Symbolization of Mobile Object Trajectories with the Support to Motion Data Mining

Transcription

1 Symbolizatio of Mobile Object Trajectories with the Support to Motio Data Miig Xiaomig Ji, Jiami Wag, ad Jiaguag Su School of Software Tsighua Uiversity, Beijig, , Chia Abstract. Extractio ad represetatio of the evets i trajectory data eable us go beyod the primitive ad quatitative values ad focus o the high level kowledge. O the other had, it eables the applicatios of vast off the shelf methods, which was origially desiged for miig evet sequeces, to trajectory data. I this paper, the problem of symbolizig trajectory data is addressed. We first itroduce a static symbolizatio method, i which typical subtrajectories are geerated automatically based o the data. For facilitatig the data miig process o streamig trajectories, we also preset a icremetal method, which dyamically adjusts the typical sub-trajectories accordig to the most recet data characters. The performaces of our approaches were evaluated o both real data ad sythetic data. Experimetal results justify the effectiveess of the proposed methods ad the superiority of the icremetal approach. Keywords: Motio data miig, spatial trajectory, symbolizatio 1 Itroductio The recet advaces i geographic data collectio devices ad locatio-aware techologies have icreased the productio ad collectio of spatial trajectories of movig objects [1, 2]. Well kow examples iclude global positioig systems (GPS), remote sesors, mobile phoes, vehicle avigatio systems, aimal mobility trackers, ad wireless Iteret cliets. Briefly, trajectory of a movig object is a sequece of cosecutive locatios of the object i a multidimesioal (geerally two or three dimesioal) space [3]. Fig. 1 shows a simple example of 2D trajectory data. Recetly, developmets of data miig techiques o trajectory data have received growig iterests i both research ad idustry fields [4, 5, 6]. For example, sequetial rules with the format if movemet A the movemet B ca be discovered by searchig the historical trajectories geerated by a vehicle user. The based o the rules, persoalized services ca be provided accordig to the user s curret movemets i a vehicle avigatio system. I aother case, by aalyzig the trajectories geerated by all vehicles i a city, we ca fid some frequet patters that reveal importat passages, crossroads, highway, or other traffic facilities that are used frequetly. The results of such aalysis ca be used for improvig the maagemet ad maiteace of the traffic system. S. Wag et al. (Eds.): ER Workshops 2004, LNCS 3289, pp , Spriger-Verlag Berli Heidelberg 2004

2 104 Xiaomig Ji, Jiami Wag, ad Jiaguag Su Fig. 1. A example of movig object trajectory i 2D space. Each diamod i the figure stads for a sigle locatio of the object at a certai time, ad arrow represets a movemet from oe locatio to aother. I this paper, we address a ovel problem, which is symbolizatio of trajectories. The problem is to map a trajectory ito a symbolic sequece, i which each symbol represets the local movemet at a time poit or durig a time period. For example, the trajectory of a car movemet trajectory ca be simply represeted by a symbolic sequece i which each symbol idicates the shape of a quarter-hourly movemet. The the symbolic sequece ca be examied for discoverig rules such as if a particular movemet A, idicatig a crossroads, occurs, the the car will go orth straightly, say movemet B, i the secod quarter-hour. Our research o this topic is maily motivated by two ideas: First, there is a broad cosesus that we are usually iterested i high level represetatio of the data, rather tha the primitive ad quatitative values. Secod, techiques for miig symbolic (evet) sequeces have bee studies extesively i various applicatio domais, such as click stream aalysis, bio-iformatics, ad so o, ad may sophisticated algorithms, models, ad data structures were desiged for hadlig symbolic sequeces. Therefore, covertig trajectory data ito symbolic sequece could eable the applicatios of the vast off the shelf methods i miig trajectory data. Obviously, a appropriate ad efficiet method for extractig ad represetig the evets i the trajectory data will achieve these goals, ad whereupo, beefits the data miig tasks over trajectory data. Symbolizatio is useful i miig spatial trajectory data. O the other had, it ca also help represetig other data objects that seem differet but are essetially with similar characters, such as features extracted from sigature image ad multiple attribute respose curves i drug therapy. Geerally, symbolizatio ca be viewed as a explaiig process that classifies (or approximates) each idividual atomic sub-trajectory ito a typical sub-trajectory movemets, e.g. go orth, circumambulate, which is retrieved or defied beforehad. A simple ad ituitive solutio for geeratig the typical sub-trajectory is to choose it maually based o the domai expert s aalysis ad explaatio. Ad the the atomic sub-trajectory at each time is represeted by a simple earest eighbor query through the give sets of typical sub-trajectory. Such ideas had bee embedded ito some ad hoc problem defiitios, as well as data miig methods, implicitly or explicitly. However, it is usually extremely expesive to fid ad uderstad all ca-

