Statistical Analysis on Multivariate Expressway Time Series Traffic Under the Different Toll Policies

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1 Aian Tranport Studie, Volume 2, Iue (213), ATS All right reerved Statitical Analyi on Multivariate Expreway Time Serie Traffic Under the Different Toll Policie Makoto TSUKAI a, Shinichi INOUE b, Maahi KUWANO c, Makoto OKUMURA d a,c Graduate School of Engineering, Hirohima Univerity, 1--1 Kagamiyama, Higahi- Hirohima, Japan a mtukai@hirohima-u.ac.jp b Port and Airport Technology Office, Kanto Region Bureau of Land and Tranportation, 2-1-, Hahimoto-cho, Kanagawa-ku, Yokohama, Japan; inoue-3ab@pa.ktr.mlit.go.jp c kuwano@hirohima-u.ac.jp d Center for Northeat Aian Studie, Tohoku Univerity, 1, Kawauchi, Aoba-ku, Sendai, Japan; mokmr@cnea.tohoku.ac.jp Abtract: Recently, variou toll policie and ocial experiment have been conducted for ETC uer of Japanee expreway. While the vat number of report refer to the change of traffic for each ection, the influence of toll policie ha rarely been evaluated from the tatitical viewpoint, due to the lack of an adequate model to deal with the multivariate time erie including many peak. The purpoe of thi tudy i to clarify the influence of toll policie on the traffic level on expreway by uing Independent Component Analyi (ICA). The continuouly oberved traffic data at everal different ite on the expreway were decompoed into everal independent erie by uing mixing coefficient (i.e. weight parameter) to convert the independent erie into the oberved erie. ICA iolate the common and table fluctuation in everal traffic erie in relation to toll policie. Moreover, the etimated independent erie under the different toll policie were tatitically compared to clarify whether the ignificant difference between the independent erie occur, or not. Key Word: Independent component analyi, Kolmogorov-Smirnov tet, Toll policy, Continuouly-oberved traffic 1. INTRODUCTION 1.1 Background and Conventional Studie In Japan, the local expreway network in each region have been interconnected by the year 2. The inter-regional acceibility ha improved ubtantially in recent year. Accordingly, the uer environment relevant to the toll policie ha remarkably changed. In term of hardware environment, number of Electronic Toll Collection (ETC) ytem uer ha rapidly increaed due to everal dicounting policie applied to ETC uer. Subequently, ETC uer hare in oberved traffic became 7% or more in 29. The increae of ETC uer ha enabled toll policy manager to adequately control the toll level correponding to the characteritic of demand, conidering regional or target pecific condition. After long dicuion about the expreway management policy from 2 to 25, the expreway toll have been dicounted for each car type, or etting the dicounting period in a day, or etting the temporal ocial experiment. It i very important to et the appropriate future toll level baed on the comparion of the different type of policie. Conidering the huge invetment in expreway up to now, le 11

2 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 effective toll dicounting policy hould not be allowed to continue conidering the need to reinvigorate regional economic activitie. For the effective ue and management of expreway network, the effect of toll policie in the pat and their peritency (how long the dicounting policy continue to increae the traffic) hould be carefully analyzed. Iida and Takayama (191) tudied the characteritic of traffic variation on expreway conidering eaon, day of the week and weather. Uing analyi of variance, or chi-quared tet, the periodic variation or interdependency among the decompoed variation wa identified. Sugie et al. (26) empirically tudied the effect of dicount toll policy at Hanhin expreway. Thi tudy analyzed that ETC data aggregated with driver peronal attribute and reported that the trip generation pattern of driver during holiday and weekday, or trip frequency had changed before and after the dicounting policy wa in effect. There are ome tudie about the influence of toll policie on oberved traffic but the number of paper i limited, becaue of it inherent difficultie. While the purpoe of analyi i to iolate the influence of toll policy from the