Mining User Similarity Using Spatial-temporal Intersection

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1 215 Mnng User Smlarty Usng Spatal-temporal Intersecton Ymn Wang 1, Rumn Hu 1, Wenhua Huang 1 and Jun Chen 1 1 Natonal Engneerng Research Center for Multmeda Software, School of Computer, Wuhan Unversty Wuhan, Hube, , Chna Abstract The boomng ndustry of locaton-based servces has accumulated a huge collecton of users locaton trajectores and also brngs us opportuntes and challenges to automatcally dscover valuable knowledge from these trajectores. In ths paper, we nvestgate the problem of measurng the smlarty between users. Such user smlarty s sgnfcant to ndvduals, communtes and busnesses by helpng them effectvely retreve the nformaton. To acheve ths goal, we frstly propose a storage structure to represent the user s trajectores, whch not only stores the sequence of user s trajectory, but also stores regons wth ndexng of trajectores whch pass the regons. After that, we gve the smlarty functon between users usng the spataltemporal ntersecton n regons whch are passed by the two users. Fnally, we develop a spatal-temporal ntersecton algorthm to measure user smlarty based on the defnton and storage structure, and we llustrate the results and performance of the algorthm by extensve experments. Keywords: Trajectory Analyss, User Smlarty, Spataltemporal Data Mnng (LCSS)[5], Dynamc Tme Warpng (DTW)[6], and Edt Dstance on Real sequence (EDR)[7] etc. These methods address the problem of dfferent length, nose and local shft, whch often appear on comparng the smlarty between trajectores. However, the complexty of the algorthm s O(n2), where n s the length of trajectory. Many exstng methods were developed based on these classcal methods, most of whch optmze the effcency of k-nearest neghbors search algorthm by usng ndex structure and prunng technques, nevertheless these optmzaton are restrcted by the complexty of computng smlarty between two trajectores. 1. Introducton Recently, the ncreasng pervasveness of locatonacquston technologes, lke GPS and GSM networks, are leadng to the collecton of large spatal-temporal trajectory data, whch brng the opportunty of dscoverng valuable knowledge about users movements [1]. A number of nterestng applcatons are beng developed based on the analyss of trajectores. For example, t s possble to determne mgraton patterns of anmals by analyzng smlar trajectores of them; n a cty traffc system, t s helpful to locate popular routes for programmng a new route by mnng the trajectores of the ctzens. The bass of these applcatons s determnng the smlarty among trajectores of movng object. The trajectory of a movng object could be defned as a sequence of postons of the movng object over a perod of tme, see Fg. 1. We wll say that A and B are more smlar, because they pass more common locatons. As a trajectory s a tme seres essentally, many sequence smlarty search technques were used to address ths problem, such as Longest Common Subsequences Fg. 1 Trajectores of Persons In ths paper, we proposed a new approach nspred by text retreval methods, n whch a document s seen as a set of many words and all the documents are organzed as an nverted fle to facltate effcent retreval. An nverted fle s a structure that has an entry for each word n the corpus followed by a lst of all the documents n whch occurs. In our methods, the persons are analogy to documents and the locatons are the words. Dfferent to words whch are dscrete, locatons are contnuous. We compensate ths dfference by regon dvdng. Specally, we dvde the whole actvty area on whch the persons move on nto many small regons, and then we use the sequence of regons to approxmately represent the trajectory. Whle computng the smlarty between two persons, we use the number of regons passed by both the two persons, wth the regon-based representaton. The major contrbutons of ths paper are as follows: 1. We propose Compresson representaton of user s trajectores and a storage structure whch not only stores

