DTS-AdapSTNet: an adaptive spatiotemporal neural networks for traffic prediction with multi-graph fusion

Traditional prediciton methods

One common approach to traffic forecasting involves the application of traditional methods, which utilize statistical or machine learning models applied to historical or real-time traffic data (Almeida et al., 2022). Such as linear regression (LR) (Alam, Farid & Rossetti, 2019), Autoregressive Integral Moving Average (ARIMA) (Ahmed & Cook, 1979), as well as ARIMA-based variant models, etc. (Chen et al., 2011; Alghamdi et al., 2019). With the development of machine learning, several typical machine learning-based models have emerged, such as random forest regression (RFR) (Liu & Wu, 2017), support vector machine (SVM) (Toan & Truong, 2021), K-nearest neighbor (KNN) (Sun et al., 2018) and Bayesian Network (AlKheder et al., 2021), etc. However, these methods also face some challenges. Such as dealing with nonlinearity, uncertainty and anomalies in traffic data, requiring a large amount of labeled data, and ignoring the spatial correlations among different locations, etc.

In addition, the accuracy of traffic forecasting is not only determined by the chosen prediction methods, but also on the application and processing of various traffic data (Jeon & Hong, 2016; Nagy & Simon, 2021).

Prediction methods based on trajectory data

In order to improve applicability to specific problems, researchers often seek to evaluate the performance of predictions on real datasets. The utilization of public transportation trajectory data, which offers convenience and wide coverage, has spurred many researchers to explore the application of deep learning techniques combined with real trajectory datasets for predicting (Jiang, 2022). In such approaches, vehicle trajectories (e.g., buses or taxis) are modeled using deep learning techniques to leverage their rich informational content. For example, various deep neural network architectures are employed to learn complex spatiotemporal dependencies between vehicle trajectories and traffic flows.

Initially, some researchers focused on capturing temporal correlations of individual historical traffic data using RNNs, Long short-term memory(LSTM), and Gated Recurrent Units (GRUs) (Lv et al., 2018; Altché & de La Fortelle, 2017; Lu et al., 2020). However, focusing on temporal correlations solely proves insufficient as traffic networks also exhibit complex spatial patterns. Inspired by the success of GNNs and sequence modeling approaches, spatio-temporal GNNs have been introduced to simultaneously capture spatial relationships and temporal dependencies. For instance, Wang et al. (2022a) propose a hierarchical traffic flow prediction method based on spatial–temporal Graph Convolutional Networks (GCNs). This method considers the spatial and temporal dependencies of historical trajectories comprehensively, resulting in more accurate traffic flow prediction. However, most existing spatio-temporal GNNs follow the practice of constructing an adjacency matrix based on predefined measurements, such as spatial distance, functional similarity, or traffic connection, etc. (Geng et al., 2019; Wang et al., 2019). This approach involves learning on a predefined adjacency matrix.

Unfortunately, predefined adjacency matrices may not capture spatial relationships adequately or describe them accurately. Recently, dynamic graph generation modules have been adopted in multi-step traffic prediction widely to learn the spatial relationships dynamically. For example, Djenouri et al. (2023) propose a spatio-temporal GCNs based on graph optimization to predict urban traffic. A dynamic adjacency matrix is used to reflect the spatial relationships in the traffic network. Zhang et al. (2022) also recognize that the traditional end-to-end training methods face challenges in controlling the learning direction of parameters. This limitation results in unclear information from the generated graph and limited improvements in prediction performance. To address this, an alternate training approach is proposed, utilizing historical trajectory flow data of public transportation. The graph learning module and prediction network are combined, enabling the prediction of future traffic flow for a specific road at a specific time. Li et al. (2023) focus on the challenges in integrating and expanding advanced end-to-end spatio-temporal prediction models due to the increasing demands of traffic management and travel planning. To overcome these challenges, they introduce a spatio-temporal pre-training framework that can integrate with downstream baselines to improve performance. This framework includes a spatio-temporal mask auto-encoder with customized parameter learners and a hierarchical spatial pattern encoding network to capture often-neglected spatio-temporal representations and region semantic relationships. Additionally, an adaptive mask strategy is proposed as part of the pre-training mechanism to help the auto-encoder learn robust representations and model different relationships in an easy-to-hard manner.

The above methods can extract the spatio-temporal features in traffic data effectively. However, these methods tend to treat each road as standard grid data or node data, overlooking important details potentially. Consequently, the predicted roads may lack sufficient detail and span a relatively long length. Moreover, these methods rely on the historical trajectories of specific vehicles to predict traffic conditions, which have strong travel characteristics.

