Traffic Flow Forecasting by a Least Squares Support Vector Machine with a Fruit Fly Optimization Algorithm

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Procedia Engineering 137 ( 2016 ) 59 – 68

Peer-review under responsibility of the Department of Transportation Engineering, Beijing Institute of Technology doi: 10.1016/j.proeng.2016.01.234

ScienceDirect

GITSS2015

Traffic Flow Forecasting by a Least Squares Support Vector

Machine with a Fruit Fly Optimization Algorithm

Yuliang Cong

a

_{*, Jianwei Wang}

a

_{,Xiaolei Li}

a

School of Communication Engineering, Jilin University, Changchun, Jilin, 130012,P.R. China

Abstract

The accuracy of traffic flow forecasting plays an important role in the field of modern Intelligent Transportation Systems (ITS). The least squares support vector machine (LSSVM) has been shown to provide a strong potential in forecasting problems, particularly by using appropriate heuristic algorithms to determine the value of its two parameters. However, the disadvantage of these meta-heuristics is that it is difficult to understand and slowly achieve the global optimal solution. As a new heuristic algorithm, the fruit fly optimization algorithm (FOA) has the advantages of easy to understand and quickly converge to the global optimal solution. Therefore, in order to improve the prediction performance of the model, this paper presents a traffic flow prediction model based on least squares support vector machine and automatically determines the least squares support vector machine model with two parameters in the appropriate value by FOA. The experiment results show that the LSSVM combined with FOA (LSSVM-FOA) perform better than other methods, namely single LSSVM model, RBF neural network (RBFNN) and LSSVM combined with particle swarm optimization algorithm (LSSVM-PSO).

Peer-review under responsibility of the Department of Transportation Engineering, Beijing Institute of Technology.

Keywords: traffic flow forecasting, least squares support vector machine (LSSVM), fruit fly optimization algorithm (FOA), optimization forecasting problem

1. Introduction

In recent years, Intelligent Transportation Systems (ITS) have achieved great developments. At the same time, intelligent traffic control and guidance system plays an important role in the research area of ITS. The traffic flow forecasting results will be directly related to traffic control and guidance. An accurate and robust traffic forecasting

* Corresponding author. Tel.:15590665518.

E-mail address:[email protected]

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model can help traffic authorities to manage traffic proactively. However, the traffic forecasting problem is very challenging. Although a wide variety of different methods have been published in the literature, such as traffic model-based methods, statistical methods and data mining-based methods, accurate, robust and reliable traffic forecasting methods for practical use are still hard to find. Accurate and real-time traffic flow forecasting is important for developing the real-time, dynamic and high efficient traffic management and control systems. In reality, for traffic managers, the traffic flow forecasting would enable them to apply traffic control management early enough to prevent traffic congestion rather than to deal with the traffic problems after the traffic congestion has already occurred [1]. For travellers, it would enable them to plan their trips in advance and adjust their way at any moment with the dynamic traffic flow forecasting. Finally, the traffic congestion and road network efficiency can be improved, which provides a reasonable basis for the planning and development of the transportation system.

With the rapid development of China's ITS, traffic flow forecasting technology has aroused wide attention in the field of practice and academia. An efficient and accurate traffic flow forecast can provide a basis for the real-time, dynamic and high efficient traffic management and control systems. The traffic flow forecast to some extent affected the development of ITS. For that reason, more accurate traffic flow forecasting is needed for maintaining the safety and stability operation of ITS. However, summarizing the existing forecasting models and considering the characteristics of the traffic itself such as nonlinearity, complexity and uncertainty, it is quite difficult to predict traffic flow accurately.