other factor, the oberved traffic include temporal hift and change in traffic generation. Therefore, the oberved data erie would be contaminated by factor other than toll policie, including eaonal fluctuation and economic condition. A a reult, the influence of toll policie on traffic i often evaluated uing a imple comparion of the total or the average traffic between the period before and after the new policie introduced. The tatitical approach to analyze the multivariate continuouly oberved traffic data i developed in ome way. Iryo et al. (27) propoed a tructural model for expreway traffic, which include hort term and long term variation of traffic. Through the application of thi model, it i clarified that Inoue et al. (26) tudied the multiple erie by uing Fourier analyi, or factor analyi (FA). In thi tudy, the regional touring trip i analyzed by uing the multivariate time erie data oberved at the Honyu-Shikoku connecting bridge. The common fluctuation feature of traffic i extracted by uing FA, and then Fourier analyi i applied to the etimated erie. In order to clarify the characteritic of multivariate erie, independent component analyi (ICA) wa developed in ignal proceing of communication engineering (Hyvarinenn et al., 21). ICA wa originally developed a an algorithm for blind ource eparation, which deal with multivariate erie of ound, including voice of everal peron. The problem here i how to eparate the individual voice from the recorded (i.e. mixture) ignal, without uing additional information. Along with the progre of tudie in it theoretical background and more effective algorithm, ICA ha been applied to variou multivariate erie (Akaho, 22), uch a climate change (Ilin et al., 26) or financial time erie (Cheung and Xu, 21). The important application for engineering field i fault detection, which will eparate the outlier or abnormal erie from the mixed multivariate erie. Zhang and Zhang (21) demontrated another algorithm to detect a fault uing the following tep: 1) Learning (etimating) by uing normally proceed obervation to extract normal independent erie; 2) Confidence interval of normal independent erie, and the mixed ignal are etimated; 3) Target obervation which could include the fault proce are compared with the etimated confidence interval; ) Detect whether fault procee exit or not. In thi procedure, the performance of the ICA algorithm in finding the normal or reference erie i important. Wand and Shi (21) propoed a new algorithm to detect the wate water proceing, even if the oberved erie follow a kewed or non-gauian ditribution. 1.2 Purpoe of the Study 12

3 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 The purpoe of thi tudy i to tatitically analyze the influence of pat toll policie on the daily oberved erie of traffic in order to gain information for etting the toll level in the future. By uing Independent Component Analyi (i.e., ICA), continuouly oberved traffic time erie data are decompoed into independent erie uing mixing coefficient, in order to clarify the ignificant difference of the ditribution of independent erie under the different toll policie and by eaonal factor. In addition, the independent erie and the mixing coefficient are analyzed by the tatitical tet, and the characteritic of the traffic at each obervation pot are clarified. The ection in thi paper are organized a follow. Section 2 how the formulation of the model. Section 3 how the ummary of data. Section how the reult of the etimated model. Section 5 how the ummary and concluion. 2. MODEL SPECIFICATION 2.1 Independent Component Analyi Independent Component Analyi i one of multivariate analyi which i a tatitical approach to ummarize the multivariate into the principal, or fewer variable. Let x i (t) be oberved traffic data at ite i, j (t) be a j th independent erie which will correpond to the term of toll policie or eaonal factor, and be a mixing coefficient. The baic model of ICA i, k xi( t) = aijj( t), ( i = 1,..., n) (1) j= 1 x = A (2) where x, A and are vector repreentation of (t) a and (t), repectively. In equation (2), j (t) are combined by the mixing coefficient a ij, and a ij i etimated o a to maximize the independence among the erie of j (t). Note that in applying the ICA model, the