344 Iteratioal Joural of Fuzzy Systes, Vol. 3, No. 4, Deeber 0 Traffi Flow Data Miig ad Evaluatio Based o Fuzzy Clusterig Tehiques Hu Chuhu, Luo Niaxue, Ya Xiaohog, ad Shi Wezhog Abstrat Effetive iig tehology a extrat the spatial distributio patter of the road etwork traffi flow. I this paper, the siilarities betwee traffi flow obets with spatial teporal harateristis were easured by itroduig the Dyai Tie Warpig (DTW) ad the shortest path aalysis ethod. We proposed a ew fuzzy lusterig algorith for road etwork traffi flow data. So that traffi flow data obets with siilar properties ad spae orrelatio are lustered ito a group, whih fid the spatial distributio patter of road traffi flow. The experietal results show that the ethod was valid ad effetive. The road etwork was lassified reasoably, ad lassifiatio results ould provide traffi zoe divisio with deisio auxiliary support. Keywords: Fuzzy lusterig, traffi flow, siilarity easure, luster validity.. Itrodutio For traffi flow o road etwork, there are differet spatial distributio patters. Suh as liear patter for aor road traffi flow, surfae patter for thrivig road ad so o. Aordig to the harateristis of the spatial distributio of traffi flow, dyai traffi zoe partitio is oe of the researh hotspots of itelliget trasportatio syste. But road traffi zoe partitio also a produe orrespodig hage with the traffi flow peak, oral ad botto period hage. We a apply effetive iig tehology to extrat the spatial distributio Correspodig Author: Hu Chuhu is with with the Shool of Geodesy ad Geoatis, Wuha Uiversity, 9 Luoyu Road,Wuha 430079, Chia. E-ail: hai@yahoo.o. Luo Niaxue is with the Shool of Geodesy ad Geoatis, Wuha Uiversity, 9 Luoyu Road,Wuha 430079, Chia. E-ail: xluo@sgg.whu.edu. Ya Xiaohog is with the Shool of Resoure ad Eviroetal Si-ee, Wuha Uiversity, 9 Luoyu Road,Wuha 430079, Chia. Shi Wezhog is with the Departet of Lad Surveyig ad Geo-Iforatis, The Hog Kog Polytehi Uiversity. Mausript reeived 5 Nov. 0; revised De. 0; aepted De. 0. patter o road etwork. It is helpful to partitio traffi zoe, aage ad otrol road etwork traffi, ad irease the road etwork apaity ad ease the traffi pressure. Clusterig aalysis is oe of the ost useful ethods i kowledge aquisitio, ad is used to disoverig uderlyig lusters ad iterest distributed patter fro data itself. Cluster aalysis tehiques i geeral a be divided ito two ategories, aely, risp ad fuzzy lusterig. Crisp lusterig is a kid of o-overlappig partitios ethod. Ad exploitig it eas that a obet either belogs to oe lass or ot aordig to soe proxiity easure ad lusterig riterio. But the risp lusterig a ut off the lik betwee obets ad ause ore deviatio for lusterig results. While the issue of uertaity support i lusterig task leads to the itrodutio of algoriths that use fuzzy logi oepts i lusterig proedure []. A oo fuzzy lusterig algorith is the Fuzzy C-Meas (FCM) []. It attepts to fid the ost harateristi poit i eah luster, whih a be osidered as the eter of the luster, ad the grade of ebership for eah obet i the lusters []. Spatial lusterig aalysis of traffi flow a fid the spatial distributio patters o road etwork, ad it ake that traffi flow data obets with siilar properties ad spatial assoiatio, are lustered ito a group. I order to ahieve our goal, we first review the related researh work about traffi flow data. Due to the oplexity of traffi etwork, it will ot be suit for easurig the siilarity of traffi flow oly opared the differee of the tie series data. Next, we defie a siilarity easure betwee traffi flow for disoverig obet groups aordig to proxiity i tie ad road etwork. We develop a algorith for lusterig traffi flow, usig the siilarity easure defied earlier for disoverig siilar traffi flow i setio 4. Fially we evaluate the proposed algorith by odutig experiets o real traffi flow data.. Related work Previous work o iig traffi flow data iludes traffi data odel, lusterig algorith ad siilarity easure of traffi flow. Gaussia ixture odels (GMM), speed, flow ad 0 TFSA