In order to improve the accuracy of traffic flow prediction, scholars and practitioners have proposed many methods over the past several decades, such as time series techniques and regression models [2–6]. In recent years, many artificial intelligence forecasting techniques have been applied to traffic flow forecasting to improve the accuracy of forecasting. Yan Chen et al. [7] proposed a kind of intelligent forecasting method based on particle swarm optimization algorithm, which improves the stability and reliability of the forecast. Wei-Chiang Hong et al. [8] proposed a hybrid model combined with differential evolution algorithm for traffic flow, which was shown to outperform the SVR default parameter prediction support vector regression, regression forecast model and BP artificial neural network (BPNN). Li et al. [9] implemented a method for chaotic time series of optimized BP neural network based on particle swarm optimization (PSO), and the practical application of this method is indicated. Xu et al. [10] presented a short-term traffic forecasting model which combines support vector regression (SVR) model with continuous ant colony optimization (SVRCACO) to forecast traffic flow. These methods improve the accuracy of traffic flow forecast to a certain extent.

Least squares support vector machine (LSSVM) is a new support vector machine (SVM) which can be used to approximate the nonlinear system with higher accuracy. Therefore, it is a powerful tool for the model of nonlinear systems [11]. The LSSVM model has been successfully used to solve the problem of prediction in many fields, such as CO concentration, gas], short-term power load, income, precipitation, wind speed, the hydropower consumption prediction, and so on. However, it is found that the LSSVM model is seldom applied to traffic flow prediction. This paper explores the feasibility of using the least squares support vector machine model for the prediction of traffic flow. The prediction performance of the least squares support vector machine model is largely determined by the value of its two parameters. At present, in order to determine the appropriate values of these two parameters, several meta-heuristic algorithms have been applied including particle swarm optimization [12], genetic algorithm [13], chaotic differential evolution approach [14], artificial bee colony algorithm [15], and simulated annealing algorithm [16]. However, the disadvantage of these algorithms is that it is difficult to understand and slowly achieve the global optimal solution. The fruit fly optimization algorithm (FOA) is a new evolutionary computation and optimization technique. The advantage of this new optimization algorithm is easy to understand, and program code is shorter than other optimization algorithms and achieves the best fast solution. Thus, this paper attempts to use FOA to determine the two parameters of the least squares support vector machine model in traffic flow prediction.

The remainder of this paper is organized as follows: Section 2 introduces the model of the LSSVM and FOA, then mixed traffic flow prediction model (LSSVM-FOA) is discussed in detail. Section 3 describes the processing procedures of sample data used in this paper, and a numerical example is calculated, compared and discussed. Section 4 summarizes this article.

2. Method of LSSVM-FOA model

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The LSSVM algorithm is an extension of the support vector machine using linear least squares criterion for the loss function instead of inequality constraints [17]. The basic principle is as following [18]: given a group of samples{ , } 1

N i i i

x y , where n i

x R is the input vector and yiRis the output value of the corresponding sample

i

. By

a nonlinear function

M

, data from the original feature space is mapped to a high dimensional transformation, and thus, in order to approximate the linear way as follows:

( ) T ( )

f x wM x b (1) where

w

represents the weight vector; and b represents the error.

In the original space, the least square support vector machine with equality constraints can be expressed as:

2 , , 1 1 1 min ( , ) 2 2 . . ( ) , 1,2, , N T i w b e i T i i i J w e w w e s t y w x b e i n J M ° ® ° ¯

¦

" (2)

where J indicates the regularization parameter; and ei indicate the slack variable.

The Lagrange function Lcan be built by:

1 ( , , , ) ( , ) { ( ) } N T i i i i i L w b eD J w e

_¦

D wMx b e y (3) where

D

_idenotes the Lagrange multiplier. The optimality conditions of the Karush-Kuhn-Tucker (KKT) are given:

1 1 0 ( ) 0 0 1,2, , 0 0 ( ) 0 N i i i N i i i i i T i i i i L w x w L b for i N L e e L w x b y e DM D D J M D w ° w ° °w °w ° ®_w ° ° w ° w ° °w ¯

¦

" (4)

The optimization problem can be transformed into linear solution by eliminating the variablewandei:

0 0 1 T Q b A Y Q K I J ª º ª º ª º « _{» x} « » « » « » _{¬ ¼} _{¬ ¼} « » ¬ ¼ (5) [1, ,1]T Q " , [ 1, , ] T N A D "D , [ ,1 , ] T N

Y y " y . Based on the Mercer’s condition, the Kernel function can be

set as:

