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1 Paper Number: 19-00525 2 3 4 PREDICTING LINK TRAVEL SPEED IN URBAN ROAD NETWORKS USING VARIATIONAL MODE DECOMPOSITION 5 6 7 8 Eui-Jin Kim 9 10 11 12 13 Department of Civil and Environmental Engineering, Seoul National University 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea Phone: +82-2-880-9154; Fax: +82-2-873-2684; E-mail: [email protected] 14 15 16 17 Institute of Construction and Environmental Engineering, Seoul National University 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea Phone: +82-2-880-7377; Fax: +82-2-873-2684; E-mail: [email protected] 18 Seung-Young Kho 19 20 21 22 23 Department of Civil and Environmental Engineering and Institute of Construction and Environmental Engineering, Seoul National University 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea Phone: +82-2-880-1447; Fax: +82-2-873-2684; E-mail: [email protected] 24 Dong-Kyu Kim, Corresponding Author 25 26 27 28 Department of Civil and Environmental Engineering and Institute of Construction and Environmental Engineering, Seoul National University 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea Phone: +82-2-880-7348; Fax: +82-2-873-2684; E-mail: [email protected] Ho-Chul Park 29 30 31 32 33 Call for Papers: ABJ70 - Standing Committee on Artificial Intelligence and Advanced Computing Applications 34 35 36 37 38 39 40 Submitted for presentation at the 98th Transportation Research Board Annual Meeting This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, South Korea (2016R1C1B1008492). TRB 2019 Annual Meeting Paper revised from original submittal. Kim et al. 1 2 INTRODUCTION 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Providing accurate link travel time to travelers is the most important component in the urban intelligent transportation system (ITS). However, it is difficult to generate reliable information due to the complexity of traffic dynamics in congestion where this information is needed. Many researchers have proposed the machine learning-based models such as the artificial neural network (ANN) (1), the support vector machine (SVM) (2, 3), and the k-nearest neighbor method (KNN) (4), which shows better performance than conventional parametric statistical methods (5). Despite their efforts to obtain models with good performance, they have been struggling with the uncertainty of urban traffic, especially in unstable congested conditions (6). Also, since the urban travel speed shows significantly different patterns across the network and during different time periods (7-9), the need for models that cope with various traffic patterns has been raised. Recently, the concept of “divide and conquer” (DC) was proposed to deal with the uncertainty of urban travel speed by decomposing complex time series data into clear signals that represent oscillatory patterns. The decomposed signals are respectively predicted and summed up to reconstruct the predicted travel speed. This data-adaptive and easy-to-use concept uses a multiresolution technique for signal processing such as Fourier transform (FT) (10), wavelet transform (WT) (11), and empirical mode decomposition (EMD) (12), but several limitations make them challenge to use for urban travel speed that includes non-stationarity, non-linearity, and stochastic feature (13). The goal of this study is to propose and evaluate the hybrid model for predicting the urban travel speed using a variational mode decomposition (VMD) (13). The VMD decomposes the travel speed into modes representing the periodic pattern of each time-scale. This technique is suitable for analyzing urban travel speed due to its robustness to sampling and noise. The decomposed modes, which are orthogonal and oscillatory, are more predictable than the original data. Therefore, performance improvement can be expected in prediction and summation process for each mode. To verify our hybrid model, we conduct a performance evaluation on various links, days of the week, and traffic condition. 29 METHODOLOGY 30 31 32 33 Figure 1 showed an overview of our hybrid model. The predicted travel speed was computed as the results of this process. We used the SVM and ANN, which are well-known non-linear regression model, as prediction models of a hybrid model using VMD. 34 35 36 FIGURE 1 Hybrid model using variational mode decomposition. TRB 2019 Annual Meeting Paper revised from original submittal. Kim et al. 1 2 3 4 3 The purpose of the VMD is to decompose an original signal into πΎπΎ modes. Each mode is required to be mostly compact around a central frequency, and the VMD algorithm solves the following minimization problem to obtain this central frequency: πΎπΎ 2 ππ min {οΏ½ οΏ½πππ‘π‘ οΏ½οΏ½πΏπΏ(π‘π‘) + οΏ½πππποΏ½ ∗ π’π’ππ (π‘π‘)οΏ½ ππ −πππππππ‘π‘ οΏ½ } {π’π’ππ },{ππππ } 2 ππ=1 πΎπΎ 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 (1) s. t. οΏ½ π’π’ππ = ππ ππ=1 where πΏπΏ is the Dirac distribution, βοβ is the vector β2 norm, πππ‘π‘ is partial derivative of time π‘π‘, π’π’ππ is the ππth decomposed mode, πΎπΎ is the number of predefined modes, ππ is the original time-series, and ππππ is the central frequency of the ππth mode. The solution to the problem is the saddle point of the augmented Lagrangian described with Lagrangian multipliers and quadratic penalty. 