) hen : max = al; : pre c i = c i ; : f c i = max; 4: Iniialize rli a an empy li; // a li of mached poin in revere order 5: c = arg max cn (f[c n ]); 6: for i = n o do 7: rli.add(c); 8: c = pre c ; 9: rli.add(c); 0: reurn rli. revere(); Figure 0 how a running example of he above procedure. The value of obervaion probabiliy, ranmiion probabiliy and emporal meauremen funcion are alo lied in he figure. V, F : P ' candidae P ' candidae P ' candidae c c c Figure 0. An example of FindMachedSequence procedure The algorihm fir fill ou he able f[ ] a follow. c c N: c c c c c c c f : c c c (0.5,0.5) (0.8,0.5) c c c (0.,0.4) (0.,0.9) (0.4,0.6) (0.9,0.9) Nex conider candidae c. The algorihm fir evaluae he pah c c. We have F c c = = 0.5. Similarly, F c c = 0.07 and F c c = Therefore f c = max{ , , } = 0.95, and c paren i c. We repea he above proce for all he candidae and complee he able f[ ]. We find ha c ha he highe overall core, and we chooe c a he maching reul for he la GPS poin. Baed on he ored F: paren informaion, he algorihm finally repor he maching reul a c c 0.8 c Localizing ST-Maching Sraegie The propoed ST-Maching algorihm i a global algorihm a we can only idenify he be pah equence afer compuing overall core for he enire rajecory. In pracice, when a rajecory ha oo many poin (i.e. n i very large), or online proceing i deired, we may localize he ST-Maching algorihm by conruc a parial candidae graph G T[w] over a liding window of he rajecory, where w(w < n) i he window ize. A illuraed in Figure, each parial candidae graph G T[w] i conruced from a ube of rajecory T. We find he be maching pah equence for hi ube imilarly a in global STmaching. Then we lide he window and repea he proce. c c c c c c c c c c c c (0.,0.6) (0.,0.7) (0.,0.) (0.5,0.9) f : c c c c c c c

7 Coun Coun c c c c G T[w] GT[w] c P ' candidae P ' candidae P n ' candidae l c c c c c n c n baed global map-maching algorihm. The reul are mainly evaluaed in erm of running ime and maching accuracy. 7. Experimenal Seing 7.. Daae Decripion Road Nework: In our experimen, we ue he road nework of Beijing a viualized in Figure. The nework graph conain 58,64 verice and 0,74 road egmen. Figure. Localizing ST-Maching (w =, l = ) Pariioning rajecory ino window can reduce he average delay and ave orage pace for online proceing, bu doe no necearily peed up he overall proceing ime becaue he mo expenive par of ST-Maching algorihm i hore pah compuaion. To reduce compuaion complexiy, we may heuriically reain candidae poin wih op l core o far, hu reducing he number of hore-pah ha need o be compued for nex ampling poin. When l =, ST-Maching degenerae ino an incremenal algorihm. 6. ALGORITHM ANALYSIS Now we analyze he ime complexiy of our propoed ST- Maching algorihm. Le n denoe he number of ampling poin on he given rajecory, and m denoe he number of road egmen in he road nework. We furher aume ha he maximum number of candidae of one ampling poin i k. Noice ha he number of hore pah in G T i (n )k. The ime complexiy for conrucing he candidae graph i herefore O(nk m log m). For he FindMachedSequence procedure, each edge in G T i viied exacly once. The ime complexiy i O(nk ). Therefore our global ST-Maching algorihm ha he ime complexiy of O(nk m log m + nk ). Noe ha for any given ampling poin, we can pracically chooe a mall value of k. Thu he ime complexiy i cloe o O nm log m. By conra, he global algorihm in [7] ha he complexiy of O nm (log mn) uing Fréche Diance and O nmlogmn uing Weak Fréche Diance. In pracice, Dijkra i ofen no a good choice o compue hore pah. We may chooe A* [0] or ALT algorihm [6] o achieve a beer running ime. A* algorihm ue eimae on diance o he deinaion o guide verex elecion in a earch from he ource. To achieve even faer running ime, ALT algorihm employ he combinaion of landmark and riangular inequaliy o ge a igher lower bound, bu i require more dik pace for pre-proceing. Moreover, he localizing raegie propoed in Secion 5.5 are alo ueful o reduce he repone ime and improve compuaion efficiency. 