Traffic Flow Forecasting Based On Combination Of .
Traffic Flow Forecasting Based on Combination of Multidimensional Scalingand SVMZhanquan Suna, Geoffrey Foxba. Key Laboratory for Computer Network of Shandong Province, Shandong Computer Science Center(19 Keyuan Road, Jinan, Shandong, 250014, China, sunzhq@keylab.net)b. School of Informatics and Computing, Pervasive Technology Institute, Indiana University Bloomington(2719 E 10th St, Bloomington, Indiana, 47408, USA, gcfexchange@gmail.com)Abstract: Traffic flow forecasting is a popular research topic of Intelligent Transportation Systems (ITS). With thedevelopment of information technology, lots of history electronic traffic flow data are collected. How to take full use ofthe history traffic flow data to improve the traffic flow forecasting precision is an important issue. More history data areconsidered, more computation cost should be taken. In traffic flow forecasting, many traffic parameters can be chosento forecast traffic flow. Traffic flow forecasting is a real-time problem, how to improve the computation speed is a veryimportant problem. Feature extraction is an efficient means to improve computation speed. Some feature extractionmethods have been proposed, such as PCA, SOM network, and Multidimensional Scaling (MDS) and so on. But PCAcan only measure the linear correlation between variables. The computation cost of SOM network is very expensive. Inthis paper, MDS is used to decrease the dimension of traffic parameters, interpolation MDS is used to increasecomputation speed. It is combined with nonlinear regression Support Vector Machines (SVM) to forecast traffic flow.The efficiency of the method is illustrated through analyzing the traffic data of Jinan urban transportation.Keywords: Intelligent transportation; Traffic flow forecasting, Multidimensional Scaling; SVM; Interpolation1. IntroductionShort-time traffic flow forecasting is a popularresearch topic of Intelligent Transportation Systems(ITS). Correct traffic flow forecasting is the preconditionof real-time traffic signal control, traffic assignment,route guidance, automatic guidance, and accidentdetection. The study of traffic flow forecasting is verysignificant in ITS. Many scholars have been studying onthe topic and many forecasting models have beendeveloped. Commonly used methods include averagemethod, ARMA, linear regression, nonparametricregression, and neural networks [1-3]. The forecastingprecisions of these methods usually can’t meet with thepractical requirement. Support Vector Machines (SVM)is proposed by V. Vapnik in 1995[4]. It is a networkmodel that is based on the principle of structure riskminimization and VC dimension theory. It can resolvesmall sample, nonlinear, high dimension, and localminimum problems efficiently [5]. SVM is mainly usedto resolve classification and regression problems.Nonlinear regression SVM has been used to forecasttraffic flow and obtained good results [6].In practical, there are many parameters are availablefor the traffic flow forecasting. Many forecastingmethods are real-time. Too many input parameters willdecrease the real-time performance. In current trafficflowing forecasting research, mostly concentrate onshort term history traffic flow data. Lots of history dataare not taken into consideration because the computationcost is expensive. For taking full use of history trafficflow data and improving the computation speed, featureextraction is an efficient means. It can decrease thedimension of input and decrease the computation costefficiently. Many feature extraction methods have beenproposed, such as Principal Component Analysis (PCA),Self Organization Map (SOM) network, and so on[7-8].Multidimentional Scaling (MDS) is a kind of Graphicalrepresentations method of multivariate data[9]. It iswidely used in research and applications of manydisciplines. The method is based on techniques ofrepresenting a set of observations by a set of points in