Prediction methods based on sensor data

Road sensor data serves as a significant source of traffic data, enabling the representation of all vehicles and providing more detailed traffic information. Consequently, it serves as an ideal dataset for conducting traffic prediction experiments. In recent years, there has been a parallel advancement in both trajectory-based and sensor-based forecasting methods (Emami, Sarvi & Bagloee, 2020). The increasing utilization of sensor data contributes to the development of more general and extensive traffic prediction methods.

Yu, Yin & Zhu (2017) introduce a full convolutional structure aimed at capturing spatio-temporal patterns in urban traffic network. Sun et al. (2020) develop a multi-view GCNs that captures multiple temporal correlations among sensors from different time intervals. Li et al. (2018) reformulate the spatial dependencies of traffic flows as a diffusion process and extend GCNs to directed graphs. Guo et al. (2019) propose an attention-based spatio-temporal GCNs to capture deep spatiotemporal correlations among sensors. Recently, graph learning modules have been used to acquire graph structures (Lee & Rhee, 2022). Wu et al. (2019) combine GCNs with dilated causal convolutional networks to reduce computational costs when processing long sequences. Additionally, an adaptive adjacency matrix is proposed to obtain more reliable spatial correlations among sensors. Considering the highly dynamic nature of urban traffic network, Ta et al. (2022) introduce a dynamic GCNs structure for predicting traffic on urban roads. By updating the adaptive adjacency matrix during the training process continuously, the predicted results are more accurate.

The above methods primarily focus on utilizing sensors on the road network as the research subject, resulting in more detailed predictions and demonstrating good performance in predicting individual sensors. However, they overlook the spatial dependencies among sensors on the same road segment. Specifically, the predicted results tend to prioritize the accuracy of individual sensor predictions rather than predicting traffic for the road segment accurately where the sensors are located. As a result, these methods may not be beneficial for road prediction.

The current forecasting methods often overlook the significance of important spatial dependencies, such as those between sensors on the same road segment, and they provide limited control over the direction of parameter learning during the adaptive learning process. Furthermore, many existing approaches inadequately handle multi-source traffic data, which leads to suboptimal traffic predictions.

To address these limitations, this article proposes a novel traffic prediction framework, DTS-AdapSTNet, which adapts dynamically to spatial relationships within the traffic network. Unlike prior models, our approach generates an adjacency matrix that evolves based on sensor data, learning spatial dependencies between road segments more precisely. By integrating multiple traffic information and utilizing a novel two-stage alternating training structure, the model ensures better control over parameter learning and the prediction process. This enables accurate and adaptive predictions of traffic conditions for specific road segments, overcoming the shortcomings of previous methods.

Motivation

The traffic network exhibits not only complex spatial patterns but also constantly changing spatial states. These variations in traffic conditions occur continuously, with each road segment having its own unique driving direction and being equipped with multiple sensors that record traffic data at various time intervals. To achieve accurate traffic prediction, it is essential to analyze large amounts of historical traffic data collected by these sensors. Additionally, precisely grasping the dynamics of traffic network changes is crucial for learning the spatial dependencies more accurately. This grasp is the key to predicting traffic conditions for each road segment at specific future moments.

The proposed DTS-AdapSTNet framework leverages traffic status data collected from sensors to adaptively learn these spatial dependencies. By integrating GCNs, the model enables accurate predictions for individual road segments. The primary goal is to improve road traffic prediction accuracy by dynamically learning the spatial dependencies among sensors in combination with historical traffic data.

Related definition

Definition 1 (Crossing sensors). Crossing sensors can be defined as sensors located at both ends of a directed road segment. As shown in Fig. 2A, the sensor at the starting position of a road segment is denoted as $S_i^{in}$, while the sensor at the end position of a road segment is denoted as $S_i^{out}$, where i represents the ith road.

Definition 2 (Sensors on the road segments). Sensors on a road can be defined as all other sensors on a road except crossing sensors, represented as $S_{i,j}^{on}$, where i represents the ith road, and j denotes the jth sensor on the ith road.

Definition 3 (Road segments composition). A road segment S_i in the road network can be defined as composed of an inflow crossing sensor $S_i^{in}$, an outflow crossing sensor $S_i^{out}$, and several sensors $S_{i,j}^{on}$ on the segment.

As shown in Fig. 2B, $S_i=\left\{S_i^{in},S_{i,1}^{on},\dots ,S_{i,j}^{on},S_i^{out}\right\}$. S_i belongs to the segment set S, $S_i^{in}$ and $S_i^{out}$ belong to the crossing sensor set S^in−out, $\left\{S_{i,1}^{on},\dots ,S_{i,j}^{on}\right\}$ belong to the non-crossing sensor set S^on.