average of each erie over the period hould be ubtracted, repectively, to decreae the calculation load, and the number of independent erie hould be le than that of oberved erie. In equation (1) the order of independent erie are not fixed during the ICA etimation procedure, becaue the permutation in independent and plu / minu manipulation among j (t) and aij can be allowed. Therefore, the obtained erie i permutated according to the order of variance contribution to the original erie, and the plu/ minu manipulation i applied in order to get many poitive coefficient on table, for eae in the interpretation of each erie. In ICA, a well a factor analyi (FA), mixing coefficient and independent erie are imultaneouly etimated. Hyvarinenn (1997) propoed a FatICA algorithm in which an independence of the erie are meaured by a kurtoi (i.e, the forth moment of the erie). An advantage of ICA over conventional PCA or FA i that no aumption i made for the tatitical ditribution of unoberved factor in ICA, while PCA or FA etimate the unoberved factor by uing the econd moment information. Another advantage in empirical analyi i that ICA can capture the outlier in the etimated factor, omething that i not clearly captured in PCA and FA. Such difference tem from the implicit aumption for the unoberved erie. x i, ij j 13

4 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 In PCA or FA, the information of data ditribution ued in the analyi i at mot the econd moment, while in ICA, at leat the fourth moment. Since the higher moment of ditribution i more enitive to outlier, i.e., a ample located around the edge of ditribution (Wang, et al., 26) can be expected to produce a better etimate. In thi tudy, the FatICA algorithm i applied to oberved traffic erie in order to extract the independent characteritic of day to day ditribution of traffic correponding to the hortterm concentration of traffic due to conecutive holiday in January, May, and Augut. 2.2 Kolmogorov-Smirnov tet An important difference between fault detection in the deigned ytem and the obervation for ocial phenomenon i that it i difficult to define the normal or reference tate in the oberved traffic. In order to avoid uch a problem, the Kolmogorov-Smirnov Tet (i.e., KS tet) i applied to check the identity of probability denity in the couple of ample erie (Feller, 19). The KS tet i applied for each couple of an independent erie to clarify a ignificant difference in the probability denity ditribution of thoe two erie under the different toll policie. The KS tet doe not aume the pecific probability denity function a the population ditribution. In other word, KS tet i a comprehenive tet for all moment. The null hypothei and alternate hypothei of KS tet are a follow: H H 1 : Two et of ample follow the identical probability denity ditribution : Two et of ample follow the different probability denity ditribution and the KS tet tatitic i = D n n χ (3) n1 + n2 1, n2 > where, n 1 and n2 are the number of band for each erie. When approximately follow a chi-quare ditribution with two degree of freedom. n, a tet tatitic χ 2 The tet tatitic i a function of the value D, which i defined a the maximum ditance between the empirical cumulative ditribution function for the different obervation period. Let F () be the probability ditribution function of the period, and G ( ) be the probability ditribution function of the other period. Then, D i defined a, D = up F( ) G( ) () FatICA wa applied to ten erie of day-to-day traffic to etimate ten independent erie with the mixing coefficient. A et of mixing coefficient correponding to each independent erie were teted for their ignificance (Shimizu et al. 26). Thi tet enable u to pecify the tatitical ignificance on the oberved traffic. By oberving the etimated independent erie, the peak period of each erie howing the characteritic of the concentrated demand i identified. 3. DATA AND TOLL POLICIES IN JAPANEASE EXPRSSWAY In thi tudy, the continuouly-oberved daily traffic erie at the three bridge connecting Honyu and Shikoku are analyzed. The obervation period i 2,69 day, from January 1, 1