2 πποΏ½ −ππππππ π‘π‘ (π‘π‘)οΏ½ L({uk }, {ππππ }, ππ) = πΌπΌ ∑πΎπΎ οΏ½ππ οΏ½οΏ½πΏπΏ(π‘π‘) + οΏ½ ∗ π’π’ ππ οΏ½ + π‘π‘ ππ ππ=1 ππππ 2 πΎπΎ 2 βπ₯π₯(π‘π‘) − ∑πΎπΎ ππ=1 π’π’ππ (π‘π‘)β2 + [λ(t), x(t) − ∑ππ=1 π’π’ππ (π‘π‘)] (2) where ππ(π‘π‘) are Lagrangian multipliers, and πΌπΌ is a balance parameter. The solution to Equation 2 is a sequence of iterative, sub-optimization algorithms, called the alternate direction method of a multiplier (ADMM), and more details are presented in (13). The ANN is a well-known non-linear regression model for predicting time series. A multi-layer perceptron (MLP), which includes an input layer, one or more hidden layers, and an output layer, can capture time series traffic pattern by training the weights and biases between the interaction of neurons in multiple layers. A standard backpropagation algorithm was used to train our MLP (14). Lagged values of speed πππ‘π‘−ππ , … , πππ‘π‘−ππ are used as input data to predict the predicted values of speed πποΏ½π‘π‘ as in Equation 3. πΌπΌ ππ ππ=1 ππ=ππ πποΏ½π‘π‘ = οΏ½ ππππ ππ( οΏ½ π€π€ππππ πππ‘π‘−ππ + ππππ ) (3) where π‘π‘ is the number of input data (π‘π‘ = 1, … , ππ), ππ and ππ are minimum, and maximum lagged time, ππππ and π€π€ππππ are weights of interaction between neurons, and ππππ is the bias. The basic idea of SVM is to map the data from the input space into a high-dimensional feature space to construct an optimal decision function. Given the observed speed, πππ‘π‘ , and its ππ lagged values, πΊπΊππ = [πππ‘π‘−ππ , … , πππ‘π‘−ππ ], the optimal decision function is presented in Equation 4: ππ πποΏ½π‘π‘ = π€π€ ππ π±π±(πΊπΊππ ) + b (4) ππ where π±π±(πΊπΊππ) is a mapping function that transforms the data from input space into feature space, π€π€ is weight, and ππ is bias. The optimal decision function is estimated by minimizing the regression risk as in Equations 5 and 6: TRB 2019 Annual Meeting Paper revised from original submittal. Kim et al. 1 2 3 4 5 6 4 ππ 1 R reg (ππ) = πΆπΆ οΏ½ πΏπΏ( πππ‘π‘ , πποΏ½π‘π‘ ) + βπ€π€β2 2 π‘π‘=1 0 ππππ οΏ½π¦π¦ − πποΏ½π‘π‘ οΏ½ ≤ ππ πΏπΏοΏ½πππ‘π‘ , πποΏ½π‘π‘ οΏ½ = οΏ½ οΏ½π¦π¦ − πποΏ½π‘π‘ οΏ½ − ππ ππππβππππππππππππ (5) (6) where L(ο) is a loss function, and C and ππ are the regularization parameters. To convert a nonlinear learning problem into a linear one, the radial basis kernel function was used, and the πποΏ½π‘π‘ can be written with the Lagrangian multipliers, πππ‘π‘ , πππ‘π‘∗ , as in Equation 7 (15): ππ ππ ππ πποΏ½π‘π‘ = οΏ½(πππ‘π‘ − πππ‘π‘∗ ) π²π²οΏ½πΊπΊππ ,π‘π‘ , πΊπΊππ οΏ½ + ππ (7) π‘π‘=1 7 8 9 10 11 12 13 14 15 We used 5-minute aggregated travel speed data collected by dedicated short-range communication (DSRC) from April 1, 2016, to June 30, 2016, for 61 links. To ensure the data reliability, only the data from 06:00 A.M. to 23:55 P.M., which have the missing rate of less than 2%, was used. To evaluate our method on a congested condition, we identified the congestion based on a variation of travel time. After the travel time was normalized to have a zero mean with one standard deviation, we identified the valleys, which have lower speed than the threshold in travel speed data, as congestion. 16 FINDINGS 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 We compared our method with benchmark models such as SVM and ANN in various traffic patterns. Hyperparameters of each model were consistently calibrated using 5-fold crossvalidation based on MAPE. The prediction performance of each model was aggregated to a linkday unit, i.e., 61 links with 7days (427 link-days). Performance measure was set as the mean absolute percent error (MAPE) for the 15-minutes ahead prediction, i.e., 3-steps ahead prediction. In the comparison between the benchmark models, the SVM showed better performance than ANN regarding robustness (standard deviation of MAPE) and accuracy (mean MAPE). With the VMD, however, the VMD-ANN performed better than the VMD-SVM, and this hybrid model outperforms the benchmark model in both overall and congestion conditions. Also, the hybrid model was found to be particularly strong in specific link-day where the benchmark model was hard to predict. These results indicated that the VMD could effectively reduce the complexity of urban traffic, and complement specific situations that are difficult to improve with existing models. Also, we computed the statistical properties of each mode such as dominant period and the percent of variance explaining the original data, and those properties examined that VMD successfully decompose the regular patterns such as daily travel demand and commuting, as well as the irregular patterns such as stochastic variation and transition of traffic state. 34 TRB 2019 Annual Meeting Paper revised from original submittal. Kim et al. 1 5 CONCLUSIONS 2 3 4 5 6 7 8 9 10 11 12 13 In order to improve the prediction models in urban network, it is essential to understand how to deal with the complex dynamics of urban traffic. In this study, we proposed the multiresolution technique for signal processing, VMD, to mitigate the complexity of the travel speed data by decomposing them into oscillatory and orthogonal modes. 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