3 Symbolizatio of Mobile Object Trajectories with the Support to Motio Data Miig 105 didate sub-trajectories i may real applicatios, whereupo it is very difficult, if ot impossible to geerate typical sub-trajectories by the above maual approach. Aother importat aspect of mobile object trajectories is that the data are usually bor with streamig property i may applicatio domais. That is, the trajectories are frequetly appeded i the ed over time. Sice may factors that impact a movig object might be time varyig, the patters of the object movemets are usually time varyig, whereupo the typical sub-trajectory should also be adjusted correspodigly. I this case, it is usually difficult to apply a static approach from a practical poit of view, where typical sub-trajectories are geerated based o a sapshot of the trajectory, because it may fail to give a good represetatio of the ew data. Alteratively, the typical sub-trajectories ca be re-geerated cocurretly with each update. However, such re-geeratio will eed all the data scaed, whereupo the time complexity might be extremely poor. The above issues challege the research o symbolizig trajectory data. Therefore, it is by o meas trivial to cosider this ovel problem ad develop effective ad efficiet methods for it. I this paper, the problem of symbolizig trajectory data is addressed. We first propose a symbolizatio method, i which typical sub-trajectory is geerated automatically based o the data. For facilitatig the data miig process o streamig trajectories, we also preset a icremetal symbolizatio method, which dyamically adjusts the typical sub-trajectories accordig to the most recet data characters, without scaig the whole date set. The performace of our approach is evaluated o both real trajectory data ad sythetic data. Experimetal results justify the effectiveess of the proposed methods ad the superiority of the icremetal approach. 2 Related Work Data maagemet ad data miig o trajectory data had bee studied i may applicatios [1, 2, 3, 4, 5]. Data miig applicatios ca be foud i [5, 6]. The most fudametal works i these cotexts is o similarity measurig ad idexig, e.g. [7, 8, 9, 10]. Related methods were developed maily by extedig the exist oes, e.g. Euclidea distace, Dyamic Time Warpig (DTW), Logest Commo Subsequece (LCSS), ad multi-dimesioal idexes. These works did ot cosider the symbolizatio process, but may ideas may serve as subrouties i the approach proposed i this paper. I some previous work, usupervised learig techiques were applied o the trajectories or similar data objects. For example, [11] preseted a approach to clusterig the experieces of a autoomous aget. Such methods were desiged for purposes fudametally differet from ours, ad were ot further developed to fit the geeral data miig tasks. Symbolizatio o time series data, as a importat preprocessig subroutie, had bee extesively studied for various data miig tasks [12,13,14], such as rule discovery, frequet patter miig, predictio, ad query by cotet, etc. A shape defiitio laguage was proposed i [15]. I [16], the time series was symbolized usig cluster method for discoverig rules with the format if evet A occurs, the evet B occurs withi time T. This method was the used i may applicatios that focus o

