Introduction

In recent years, the phenomenon of vehicle congestion has been occurring more and more frequently, which not only brings trouble to people's travel [1], but also increases the workload of traffic police departments. Intelligent transportation system came into being, and how to predict urban traffic flow quickly and accurately has become a key issue in building an intelligent transportation system. Most of the traditional traffic flow prediction models are highly dependent on experienced experts, lack of independent learning ability and low prediction accuracy, so they have been gradually eliminated. Neural networks are widely used in traffic flow prediction because of their high operational efficiency and independent learning ability [2]. Therefore, this paper starts from the neural network to improve the traffic prediction accuracy as the goal of research.

With the emergence of intelligent optimization algorithms, more and more scholars have improved the prediction ability of network models by combining algorithms with neural networks. In the literature [3], a based LSTM-RF traffic flow prediction model is proposed, which uses Long Short-Term Memory (LSTM) to obtain the temporal characteristics of the target road and combines them with the upstream and downstream data of neighboring road sections to be incorporated into a random forest model to predict traffic flow. The superior memory function of LSTM network solves the problem that Recurrent Neural Network (RNN) is prone to gradient disappearance and gradient explosion, but the training speed is still slow. The literature [4][5][6][7] compared GRU with other prediction models, such as RNN, LSTM and Auto Regressive Integrated Moving Average (ARIMA) statistical technology models, where GRU showed superior prediction results. In the literature [8], the adaptive nonlinear inertial weight particle swarm optimization (APSO) algorithm was proposed, and the APSO-GRU model shows significant stability.

Related work 2.1. GRU Neural Network

To balance the input and forgetting gates in LSTM [9], GRU improves on the LSTM network by adding update gates z to provide memory of past history information [10]. This not only optimizes its internal structure, but also increases its learning speed. The GRU model is defined in the way shown in Equations ( 1)-( 6) [11]. The inputs to the update gate 𝑧 𝑡 and reset gate 𝑟 𝑡 in GRU are 𝑥 𝑡 , ℎ 𝑡−1 , where 𝑥 𝑡 is the t th component of the input sequence 𝑥 and ℎ 𝑡−1 is the hidden state of the previous time step [12]; 𝑊 𝑧 , 𝑈 𝑧 , 𝑊 𝑟 , 𝑈 𝑟 , 𝑊 ℎ , and 𝑈 ℎ are the weight matrices [13]; 𝑏 𝑧 , 𝑏 𝑟 and 𝑏 ℎ are the bias matrices.  is the Sigmoid function. * denotes the corresponding elements in the matrix are multiplied. The GRU always calculates a candidate state before the current hidden state. Finally, the network calculates the final state ℎ 𝑡 at the current moment.

 

1 t z t z t z z U x W h b     (1)

 

1 t r t r t r r U x W h b      (2)     1 tanh * t h t h t t h h U x W r h b     (3)   1 1 * * t t t t t h z h z h    (4)

 

1 x 1 x e    (5)

 

tanh x x x x ee x ee      (6)

Convolutional Neural Network

The traffic flow of a site is not only related to the past traffic flow of that site, but also influenced by the traffic flow of other sites, that is, by the spatial characteristics. CNNs use both local connectivity and shared weights to identify spatially relevant features, which can effectively reduce the complexity and errors such as over-fitting during data processing [14]. Therefore, this paper uses CNN for spatial feature mining of traffic flow. As shown in Figure 1. Its structure consists of an input layer, a convolutional layer, a pooling layer, and a fully connected layer [15]. The convolution layer mainly performs convolution operations on the input data to extract local feature information and obtain the feature matrix. After feature extraction in the convolutional layer, the number of features is not reduced, so it needs to go through the pooling layer to reduce the amount of data processing, and avoid too many parameters in the fully connected layer.

Slime Mould Algorithm

In 2020, Li et al [16] proposed the Slime Mould Algorithm (SMA) based on the oscillatory foraging activity of slime mould. The higher the concentration of food odor in the air, resulting in the accumulation of more slime mould in that domain; when the concentration of food in that range is low, slime mould will be lured to forage in other domains. When the slime mould finds a superior quality food source, they will also separate some individuals to explore better food in other areas.

The equation of slime mould population renewal location is shown in (7). 𝑡 is the number of iterations; 𝑥 𝑏 (𝑡) is the current best position; 𝑥 𝐴 (𝑡) and 𝑥 𝐵 (𝑡) are the positions of two arbitrarily selected individuals. 𝑤 is the slime mould quality, which denotes the fitness weight. 𝑣 𝑏 and 𝑣 𝑐 are the control parameters; 𝑣 𝑏 lies within the range [-a, a]; r and rand are random numbers between [0,1]; 𝑢𝑏 and 𝑙𝑏 represent the upper and the lower bounds of the exploration domain, respectively; and 𝑧 is a custom parameter.