2 216 the grd sequence of a trajectory, but also stores regons wth ndexng of trajectores whch pass the regons. Wth the storage structure, we can easly determne whether and when a trajectory passes through a certan grd. 2. We gve three defntons of smlarty functon, named Maxmum Co-occurrence Tme (MCT): Absolute Smlarty, Relatve Smlarty and Partal Smlarty based on the compresson representaton. The detal descrpton of these defntons s gven n Secton Based on the storage structure, we propose a fast k- nearest neghbor search algorthm. The detaled descrpton of these algorthms s gven n Secton 3. The rest of the paper s organzed as follows. Secton 2 descrbes the grd representaton of trajectory, a storage structure and a novel smlarty functon. In Secton 3, we gve the k-nearest neghbor search algorthm. In Secton 4, we evaluate the accuracy and effcency of MCT by comparng t wth other algorthms. Secton 5 s a bref concluson. 2. Related Work Smlarty search has been well studed n the context of tme seres and trajectory data. The smplest approach to defne the smlarty between two sequences s to convert sequence nto a vector and then use a p-norm dstance to defne the smlarty measure. The p-norm dstance between two n-dmensonal vectors x and y s defned as L ( x, y) = ( n ( x y ) p ) p. For p=2 t s the well p = 1 known Eucldean dstance. Agrawal et al. frstly proposed an approach for sequence smlarty measure. Ther method transforms the sequence nto vector wth Dscrete Fourer Transformaton and uses Eucldean dstance to determne the smlarty [2]. Chen et al. [3] took Dscrete Wavelet Transformaton to convert the sequence, whle Ca et al. [4] transformed the sequence wth Chebyshev Polynomals. However, these methods based on p-norm dstance requre the trajectores wth the same length.several typcal smlarty functons for dfferent length sequence nclude Longest Common Subsequence (LCSS) [5], Dynamc Tme Warpng (DTW) [6], Edt Dstance on Real Sequences (EDR) [7]and Edt Dstance wth Real Penalty (ERP) [8]. Tme warpng technque frst has been used to match sgnals n speech recognton. Berndt and Clfford[6] proposed to explot ths technque to measure the smlarty of tme-seres data, and ts basc dea s to allow repeatng some ponts as many tmes as needed to acheve the best algnment. Sakura et al. [9] mproved DTW by usng an ndex structure wth segmentaton and lower bounded dstance measure. 1 Vlachos et al. [5] used LCSS to compare two trajectores wth the consderaton of spatal space shftng. Longest Common Subsequence s a classcal dstance functon for two strngs; another dstance functon s Edt Dstance. The edt dstance between two strngs of characters s the number of operatons requred to transform one of them nto the other. The allowed operatons nclude replace, delete, nsert and so on. Each operaton s cost s 1. EDR and ERP are both based on Edt Dstance and proposed by Chen et al. EDR drectly utlze Edt Dstance to measure smlarty of two trajectores. Compare to EDR, ERP use the Eucldean dstance as operaton s cost nstead of 1[8]. These method could be used n both tme seres and trajectory data and the tme complexty s bascally O(n 2 ). Ln et al. [10] use OWD to compute two trajectores smlarty. They only consder the spatal shape of trajectory wthout the tme orderng, and utlze grd to approxmate represent trajectores. Grd representaton s also used n ths paper. Frentzos et al. [11] defne a dssmlarty metrc (DISSIM) for the measurement of the spatotemporal dssmlarty between two trajectores. Peleks et al. [12] ntroduce the Localty n-between Polylnes (LIP) functon. The dea s to calculate the area of the shape formed by two 2D polylnes. Ths method requres that the area s fnte. Recently, Takas et al. [13] consder smlarty search for movng object trajectores n spatal networks. They use the dstance n networks nstead of the Eucldean dstance between two ponts. Chen et al. [14] study a problem of searchng trajectores by locatons, n whch the query s a set of locatons. 3. Representaton and Storage of Trajectory Ths secton frst ntroduces a regon-based representaton for user s trajectores, and then proposes a storage structure for the regon-based trajectores. 3.1 Dvdng Regon For transfer contnuous locaton nformaton to dscrete regons, the cluster method could be used for creatng the regons from locaton nformaton by mergng the locatons wth a small dstance n spatal [1]. In real world, the regon usually dvdng by manual or auto dvded though the semantc functon of an area, such as super market, restaurant, park and Stadum etc. For easy to explan, we use a grd-based regon dvdng methods n ths paper, wth whch the whole area s dvded to many grds wth the same sze, and a grd s seen as a regon, see Fg. 2. Ths method was also exploted by [10]