4 106 Xiaomig Ji, Jiami Wag, ad Jiaguag Su miig time series [13,14]. [17] Claims that the method i [16] is meaigless if the step of slidig widow is set to be 1. Actually, this problem could be solved by icreasig the step of slidig widows. The above works deal with oe-dimesioal time series ad the further extesio for miig trajectory data have ot bee well cosidered. Our approach ca be viewed as a expasio of the method itroduced i [16] to facilitate the data miig process o trajectory data. 3 Problem Descriptios A trajectory of a movig object is a sequece of cosecutive locatios i a multidimesioal (geerally two or three dimesioal) space. I this paper, we oly address the trajectories i 2D for clearess. A trajectory is deoted as: T=T(1),,T(N) ad T()=(Tx(), Ty()) stads for the locatio of the movig object at time. T =N deotes the legth of T. The projectio of T i x-axis ad that i y-axis are represeted as Tx = Tx(1),, Tx(N) ad Ty = Ty(1),, Ty(N) respectively. The sub-trajectory T(m), T(m+1),,T() is deoted by T[m,]. As itroduced i sectio 1, symbolizatio is a process to represet the object s behaviors at each idividual time poit. The it is ituitive to first divide the trajectory to extract the sub-trajectories sequetially at various time poits, ad the to symbolize each extracted sub-trajectory idividually by comparig it with a group of typical sub-trajectories. A typical sub-trajectory is a prevalet sub-trajectory that represets a typical form of sub-trajectory movemets. Here we use the ituitive slidig widow approach: Give trajectory T, widow sub-trajectories of T are cotiguous sub-trajectories W =T[s(),e()] or (Wx, Wy )=(Tx[s(),e()], Ty[s(),e()]) extracted sequetially by slidig the widow through the trajectory, where s()=k-k+1 ad e()=k-k+w-1 stad for the startig poit ad edig poit of the -th widow respectively, deotes the order umber of the widows, parameter w cotrols the size of each widow, ad k cotrols the offset of positios of two cosecutive widows. Based o the above otios, the problem of symbolizig trajectory ca be defied as follows: Give a trajectory T ad all its widow sub-trajectory W, symbolizatio is to covert T ito a temporal sequece S=(S(1),,S(M)), which is a ordered list of symbols where each symbol S() comes from a predefied alphabet Σ ad represets the movemets i the -th sub-trajectory W. 4 Symbolizatio Approach The key idea of our symbolizig approach is to cluster all the sub-trajectories, ad the each sub-trajectory is represeted by the idetifier of the cluster that cotais it. The overall approach is formally illustrated as follows: 1) Extract all sub-trajectories W i trajectory S (W is defied i sectio 3). 2) Normalize each sub-trajectory W to W. 3) Cluster all W ito sets C 1,, C H, that is, W C j(). 4) For each cluster C h, a uique symbol a h from Σ is iducted.

5 Symbolizatio of Mobile Object Trajectories with the Support to Motio Data Miig 107 5) The symbol sequece S is obtaied by lookig for each W the cluster C j(), ad usig the correspodig symbol a j() to represet the sub-trajectory at that poit, i.e. S = a j(1), a j(2),, a j(m), M= N/k. Though the overall strategy seems similar with that for time series data, the symbolizatio process for spatial trajectory is differet i several importat aspects, e.g. ormalizatio process, similarity measuremet, ad cluster process. The ormalizatio step is applied for precise represetatio by removig the impacts of the absolute locatio value ad the scalig factors i both dimesios. For time series X=(X(1),,X(N)), ormalizatio ca be easily doe by X =(X-E(X))/D(X) where E(X) is the mea of X ad D(X) is the stadard deviatio of X. However, this problem is somewhat less straightforward for trajectory data, because the scale of differet dimesio may differ. For example, give a trajectory T, it may be uadvisable to simply use the above strategy o Tx ad Ty respectively, sice this process zooms the movemets i both dimesios ito exactly the same scale, whereupo the shape of the origial trajectory may be demolished. We ormalize a sub-trajectory W = (Wx, Wy ), as follows: ( Wx ( Wx )/ max( D( Wx ), D( Wy ) ( Wy ( Wy )/ max( D( Wx ), D( Wy ) Wx = E Wy = E This ormalizig process zooms a sub-trajectory ito a cell with uit size without chagig its shape. The differece of the above two ormalizatio processes is demostrated i Fig. 2. (A) (B) (C) Fig. 2. Example trajectory (A) ad the ormalized sub-trajectories with the origial shapes modified (B), ad ormalized sub-trajectories with the shape preserved (C). I some applicatios, the iformatio of iterest is the relative movemets that are irrelevat to its absolute directios, e.g. go left or forward, rather tha go orth or go alog a meridia 45 couterclockwise from east. O such occasios, a rotatig trasform eed to be applied based o the startig directio, i.e. each movemet (represeted by two cosecutive locatios) i a sub-trajectory is rotated to its relative directio to the last movemet i the precedig sub-trajectory. Clusterig is the process of groupig a set of objects ito classes of similar objects. Our symbolizatio approach has o costrait o the clusterig algorithm, ay commo distace based clusterig method could be used here, e.g. greedy method, recursive k-meas, agglomerative ad divisive hierarchical clusterig, BIRCH, CUBE,