( , ) ( )T ( ), , 1, 2, ,

i j i j

K x x M x M x i j " N (6) And then, the regression LSSVM model becomes:

1 ( ) ( , ) N i i i f x

¦

DK x x b (7) Mercer kernel functionK x x( , i)has several different types, for instance sigmoid, polynomial and radial basis function (RBF). RBF is a common choice for the kernel function because of the need to set very few parameters and excellent overall performance [19]. So, RBF is selected as the kernel function in this paper:

2 2 ( , ) exp 2 i i x x K x x V § · ¨ ¸ ¨ ¸ © ¹ (8)

Thus, two parameters are required in the least squares support vector machine model selection, which are the bandwidth of the Gauss radial basis kernel“V” and the regularization parameter “J”. The optimal values of these two parameters are determined by FOA in this paper.

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2.2. Fruit Fly Optimization Algorithm (FOA)

FOA is a calculation method of interactive evolution. By simulating the behavior of the fruit fly swarm, the FOA can achieve the global optimal solution. Fruit fly is an insect that lives in temperate and tropical climates, rotten fruits are its food. Fruit flies have strong sense of vision and sense of smell. The fruit fly food discovery process is as follows: First it smells the food source with its olfactory organs and flies to that position; after it is close to the position of the food, and its sensitive eyes are also used to find food and other fruit fly's gathering place, and then it flies in that direction. Problems in many areas can be solved by FOA, including traffic events [19], forecasting of export trade [20], and the design of analog filters [21]. It can be divided into several steps to describe the FOA, as following:

Step 1: Initialization Parameters

The maximum number of iterations of maxgen, the population size sizepop , the initial location (X_axis,Y_axis), and the random flight distance range FR are the main parameters of FOA.

Step 2: Initialization Population

Give random flight directions and distance of discovering food by using osphresis:

_ * ()

i i

X X axis Value random Y Y axis Value random

® ¯ (9)

Step 3: Evaluation Population

First, distance (Dist) of the fly to origin is required for the calculation; second, the smell concentration distinguish value (S) is required for the calculation. Assuming that the reciprocal of Dist is S:

2 2 i i i Dist X Y (10) 1 / i i S Dist ₍₁₁₎

Then, we calculate the smell concentration (Smelli) of the individual fruit fly location by substituting the smell

concentration judgment value (Si) into the smell concentration judgment function (also called fitness function).

Finally, find out the individual fruit fly with the maximal smell concentration (the maximal value ofSmell_i) among the fruit fly swarm:

= ( )

i i

Smell Function S (12) [bestSmell bestIndex] max(Smell_i) (13)

Step 4: Selection Operation

Maintain xand ycoordinates of the maximal smell concentration value. Then, the fruit fly through the use of the visual to fly to the location of maximal smell concentration value. Enter iterative optimization to repeat the step 2~3. When the smell concentration is not better than the previous iteration smell concentration, or the number of iterations to reach the maximum number of iterations, the loop stops:

_ ( ) _ ( ) Smellbest bestSmell X axis X bestIndex Y axis Y bestIndex ° ® ° ¯ (14)

2.3. LSSVM-FOA Prediction Model

The parameters of the LSSVM model are determined as follows:

Step1: Parameters Initialization

The maximum number of iterations of maxgen, the population size sizepop , the initial location (X_axis,Y_axis), and the random flight distance range FR are the main parameters of FOA, and this parameters should be determined first. In this research, we assume maxgen = 100, sizepop = 20, FR ( 10,10). In the LSSVM-FOA program, we setgen,Y_axis rands(1, 2) , where rands()represents of the random number of generating functions.

Step2: The Evolution Begins

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flyi. In the LSSVM-FOA program, we use two variables [ ( ,:), ( ,:)]X i Y i to represent the flight distance which is needed for an individual fruit flyi, and setX i( ,:) X_axis20 *rand() 10 ,Y i( ,:) Y_axis20 *rand() 10 , respectively.