3 Regon-based Representaton We dvde the whole actvty area of movng object nto dsjoned small regons, and then we merge the contnuous ponts n a regon, formulated as ( R, t, τ ), where R means the regon, t means the tme of the frst pont of the contnuous ponts, and τ means the nterval between the last pont and the frst pont. Therefore, the user s trajectory can be compressed represented as: Co ( ) = {( R0, t0, τ0),( R1, t1, τ1), L (1),( Rm, tm, τm)}, t + τ < t+ 1,0 m Further, ( t, τ ) s denoted by T for short, that s T = ( t, τ ) = [ t, t + τ ], and then ( R, t, τ ) s denoted as ( R, T ), and we haveτ = T. The dagram of compressed representaton s shown as Fg 2. We don t restrct the way of dvdng the actvty area, but use the grd wth the same sze n ths paper merely. 3.3 Storage Structure Fg. 2 compressed representaton Storage structure s dvded nto two parts, shown n Fg 3. The frst part s a users trajectores table for storng the users trajectores whch store ndex of regons whch are passed by the users. The second part stores a set of regons, n whch ndexng of trajectores and the ntervals of users passng the regon are stored. For example, as shown n the Fg 3, User3 passes three regons, R3, R5 and R7; and R7 s passed by three users, User1, User3 and User5, whle User5 passes R7 at three nterval [t1,t2], [t3,t4], [t5,t6]. Fg. 3 The dagram for storage structure

4 Smlarty Functon and KNN Search Ths secton frst gves three smlarty functons between trajectores and then provdes an algorthm for KNN search wth the storage structure proposed n Secton Smlarty Functon Before defnng the smlarty functon, we frst ntroduce a varable to measure the user's actvty scope n a regon wth several operatons. Defnton 1:The user's actvty scope n the regon R s defned as V( R, t, τ ), and V( R, t, τ ) = SR ( ) τ, where SR ( ) represents the area of the regon R. Intersecton operaton: V(( R, t, τ ) ( R, t, τ )) j j j 0 R Rj (2) = V( R, T Tj) = SR ( ) T Tj R = Rj Unon operaton: V(( R, t, τ) ( Rj, tj, τ V( R, t, τ) + V( Rj, tj, τ j) R Rj (3) = V( R, T Tj) = SR ( ) T Tj R = Rj Based on the above operatons, we defne three smlarty functons: Absolute Smlarty, Relatve Smlarty and Partal Smlarty. Defnton 2: we defne the Absolute Smlarty as the absolute value of ntersecton of the user's actvty scope. The formula s as follows. asm( o, oj) = VCo ( ( ) Co ( (4) Defnton 3: we defne the Relatve Smlarty as the rato of the ntersecton and unon of the user s actvty scope. The formula s as follows: VCo ( ( ) Co ( rsm( o, oj) = (5) VCo ( ( ) Co ( )) From the defnton of Relatve Smlarty, we can easly obtan that the smlarty between two trajectores s between 0 and 1. Defnton 4: we defne the Partal Smlarty as the rato of the ntersecton and one user s actvty scope. The formula s as follows: VCo ( ( ) Co ( psmo ( o ) j = (6) VCo ( ( )) Obvously, Absolute Smlarty and Relatve Smlarty functon s symmetrcal, where Partal Smlarty s not, and Relatve Smlarty can be seen as Normalzed form of Absolute Smlarty. In some applcatons, Partal j Smlarty s valuable. Ths paper focuses on Absolute Smlarty. 4.2 Smlarty Search Problems K-nearest neghbors search problem can be descrbed as follows: Gven a set of Users, S={O 1,O 2,, O n }, a query User Q, and a postve nteger k, fnd the k Users n S to form a new set S, meetng for any User O n S and any User O j n S-S, Sm(O,Q)>=Sm(O j,q). 4.3 KNN-search Algorthm Ths subsecton gves the k-nearest neghbor search algorthm, see Algorthm 1. Inputs nclude target user, the number of the nearest neghbor and user set wth compressed trajectores. And the output s the k nearest users. Lnes 1-2 are used to ntalze memory space used for storng smlarty values, Lnes 4-10 are used to sum up the co-occurrence tme of user at each regon to calculate the smlarty of users. The Com_Tme n Lne 6 s a functon to calculate ntersecton of two nterval lsts. 5. Expermental Evaluaton In ths Secton, we present expermental results to evaluate our technques, Maxmum Co-occurrence Tme (MCT) by comparng t wth well known method: LCSS, DTW and EDR. For expermental purposes, we used the synthetc datasets whch are generated by Network generator a movng object dataset generator developed by Brnkhoff [15]. Ths trajectory generator s also used by [10]. Our experments were run on a PC AMD Athlon at 2.09GHz wth 1.87G RAM and 160G hard dsk.

