A hybrid prediction method for short-duration traffic flow on urban roads based on integrated neural networks

2.1. Analysis of short-term traffic flow forecasting problems

The short-term traffic flow forecasting problem is defined as: forecasting the traffic flow of an observation point in the next time interval within a certain time interval [12]. Short-term traffic flow prediction on urban roads generally uses intervals of 5-30 minutes. the traffic flow of the $i$th observation point in the $t$th time gap is recorded as $f_{i,t}$, and at the gap $T$, the urban road traffic flow sequence is $F=\left\{f_{i,t}\left|i\in D,t=\mathrm{1,2},\cdots ,T\right.\right\}$. Among them, $D$ is all the observation points in the urban road traffic flow prediction area. At this time, the urban road traffic flow prediction task is the predicted gap $T+1$ traffic flow $f_{i,T+1}$ or $T+n$ traffic flow $f_{i,T+n}$. Let the traffic flow on the city roads during a certain time gap be $X$, and the traffic flow on urban roads has the following characteristics:

(1) Continuity: The traffic flow on urban roads exhibiting proximity characteristics is $X_c=(X_{t_0-T_h+3},X_{t_0-T_h+2},X_{t_0-T_h+1},\cdots ,X_{t_0})\in R^{2\times M\times K}$, of which $R$ is a collection of traffic trajectories. That is, the traffic flow of each time gap, often affected by the flow of the previous time gap [13], but also affect the traffic flow of the next time gap. Usually in a day, the most active time periods of urban road traffic flow are morning and evening peaks.

(2) Correlation: urban road traffic flows that exhibit trend characteristics is $X_t=(X_{t_0-T_m+3},X_{t_0-T_m+2},X_{t_0-T_m+1},\cdots ,X_{t_0})\in R^{2\times M\times K}$.

(3) Periodicity: The traffic flow on urban roads showing periodic characteristics is $X_p=(X_{t_0-T_d+3},X_{t_0-T_d+2},X_{t_0-T_d+1},\cdots ,X_{t_0})\in R^{2\times M\times K}$. Periodic urban road traffic flow consists of a sequence of multiple days prior to the predicted time gap that is the same as the predicted time gap [14]. Traffic flow in urban areas is cyclical, i.e., it is approximately the same for each day of the same month with the same time gap.

In order to better take into account the influence of spatial and temporal factors on traffic flow [15], urban area roads were divided into a, based on latitude and longitude $M\times N$ of a grid map, where 1 grid represents 1 region. The inbound and outbound traffic flows of the $m$th row and $k$th column grid($m,k)$ are defined as:

Among them, $T_r:h_1\to h_2\to \cdots \to h_{T_r}$ is one of the trajectories in $R$, the $h_n$ is the coordinates of the geographic location, the $h_i\in (m,k)$ is the point $h_n$ at which in grid$(m,k)$. For each time gap $t$, $M\times N$ area to the traffic inflow $(X_t{)}_{0,m,k}=x_t^{in,m,k}$ and outflow $(X_t{)}_{1,m,k}=x_t^{out,m,k}$ is $X_t$, $X_t\in R^{2\times M\times K}$. According to the inflow and outflow of all grids in the urban road area, the space-temporal matrix $P_t$ is established as follows:

On this basis, a new and fast method for predicting urban road traffic volume is proposed. i.e., the use of integrated neural networks to obtain the corresponding moments of $P_t$ value.

2.2. Integrated neural network construction for short-term traffic flow hybrid prediction on urban roads

Based on the traffic flow characteristics of urban roads analyzed in the previous subsection, according to the traffic flow data with different characteristics, the neural networks suitable for them are used to predict the traffic flow, using the integration strategy to mix multiple neural networks, and designing the structure of the traffic flow fusion prediction model based on the integrated neural network as shown in Fig. 1.