Step3: Preliminary Calculation

Calculate the distanceDisti, and then calculateSi. In the LSSVM-FOA program, we use ( ( ,1), ( , 2))D i D i to

representDisti, and setD i( ,1) ( ( ,1) ^ 2X i Y i( ,1) ^ 2) ^ 0.5, D i( , 2) ( ( , 2) ^ 2X i Y i( , 2) ^ 2) ^ 0.5, respectively.

The same, in the LSSVM-FOA program, we employ [ ( ,1), ( , 2)]S i S i to represent Si , and set

( ,1) 1 / ( ,1), ( , 2) 1 / ( , 2)

S i D i S i D i , respectively. Then, for traffic flow forecasting, input Si into the LSSVM

model. The parameters [ , ]J V of LSSVM model are represented by [ ( ,1), ( , 2)]S i S i in the LSSVM-FOA program, and we set J 20 * ( ,1)S i and 2

( , 2)

S i

V , respectively. According to the traffic flow forecasting result, we can calculate the smell concentrationSmelli. The Smelli is employed by the root-mean-square error (RMSE), as shown

in Equation (15), which measures the deviations between the forecasting values and actual values:

2 1 1 ( ) n i i i RMSE f f n

¦

(15) where n is the number of data points; fi is the actual value at period i; fi

denotes the forecasting value at period i'.

Step4: Filial Generation

The filial generation is generated based on Equations (9~13). Then input the filial generation into the LSSVM model and calculate the smell concentration value again. Setgen gen1.

Step5: Circulation Stops

When the maximum number of iterations is reached, the stopping criterion is satisfied, and the optimal parameters of LSSVM model are obtained. Otherwise, return Step2.

3. Simulation and analysis

3.1. Pretreatment of sample data

According to the changes of time series of the traffic flow, the current traffic flow is certainly linked with that of several minutes before on the roads. Thus the previous traffic flow sequence data can be used to predict the future traffic flow.

The sample traffic data used in this paper are from the Renmin Street in the centre of Changchun. Renmin Street is a frequently congested road that is characteristic of many similar roads in Central Changchun. Moreover, it is a street that we have studied in detail in the past and for which we have relevant readily accessible data. Before the calculation, use the following formula to normalize the sample data, and make them in the range from 0 to 1:

min max min t t x x x x x c (16)

where

x

min_and

x

max_{present the minimal and maximal value of each input factor, separately.}

The sample data were divided into the training data and testing data (the traffic flow data were sampled per 10 minutes from 6:00 a.m. to 6:00 p.m. in every day). Furthermore, 240 traffic flow data are used as training data between June 3 and June 5 of 2010. The testing sample includes 80 sample data on June 6 of 2010. The training data set is used exclusively for model development and then the test sample is used to evaluate the established model.

3.2. The selection of comparison models

Several traffic flow forecasting models are selected to compare the results of traffic flow forecast. According to traffic flow data, it is discerned that the traffic flow series shows great randomness, discrete and nonlinear. Therefore, the single LSSVM model, LSSVM-PSO (particle swarm optimization algorithm) model, and RBFNN model were also employed for comparison. In RBFNN model, only one parameterV needs to be determined.

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There are different types of performance evaluation that have been documented in the literature. The performance evaluation for each model should have at least a measure of absolute error, such as mean absolute percentage error (MAPE) or mean square error (MSE). Wang et al. stated that MSE is a good performance evaluation measurement because it is very sensitive to even small errors, in which case it is better to compare the small differences in the model’s performance. The MAPE and MSE are defined as follows:

1 1 , , 0 n t t t t t e MAPE e y y y n y z

¦

(17) 2 1 1 ( ) n t t t MSE y y n

¦

(18) whereytis the actual traffic flow value at time t;andyt

is the forecasting value at timet.

The criteria to judge the best model are relatively small for MAPE and MSE in modelling and forecasting. Other than these, the equal coefficient (EC) was also used as a performance measurement. EC was also used to test the ability of the model to capture the complex nature of the process that was being modelled. It is a measure of how well the future outcomes are likely to be predicted by the model, where the forecasting flow correlates with the actual flow. EC is defined as

2 2 2 ( ) 1 t t t t y y EC y y

¦

(19) whereytandyt

are the actual and forecasting traffic flow data, respectively, and n is the number of data points. The EC value is used to evaluate the linear correlation between the actual and forecasting flow. Clearly, an EC value close to unity indicates a satisfactory result, while a low value or one close to zero implies an inadequate result.