5 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), to October 31, 2. The location of ten cro-ection are a follow: even ection on the Nihi-Seto expreway, a ection on the Seto-Chuou expreway, and two ection on the Kobe-Awaji-Naruto expreway. Although the oberved traffic wa recorded acro five type of vehicle, the tudy ued the total amount of traffic. Although many toll policie were applied during the obervation period, we focu on two policie becaue the major policy change applied for all cro-ection are twice a follow: 1) Dicounting toll policy for all ection tarted from July 1, 23, and 2) Dicounting toll policy for ETC uer tarted from July 1, 25. A dicued in 2.2, we do not aume the normal tate among the oberved period. Intead, ome comparion between the different term are made. In order to compare whether each independent erie ha changed among the period under the different toll policie or not, the oberved data erie devided into everal uberie a follow. Conidering the independent erie are divided into four period uch a follow: (a) January 1, 22~June 3, 23 (57 day) (b) July 1, 23~June 3, 25 (731 day) (c) July 1, 25~June 3, 27 (73 day) (d) July 1, 27~October 31, 2 ( day) Though the toll policie did not change between period (c) and (d), period (d) wa et to approximately equalize the number of day at each period. Uing the KS tet, a ignificant difference of the probability denity ditribution between the all combination of ub-period (a) to (d) wa clarified. When the ignificant difference i detected, we can deduce that the oberved traffic under the deferent toll policie had changed.. MIXING COEFFICIENTS AND INDEPENDENT SERIES ICA i applied to the ten oberved erie in order to etimate the independent erie with the mixing coefficient. The reult of the tatitical tet for the mixing coefficient i hown in Table 1. In thi tet, we adjuted the ignificance level to.5% (=5%/1) by the Bonferoni Collection tandard in order to keep the ize of tet for a et of coefficient to individual erie to 5%. By oberving the temporal characteritic of the etimated erie, we identified the peak period of each independent erie. Each erie are named baed on the peak appearing in each of the independent erie. The ummary of thee erie i hown in Table 2 with kewne and kurtoi. Skewne and kurtoi are the characteritic of probability ditribution, uch a aymmetricity around the average and the thickne of both ide of ditribution edge, repectively. Note that the each independent erie i tandardized with mean and unit variance. From table 2, we can undertand that kewne and kurtoi correpond to the peak frequency and the range of ditribution. Among the independent erie ome of characteritic erie a 1, 3,, 7,and out of ten independent erie are hown in figure 1. Note that the ubcript of an independent erie indicate the decending hare of the oberved erie calculated from the mixing coefficient. 15

6 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 In Table 1, all of mixing coefficient of 1, 2, and 6 are hown to be ignificant. Table 1. Mixing coefficient a and the reult of the tatitical tet ij S1 S2 S3 S S5 <Nihi-Seto Expreway> Mukaihima-honen to Mukaihima * * 73.9 * 52.5 * * Mukaihima to In-no-hima-kita * * * 57. * 35.1 * In-no-hima-kita to Ikuchijima-kita * 115. * * 25.6 * * In-no-hima-minami to Ohmihima 32.9 * * * * 176. * Ohmihima to Hakatajima 2. * * 51.5 *.7 * 16.6 * Hakatajima to Ohhima 1.6 * * 59. * 5.19 * * Umahima to Imabarikita 79.5 * * 56. * * * <Seto-chuoh Expreway> Kojima to Hituihi * * * 62.2 * * <Kobe-Awaji-Naruto Expreway> Tarumi to Awaji * * * 2. * * Awajihima-minami to Narutokita * 32.6 * * * * S6 S7 S S9 S1 <Nihi-Seto Expreway> Mukaihima-honen to Mukaihima 25.9 * 2.36 * * *.9 Mukaihima to In-no-hima-kita * 2.7 * 92.3 * In-no-hima-kita to Ikuchijima-kita * * * * In-no-hima-minami to Ohmihima 36.9 * * * * 71.9 Ohmihima to Hakatajima * * Hakatajima to Ohhima 71.1 * 7.3 * 1.27 * Umahima to Imabarikita * * 2.7 * 1.59 * * <Seto-chuoh Expreway> Kojima to Hituihi * * -.57 <Kobe-Awaji-Naruto Expreway> Tarumi to Awaji * 16.5 * * Awajihima-minami to Narutokita 13.3 * 2.31 * * 7.97 *:adjuted ignificance at 99.5% level by Bonfferoni Correction Table 2. Summary of the independent erie Characteritic Seaon, Location Skewne Kurtoi S1 Peak at the ucceive holiday New year, May and Augut S2 Peak at May holoday May S3 Ditinct peak at May 5 May in S Peak at Augut holiday Augut S5 Negative peak July 3, Dec. 3 in 2, Sept. 5 in S6 Peak at Newyear holoday Newyear S7 Temporal peak in ummer vacation Lat week in July to firt ween in Augut S Increaing trend Nihi-eto Expreway S9 Periodical peak Sunday S1 Illegal peak Third Sunday of May The temporal ditribution of 1, 2, and 6 have conecutive peak of the correponding holiday in every year. Epecially, in, the peak of the Augut holiday after 25 are higher than thoe of the previou year. Such the traffic change at Augut would be influenced 16