A neural network-based short-term mixed prediction method for urban road traffic flow is proposed. Each neural network is independent yet cooperates with one another. Collect and preprocess urban road traffic data to obtain historical traffic flow data. When short-term traffic flow hybrid prediction of urban roads is conducted, the collected historical data of urban road traffic flow is preprocessed, and the correlation sequence set is constructed according to the characteristics of historical data of urban road traffic flow. On this basis, correlation analysis and comprehensive evaluation of road network traffic at each intersection in the road network are conducted to determine the network structure and initial parameters of each node. Provide urban road traffic flow training samples for each neural network. The local prediction neural network based on the correlation series of urban road traffic flow includes three types: periodic prediction sub network, continuous prediction sub network and correlation prediction sub network. According to the characteristics of various urban traffic flow sequences, the periodic prediction sub network is realized by Elman neural network, the continuity prediction sub network is realized by long short memory neural network, and the correlation prediction sub network is realized by ELM neural network. The local prediction of the prediction sub network is divided into two parts, one is how to train the network, and the other is how to implement the prediction. The sub neural network can run offline and has learning ability. For the convenience of use and management, each neural network will establish a file. The file is composed of two parts, one part contains the structural characteristics of neural network and the significance of input and output units. The other part is the standard sample of neural network learning. In the learning process of neural networks, automatic recognition of the road network is achieved by calling traffic flow information in the network. The learned results, namely weights and weights, are also saved in the form of files to form a knowledge base integrated with the network. Each sub neural network implements the urban road traffic flow forecast under the condition of sound archives and knowledge. The mixed prediction network is used to receive the traffic flow prediction results of each sub neural network, and the integrated algorithm is used to carry out the mixed prediction of the prediction results of each neural network. The integrated neural network includes three predictive neural networks and one hybrid network. Each output in the prediction sub network corresponds to the input of the hybrid network.

Fig. 1Integrated neural network for mixed prediction of short-term traffic flow on urban roads

2.3. Periodic prediction of traffic flow based on Elman neural network

Elman neural network adds a receiving layer in the hidden layer of the feedforward network as a delay operator to achieve the purpose of memory, therefore, neural networks have strong adaptability to time-varying characteristics [16], which can more intuitively reflect the dynamic characteristics of urban road traffic. An artificial neural network consists of an input layer, a hidden layer, a receiving layer, and an output layer. The input layer unit only plays the role of transmitting urban road traffic flow data, while the output layer performs linear weighting on the input signal. Linear or nonlinear functions [17] can be used to transmit hidden layer elements, and the receiving layer is also called the state layer, which is used to remember the output value of the urban road traffic flow at the previous time of the hidden layer unit and return it to the input as a one-step delay operator. The Elman neural network is characterized by the correlation between the output of the hidden layer and the input of the hidden layer during the receiver layer delay and storage process [18]. This self linking mode makes it sensitive to the data of the historical state of urban road traffic flow. The addition of internal feedback network increases the network’s ability to process the dynamic information of urban road traffic flow [19], thus achieving the purpose of dynamic modeling.

The nonlinear state space expression of short-term traffic flow prediction of urban roads based on Elman neural network is shown in Eqs. (4-6):

Among them, $y$, $x$, $q$, $x_c$ respectively represents $m$ dimensional output node vector, $n$ dimensional intermediate layer node unit vectors, $r$ dimensional time series of multisection traffic flow on a subway network processed by phase-space reconstruction techniques and $n$ dimensional feedback state vector. $w^3$, $w^2$ and $w^1$ respectively describe the transition from the intermediate layer to the output layer, from the input layer to the intermediate layer, and from the takeover layer to the intermediate layer. Among them, $g(\cdot )$ is the transfer function of the output neuron, which is formed by the linear composite of the intermediate output signal. Among them, $f(\cdot )$ is the transfer function of the middle layer cells, usually using a $S$ shape. Elman network uses the sum of error squares as the learning index function, and its expression is as follows:

Among them, ${\tilde{y}}_k\left(w\right)$ is the predicted output value of short-term traffic flow on periodic urban roads.