6 108 Xiaomig Ji, Jiami Wag, ad Jiaguag Su Chameleo, etc. We use greedy method as the cluster subroutie i presetig our method ad experimets, because the time complexity is soud, it ca be easily implemeted, ad it does a good job i supportig the icremetal symbolizatio process. For each W, greedy method first fids the cluster ceter q such that the distace betwee W ad q is miimal. If the distace is less tha a predefied threshold d max, W is added to the cluster whose ceter is q ad the cluster ceter of q is regeerated as the poit-wise average of all the sub-trajectory cotaied i it, otherwise a ew cluster with ceter W is created. I the cluster process, Euclidia distace is used as the distace measuremet. Give two trajectories A=(Ax(1),Ay(1)),, (Ax( A ),Ay( A )) ad B=(Bx(1),By(1)),, (Bx( B ),By( B )) with the same legth M, the Euclidia distace betwee A ad B is defied as: D M ( ) 2 2 ( A, B) = ( Ax() i Bx() i ) + ( Ay() i By() i ) i= 1 Note that, other more sophisticated similarity measuremets ca also be applied i our method, such as Dyamic Time Warpig (DTW) or Logest Commo Subsequece (LCSS). DTW is defied as: DTW ( A, B) = Lp( Ax( A), Ay( A ), ( Bx( B ), By( B ) DTW Head + mi DTW DTW Head ( () A, B), ( A,Head() B ), ( () ()) A,Head B where Lp stads for p-orm, i.e. Lp(V1,V2) =( k V1(K)- V2(K) p ) 1/p for vector V1 ad V2, ad Head (A) = (Ax(1),Ay(1)),, (Ax( A -1),Ay( A -1)). LCSS is defied as follows: Give parameters δ ad 0<ε<1, LCSS δ, ε max ( A, B) = 1+ LCSS ( Head( A),Head( B ) 0 ( LCSS ( Head() A, B),LCSS ( A,Head( B ) δ, ε δ, ε δ, ε A = φ B = φ Ax < ε Ay ( A) Bx( B ) ( A) By( B ) A B < δ otherwise < ε Aother importat problem is the symbol mappig model, i.e. Σ. The symbol set is usually defied by a group of simple symbols, e.g. the lower case alphabet a z. Sice our symbolizatio approach geerates the typical sub-trajectories automatically, the meaig of each symbol i Σ caot be explaied or defied a priori. However, after the whole trajectory has bee symbolized, the meaig of symbol a h Σ ca be extracted by maual reviews of the correspodig cluster ceter of C h. Therefore, if meaigful represetatios are more favorable, a simple post-processig procedure ca be applied by itroducig a ew alphabet Σ that is geerated by first replacig each symbol a h i Σ by a meaigful oe based o the maual explaatio of the cluster C h, ad the rewritig the resultig symbolic sequece based o Σ.