alow-dimensional real (usually) Euclidean vector space,so that observations that are similar to one another arerepresented by points that are close together. It is anonlinear dimension reduction method. But thecomputation complexity is O(n 2) and memoryrequirement is O(n 2). With the increase of sample size,the computation cost of MDS increase sharply. Forimproving the computation speed, interpolation MDS areintroduced in reference [10]. It is used to extract featurefrom large scale traffic flow data. Nonlinear SVM isused to forecast traffic flow.The following of the paper is organized as follows.Interpolation MDS method is introduced in part 2.Nonlinear SVM is introduced in part 3. Traffic flowforecasting procedure based on MDS and nonlinearSVM is introduced in part 4. A practical example isanalyzed with the proposed model in part 5. At last someconclusions are summarized.2. Interpolation MDS2.1 Multidimensional Scaling
MDS is a non-linear optimization approachconstructing a lower dimensional mapping of highdimensional data with respect to the given proximityinformation based on objective functions. It is anefficient feature extraction method. The method can bedescribed as follows.Givenacollectionof n objects D {x1 , x2 , , xn }, xi RN (i 1,2, , n) on which adistance function is defined as δi,j , the pairwise distancematrix of the n objects can be denoted byδ1,1 δ1,2δ1,n δδ2,2δ2,n 2,1 δn,1 δn,2 δn,nwhere δi,j is the distance between xi and xj . Euclideandistance is often adopted.The goal of MDS is, given Δ, to find n vectorsp1 , , pn RL (L N) to minimization the STRESS orSSTRESS. The definition of STRESS and SSTRESS areas follows.2(1)σ(P) i j wi,j di,j (P) δi,j 22σ2 (P) i j wi,j (di,j (P))2 δi,j (2)where 1 i j n, 𝑤𝑤𝑖𝑖,𝑗𝑗 is a weight value (𝑤𝑤𝑖𝑖,𝑗𝑗 0),𝑑𝑑𝑖𝑖,𝑗𝑗 (𝑃𝑃) is a Euclidean distance between mapping resultsof 𝒑𝒑𝑖𝑖 and 𝒑𝒑𝑗𝑗 . It may be a metric or arbitrary distancefunction. In other words, MDS attempts to find anembedding from the 𝑛𝑛 objects into 𝑅𝑅 𝐿𝐿 such thatdistances are preserved.2.2 Interpolation Multidimensional ScalingOne of the main limitations of most MDS applicationsis that it requires 𝑂𝑂(𝑛𝑛2 ) memory as well as O(n2 )computation. It is difficult to process MDS with largescale data set because of the limitation of memorylimitation. Interpolation is a suitable solution for largescale MDS problems. The process can be summarized asfollows.Given n samples data 𝐷𝐷 {𝒙𝒙1 , 𝒙𝒙2 , , 𝒙𝒙𝑛𝑛 }, 𝒙𝒙𝑖𝑖 𝑅𝑅𝑁𝑁 (𝑖𝑖 1,2, , 𝑛𝑛) in N dimension space, m samples𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠 {𝒙𝒙1 , 𝒙𝒙2 , , 𝒙𝒙𝑚𝑚 }, are selected to be mapped intoL dimension space 𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠 {𝒑𝒑1 , 𝒑𝒑2 , , 𝒑𝒑𝑚𝑚 } with MDS.The other samples 𝐷𝐷𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 {𝒙𝒙1 , 𝒙𝒙2 , , 𝒙𝒙𝑛𝑛 𝑚𝑚 }, will �𝑟𝑟𝑟𝑟 {𝒑𝒑1 , 𝒑𝒑2 , , 𝒑𝒑𝑛𝑛 𝑚𝑚 } with interpolation method. Thecomputation cost and memory of interpolation MDS isonly 𝑂𝑂(𝑛𝑛) . It can improve the computing speedmarkedly.Select one sample data 𝒙𝒙 𝐷𝐷𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 , calculate thedistance 𝛿𝛿𝑖𝑖𝑖𝑖 between the sample data 𝒙𝒙 and the premapped samples 𝒙𝒙𝒊𝒊 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠 (𝑖𝑖 1,2, , 𝑚𝑚). Select the 𝑘𝑘nearest neighbors 𝑄𝑄 {𝑞𝑞1 , 𝑞𝑞2 , , 𝑞𝑞𝑘𝑘 }, where 𝒒𝒒𝑖𝑖 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠 ,who have the minimum distance values.After data set 𝑄𝑄 being selected, the mapped value ofthe input sample is calculated through minimizing thefollowing equations as similar as normal MDS problemwith 𝑘𝑘 1 points.22 𝜎𝜎(𝑋𝑋) 𝑖𝑖 𝑗𝑗 𝑑𝑑𝑖𝑖,𝑗𝑗 (𝑃𝑃) 𝜹𝜹𝒊𝒊,𝒋𝒋 𝐶𝐶 𝑘𝑘𝑖𝑖 1 𝑑𝑑𝑖𝑖𝑖𝑖𝑘𝑘2 𝑖𝑖 1 𝑑𝑑𝑖𝑖𝑖𝑖 𝛿𝛿𝑖𝑖𝑖𝑖(3)In the optimization problems, only the position of themapping position of input sample is variable. Accordingto reference [10], the solution to the optimizationproblem can be obtained as1𝛿𝛿 𝑘𝑘𝑖𝑖 1 𝑖𝑖𝑖𝑖 𝑥𝑥 [𝑡𝑡 1] 𝒑𝒑𝑖𝑖 𝑥𝑥 [𝑡𝑡] 𝒑𝒑(4)𝑑𝑑𝑖𝑖𝑖𝑖𝑘𝑘 is the average of k prewhere 𝑑𝑑𝑖𝑖𝑖𝑖 𝒑𝒑𝑖𝑖 𝑥𝑥 [𝑡𝑡 1] and 𝒑𝒑mapped results. The equation can be solved