              , 1 , , b b A B c rand ub lb lb rand z x t x t v w x t x t r p v x t r p                   (7)

The mathematical model to describe the control variable 𝑝 and parameter 𝑎 is shown in Equations ( 8) and ( 9). 𝑖∈ 1, 2, 3,…, n; 𝑠(𝑖) is the current individual fitness value; DF is the current best fitness value.

 tanh | | p s i DF  (8)

max arctan 1

t ah t     (9)

The weight parameter 𝑤 is described as follows in Equation (10). 𝑏 𝑓 denotes the best fitness of the current iteration; 𝑤 𝑓 denotes the worst; 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 denotes the top half of individuals in terms of fitness; 𝑒𝑙𝑠𝑒 denotes the remaining individuals; and 𝑆 𝑖𝑛𝑑𝑒𝑥 (𝑖) is the fitness ranking and denotes the odor index.

            1 log 1 , 1 log 1 , f f f f index f f index b s i r condition bw b s i w s i r else bw s i sort s                                        (10)

Proposed methodology 2.4.1. Improved Slime Mould Algorithm

It was found that the synergistic effect of 𝑣 𝑏 and 𝑣 𝑐 causes the slime mould to not only shrink toward the optimal position, but also separate a part of organic matter to explore other fields, and the oscillatory effect of 𝑣 𝑏 increases the possibility of global exploration [17]. But the oscillatory effect of 𝑣 𝑏 weakens significantly in the late iterations, which leads to the fact that the algorithm falls into a local optimum easily [18]. Therefore, the improved slime mould algorithm (ISMA) which improved by Tent chaotic mapping and the adaptive inertia weighting strategy makes the search of slime mould more effective.

Among the many chaotic mapping functions, Tent chaotic mapping is chosen in this paper because of its uniform distribution of chaotic sequences, better traversal uniformity and higher iteration speed [19]. The expressions are shown in (11) and (12). 𝑖 denotes the population size; 𝑗 denotes the chaotic sequence number; 𝑌 𝑖,𝑗 is the chaotic sequence of [0,1]; the initial position of the population is obtained using the 𝑋 𝑖,𝑗 inverse mapping; and [𝑙𝑏, 𝑢𝑏] denotes the search range of the individual position 𝑋 𝑖,𝑗 . ,

.7 1 , 0.7 0.3 ij ij ij ij ij Y Y Y Y Y            , , , ,1 , 0.7 0

, , ( )

i j i j X Y ub lb lb    (12)

It is found that at the beginning of each iteration the slime mould is prone to a large forward state, which leads to a lack of search range and makes the algorithm easily fall into local extrema. We propose the adaptive inertia weight strategy as shown in (13) (14) to help the slime mould jump out of the local extrema. w1 and w2 are weight adjustment parameters. The pseudo-code for ISMA is shown in Table 1.

              , 1 m , , b b A B c rand ub lb lb rand z x t x t v w x t x t r p m v x t r p                     (13) 2 1 1 2 1 2 max max ( )( ) 2( )( ) t t m w w w w w t t     (14)

ISMA-CNN-GRU Prediction Model

We propose to combine CNN and GRU to form a model. CNN mines spatial association features by using local connectivity when processing time series data, while GRU can extract deep features before and after time series data; then the ISMA algorithm is used to solve for the best combination of hyperparameters in the network, and an ISMA-CNN-GRU prediction model is established. The input to the model will be a two-dimensional spatio-temporal matrix constructed using the traffic flow by time as the horizontal coordinate of the matrix and the site ID as the vertical coordinate of the matrix. The spatio-temporal matrix is equation (15). 𝑎 𝑖 ,𝑏 𝑖 ,𝑚 𝑖 denote the traffic flow at different sites.

m m m             (15)

The ISMA-CNN-GRU model framework is illustrated in Figure 2. First, the traffic flow is preprocessed and transformed into a two-dimensional spatial and temporal matrix, and the training set is fed into the convolutional neural network to extract the spatial distribution features of the traffic flow.

After the convolutional operation, the data are transferred to the pooling layer, which reduces the dimensionality of the model to improve the computational speed of the model. The final data are convolved by the convolutional neural network to obtain data samples with spatial characteristics.