3.4. FOA result for parameter determination of the LSSVM Model

In LSSVM-FOA model, the least squares support vector machine model's two parameter values were dynamically adjusted by the FOA. The fruit fly swarm flight route for parameter optimization is shown in Fig.1(a). The iterative RMSE trend of the LSSVM-FOA model is shown in Fig.1(b) when searching for the optimal parameters. After 100 evolution iterations, the convergence can be seen in generation 24, and the optimal values of the parameters V and J are 2.3103,33.5087,respectively.

3.5. Forecasting result and discussion

According to the result of the FOA, the values of V and J were set as 2.3103 and 33.5087, separately. The values of V and J were set as 0.1 and 500 in the single LSSVM model, separately. According to the result of PSO, the optimal values of V and J were set as 5.7335 and 15.2253 in the LSSVM-PSO model, separately. The spread parameter value was set as 0.2 in the RBFNN model.

With the LSSVM-FOA, single LSSVM, LSSVM-PSO and RBFNN model, the corresponding training times of the data are 43, 3, 83 and 4s, separately. The training time of the four models is different in training data processing. LSSVM-FOA and LSSVM use longer time because they need to determine the two parameters for each generation. Yet, the LSSVM-FOA uses 43s less than the LSSVM-PSO computation.

Table 1 shows the comparisons of the values of MAPE, MSE, and EC for the FOA, LSSVM, LSSVM-PSO and RBFNN model. It can be drawn that the MAPE value of LSSVM-FOA model is 4.9960%, which is much smaller than that obtained by single LSSVM, LSSVM-PSO and RBFNN model (7.5715%, 5.6151% and 8.4454%, separately). The MSE value of LSSVM-FOA model is 46.8830, which is dramatically smaller than that obtained by another three models (73.8215, 55.5334 and 80.6750, separately). The EC value of LSSVM-FOA model is 0.9720, which is much larger than that obtained by single LSSVM, LSSVM-PSO and RBFNN model (0.9569, 0.9667 and 0.9521, separately). At the same time, the values of MSE and MAPE of LSSVM-PSO model are much smaller than

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that of single LSSVM and RBFNN models. These show that the parameter selection of the meta-heuristic algorithm has been used to improve the prediction accuracy of the traffic flow forecasting model based on the LSSVM. The LSSVM-FOA model has better forecasting prediction than the LSSVM-PSO model in this paper. Furthermore, the RBFNN model has the lowest forecasting accuracy because the values of MAPE and MSE are the largest. The single LSSVM model’s MAPE and MSE values are smaller than that of RBFNN model, but the EC value is much larger. So, in this paper, it is clear that the LSSVM-based traffic flow forecasting model performs better than the RBFNN-based traffic flow forecasting model.

-1 -0.5 0 0.5 1 1.5 2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Fruit fly flying route

X-axis Y-a x is 0 10 20 30 40 50 60 70 80 90 100 5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.75 5.8 5.85 5.9 Optimization process Iteration Number RM SE ⧊a⧋ (b)

Fig.1.(a) The fruit fly swarm flying route for parameter optimization; (b) The iterative RMSE trend of the LSSVM-FOA model searching for optimal parameters.

Table.1. The values of MAPE, MSE, and EC for LSSVM-FOA, single LSSVM, LSSVM-PSO and RBFNN model.

Model LSSVM-FOA LSSVM LSSVM-PSO RBFNN

MAPE (%) 4.9960 7.5715 5.6151 8.4454 MSE 46.8830 73.8215 55.5334 80.6750 EC 0.9720 0.9569 0.9667 0.9521 6:00 7:30 9:00 10:30 12:00 13:30 15:00 16:30 18:00 0 200 400 600 800 1000 1200 1400

Traffic flow forecasting based on LSSVM-FOA model

Sampling time T ra ffi c fl o w

Actual traffic flow Predicted traffic flow

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90

100 Relative error of the LSSVM-FOA model

Time interval/10min R e la ti v e e rr o r/%

(a)Relative error (b) Graph of actual and forecasting flow Fig.2. Results of traffic flow forecasting based on LSSVM-FOA model.