5 Accuracy Evaluaton Accuracy s one of the most mportant aspects of smlarty functon. In ths work, we use an objectve evaluaton method recently exploted by [5-7,10] to evaluate the accuracy of our technques. The dea s to use a k-nearest neghbor classfer on labeled data to evaluate the effcacy of the dstance measure used. We used the trajectores generator create 1000 trajectores wth 500 tme steps and 20 Categores. We contnue to adjust the sze ofε and select the optmal value for LCSS and EDR, and set the parameterδ for LCSS to be to get the optmal accuracy. For our method, we set the sze of grd to be 2 tmes ofε for LCSS. The rato of tran set s 20%. In order to avod possble random, we repeated the experments 20 tmes and took the average. Fg. 4 shows the results. Our method-mcg has nearly classfcaton accuracy wth LCSS, whle has hgher accuracy than DTW and EDR lcs dtw edr mct grd cells may lead to greater errors and more Space consumpton. Therefore, we should set the grd sze dependent on the applcaton and the dataset. 0.5 Accuracy K Fg. 4 Classfy accuracy wth dfferent tran rato 5.2 The number of Trajectores n grd By algorthm 3, the key of KNN search effcency s the number of trajectores n grd. It s easy to know, the number s relatve to the sze of grd cells and dataset. In ths secton, we analyze the mpact of the two factors Impact of grd sze In ths subsecton, we do experments on Trucks dataset, changng the sze of grd cell from 1 to 20, and recordng the maxmum and average of the number of trajectores n grd cells. Fg. 5(a) shows changes of maxmum value. When the sze s smaller than 9, maxmum value grows as a lnear almost, whle when the sze s greater than 9, the value s steady. Fg. 5(b) shows average value grows as a lnear as sze of grd s ncrease. Ths s consstent to our conscous. By ncreasng the sze of grd, two trajectores whch were n dfferent grds orgnally, may be n the same grd now, so the maxmum and average value may ncrease. To reduce the computaton, t should be reduce grd sze, but too small Fg. 5 Impact of grd sze to the number of trajectores n one grd Impact of dataset sze In ths secton, we use Network generator to create datasets wth dfferent sze, concludng 1000, 2000, 3000, 4000, 6000, and Fg. 6 shows that the maxmum and average value s almost steady, when the sze of datasets ncreases. The reason s that new trajectores ncrease the number of grd cells, thus maxmum and average value change lttle. Further, the average number of trajectores n one regon s much smaller than the total number of trajectores set. Ths property makes our methods sgnfcantly faster than other methods wth matchng trajectory drectly. In a sense, we use a hash technology when we use an nverted fle.

6 220 Fg. 7: Effcency of KNN-search 6. Concluson Fg. 6 Impact of dataset sze to the number of trajectores n one grd 5.3 Effcency Evaluaton Effcency s the other mportant aspect. We use Network generator to create datasets wth dfferent sze, concludng 1000, 2000, 3000, 4000, 6000, and Fg. 7 presents the results. The tme of DTW s smlar to EDR, and s longer than LCS. Our method s much faster than other methods: LCS, DTW and EDR, even not n an order of magntude. There are two factors to get the fast speed for our method. Frst, the number of trajectores n grd cells s much smaller than the number of trajectores; and second, we only compute the grd cells passed by the target users, whch s also much smaller than the number of grds, nstead of comparng them wth all users. In ths paper, we present effcent technques to compute the smlarty between users by mnng the hstorcal trajectores. We frst dvde the space nto small regons and compressed represent user s trajectores. Then, we presented a storage structure whch not only stores the sequence of user s trajectory, but also stores regons wth ndexng of trajectores whch pass the regon. Based on compressed representaton, we defned three smlarty functons between users. At last, we gve the k-nearest neghbor search algorthm and evaluaton the accuracy and effcency. Our experments ndcate that our smlarty functon has good accuracy comparng wth LCSS, DTW and EDR and our algorthm s sgnfcantly faster than other algorthms. Acknowledgments Acknowledgments The research was supported by the major natonal scence and technology specal projects (2010ZX , 2010ZX ), the Natonal Basc Research Program of Chna (973 Program) (2009CB320906), the Natonal Natural Scence Foundaton of Chna ( , , , , , ). References [1] Zheng, Y., et al., Recommendng frends and locatons based on ndvdual locaton hstory. ACM Trans. Web, 2011,5(1), pp5-42. [2] Agrawal, R., C. Faloutsos and A.N. Swam. Effcent Smlarty Search In Sequence Databases. n FODO ' London, UK: Sprnger-Verlag.