Hu Chuhu et al.: Traffi Flow Data Miig ad Evaluatio Based o Fuzzy Clusterig Tehiques 345 oupay are used together i the luster aalysis of traffi flow data i the literature [3]. Che et al. studied ulti-diesioal traffi flow tie series aalysis with self-orgaizig aps [4]. I urba ity traffi flow iig ethods, disrete wavelet trasfor is adopted for flow feature extratio for it is isesitive to disturbae/salig, ad zoos i ultiple fier graularities [5]. For high-diesioal traffi data lusterig, a two stage fuzzy lusterig ethod was exploited. The optial partitio idetified fro the first lusterig results is used as the iitial partitio i the seod stage lusterig based o full diesioal data, thus effetively redues the possibility of loal optiu [6]. Siilarity easure of traffi flow tie series was studied i the literature [7], ad it ahieved the effetive separatio of traffi flow tie series. I the literature [8], it lustered traffi flow sequee by eployig partitio lusterig tehique, ad ould idetify road traffi flow of the TOD ( tie of day) iterval aordig to differet flow. However, aforeetioed researh about traffi flow lusterig aalysis oly osidered the tie attribute of road traffi flow, without osiderig the spatial distributio harateristis o road etwork. Traffi flow o road etwork is related to teporal iforatio ad road seget, so the spatial lusterig is differet fro the oo lusterig ethod. Siilarity easure of traffi flow a be defied by exploitig tie series data with dyai harateristi ad topology relatio betwee road segets o road etwork. A ew lusterig algorith will a be proposed whih a well easure the siilarity of traffi flow obet, ad further fid potetial traffi patter fro a large uber of road etwork traffi flow series. 3. Siilarity Measure betwee the Traffi Flow Obet A good spatial lusterig algorith a group road segets aordig to the data harateristis of road etwork. Traffi flow series o road etwork is ultidiesioal, so a suitable distae futio is required to express the siilarity betwee two road setios. The Lp paradig [9] ad Dyai Tie Warpig (DTW) [0] are distae futio for tie series siilarity aalysis. Siilarity distae easure algorith is siple ad easy to ipleet based o the Lp paradig, but it a oly be dealt with equal legth of tie sequees. Whereas DTW is a kid of dyai prograig ethod for siilarity easure of tie series, ad it is ot subeted to legth liit of tie series. A. DTW Give two tie sequee Q ad C whih their data legths are ad respetively, we a alulate the distaes betwee the i order to opare the siilarity. Saller distae express greater the siilarity. I distae atrix of two differet tie series, a group of otiguous atrix eleets set, whih defied dissiilarity relatios betwee series, is alled a urved path. The ai of DTW ethod is to searh the iiu total legth of the urved path. The iiu total legth of urved path a be alulated by dyai prograig ethod usig the forula as show i (). If the poit (i, ) is loated o the optial path, the sub path fro poit (, ) to (i, ) is also a loal optial solutio. The best path a be obtaied by reursive searh the loal optial solutio betwee tie startig poit (, ) ad the ed of (, ). S, = d ( q, ) S = d( qi, i ) + i{ S( i, ); S( i, ); S( i, )} () B. The Shortest Path Spatial teporal properties are ai portio of road traffi flow iforatio obtaied by real-tie. Ad the spatial relatios betwee eah road seget o road etwork are also sigifiat. So siilarity easure of the traffi flow obet also osidered the topology relatio of road etwork, ad spatial siilarity degree is higher betwee oetive ad reahable road setios. The oetivity ad aessibility of road ould be easured by the shortest path aalysis o road etwork. Ad the dyai shortest path legth betwee two road setios is defied as the siilarity easureet futio. C. Defiig Siilarity easures For the graphi struture of G = {V, E} orrespodig to road etwork, V represets a road ode ad E represets a road edge o etwork. Ad traffi flow series was set to T i = {T i, T i,..., T i } ( was used to express the th tie period of traffi flow series) geerated by eah edge E i = <v i0, v i >. Spatial teporal siilarity easure futio of the traffi flow is defied as follows: TFSM E, E ) = DTW ( T, T ) + Shorest Path( v, v ) ( i i i0 () I (), DTW (T i, T ) express siilarity distae of two roads E i ad E i traffi flow series, Shorest-Path (v i0, v ) is spatial siilarity distae betwee the begiig ode of the road E i ad the edig ode of the road E. The sall value of TFSM (E i, E ) is expeted. Thus, the sall value represets that the E i ad E are ore siilar. 4. Traffi Flow Clusterig ad Evaluatio A. The Fuzzy Algorith A oo fuzzy lusterig algorith is the Fuzzy