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6:000 7:30 9:00 10:30 12:00 13:30 15:00 16:30 18:00 200 400 600 800 1000 1200 1400

Traffic flow forecasting based on LSSVM model

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 100

Relative error of the LSSVM model

Time interval/10min R e la ti v e e rro r/%

(a)Relative error (b) Graph of actual and forecasting flow Figure.3. Results of traffic flow forecasting based on LSSVM model.

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 100

Relative error of the LSSVM-PSO model

Time interval/10min R e la ti v e e rro r/% 6:000 7:30 9:00 10:30 12:00 13:30 15:00 16:30 18:00 200 400 600 800 1000 1200 1400

Traffic flow forecasting based on LSSVM-PSO model

(a)Relative error (b) Graph of actual and forecasting flow Fig.4. Results of traffic flow forecasting based on LSSVM-PSO model.

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 100

Relative error of the RBFNN model

Time interval/10min R e la ti v e e rro r/% 6:000 7:30 9:00 10:30 12:00 13:30 15:00 16:30 18:00 200 400 600 800 1000 1200 1400

Traffic flow forecasting based on RBFNN model

(a)Relative error (b) Graph of actual and forecasting flow Fig.5. Results of traffic flow forecasting based on RBFNN model.

Fig.2, Fig.3, Fig.4 and Fig.5 show the traffic flow forecasting results and the relative errors with the LSSVM-FOA, LSSVM, LSSVM-PSO and RBFNN model. Fig.2, Fig.3, Fig.4 and Fig.5, it can get the deviations between the actual value and the predicted results of the four models.It can be seen that the forecasting data of Fig.3 fit the actual data well. The gap between the actual and forecasting results throughout the whole domain is small. Therefore, the

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proposed model, the LSSVM combined with FOA (LSSVM-FOA), can be seen to be a very accurate and practical forecasting method for traffic flow forecasting.

In a word, the LSSVM-FOA model proposed in this paper greatly reduces the deviation between the predicted value and the actual value, and better than the single LSSVM, LSSVM-POS and RBFNN model in the traffic flow forecasting.

4. Conclusions

ITS technologies may lead to more efficient use of existing road network systems resulting in reduced traffic congestion, delays, emissions, energy consumption and improved safety. The success of these strategies depends on traffic forecasting for parameters such as traffic flow, travel speed and occupancy. However, the traffic flow forecasting is very complicated because of its nonlinear relationship and its influence factors. Therefore, how to improve the accuracy of traffic flow forecasting is very meaningful. Although the LSSVM has been widely used in various fields, LSSVM is rarely used in the prediction of traffic flow. The fruit fly optimization algorithm (FOA) is a new swarm intelligence algorithm. Compared with other meta-heuristic algorithms, its program code is short, and it can reach the global optimal solution quickly. In this paper, we combine LSSVM and FOA, which is called the LSSVM-FOA model, to study its potential of the traffic flow forecasting. To verify the effectiveness of the method, three other alternative models (single LSSVM, LSSVM-PSO and RBFNN models) were employed. Instance calculation results show that the relative errors of LSSVM-FOA model for traffic flow prediction are acceptable and the MAPE and MSE values are smaller than those of single LSSVM, LSSVM-PSO and RBFNN models. These show that the LSSVM-FOA model has obvious advantages in traffic flow forecasting accuracy. The time of the LSSVM-FOA model is less than LSSVM-PSO model, which shows the advantage of the FOA algorithm to achieve the global optimum quickly. Although the LSSVM-FOA model is more time-consuming than the single LSSVM, this new hybrid forecasting model can greatly improve the traffic flow forecasting precision. It also can be concluded that artificial intelligence prediction model has better performance than the RBNN model and has good nonlinear fitting ability. In short, the LSSVM-FOA model is very suitable for traffic flow forecasting. Hope that it can be applied to the actual traffic flow forecasting in the future.

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