7 221 [3] Effcent Tme Seres Matchng by Wavelets. n ICDE ' Washngton, DC, USA: IEEE Computer Socety. [4] Ca, Y. and R. Ng. Indexng spato-temporal trajectores wth Chebyshev polynomals. n SIGMOD ' New York, NY, USA: ACM. [5] Vlachos, M., D. Gunopoulos and G. Kollos, Dscoverng Smlar Multdmensonal Trajectores, n 18th Internatonal Conference on Data Engneerng (ICDE'02). 2002: Los Alamtos, CA, USA. p [6] Berndt, D.J. and J. Clfford. Usng Dynamc Tme Warpng to Fnd Patterns n Tme Seres [7] Chen, L., et al. Robust and fast smlarty search for movng object trajectores. n Proceedngs of the 2005 ACM SIGMOD nternatonal conference on Management of data Baltmore, Maryland: ACM. [8] Chen, L. and R. Ng. On the marrage of Lp-norms and edt dstance. n VLDB ' : VLDB Endowment. [9] Sakura, Y., M. Yoshkawa and C. Faloutsos. FTW: fast smlarty search under the tme warpng dstance. n Proceedngs of the twenty-fourth ACM SIGMOD-SIGACT- SIGART symposum on Prncples of database systems Baltmore, Maryland: ACM. [10] Ln, B. and J. Su, One Way Dstance: For Shape Based Smlarty Search of Movng Object Trajectores. GeoInformatca, (2): p [11] Frentzos, E., K. Gratsas and Y. Theodords. Index-based Most Smlar Trajectory Search. n Data Engneerng, ICDE IEEE 23rd Internatonal Conference on [12] Peleks, N., et al. Smlarty Search n Trajectory Databases. n Proceedngs of the 14th Internatonal Symposum on Temporal Representaton and Reasonng. 2007: IEEE Computer Socety. [13] Takas, E., et al., Searchng for smlar trajectores n spatal networks (5): p [14] Chen, Z., et al. Searchng trajectores by locatons: an effcency study. n SIGMOD ' New York, NY, USA: ACM. [15] Brnkhoff, T. Generatng Traffc Data, IEEE Data Eng. Bull., 2003, 26(2), pp Wenhua Huang receved the B.S. degree n computer school of Wuhan Unversty, Wuhan, Chna, n He s currently workng as a Graduate at Natonal Engneerng Research Center for Multmeda Software, Wuhan, Chna. Hs research nterests nclude publc safety and multmeda content analyss. Jun Chen receved the Ph.D degree n computer school of Wuhan Unversty, Wuhan, Chna. He s currently a professor at Natonal Engneerng Research Center for Multmeda Software, Wuhan, Chna. Hs research nterests nclude publc safety and multmeda content analyss Ymn Wang receved the B.S. degree n computer school of Wuhan Unversty, Wuhan, Chna, n He s currently workng as a doctor at Natonal Engneerng Research Center for Multmeda Software, Wuhan, Chna. Hs research nterests nclude publc safety, multmeda content analyss, machne learnng, and Data Mnng. Rumn Hu receved the B.S and M.S degrees from Nanjng Unversty of Posts and Telecommuncatons, Nanjng, Chna, n 1984 and n 1990, and Ph.D degree n Communcaton and Electronc System from Huazhong Unversty of Scence and Technology, Wuhan, Chna, n Dr. Hu s the drector of Natonal Engneerng Research Center for Multmeda Software, Wuhan Unversty and Key Laboratory of Multmeda Network Communcaton Engneerng n Hube provnce. He s Executve Charman of the Audo Vdeo codng Standard (AVS) workgroup of Chna n Audo Secton. He has publshed two books and over 100 scentfc papers. Hs research nterests nclude audo and vdeo codng and decodng, vdeo survellance and multmeda data processng.

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