346 Iteratioal Joural of Fuzzy Systes, Vol. 3, No. 4, Deeber 0 C-Meas (FCM), a extesio of lassial C Meas algorith for fuzzy appliatios []. It uses fuzzy tehiques to luster data. Ad i the algorith, a obet a be lustered to ore tha oe luster, whih opatible with the status of real data. The FCM lusterig algorith has bee widely used to obtai the fuzzy -partitio. It is a kid of fuzzy lusterig algorith based o obet futio. Give a dataset X={X, X,, X } with s diesio, the obet of FCM is to partitio dataset X ito hoogeeous fuzzy lusters by iiizig the futio J. J = ( u i ) d ( X, V ) i= = (3) I (3), is the uber of luster, is uber of data ad u i is the ebership degree of data poit X belogig to the fuzzy luster C i. V i is the i th luster etroid, is weightig expoet ad otrols the fuzziess of ebership of eah datu []. The d ( X, V i ) represets the Eulidea distae betwee X ad V i. The FCM solutio is a atheatial plaig proble, ad the data set X a be divided ito differet ategories by iiizig the obetive futio. The liited oditio of futio J is that the su of ebership degree (u i ), whih X belogig to eah of luster i, equals to. It is desribed as follow. u i i= = i 0 u i, (4) The FCM algorith is arried out i the followig steps: Iitialize threshold ε ad luster etroids V(0), set k=0. Give a predefied uber of luster, ad a hose value of. Copute atrix of the ebership degree U(k) =[ u i ] for i=,,, i(5). ( ) ( d ( X, Vi )) ui = ( ) ( d ( X, Vi )) i= (5) Update the fuzzy luster etroid V i (k+) for i=,,, i (6). ui ( = + = ui = k) X Vi ( k ) ( k) (6) If eet (7), the iteratio halts; Otherwise retur the third step. v ( k) v( k + ) ε (7) The FCM algorith always overge a loal axiu value through above iteratio alulatio []. B. New Fuzzy Clusterig Algorith I order to extrat eaigful road traffi distributio patter, the forula () is exploited as a fuzzy siilarity easure futio, ad the ew obetive futio of FCM is defied as follows: J = i= = ( u ) i TFSM ( E, E ) (8) I (8), TFSM (E i, E ) is the siilarity easure futio of the traffi flow o road etwork. So ew fuzzy lusterig algorith was desribed as follows: Step: To build the topology struture of the road etwork, alulate the shortest path betwee start ode ad ed ode of two road segets by Dikstra algorith o the road etwork. Step: To radoly selet road traffi flow sequee as the iitialized luster eter fro the road etwork. Step3: To alulate the degree of ebership atrix U(k) aordig to (5). I order to fiish this step, we eed to alulate the iiu dyai bedig path betwee eah road seget ad lusterig eter traffi flow sequee by DTW algorith o the road etwork aordig to () ad opute d ( X, V i ). Step 4: To adust the fuzzy lusterig eter aordig to (6). For eah luster eter, We fid out road segets with the iial dyai urved path of traffi flow sequee opared with lusterig eter by DTW algorith. Ad these road segets with traffi flow series will be ew the fuzzy lusterig eter. Step5: Repeat Step 3 ad Step 4 util the axiu uber of iteratios t ax or (7) to eet. C. Evaluatio Methods Sie fuzzy lusterig is a usupervised ahie learig tehique, there is o set of orret aswers that a be opared to the results. Ad it requires tuig the iput paraeters aordig to soe way for obtaiig the optial luster results [3]. It is quite eessary to validate the lusterig results produed by the Fuzzy lusterig algorith. Measurig lusterig results ad idetifyig the optial partitio or the optial uber () of lusters are alled luster validity evaluatio. The obet of the luster idies is to seek optial lusterig shees where ost of data sets preset high degree of ebership withi a luster [4]. The Xie-Bei idex ivolves the ebership values ad the dataset itself. It is a opat ad separate fuzzy validity futio [4] ad defied as i
Hu Chuhu et al.: Traffi Flow Data Miig ad Evaluatio Based o Fuzzy Clusterig Tehiques 347 v XB ui x vi = i= = i vi vk i k (9) Equatio (9) is explaied as the ratio of total opatess to the separatio of the fuzzy -partitio. For opat ad well-separated lusters, the sall values of v XB are expeted. Therefore, the optial luster is obtaied by fidig the fuzzy -partitio with the sallest value of v XB. I additio to luster validity evaluatio, we easure ruig ties of the ew fuzzy lusterig algoriths with ertai paraeter values. The algoriths salability will be tested by easurig ruig ties of the sae fuzzy lusterig versus differet uber of obets that are beig lustered. 