7 Symbolizatio of Mobile Object Trajectories with the Support to Motio Data Miig Icremetal Symbolizatio Approach Streamig trajectory is a trajectory with ew data items geerated ad appeded i the ed frequetly. O such occasios, the icremetal symbolizig process ca be formalized as follows: Wheever a trajectory T is updated to TU (TU=T(1),, T(N),U(1),,U(K) is the direct coectio of T=T(1),,T(N) ad U=U(1),,U(K)), update the represetig symbol sequece from S T to S T S U, where S T, S U correspod to origial trajectory T ad the update U respectively. I this process, both typical subtrajectories ad symbol mappig model eed to be dyamically updated with the data collectios. Therefore, the static symbolizatio approaches are ot applicable for streamig trajectories from a practical poit of view as itroduced i sectio 1. The key idea of our dyamic symbolizig approach is that istead of usig the static cluster ceters as the typical sub-trajectories, we dyamically maitai the cluster iformatio, ad the the symbol sequece of ew sub-trajectories is geerated based o the up-to-date versio of typical sub-trajectories. Assume that, the method itroduced i sectio 4 has already bee applied o a iitial trajectory, ad the resultig cluster ceters are saved as the iitial typical subtrajectories. The, wheever the trajectory is updated, the cluster iformatio is updated by itroducig the ew sub-trajectories ad removig the old sub-trajectories with the geeratio time t such that t<t ow -t max where t ow deotes the curret time ad t max is a predefied threshold maximal time rage. Here it is assumed that the typical sub-trajectories will evolve with the updatig process, whereupo the sub-trajectories that are geerated too log ago will have mior effect o represetig the curret subtrajectories. The detailed method is illustrated as follows: First, extract all sub-trajectories W cotaied i clusters ad with the geeratio time t<t ow -t max. Remove each W from the correspodig clusters C j(), ad the cluster ceter q j() is re-computed as the poit-wise average of all the sub-trajectories remaied i C j(). If C j() becomes empty after the deletio, remove it. The extract ad ormalize all sub-trajectories cotaied i U, let the resultig sub-trajectories be W m. For each W m, add it to a cluster C j(m) ad geerate symbol a j(m) for W m. The method used i this step is same as that itroduced i sectio 4. Whe a trajectory T is updated to TU, oly the sub-trajectories i update part U or the old oes eed to be cosidered. The the total umber of affected subtrajectories is roughly O( U + U ). Each ew sub-trajectory ca be iserted ito the proper cluster i O(H) time where H is the umber of clusters that is depeded o predefied threshold d max. Ad the update of typical sub-trajectories ca be simply doe by a weighted sum of the origial typical sub-trajectories ad the affected subtrajectories. Therefore, the overall time complexity of the icremetal symbolizatio approach is O(H U + U ). This time complexity, i our opiio, ca meet the requiremets of real applicatios.

8 110 Xiaomig Ji, Jiami Wag, ad Jiaguag Su 6 Experimetal Results I this sectio, we preset a empirical study of the proposed methods. The objectives of this study are: 1) to evaluate the effectiveess of the proposed method, ad 2) to compare the performace of icremetal symbolizatio method with the static oe. I the experimets, two sets of data were used: Real Data: The real data were a combiatio trajectory of a group of aimal movemets collected by satellite trackig. The whole trajectory cosists of 238 locatios (i.e. data poits), each of which idicates the logitude ad latitude of the observed object at a time poit. Sythetic Data: The real data is relatively small. To evaluate our approach o varyig size data, we used sythetic data that were geerated as T()=L()+Rg, where L was a trajectory that was geerated based o the real data itroduced above, but with varyig size, each L() was radomly selected aroud the relative positio i the real trajectory. g was Gaussia oise with zero mea ad uit variace. Ad parameter R cotrolled the oise-to-sigal ratio. The performace of a symbolizatio approach was evaluated by the average distace betwee origial sub-trajectories ad the correspodig typical sub-trajectories. That is, a represetatio that is more similar to the origial sub-trajectory is a better symbolizatio result. Give the trajectory T, symbolizatio results is S=a j(1), a j(2),, a j(m), the the performace of this symbolizatio process was evaluated as: P ( C j, W ) S 1 () S = D ( m) S m= 1 where D is Euclidia distace, W stads for the ormalized sub-trajectories of T, ad C stads for the cluster ceters (detailed defiitios ca be foud i sectio 4). It should be oted that the less the measurig result, the better the performace. m Fig. 3. Typical sub-trajectories ad correspodig symbol represetatios, geerated by static symbolizatio approach, o real data set.