The local features extracted by the convolutional neural network are then used as input to the GRU to further extract the temporal features of traffic flow.

Finally, the predicted results are output through the fully connected layer, which is the output layer. The specific steps are as follows.

Step 1: pre-processing of traffic flow data and partitioning of the data set.

Step 2: initialize the structure of the CNN-GRU neural network and initialize the basic parameters.

Step 3: initialize the slime by the Tent chaos mapping.

Step 4: construct a spatio-temporal matrix of the traffic flow based on site and time traffic flows.

Step 5: input the matrix data into the CNN to extract the spatial features of the traffic flow.

Step 6: input the CNN-processed data to the GRU layer.

Step 7: calculate the fitness of the slime mould using the fitness function, retain the best value.

Step 8: calculate the weight parameter w according to equation (10) and generate adaptive inertia weight factors by equation (13) (14) to update the position of the slime mould.

Step 9: calculate and re-ranking the slime adaptation, update the best position and the fitness value.

Step 10: determine whether the algorithm has reached the maximum number of iterations, and if so, end and output the optimal solution, otherwise return to step 6 to continue execution.

Step 11: the hyperparameters optimized by the slime mould algorithm are used as parameters for the CNN-GRU network, and the optimized model is used to train and predict traffic flow.

We choose the convolutional kernel size of 3×3, the number of convolutional layers is 1, the number of convolutional kernels is 32, and the number of pooling layers is 1, using mean pooling. The GRU contains two layers. The activation function is the Relu. The loss function is MSE. Setting the slime population size to 30 and the maximum number of iterations to 10. The adaptive inertia weights 𝑤 1 =0.9, 𝑤 2 =0.5. Using MSE as the fitness function, the formula is shown in ( 16), where n is the number of predicted samples, 𝑌 𝑖 is the sample output value and 𝑦 𝑖 is the actual output value.

n i i i Yy f i MSE n      (16)

ISMA-CNN-GRU Based Traffic Flow Prediction 3.1. Experiment Dataset

The experimental running environment in this section is a 64-bit Windows 10 operating system with an Intel Core i5-7200U CPU @ 2.50GHz and a host installation with 4+8G of RAM. The simulation software used for testing is MATLAB version R2021b. The traffic flow data was taken from the British Motorways Dataset website. As shown in Figure 3, the selected dataset contains information from seven sites on the M25 freeway near Heathrow Airport, UK. The seven sites in the dataset are used as the vertical coordinates of the spatio-temporal matrix, and the horizontal coordinates are the data of each site divided by 15 minutes. In order to observe the prediction effect of the ISMA-CNN-GRU model more intuitively, this subsection takes the flow of site P as an example for prediction. The traffic volumes from August 1 to 25 in summer and September 1 to 25 in autumn were set as the training set, and August 25 to 30 and September 25 to 30 were set as the test set.

Data Pre-processing

In order to facilitate data processing, the commonly used normalization method was chosen to normalize the historical data before training the network, scaling the data to the range of [0,1]. The calculation formula is shown in (17), where 𝑥 is the original data; 𝑥′ is the normalized data; 𝑥 𝑚𝑎𝑥 is the maximum values of the input quantities; and 𝑥 𝑚𝑖𝑛 is the minimum.

Prediction Evaluation Indicators

In order to facilitate us to compare the prediction accuracy of each model more clearly, we choose the commonly used error evaluation metrics of mean absolute error (MAE) [20,21], root mean square error (RMSE) [22], mean absolute percentage error (MAPE) [23] and coefficient of determination (R 2 ) [24,25] to evaluate the prediction effectiveness and goodness of fit of the models. The formulas are shown in ( 18)- (21). n indicates the number of predicted data; 𝑥 ̂𝑖 indicates the predicted data of the i th ; and 𝑥 𝑖 indicates the true data of the corresponding.

1 1 n i i i MAE x x n   (18) 2 1 1 (

)

n i i i RMSE x x n     (19) 1 1 100% n i i i i xx MAPE n x     (20) 2 2 1 2 1 ( ) R 1 ( )n i i i n i i i x x x x       (21)

The metrics used to evaluate the prediction intervals are selected as the prediction intervals coverage probability (PICP) [26], prediction interval normalized average width (PINAW) [27] and coverage width criterion (CWC) [28]. n denotes the total number of samples; 𝑐 𝑖 is the reliability metric of the upper interval 𝑈 𝑖 and the lower interval 𝐿 𝑖 against the true value 𝑦 ̂𝑖;μ is the confidence level and η is the penalty parameter [29]. Here μ = 0.95, η = 20. The formulas are shown in ( 22)-( 26). 100%

n i i PICP c n    (22) 1, [ , ] 0, [ , ]i i i i i i i y L U c y L U       (23) 1 1 100% n i i i i UL PINAW n y     (24) () (1 (