throughiteration. The iteration will stop when the differencebetween two iterations is less than the prescribedthreshold values. The difference between two iterationsis denoted by𝛿𝛿 ( 𝑥𝑥 [𝑡𝑡] 𝑥𝑥 [𝑡𝑡 1] )(5) 𝑥𝑥 [𝑡𝑡 1] 3. Support Vector MachinesSVM first maps the input points into a highdimensional feature space with a nonlinear mappingfunction Φ and then carry through linear classificationor regression in the high-dimensional feature space. Thelinear regression in high-dimension feature spacecorresponds to the nonlinear classification or regressionin low-dimensional input space. The general SVM canbe described as follows.Let l training samples be T {( x1 , y1 ), , ( xl , yl )} ,where xi Ω X R n , yi ΩY R , i 1, , l .Nonlinear mapping function is k ( xi , x j ) Φ ( xi ) Φ ( x j ) .Nonlinear regression SVM can be implemented throughsolving the following equations.𝑙𝑙1min (𝛼𝛼𝑖𝑖 𝛼𝛼𝑖𝑖 ) 𝛼𝛼𝑗𝑗 𝛼𝛼𝑗𝑗 𝑘𝑘 𝑥𝑥𝑖𝑖 . 𝑥𝑥𝑗𝑗 2𝑙𝑙𝛼𝛼 𝑅𝑅 2𝑖𝑖,𝑗𝑗 1𝑙𝑙 𝜀𝜀 (𝛼𝛼𝑖𝑖 𝑖𝑖 1𝑙𝑙 𝛼𝛼𝑖𝑖 ) 𝑦𝑦𝑖𝑖 (𝛼𝛼𝑖𝑖 𝛼𝛼𝑖𝑖 )𝑖𝑖 1𝑠𝑠. 𝑡𝑡. 𝑙𝑙𝑖𝑖 1(𝛼𝛼𝑖𝑖 𝛼𝛼𝑖𝑖 ) 0(6)𝛼𝛼𝑖𝑖 , 𝛼𝛼𝑖𝑖 0 𝑖𝑖 1, , lThroughoptimization,optimumsolution(*)**α (α1 , α1 , , α l , α l ) can be solved.Select the positive sub-vector α j 0 of α or thepositive sub-vector α * of α j* 0 and calculate theparameterlb y j (α i* α i ) K ( xi , x j ) ε(7)i 1After getting the optimum parameters, the decisionfunction can be denoted aslf ( x) (α i α i* )k ( xi x) b(8)i 1It is very important to choose appropriate kernelfunction of SVM. The kernel function must satisfy theMercer condition. At present, many kernel function
model have been developed. Commonly used kernelfunctions include(1) linear: 𝐾𝐾 x𝑖𝑖 , x𝑗𝑗 x𝑖𝑖 𝑇𝑇 x𝑗𝑗𝑑𝑑(2) polynomial: 𝐾𝐾 x𝑖𝑖 , x𝑗𝑗 𝛾𝛾x𝑖𝑖 𝑇𝑇 x𝑗𝑗 𝑟𝑟 , 𝛾𝛾 0(3) radial basis function (RBF): 𝐾𝐾 x𝑖𝑖 , x𝑗𝑗 2exp( 𝛾𝛾 x𝑖𝑖 x𝑗𝑗 ), 𝛾𝛾 02(4) sigmoid: 𝐾𝐾 x𝑖𝑖 , x𝑗𝑗 exp( 𝛾𝛾 x𝑖𝑖 x𝑗𝑗 ), 𝛾𝛾 0Here, 𝛾𝛾, 𝑟𝑟, 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑 are kernel parameters.4 Traffic Flow ForecastingIn intelligent transportation system, many traffic flowparameters are useful in identifying the traffic state, suchas speed, traffic flow volume, and time occupancy andso on. Short term forecasting of the parameters is theprecondition of providing traffic information services. Inthe forecasting of the traffic flow parameters, manytraffic flow data can be used, such as the previoussampling data, history cycle data and so on. Theincluded data should be prescribed previously accordingto practical requirement and experience. Traffic flowforecasting model is built according to history trafficflow data. Training samples can be generated accordingto the model. Sample data are mapped into lowdimension space with MDS method. Traffic flow dataare forecasted based the mapped data with SVM. Themethod is summarized as follows.1) Generate samplesFirstly,determinethefeaturevector x [x1 , x2 , , xN ], N is the number of selected traffic flowdata. Current time traffic flow data to be forecasted isdenoted by y. Samples can be generated according to themodel with history traffic flow data.2) Dimension reductionSelect some samples and mapped them into lowdimension space with MDS methods introduced as insection 2.1. Prescribe the number k of nearest neighbors.The other samples are mapped into low dimensions withinterpolation method introduced as in section 2.2.3) Traffic flow forecasting with SVMAll the mapped samples are divided into two parts.One part is used to train nonlinear SVM model. Theother is used to test the trained model. Some indices canbe used to evaluate the training model in quantitatively.Commonly used are following three indices.