5. Experietal Results The experietal data set iludes 3648 road segets of ertai ity road etwork. Traffi flow data were olleted at 5 iutes iterval. I the experiet, the traffi flow series with the traffi flow peak (betwee seve o'lok to ie thirty i the orig) aog a day were exploited. Ad the traffi flow series of eah road seget otaied thirty traffi flow data. Table desribed the traffi flow series data. For obtaiig the reliable experietal results, the paraeters of FCM as fitess futio are set to the weightig expoet =, whih is oo hoie of the FCM algorith, i the rage of [.5,.5] [5]. Ad the axiu uber of iteratios tax is equal to 40 ad iteratio teriatio oditio ε is equal to 0.000. For the fuzzy lusterig was a usupervised lassifiatio ethod, we took the luster uber = 3, 4, 5, 6 to observe the results of the experiet. Road seget ID Table. traffi flow sequee data. Road seget startig ode Road seget edig ode Average ost (tie) Tie 000-00 000 00 0.9375 009-0-4 07:00-07:05 000-00 000 00 0.9375 009-0-4 07:05-07:0 000-00 000 00. 009-0-4 07:0-07:5 098-049 098-049 098-049 098 098 098 049 049 049.87.9.4 009-0-4 07:05-07:0 009-0-4 07:5-07:0 009-0-4 07:5-07:30 A. Cluster Validity Results We opared the ew FCM ad fuzzy lusterig versios of our suggested algorith aordig to XB idex. The opared results betwee the FCM ad ew fuzzy lusterig algorith are listed i table whe the legth of tie series is equal to 0. The XB idex of FCM poit to the optial luster uber =6 or =5, while the ew fuzzy lusterig algorith poit to luster uber =3 or =5. Table. XB idex values (The legth of tie series=0). C FCM New fuzzy lusterig algorith 3 0.978573 0.43588 4 0.0309445 0.335494 5 0.8707857 0.3056 6 0.76867 0.58764 I additio, the traffi flow data with the legth of tie series equal 0 ad 30 are ru i the experietal i order to test validity of the ew algorith. The results as show i table 3 ad table 4 preset that the optial luster uber is equal to 3 or 5 whe usig the ew algorith, ad that the luster result is =3 or =6. Table 3. XB idex values (The legth of tie series=0). C FCM New fuzzy lusterig algorith 3 0.546607 0.4557453 4 0.764036 0.878994 5 0.8708834 0.47488 6 0.768434 0.7733046 Table 4. XB idex values (The legth of tie series =30). C FCM New fuzzy lusterig algorith 3 0.54653 0.5565906 4 0.0309345 0.55383 5 0.876977 0.46049 6 0.768503 0.780607 Figure show that the partial luster results by exploitig the FCM algorith whe =6, ad that the results of the ew fuzzy lusterig algorith are preseted i figure whe =5.
348 Iteratioal Joural of Fuzzy Systes, Vol. 3, No. 4, Deeber 0 Figure. The lusterig results of FCM (=6). We a see the luster eters redered by red bold lie i figure ad figure, ad the road segets are redered by differet olors. Eah olor idiates a group of traffi flow data with high siilarity. Ruig ties of the lusterig algorith is aily related to the legth of the tie series ad luster ubers i experietal. I order to estiate the effiiey of the ew lusterig algorith, we easure its ruig tie opared to the FCM algorith. The experietal results of easurig ruig ties are showed i the figure3, figure 4 ad figure 5. We a see the ruig ties of the ew algorith are loger tha the FCM algorith. Ad it will be goig up with irease of the luster aout ad the legth of series goig up. The alulatio of the iiu dyai bedig path ad the shortest path aused the expese tie ost. But the quik ruig ties of the ew algorith idiate its effiiey. Figure 4. Ruig ties opariso (the legth of series=0). Figure. The lusterig results of ew algorith (=5). Copared the two figure, the two luster eters are too adaet i figure. The area is dese traffi zoe fro real traffi situatio, ad the luster eters are ore fit to the real traffi state i figure. To su up, the experietal results show that the ew algorith is valid. B. Clusterig effiiey results Figure 5. Ruig ties opariso (the legth of series=30). 5. Colusios Figure 3. Ruig ties opariso (the legth of series=0). Fuzzy luster aalysis is a iportat data aalysis tool, it is a o-supervised lassifiatio ethod. Based o the fuzzy lusterig algorith ideas, osidered the road etwork traffi flow distributio, the paper propose a ew lusterig algorith whih a well easure the siilarity betwee traffi flow obets, ad further fid potetial traffi patter fro a large uber of road etwork traffi flow series. Experiet results show that the ew algorith is valid ad effiiey. Speially, the luster results a provide traffi zoe divisio with deisio auxiliary sup-port. Clusterig aalysis is largely a data-drive tool [6].