)

) PICP CWC PINAW PICP e    (25) 0, = 1, PICP PICP         (26)

Prediction Results of ISMA-CNN-GRU

ISMA searches for hyperparameter combinations of CNN-GRU networks, using MSE as the judging criterion. The trend of the fitness value of the ISMA iteration process is shown in Figure 4. The fitness value decreases rapidly with the increasing number of iterations and then remains stable, and reaches a steady state with the minimum error at the maximum number of iterations. Finally, the number of neurons in the first hidden layer is L1=88, the number of neurons in the second hidden layer is L2=78, the batch size is Batchsize=27, the learning rate is lr=0.0055, and the number of iterations is K=48. In order to observe the prediction effect of the ISMA-CNN-GRU model more intuitively, this subsection takes the flow of site P as an example for prediction, with a sample size of 960 for the traffic flow from the test set. This experiment compares the prediction results of the ISMA-CNN-GRU model with the APSO-GRU [8], the CNN-GRU and SMA-CNN-GRU models, as shown in Figure 6. The ISMA-CNN-GRU model and SMA-CNN-GRU model perform better than the CNN-GRU model and the APSO-GRU model in prediction performance, and the ISMA-CNN-GRU model can fit the true value curve better. It can be seen that although the prediction results do not show a significant difference between summer and autumn, and the seasonality is not obvious, the ISMA-CNN-GRU still shows high prediction accuracy.

Figure 6: Comparison curves of prediction results of four models on P-site

The comparison results based on the error estimation evaluation metrics are shown in Table 2. The prediction accuracy of the ISMA-CNN-GRU model is 98.4787%, that of the SMA-CNN-GRU model is 98.16326%, and that of the CNN-GRU model is 97.6305%. the ISMA-CNN-GRU model is higher than the SMA-CNN-GRU and CNN-GRU by 0.3161% and 0.8482%, respectively. The prediction error of the APSO-GRU model is relatively large and the fit is poor. In addition, the highest R 2 of ISMA-CNN-GRU on this dataset is 0.9661, which is closer to 1 compared with several other models, which reflects that this model can learn the changing pattern of traffic flow well and has high prediction accuracy and precision, and verifies its feasibility in traffic flow prediction. The experiment was also done based on the interval prediction evaluation index. Figure 7 shows the prediction results of the ISMA-CNN-GRU model at 95% confidence interval. As can be seen from Table 3, the true values of all models fall within the interval when the confidence level is 95%. Because PICP=1 indicates that the true value falls within the constructed interval. The minimum value of CWC for the interval composite evaluation index of ISMA-CNN-GRU model is 0.628. Compared with APSO-GRU, CNN-GRU and SMA-CNN-GRU models, both PINAW and CWC of ISMA-CNN-GRU model have been reduced, and these two indicators fully illustrate the superiority of the ISMA-CNN-GRU model, because when the PICP is determined, the smaller the PINAW and the smaller the value of CWC, the narrower the prediction interval, indicating better prediction.

Conclusion

In recent years, how to forecast traffic flow quickly and accurately has become a key issue in building an intelligent transportation system. We adopt a combination of CNN and GRU network and ISMA algorithm to form a model. CNN is used to obtain the traffic flow distribution characteristics between sites, and then the memory functions of GRU are used to obtain the temporal distribution characteristics of traffic flow. Secondly, for the shortcomings of the slime mould algorithm with low initial population quality, this paper adopts the Tent chaos mapping and adaptive inertia weighting strategy to improve the slime mould algorithm. The hyperparameters of the GRU model are optimized using the proposed ISMA algorithm. The simulation results show that the ISMA-CNN-GRU model exhibits higher prediction accuracy compared with the APSO-GRU, the unoptimized CNN-GRU and the SMA-CNN-GRU models. The accurate prediction of traffic flow by the model in this paper contributes to the operational capacity of the transportation system as well as its operational efficiency, and is of practical value to citizens, traffic management, road operations and infrastructure participants. Future research will be extended in the following. The computational efficiency of the ISMA algorithm in training out the optimal hyperparameters need to be improved. The proposed ISMA-CNN-GRU model only selects the optimal parameter combinations, future research can controlled experiments to study the degree of influence of each parameter on the overall model performance.