7 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 by the change in the toll policy introduced after July 1, 25. Concerning the ditribution of 3, the larget peak appear May 2, 26. Thi peak would be influenced by the dicounted toll policy introduced after April 1, July 1,22 July 1,23 July 1,2 July 1,25 July 1,26 July 1,27 July 1,2 October July 1,22 July 1,23 July 1,2 July 1,25 July 1,26 July 1,27 July 1,2 October July 1,22 July 1,23 July 1,2 July 1,25 July 1,26 July 1,27 July 1,2 October July 1,22 July 1,23 July 1,2 July 1,25 July 1,26 July 1,27 July 1,2 October July 1,22 July 1,23 July 1,2 July 1,25 July 1,26 July 1,27 July 1,2 October 3 Figure 1. Independent erie (S1, S3, S, S7, and S) 17

8 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 On the other hand, the mixing coefficient of 7 are ignificant over almot all the croection of Nihi-Seto expreway except Ohmihima to Hakatajima and Kobe-Awaji-Naruto expreway. The ditribution of 7 how the conecutive peak from the fourth week of July to firt week of Augut, therefore the peak would correpond to the demand in ummer vacation. Table 3. The reult of KS tet for Table. The reult of KS tet for period (a) period (b) period (c) period (d) period (a) period (b) period (c) period (d) period (a) period (a) period (b) ** period (b) ** period (c) **.7 period (c) 52. ** ** period (d) 21.2 ** 1.76 ** 7.77 period (d) 97.7 ** ** 293. ** Probability (a) (b) (c) (d) Cla value Probability (a) (b) (c) (d) Cla value Figure 3. The probability ditribution for Figure. The probability ditribution for ub-period of ub-period of Note that the period i not outtanding at firt glance of the original erie. The mixing coefficient of are ignificant only for Nihi-Seto expreway. The ditribution of doe not have a ditinctive peak, but ha the increaing trend in the whole obervation term. The mixing coefficient of 9 are ignificant for ome ection in Nihi-Seto expreway and all ection of Kobe-Awaji-Naruto expreway. The ditribution of 9 how a periodical peak every Sunday. The mixing coefficient of 1 are ignificant only for the ection from Umajima to Imabari-kita in Nihi-Seto expreway. The ditribution of 1 ha the ditinct peak at the third Sunday in May in every year. A dicued in ection3, each independent erie are divided into four ub-period correponding with different toll policie, and applied KS tet in order to tet the ignificant difference among them. In thi tet, the ignificance level i et at 1%. A a reult of thi tet, the combination of ub-period in 1, 2, 5, 6,, 9 and 1 were found ignificant. The probability ditribution function of thee independent erie are different for different toll policie. On the other hand, the combination of ub-period in 7 were not ignificant. Therefore, the temporal peak demand in ummer vacation eem table and unchanged in term of probability denity function. Concerning, we oberved another tendency for temporal demand change. For comparion, the reult of the KS tet for and are hown 1

9 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 in Table 3 and Table, repectively. In addition, the cumulative probability ditribution of ub-period for and are hown in Figure 3 and Figure. The reult of the KS tet for how a ignificant difference in the probability denity ditribution between period (d) and period (a), and (b). On the other hand, the difference found between period (d) and period (c) i not conidered to be ignificant becaue the toll policy wa not changed in thoe period. The probability ditribution of ub-period for (b), (c), and (d) were kewed in poitive area when compared with period (a). On the other hand, the ditribution of thee three period were imilar. The reult of the KS tet for how that all probability denity