(1) Mean absolute percentage error (MAPE)𝑛𝑛𝑦𝑦 𝑖𝑖 𝑦𝑦𝑖𝑖1 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑦𝑦𝑖𝑖𝑛𝑛𝑖𝑖 1(2) Mean absolute error (MAE)𝑛𝑛1𝑀𝑀𝑀𝑀𝑀𝑀 𝑦𝑦 𝑖𝑖 𝑦𝑦𝑖𝑖 𝑛𝑛𝑖𝑖 1(3) Mean square error (MSE)𝑛𝑛1𝑀𝑀𝑀𝑀𝑀𝑀 (𝑦𝑦 𝑖𝑖 𝑦𝑦𝑖𝑖 )2𝑛𝑛𝑖𝑖 1where n is the number of test samples, ŷi is theforecasting value, and yi is the detected value.5. Example5.1 Data SourceJinan traffic police branch provides us with traffic flowdata and video data of Jingshi Road expressway.Through the express way, there are about 14intersections. The traffic flow data are collected byinductance loop vehicle detectors. We select traffic flowdata of the cross between Jingshi road and Lishan Roadfrom June 1, 2007 to July 1, 2007 to study. In theintersection, there are four directions. We select thedirection from west to east. Data collecting equipment isloop detectors which can detector three trafficparameters, i.e. volume, average speed and occupancy.Collecting interval is 5 minutes. There are 53187 trafficflow data in total.5.2 Generate samplesTraffic flow parameter value to be forecasted isdenoted by variable Y . Traffic flow parameter value ofcurrent cross at previous sampling times are denoted bywherevariablevectorX ( X 1 , X 2 , , X N )1X i , i 1,2, , N1 denotes previous i sampling timevalue. Traffic flow parameter value of current cross athistory times are denoted by variable vectorH ( H 1 , H 2 , , H N 2 ) where H i , i 1,2, , N 2 denotesprevious i days’ time value. 𝑿𝑿 [𝑿𝑿, 𝑯𝑯] is taken as thefeature vector.In this example, N1 10 previous sampling time dataand N2 5 history sampling data are prescribed. Thehistory cycle is set 1 day. For generating samples, 5 dayshistory data should be retained. 51747 samples aregenerated in the end.5.3 Dimension reductionIn this example, 4000 samples are selected to be premapped into low dimension space. Firstly, calculate thedistance matrix. Euclidean distance is adopted here.Then calculate the mapped vector according to thedistance matrix with MDS method. The others aremapped into low dimension with interpolation MDSmethod. The number of nearest neighbor is set k 10.For comparison, the dimension number is set as 2, 3, and5 respectively.5.4 Forecasting with SVMAfter selecting the independent variables, we takethem as the input and variable Y as the output of SVMrespectively. We select 31048 samples randomly as thetraining set to train the SVM and 20699 samples as thetesting set.For improving the training precision of SVM, all thetraffic flow samples and corresponding transformed
values are scaled to [-1,1]. The normalization is asfollowing equation.2𝑥𝑥 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 ��𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚where 𝑥𝑥𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 denotes the scaled values of 𝑥𝑥, 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚are the maximum and minimum values of 𝑥𝑥 . Scaledtraffic flow parameters’ values are taken as the input ofSVM.The computation configuration is as follows. Theoperation OS is Ubuntu Linux. The processor is 3GHzIntel Xeon with 8GB RAM. Based on different featuredimension number, the training time of SVM iscompared. For illustrate the efficiency of featureextraction, we train the SVM with all feature variables,i.e. no feature extraction. The training time based on 2,3 ,5 and 15 feature dimensions are listed in table 1, table2 and table 3. They corresponds to traffic parametervolume, speed and time occupancy respectively.Table 1 training time and MDS processing time of volumeTotalSupportDimensionInterpolation TrainingMDS timecomputation 153.091153.09130811Table 2 training time and MDS processing time of speedTotalSupportDimensionInterpolation TrainingMDS timecomputation .21429807294442943929496Table 3 training time and MDS processing time occupancyTotalSupportDimensionInterpolation TrainingMDS timecomputation .59130318302553037430422After training SVM model, the left samples are used totest. The test results of traffic parameter volume, speed,and occupancy are listed in table 4, table 5 and table 6respectively.5.5 Forecasting with common used methodFor comparison, the samples are analyzed withaverage value and multiple linear regression methods.5.5.1 Average value methodWe take used of previous 15 time point’s traffic flowparameters X 1 , X 2 , , X 15 to forecast the current timepoint’s traffic flow parameter Y . The