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Paper Number: 19-00525
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PREDICTING LINK TRAVEL SPEED IN URBAN ROAD NETWORKS USING
VARIATIONAL MODE DECOMPOSITION
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Eui-Jin Kim
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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: kyjcwal@snu.ac.kr
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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: hochul.park@snu.ac.kr
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Seung-Young Kho
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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: sykho@snu.ac.kr
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Dong-Kyu Kim, Corresponding Author
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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: dongkyukim@snu.ac.kr
Ho-Chul Park
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Call for Papers: ABJ70 - Standing Committee on Artificial Intelligence and Advanced
Computing Applications
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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.
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INTRODUCTION
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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.
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METHODOLOGY
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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.
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FIGURE 1 Hybrid model using variational mode decomposition.
TRB 2019 Annual Meeting
Paper revised from original submittal.
Kim et al.
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The purpose of the VMD is to decompose an original signal into 𝐾𝐾 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:
𝐾𝐾
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𝑗𝑗
min {� �𝜕𝜕𝑡𝑡 ��𝛿𝛿(𝑡𝑡) + �𝜋𝜋𝜋𝜋� ∗ 𝑢𝑢𝑘𝑘 (𝑡𝑡)� 𝑒𝑒 −𝑗𝑗𝜔𝜔𝑘𝑘𝑡𝑡 � }
{𝑢𝑢𝑘𝑘 },{𝜔𝜔𝑘𝑘 }
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𝑘𝑘=1
𝐾𝐾
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(1)
s. t. � 𝑢𝑢𝑘𝑘 = 𝑓𝑓
𝑘𝑘=1
where 𝛿𝛿 is the Dirac distribution, ‖‖ is the vector ℓ2 norm, 𝜕𝜕𝑡𝑡 is partial derivative of time
𝑡𝑡, 𝑢𝑢𝑘𝑘 is the 𝑘𝑘th decomposed mode, 𝐾𝐾 is the number of predefined modes, 𝑓𝑓 is the original
time-series, and 𝜔𝜔𝑘𝑘 is the central frequency of the 𝑘𝑘th mode. The solution to the problem is the
saddle point of the augmented Lagrangian described with Lagrangian multipliers and quadratic
penalty.
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𝑗𝑗�
−𝑗𝑗𝜔𝜔𝑘𝑘 𝑡𝑡
(𝑡𝑡)�
L({uk }, {𝜔𝜔𝑘𝑘 }, 𝜆𝜆) = 𝛼𝛼 ∑𝐾𝐾
�𝜕𝜕
��𝛿𝛿(𝑡𝑡)
+
�
∗
𝑢𝑢
𝑒𝑒
�
+
𝑡𝑡
𝑘𝑘
𝑘𝑘=1
𝜋𝜋𝜋𝜋
2
𝐾𝐾
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‖𝑥𝑥(𝑡𝑡) − ∑𝐾𝐾
𝑘𝑘=1 𝑢𝑢𝑘𝑘 (𝑡𝑡)‖2 + [λ(t), x(t) − ∑𝑘𝑘=1 𝑢𝑢𝑘𝑘 (𝑡𝑡)]
(2)
where 𝜆𝜆(𝑡𝑡) are Lagrangian multipliers, and 𝛼𝛼 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 𝑆𝑆𝑡𝑡−𝑚𝑚 , … , 𝑆𝑆𝑡𝑡−𝑝𝑝 are used as input data to predict the
predicted values of speed 𝑆𝑆�𝑡𝑡 as in Equation 3.
𝐼𝐼
𝑝𝑝
𝑖𝑖=1
𝑗𝑗=𝑚𝑚
𝑆𝑆�𝑡𝑡 = � 𝑊𝑊𝑖𝑖 𝑓𝑓( � 𝑤𝑤𝑖𝑖𝑖𝑖 𝑆𝑆𝑡𝑡−𝑗𝑗 + 𝜃𝜃𝑖𝑖 )
(3)
where 𝑡𝑡 is the number of input data (𝑡𝑡 = 1, … , 𝑛𝑛), 𝑚𝑚 and 𝑝𝑝 are minimum, and maximum
lagged time, 𝑊𝑊𝑖𝑖 and 𝑤𝑤𝑖𝑖𝑖𝑖 are weights of interaction between neurons, and 𝜃𝜃𝑖𝑖 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, 𝑆𝑆𝑡𝑡 , and its
𝒑𝒑
lagged values, 𝑺𝑺𝒎𝒎 = [𝑆𝑆𝑡𝑡−𝑚𝑚 , … , 𝑆𝑆𝑡𝑡−𝑝𝑝 ], the optimal decision function is presented in Equation 4:
𝒑𝒑
𝑆𝑆�𝑡𝑡 = 𝑤𝑤 𝑇𝑇 𝜱𝜱(𝑺𝑺𝒎𝒎 ) + b
(4)
𝒑𝒑
where 𝜱𝜱(𝑺𝑺𝒎𝒎) is a mapping function that transforms the data from input space into feature space,
𝑤𝑤 is weight, and 𝑏𝑏 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.
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𝑛𝑛
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R reg (𝑓𝑓) = 𝐶𝐶 � 𝐿𝐿( 𝑆𝑆𝑡𝑡 , 𝑆𝑆�𝑡𝑡 ) + ‖𝑤𝑤‖2
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𝑡𝑡=1
0 𝑖𝑖𝑖𝑖 �𝑦𝑦 − 𝑆𝑆�𝑡𝑡 � ≤ 𝜀𝜀
𝐿𝐿�𝑆𝑆𝑡𝑡 , 𝑆𝑆�𝑡𝑡 � = �
�𝑦𝑦 − 𝑆𝑆�𝑡𝑡 � − 𝜀𝜀 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
(5)
(6)
where L() is a loss function, and C and 𝜀𝜀 are the regularization parameters. To convert a nonlinear learning problem into a linear one, the radial basis kernel function was used, and the 𝑆𝑆�𝑡𝑡
can be written with the Lagrangian multipliers, 𝑎𝑎𝑡𝑡 , 𝑎𝑎𝑡𝑡∗ , as in Equation 7 (15):
𝑛𝑛
𝒑𝒑
𝒑𝒑
𝑆𝑆�𝑡𝑡 = �(𝑎𝑎𝑡𝑡 − 𝑎𝑎𝑡𝑡∗ ) 𝑲𝑲�𝑺𝑺𝒎𝒎 ,𝑡𝑡 , 𝑺𝑺𝒎𝒎 � + 𝑏𝑏
(7)
𝑡𝑡=1
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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.
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FINDINGS
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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.
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TRB 2019 Annual Meeting
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Kim et al.
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CONCLUSIONS
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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. Our results show that the hybrid
model using VMD can contribute to improving prediction performance, which is explained by
the fact that the decomposed modes well-represented the regular and irregular patterns of travel
speed. Therefore, future studies need to leverage the properties of these modes to account for
complex traffic phenomena as well as to enhance the prediction performance. For performance,
more advanced model can be available in the process of predicting modes and summing up the
predicted modes into the predicted speed. For interpretation, exploring the physical meaning of
modes may contribute to expanding the understanding of urban traffic dynamics
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TRB 2019 Annual Meeting
Paper revised from original submittal.