ditribution are ignificantly different. The probability ditribution become kewed in poitive area which indicate that the oberved traffic in Nihi-Seto expreway tend to increae a time paed. 5. SUMMARY AND CONCLUSIONS The application of Independent Component Analyi to continuouly-oberved expreway traffic data, allowed the eparation of multiple traffic erie into independent erie and mixing coefficient in order to analyze the temporal characteritic of the oberved traffic. The Komolgrov-Smirnov tet wa applied to each combination of independent erie to determine the level of ignificance of the difference between the probability denity ditribution of four ub-period that were et in relation to the hitorical application of variou toll policie. While ome peak were not o clear in the original oberved traffic erie, baed on the reult of Independent Component Analyi, it wa found that the obervation erie can indeed be eparated into characteritic independent erie with the ditinctive peak, correponding with the conecutive holiday demand in annual period, or the weekly period. Further, through the application of the Komolgrov-Smirnov tet, almot all probability denity ditribution of the ub-period are evaluated to be ignificantly different, uch that the oberved traffic erie are tatitically different under the different toll policy. In ummary, the toll dicounting policy introduced in 23 had a great influence on the traffic, while the dicounting policy introduced in 25 did not greatly influence traffic level. In relation to the peak period, Komolgrov-Smirnov tet could pick up the ignificant difference between the ub-period. The remaining iue i that it i till neceary that an analytical procedure to avoid the arbitrarine in interpretation of temporal characteritic for independent erie hould be developed. Concerning the peak demand difference, more enitive tatitical tet to find the difference till likewie need to be developed. ACKNOWLEDGEMENTS Our reearch i ubidized by EHRF in Japan (Expre Highway Reearch Federation). The expreway traffic data i provided by the NEXCO Nihi-Nihon. 19

10 TSUKAI, M., et al. / Aian Tranport Studie, Volume 2, Iue (213), 11-2 REFERENCES Akaho, S. (22) Conditionally independent component analyi for upervied feature extraction. Neurocomputing, 9, Ilin, A., Valpola, H., Oja, E. (26) Exploratory analyi of climate data uing ource eparation method. Neural Network, 19, Iida, T., Takayama, J. (191) Statitical analyi on traffic variation on expreway. Expreway and car, 2, (in Japanee). Inoue, H., Tukai, M., Okumura, M. (23) Claification of traffic pattern by uing croectional traffic between Honyu-Shikoku bridge. Journal of Traffic Engineering in Japan, 23, (in Japanee). Iryo, T., Iwanani, A., Aakura, Y. (27) Analyi on weekly variation of inflow traffic in urban expreway. Journal of Traffic Engineering in Japan, 27, (in Japanee). Cheung, M., Xu, L. (21) Independent component ordering in ICA time erie analyi. Neurocomputing, 1, Feller, W. (19) On the KOLOGOROV-SMIRNOV limit theorem for empirical ditribution. The Annal of Mathematical Statitic, 19, Hyvarinen, A., Oja, E., (1997) A fixed-point algorithm for independent component analyi. Neural Computaion, 9, 13. Hyvarinen, A., Karuhnenn, J., Oja, E., (21) Independent component analyi. Wiley & Son. Shimizu, S., Hyvarinen, A., Kano, Yutaka., Hoyer, P., Kerminen, A. (26) Teting ignificance of mixing and demixing coefficient in ICA. Proceeding of the International Conference on Independent Component Analyi and Blind Source Separation (ICA26), 91-9, Charleton SC, USA, March 5-. Sugie, I., Yamamoto, M., Kanno, H., Yohioka, M. (26) Empirical analyi on ocial experiment in toll policie of expreway uing ETC data. Journal of Japan Society of Traffic Engineer in Japan, 26, (in Japanee). Wang, L., Shi, H. (21) Multivariate tatitical proce monitoring uing an improved independent component analyi. Chemical Engineering Reearch and Deign,, 3-1. Wang, Z., Wang, J., Calhoun, V., Rao, H., Detre, J., Childre, A. (26) Strategie for reducing large fmri data et for independent component analyi. Magnetic Reonance Imaging, 2, Zhang, Y., Zhang, Y. (21) Fault detection of non-gauian procee baed on modified independent component analyi. Chemical Engineering Science, 65,