7. EXPERIMENTS In hi ecion, we fir preen he experimenal eing, including he daae we ued and ome parameer we eleced in he experimen. Nex we decribe he evaluaion approache ha we applied. Then we repor he major reul wih ome dicuion. Our experimenal reul are compared wih boh incremenal algorihm and Average-Fréche-Diance (AFD) Figure. Road nework of Beijing Synheic Trajecory Daa: The ynheic daa ued in our experimen conain rajecorie randomly generaed by our own imulaor. The imulaor work a follow. I fir randomly elec wo verice in he road nework and compue op K hore pah beween hem. Then i randomly elec a rajecory from he K pah a he ground ruh, denoed a G: e, e,, e n. The moivaion behind hi i ha moving objec generally follow he direcion from ource o deinaion, bu no necearily follow he hore pah ricly. Noe ha he ime inerval beween any wo neighboring poin i no uniform. To rerieve a rajecory wih deired ampling inerval, he imulaor elec one road egmen from every k egmen on G, i.e., i elec e, e +k,, e +mk, where m = n. The adjumen of ampling i rae i herefore achieved by changing he value of k. The imulaor generae one GPS poin wih eimaed imeamp informaion for each eleced road egmen. The poin are produced o follow he zero-mean normal diribuion wih he andard deviaion of 0 meer. Real Trajecory Daa: Differen from exiing approache, we alo ue human labeled rue pah daa a he ground ruh in our experimen. From hundred of real rajecorie colleced from GeoLife yem [], we elec 8 rajecorie ha divere in paial coverage, lengh, ime inerval and number of road egmen. They are manually labeled wih he rue pah. Figure how he hiogram of he real daae characeriic Number of Poin (k) Figure. Hiogram of real daae Figure 4 how an example of real rajecory viualized uing Bing Map API. The blue line conneced by green marker repreen he GPS rajecory, and he red line conneced by red marker repreen he rue pah of he uer Road Lengh (km)

8 Running Time (m) Acuracy (%) Figure 4. An example of labeled rajecory daa in GeoLife[] 7.. Parameer Selecion Daa Sampling Rae: Thi paper focue on low-ampling-rae GPS rajecorie. The ampling inerval range from minue o nearly 6 minue for ynheic daa, and range from 0 econd o 5 minue for real rajecory daa. Parameer for ST-Maching Algorihm: In our experimen, we chooe k = 5 empirically a he maximum number of candidae of any ampling poin. The candidae earch radiu i eleced a r = 00m. For obervaion probabiliy eimaion, we ue a normal diribuion wih μ = 0 and δ = 0m. When evaluaing localized raegie, he window ize range from o 40, and we chooe o reain op,, 5 node. Parameer for Baeline Algorihm: We compare our experimen wih wo baeline algorihm in hi paper. For he incremenal algorihm, we ue he empirical eing μ d = 0, a = 0.7, n d =.4, μ α = 0, and n α = 4 a uggeed in []. For AFD-baed algorihm, we e λ = 0 meer. 7. Evaluaion Approache Ground Truh: In ynheic daa, he ground ruh i randomly eleced among he op K hore pah from he ource o he deinaion. In real daa colleced from GeoLife, he ground ruh i eablihed uing human labeled rue pah. During he labeling proce, if a uer canno idenify he acual road egmen ha he ampling poin reide, uch poin i labeled wih a pecial ag o ha laer on i will no be couned in he evaluaion. Evaluaion Crieria: Our ST-Maching algorihm i evaluaed boh in erm of running ime and maching qualiy. The running ime i meaured uing he acual program execuion ime. The maching qualiy i meaured uing wo accuracy meric Accuracy by Number (A N ) and Accuracy by Lengh (A L ) defined a follow. # correcly maced road egmen A N = # all road egmen of e rajecory A L = Te leng of maced road egmen Te leng of e rajecory Baeline: We mainly compare our ST-Maching algorihm wih wo baeline: an incremenal algorihm ha perform maching baed on a poin previou maching reul; and a global maching algorihm ha aim o minimize Average-Fréche- Diance (AFD). To evaluae he effec of emporal analyi, we developed an S-Maching algorihm by removing emporal analyi from ST-Maching. S-Maching i alo compared wih he original ST-Maching mehod. 