9 Symbolizatio of Mobile Object Trajectories with the Support to Motio Data Miig 111 Fig. 3 shows the experimetal results o real data, which icludes all the resultig typical sub-trajectories geerated by the static approach. I the experimets, the width w ad the step k of the slidig widow were set to be 4 ad 3 respectively, because this settig (k=w-1) ca help fully avoidig the correlatio betwee cosecutive sub-trajectories without losig ay iformatio. The results idicated that there were 8 mai typical movemets i the trajectory, i which (E) - (H) represeted straight motios roughly, (A), (B) represeted roud-motios i certai directios, ad (C), (D) represeted zigzag motios i two directios respectively. By represetig each sub-trajectory with a symbol idicatig the correspodig typical sub-trajectory, the symbolizatio process gave us a cocise ad meaigful represetatio of the trajectory data, whereupo a data miig problem could be easily solved by simply applyig a off the shelf symbol-sequece-orieted data miig approach. Recall from sectio 1 that such approaches ca be easily foud i may applicatio domais. Sice each symbol i the symbolic sequece represets the object s behaviors at a idividual time poit, the emphasis of this data miig process is o the high level represetatio of the data, rather tha the quatitive values. I additio, the borrowed data miig method ca be applied directly without ay modificatio required. For example, we could apply a patter miig method to discover frequet patters with the form a period of roud-motios was always followed by a direct motio to the orth-east i two hours. I aother group of experimets, sythetic data with various sizes ad various amouts of oise were geerated respectively for performace comparisos, ad the both static symbolizatio approach ad icremetal oe were used o the geerated data alteratively. Durig these experimets, the widow width ad widow step were also set to be 4 ad 3 respectively. Sice the static approach eed all the data ivolved for re-computatio, whereas i the icremetal approach oly the updated sub-trajectories eed to be examied, the efficiecy issue for the two approaches is quite obvious. Therefore, here we oly give the results o effectiveess comparisos by cosiderig the performace measure itroduced above. Fig. 4 shows the performaces of the two approaches o trajectories with various legths. By visual aalysis, we could fid that the icremetal approach outperformed the static approach. This is partly because the static approach grouped may subtrajectories ito relatively icompact clusters, represeted by out-dated typical subtrajectories. Fig. 5 shows the performaces of the two approaches o trajectories with various amouts of oise ivolved, geerated by varyig the parameter R. This group of results also justifies the superiority of the icremetal approach over the static oe. I additio, the improvemet i performace of the icremetal approach decreased with the icrease of the oise. Fially, the performaces of the two approaches teded to be same. This is because the potetial movemet patters implied by the origial real data were iflueced by the oise. Whe the amplitude of the oise was set to be large eough, the oise flooded the origial real data, whereupo the geerated data became a completely radom oe. For fully radom data, both the static ad the icremetal approaches will perform like a meaigless radom selectio.

10 112 Xiaomig Ji, Jiami Wag, ad Jiaguag Su Fig. 4. Comparisos betwee the static ad icremetal symbolizatio approaches o trajectories with various legths. Fig. 5. Comparisos betwee the static ad icremetal symbolizatio approaches o trajectories with various amouts of oises. 7 Coclusios I this paper, the problem of symbolizig trajectory data is addressed. We first itroduce a static symbolizatio method, i which typical sub-trajectories are geerated automatically based o the data. For facilitatig the data miig process o streamig trajectories, we also preset a icremetal symbolizatio method, which dyamically adjusts the typical sub-trajectories to fit the up-to-date trajectory characters. The performaces of our approaches were evaluated o both real data ad sythetic data. Experimetal results justify the effectiveess of the proposed methods ad show that the icremetal approach outperformed the static oe o trajectories with various legths ad various amouts of oise. I future, we ited to geeralize our symbolizatio approaches by itroducig other similarity measuremets ad cluster models that are more sophisticated. Ackowledgemets The work was partly supported by the NSFC ad the 973 Program 2002CB We thak the aoymous reviewers for the valuable commets. Refereces 1. N. Priyatha, A. Miu, H. Balakrisha, S. Teller. The cricket compass for cotext-aware mobile applicatios. I MOBICOM2001 Coferece Proceedigs, pages 1 14, G. Che, D. Kotz. Categorizig biary topological relatios betwee regios, lies, ad poits i geographic databases. Techical Report TR Dept. of Computer Sciece, Dartmouth College, M. Vlachos, G. Kollios, Dimitrios Guopulos. Discoverig Similar Multidimesioal Trajectories. I Proc. of the 18th Iteratioal Coferece o Data Egieerig (ICDE'02). Sa Jose, Califoria, Y. Yaagisawa, J. Akahai, T. Satoh. Shape-based Similarity Query for Trajectory of Mobile Objects. I Proc. of the 4th Iteratioal Coferece o Mobile Data Maagemet, pages

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