forecastingequation isY ( X 1 X 2 X 15 ) / 15It doesn’t need history data to determine thecalculation model. The forecasting errors are shown as intable 7.Table 7 forecasting result of Volume based on average 234.9440.48speed0.34407.85124.89occupancy5.5.2 Forecasting with multiple linear regressionLet variable Y be dependent variable and, be independent variables. The regressionequation is31048 samples are used as training samples to determinewith minimum leastthe regression parameterssquare methods. The left 20699 samples are used to test.The forecasting errors are shown as in table 8.Table 8 forecasting result of occupancy with SVM withmultiple regression 4.646.09MSE2366.335.9968.165.6 Results analysisThe computation cost of training time and MDSprocessing time are shown as in figure 1. From theanalysis results we can find the computation cost can bedecreased markedly with the decrease of dimensionnumber. It illustrates that feature extraction is efficient intraffic flow forecasting.300Table 4 forecasting result of Volume with 8.12computation 00Table 5 forecasting result of speed with 20.11814.604.514.443.9737.4036.7835.3729.93Table 6 forecasting result of occupancy with 5dimensionFigure 1 computation time based on different dimensionnumberThe test results of different traffic parameters areshown as in figure 2, 3 and 4. From the results we canfound that forecasting precision based on SVM is higherthan that of classical forecasting methods. Although the
forecasting precision based on dimension reduction isdecreased, it is still higher or similar to that of classicalmethod. The affection of the reduction method is notmarked in the sample is because that the dimension ofinput is not very higher.markedly and the scale of traffic flow data will becomemore large. The effective of the method will be more andmore important to large scale traffic flow forecasting.AcknowledgementsThis work is partially supported by national youthscience foundation (No. 61004115), national sciencefoundation (No. 61272433), and Provincial Fund forNature project (No. ZR2010FQ018).ReferencesFigure 2 MAPE of forecasting resultsFigure 3 MAE of forecasting results1 Yang Z S. Basis traffic information fusion technology andits application. Beijing, China Railway Publish House,2005.2 Wang, F, Tan G Z, Deng C. Parallel SMO for Traffic FlowForecasting. Applied Mechanics and Materials, 2010, 20(1):843-8483 Stephen C. Traffic Prediction Using MultivariateNonparametric Regression. Journal of TransportationEngineering, 2003, 129(2): 161-168.4 Cortes C, Vapnik V. Support Vector Networks[J]. MachineLearning, 1995, 20: 273–297.5 Chang C C, Lin C J. LibSVM: a library for support vectormachines. ACM Transactions on Intelligent Systems andTechnology, 2001, 2(3): 1--27.6 Hong, W C. Application of seasonal SVR with chaoticimmune algorithm in traffic flow forecasting. NeuralComputing & Applications; 2012, 21(3): 583-5937 Jolliffe, I. T. Principal component analysis. New York:Springer, 2002.8 George K M. Self-Organizing Maps. INTECH, 20109 Borg I, Patrick J F. Modern Multidimensional Scaling:Theory and Applications. New York : Springer, 2005: 207–21210 Seung-H B, Judy Q, Geoffrey F. Adaptive Interpolation ofMultidimensional Scaling. International Conference onComputational Science, 2012: 393-402Zhanquan Sun, Ph.D, associated professorof Shandong Computer Science Center.Major on intelligent transportation systems,data mining and cloud computing. Haspresided and attended 10 research projectsand published about 40 academic papers.GeoffreyFigure 4 MSE of forecasting results6
Abstract: Traffic flow forecasting is a popular research topic of Intelligent Transportation Systems (ITS). With the development of information technology, lots of history electronic traffic flow data are collected. How to take full use of the history traffic flow data to improve the traffic flow forecasting precision is an important issue.
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