7. Experimenal Reul 7.. Reul on Synheic Daa Impac of Max. Number of Candidae Poin: Figure 5 and 6 how he impac of maximum number of candidae poin on maching accuracy and running ime of ST-Maching. In he figure, Inf mean ha we do no limi he number of candidae Figure 5. Accuracy w.r.. number of candidae Inf Number of Candidae Figure 6. Running ime w.r.. number of candidae A hown in Figure 5, he accuracy generally increae a he algorihm ake more candidae poin ino conideraion. However, a large number of candidae poin for one GPS poin would lead o a huge amoun of hore pah compuaion, which will ignificanly increae running ime. Figure 6 how ha if we limi he number of candidae o 5, he average running ime of our map-maching algorihm i accepable. ST-Maching v. S-Maching: Nex we evaluae he effec of emporal analyi by comparing he maching qualiy of ST-Maching and S-Maching w.r. o ampling inerval. In ynheic GPS rajecory generaion, he change of ampling inerval i achieved hrough he adjumen of parameer k a dicued in Secion 7... We chooe k a 9,,, 5, and 7 repecively. The value of ampling inerval for each value of i i lied in Table. Table. Sampling inerval in ynheic daa k Inerval (min) We compue op 5 hore pah. For each k, we randomly generae 0 GPS rajecorie and e our map-maching algorihm. We hen ake he average accuracy. AN AL 4 5 Inf Number of Candidae

9 A hown in Table, he accuracy generally goe down a ampling rae decreae. Thi i becaue he hore pah aumpion will no longer hold. We alo oberve ha ST- Maching ouperform S-Maching mo of he ime, indicaing ha emporal conrain are indeed ueful in mo cae. Table. ST-Maching v. S-Maching (accuracy) k A N A L S-Maching ST-Maching S-Maching ST-Maching ST-Maching v.. Incremenal v.. AFD-baed global algorihm Nex we compare our ST-Maching algorihm wih incremenal and AFD-baed global map-maching algorihm. A Figure 7 and 8 how, ST-Maching ouperform boh incremenal and ADFbaed algorihm ignificanly. Meanwhile we noice ha he performance of incremenal algorihm degenerae harply when ampling rae decreae while global algorihm are more robu o he change of ampling rae. Running Time () A L Figure 7. A L (%) w.r. ampling inerval A N Sampling Inerval (k') Sampling Inerval (k') ST-Maching Incremenal Figure 8. A N (%) w.r. ampling inerval 7.. Reul on Real Daa Now we evaluae ST-Maching algorihm on real daa colleced from GeoLife. ST-Maching v.. Incremenal v.. AFD-baed global algorihm Figure 9 and 0 how he change of maching qualiy w.r. o he ampling rae on real daa. I can be een clearly ha our ST- Maching algorihm ouperform boh incremenal and AFD-baed algorihm on real daa. A he ampling rae decreae, he accuracy of incremenal algorihm drop harply, while wo global algorihm again exhibi more conien performance. We alo noice ha he maching accuracy of wo global algorihm end o be very cloe when he ampling inerval i high. Thi implie AFD ST-Maching Incremenal AFD ha emporal conrain become le effecive when neighboring poin are far away from each oher. A L Incremenal ST-Maching AFD Sampling Inerval (min) Figure 9. A L w.r. ampling inerval on real daa A N Incremenal ST-Maching AFD Sampling Inerval (min) Figure 0. A N w.r. ampling inerval on real daa Running Time In order o evaluae he efficiency of ST-Maching algorihm, we compare he running ime of our algorihm wih he AFD-baed global mehod. A demonraed by Figure, when number of poin i mall, he wo mehod exhibi imilar running ime. However, a he number of poin increae, he running ime of ADF-baed approach increae dramaically. 0 8 ST-Maching 6 AFD Number of Poin (n) Figure. Running ime comparion on real daa Localizing Sraegie Finally we evaluae he effec of localizing raegie propoed in Secion 5.5 wih one real rajecory of over 400 poin. From Figure i can be een ha in general when he window ize reache a cerain value, he furher increae of window ize ha lile effec in improving he accuracy. In ome cae, he increae of window ize may even have advere effec on maching accuracy. Thi implie ha localized ST-Maching can be pracically ued a long a we elec an appropriae window ize. On he oher hand, reaining op l node will ignificanly reduce he maching accuracy epecially when window ize i large.

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