Using traindx as the training function, i.e. momentum, and adopting an adaptive gradient descent method, the stability of the neural network is improved. The gradient decreasing momentum learning function Learngdm is used as the rule learning. Using input and neuron errors to calculate the rate of weight or threshold change, the learning rate and momentum constant of the weight or threshold; The Tansig function and Purelin function were used to analyze the hidden layer nodes and output layer.

2.4. Traffic flow continuity prediction based on long-term short-term memory neural network

The traffic flow is random, and the relevant parameters such as flow and speed are easily affected by the actual road conditions, traffic control and other factors, presenting non-linear characteristics. Long short-term memory neural networks have high accuracy, wide storage space distribution, strong learning ability, strong noise resistance, and good robustness to noisy neural networks, can fully approximate the complex nonlinear relationship [20], and also has the function of associative memory, this method can effectively solve the short-term traffic volume prediction problem of continuous urban road sections. Long short-term memory neural networks (LSTM) are a type of recursive neural network that shares the same characteristics as other neural networks (input layer, loop hidden layer, output layer). The difference between them is the hidden layer, and LSTM consists of a hidden layer composed of storage units. The memory unit is a powerful structure, which determines and stores the short-term traffic flow status of urban roads through the memory unit, mainly including input gate, forgetting gate and output gate. On this basis, using input gate circuits to achieve new input of short-term traffic flow data on urban roads and controlling it; Forgetting gate determines the amount of cell state information discarded at the last moment; The output gate controls which information of the current unit state is exported. Set the short-term traffic flow sequence of urban road input LSTM as $(x_1,x_2,\cdots ,x_T)$, the hidden layer state as $(h_1,h_2,\cdots ,h_T)$. At the time $t$, the calculation process of using LSTM to predict the traffic flow of continuous urban roads is as follows:

where, the $i_t$, $f_t$, $o_t$ represent input gates, forget gates, and output gates, respectively; the $c_t$, $h_t$ respectively represent the output of the memory cells as well as the hidden layer at the moment $t$; the $W_{hi}$, $W_{xi}$ and $W_{ci}$ are the weights of the input gate, implicit layer, and memory cell, respectively; the $W_{xf}$, $W_{hf}$ and $W_{cf}$ the weights of the forgetting gate, the implicit layer, and the memory cell, respectively; the $W_{xc}$ and $W_{hc}$ denote the weight matrices of the memory cells as well as the implicit layer, respectively; the $W_{ox}$, $W_{ho}$ and $W_{co}$ are the weights of the output gate, the implicit layer, and the memory cell, respectively; the $b_i$, $b_f$, $b_o$ and $b_c$ represent the bias of each function, respectively; the $\sigma \left(\right)$ and $\tanh ()$ are all activation functions, the$\cdot$denotes the vector inner product.

The training process of the LSTM model for predicting urban road traffic flow is as follows: Randomize the weight matrix and the offset. In this model, an input vector is introduced to the input layer. Then, calculate the values of the forgetting gate, output gate of the first hidden layer, as well as the input vector and state value of the memory unit. Obtain the output vector of the memory module and take it as the input vector of the next hidden layer. Repeat this process to get the prediction value of continuous urban road traffic flow. According to the predicted value of urban road traffic flow, the error function is calculated, and the weight value of the error back-propagation is updated. The training ends when the maximum iteration number is reached. The trained model is used to predict the final continuous urban road traffic flow prediction results.