Hu Chuhu et al.: Traffi Flow Data Miig ad Evaluatio Based o Fuzzy Clusterig Tehiques 349 So further work is eed for tuig the paraeters of the algorith i order to obtai the better luster results. Visualizig the lusterig proess, iprovig the siilarity easure ad perforae are also iportat work. Akowledget This researh work was supported by the grats fro the Natioal Natural Siee Foudatio of Chia uder Grat uber 90403. Referees [] M. Halkidi, Y. Batistakis, ad M. Vazirgiais, O lusterig validatio tehiques, Joural of Itelliget Iforatio Systes, vol. 7, o. -3, pp. 07-45, 00. [] J. C. Beadek, Patter reogitio with fuzzy obetive futio algoriths, Pleu Press, New York, 98. [3] Su Lu, Zhag Huii, et.al, Gaussia ixture odels for lusterig ad lassifyig traffi flow i real-tie for traffi operatio ad aageet, Joural of Southeast Uiversity, vol. 7, o., pp. 74-79, 0. [4] C. Yudog, Z. Yi, ad H. Jiaig, Multi-diesioal traffi flow tie series aalysis with self-orgaizig aps, Tsighua Siee ad Tehology, vol. 3, o., pp. 0-8, 008. [5] C. Yudog, Z. Yi, H. Jiaig, ad et.al, Miig for siilarities i urba traffi flow usig wavelets, I Pro. of the IEEE Iteratioal Coferee o Itelliget Trasportatio Systes, pp. 9-4, Seattle, Washigto, USA, 007. [6] Z. Pegu ad M. Mike, A algorith for high-diesioal traffi data lusterig, I Pro. of the Iteratioal Coferee of Fuzzy Syste ad Kowledge Disovery, pp. 59-68, Xi a, Chia, 006. [7] R. Jiagtao, X. Qiogqiog, ad Y. Jia, Traffi flow tie series separatio ethods, Coputer Appliatios, vol. 5, o. 4, pp. 937-939, 005. [8] A. Hauser Trisha ad T. Sherer Willia, Data iig tools for real-tie traffi sigal deisio support&aiteae, I Pro. of the IEEE Iteratioal Coferee o Systes, Ma ad Cyberetis, pp. 47-477, Tuso, Arizoa, USA, 00. [9] C. Faoutsos, M. Ragaatha, ad Y. Maolopouos, Fast subsequee athig i tie-series database, I Pro. of ACM SIGMOD Coferee, pp. 49-49, Mieapolis, Miesota, USA, 994. [0] D. J. Berdt ad J. Clifford, Fidig Patters i tie series: A dyai prograig approah, Melo Park, CA: AAAI Press, 996. [] J. C. Bezdek, R. Ehrlih, ad W. Full, FCM: Fuzzy -eas algorith, Coputers ad Geosiee, vol. 0, pp. 9-03, 984. [] D. W. Ki, K. H. Lee, ad D. Lee, Fuzzy luster validatio idex based o iter-luster proxiity, Patter Reogitio Letters, vol. 4, o. 5, pp. 56-574, 003. [3] M. Ki ad R. S. Raakrisha, New Idies for Cluster Validity Assesset, Patter Reogitio Letters, vol. 6, o. 5, pp. 353-363, 005. [4] M. Halkidi, Y. Batistakis, ad M. Vazirgiais, Clusterig Algoriths ad Validity Measures, I Pro. of the Iteratioal Workig Coferee o Sietifi ad Statistial Database Maageet, pp. 3-, Fairfax, VA, USA, 00. [5] N. R. Pal ad J. C.Bezdek, O luster validity for the fuzzy -eas odel, IEEE Trasatios o Fuzzy Systes, vol. 3, o. 3, pp. 370-379, 995. [6] J. Ha ad M. Kaber, Data Miig: Coepts ad Tehiques. Sa Fraiso: Morga Kaufa Publishers, 00. Hu Chuhu reeived her PhD degree of photograetry ad reote sesig fro Wuha Uiversity. Her researh iterests ilude geographi iforatio syste, spatial data aalysis, spatial database ad spatial data iig.