2.5. Traffic flow correlation prediction based on ELM neural network

Short term traffic flow is highly time-varying, nonlinear and uncertain. However, for a specific observation point, the longer the observation time is, the more certain and regular the traffic flow statistical behavior will be. Due to the short observation time, short-term traffic flow is greatly affected by random factors. The statistical behavior is often unsteady, aperiodic or quasi periodic, or even random, which makes prediction difficult. The grey model is used to accumulate the short-term traffic flow data, change the short-term traffic flow into the long-term traffic flow, use the characteristics of the long-term traffic flow with obvious regularity and strong certainty to predict the long-term traffic flow, and then restore it to the short-term traffic flow prediction results, so as to improve the prediction accuracy. Grey theory uses limited data to evaluate uncertain behavior, builds first-order differential equation by accumulating original data, and uses exponential curve to fit data to reveal the change trend of traffic flow and predict traffic flow. However, the data series fitted by grey theory is exponential smooth curve, and the prediction will often be distorted when the fluctuation of traffic flow is large. ELM neural network is a feedforward neural network with a single hidden layer. By randomly selecting the weights and thresholds of the hidden layer of the network, the parameter training problem is transformed into the problem of solving incompatible linear equations. Compared with the traditional BP network, it has the characteristics of simple training and fast training speed. In order to solve the distortion phenomenon when the grey model fits the fluctuating data, the original correlation urban road traffic flow data is grey processed, input into the ELM neural network training, and get the relationship between input and output. This paper proposes a method based on least squares support vector, which can effectively solve the problem of linear whitening differential equations.

Let the collected correlated urban road traffic flow dataset be $Q=(q_1,q_2,\cdots ,q_m)$, $m\in N_{+}$, dividing it into $n$ groups, each $M+1$ data group and satisfy $n+M=m$, $n\in N_{+}$, $M\in N_{+}$. The first of the $p$th group of urban road traffic flow data is $X_p^0={\left[q_p,q_{p+1},\cdots ,q_{p+M}\right]}^T$. Data in $X_p^0$ is processed in grey, to obtain one time accumulated sequence of $X_p^1=\left[X_p\left(1\right),X_p\left(2\right),\cdots ,X_p(M+1)\right]$. Among them, $X_p\left(k\right)={\sum }_{j=1}^k{X}_p^0\left(j\right)$. Select the first $M$ term of $X_p^1$ as the input of ELM neural network, and the first item $M+1$ as the expected output of ELM neural network with:

The input matrix set $X$ and the target output matrix set $Y$ of the ELM neural network are formed by the above urban road traffic flow data divided into $n$ groups, thus, the true output matrix $T=\left[T_1,T_2,\cdots ,T_n\right]$ of the network was obtained. $W_{ij}$ is the connection weight between the input layer and the hidden layer, the $W_i$ is the number of neurons in the hidden layer, the $B_i$ is the threshold of a hidden layer of node $i$, the $\beta$ is the weights of the hidden and output layers, among them, ${\beta }_i$ represents the connection weight between the hidden layer $i$ node and the output layer. Set sigmoid function as the hidden excitation function, and its expression is as follows:

When urban road traffic flow inputs is $X_p$, according to the principle of neural network, the expression of the hidden layer $i$ for the inputs of the nodes $ne{t}_i$ is as follows:

The output $s_i$ of the $i$th node of the hidden layer is expressed as follows:

The expression of the urban road traffic flow correlation prediction result $T_p$ output by the network output layer is as follows:

Perform short-term correlation prediction on urban roads through the above ELM neural network operations.

2.6. Integration of neural networks for global hybrid prediction of short-duration traffic flow on urban roads

The outputs of the three neural networks are blended through a global hybrid prediction process using an integrated method based on a parameter matrix. The influence weights of each neural network are different in the hybrid prediction output. The influence weights of different sub-networks need to be taken into account when mixing the outputs of three neural networks for short-term traffic flow prediction on urban roads. Finally, the integrated neural network was used to predict short-term traffic flow on urban roads and simulated:

Among them, $\times$ is Hadamarco. $W_h$, $W_d$ and $W_w$ is the weighting parameter, which reflects the degree of influence of the three sub-neural networks on the short-term traffic flow prediction results of urban roads.

The total loss value of a comprehensive neural network is the average of the predicted value and the true value, and based on this, it is backpropagation to minimize it. The loss function of this integrated neural network is represented as follows:

Among them, $y_t$ is the value of the real traffic flow, the $Y_t$ is the value of predicted traffic flow, the $\theta$ is the learning parameter.

Using the above process, the integration method of parameter matrix is used to mix the short-term traffic flow prediction results of the three neural networks and output the final short-term traffic flow prediction results of urban roads.