Definition 4 (Traffic network graph). The traffic network graph can be represented as a weighted directed graph $g=\left\{S,V,W\right\}$ of the road network, where S represents the road segment set in the road network and $\left|S\right|=N_s.V$ represents the set of all sensors and $\left|V\right|=N.W\in {\mathbb{R}}^{N\times N}$ is a weighted adjacency matrix representing the spatial correlations of sensors. In general, when W(i, j) = 0, it indicates no correlation between sensors i and j. However, in this article, some new weighted adjacency matrices among sensors will be defined where this property does not necessarily hold.

Problem formalization

If the graph signal X^(t) represents the historical traffic data observed by each sensor at the t −th moment. Then $\mathcal{X}=\left[X^{\left(t-T^{{}^{\prime }}+1\right)},\dots ,X^{\left(t\right)}\right]$, 𝒳 ∈ ℝ^T′×N×M is used to represent T′ historical graphic signals.

Definition 5 (Traffic prediction problem). The traffic prediction problem aims to use the learned traffic network adjacency matrix A^∗ ∈ ℝ^N×N and $\mathcal{X}$ as the input of the prediction function $g\left(\cdot \right)$, so as to predict T future graph signals $\mathcal{Y}=\left[X^{\left(t+1\right)},\dots ,X^{\left(t+T\right)}\right]$, 𝒴 ∈ ℝ^T×N×M. As shown in Eq. (1), since $\mathcal{Y}$ represents the traffic conditions predicted in the future. In order to obtain the traffic conditions of each road segment $\tau =\left[P^{\left(t+1\right)},\dots ,P^{\left(t+T\right)}\right]$, τ ∈ ℝ^T×N_s×M, $\mathcal{Y}$ needs to be input into the road segment traffic prediction function, which is shown in Eq. (2). (1)${\displaystyle \mathcal{Y}=g\left(\mathcal{X},A^{\ast }\right)}$ (2)${\displaystyle \tau =h\left(\mathcal{Y}\right).}$

In general, the size of X can be ℝ^N×M, where M is the number of features observed by each sensor. Similarly, the size of P can be ℝ^N_S×M, where N_S is the number of road segments on the road network, N_S ≤ N.M is the number of features observed for each road segment. The datasets used in the experiments only include speed feature, i.e., M = 1. However, all results are directly applicable to problems with M > 1. A summary of key notations used in our model is shown in Table 1.

Architecture of DTS-AdapSTNet

The overall framework of DTS-AdapSTNet is shown in Fig. 3, which consists of five steps:

Step 1: DTS Relationship Matrix Generation module (DTSRMG-module). The position and distance data of each sensor on the road network are put into the DTSRMG-module to generate the distance relationship matrix, transfer relationship matrix and same-road relationship matrix, respectively.

Step 2: Adjacency Matrix Predefined module (AMP-module). The three matrices obtained in Step 1 are put into the matrix set A. They are fused through the AMP-module to obtain the predefined adjacency matrix A^∗.

Step 3: Prediction module based on improved loss function. The relationship matrix A^∗ is utilized as the optimal input for the prediction module, while incorporating the historical traffic data to train the prediction network simultaneously. A well-designed loss function is employed to facilitate optimization. The parameter θ^∗ that maximizes the likelihood estimation of the prediction model’s excellence is determined under the condition of the current relationship matrix A^∗.

Step 4: Adaptive Matrix Generation module (AMG-module). The parameter value of the prediction network in the AMG-module is designated as θ^∗, which is obtained in Step 3 and remains fixed. The same prediction network is utilized for training, thereby allowing the AMG-module to generate the relationship matrix repeatedly. As a result, a new relationship matrix M_new that improves the prediction outcome under the current parameter θ^∗ can be obtained. The spatial dependencies among sensors are thus reweighted by this matrix, leading to more accurate predictions ultimately.

Step 5: Multi-matrix Fusion module (MF-module). The M_new generated in Step 4 is incorporated into the matrix set A of the MF-module, which utilizes the matrices generated in Step 1 as its elements. Through the MF-module, weight calculation and distribution are performed on all matrices in the matrix set. Subsequently, the optimal adjacency matrix is obtained by fusing with the new weights, and it replaces the original A^∗.

By repeating steps 3-5, a relatively better spatial adjacency matrix and a relatively better road segment traffic prediction model can be obtained as the preset maximum number of training iterations is reached. The pseudocode for DTS-AdapSTNet is described in Algorithm 1 .

 
 
_____________________________________________________________________________________ 
 Algorithm 1: The implementation of DTS-AdapSTNet 
    Input: original road network information R, parameter set P, constant 
               set C, iterations I 
    Output: a model with good prediction performance DTS-AdapSTNet, 
                  the optimal adjacency matrix A∗ 
 1  Execute Procedure 1 DTS Relationship Matrix Generation: 
      WD,WT,WS ← Generate three adjacency matrix according to 
      different dependencies ; 
  2  Execute Procedure 2 Generation of initial input matrix: 
      A∗ ← Fusing three adjacency matrices for initialization; 
  3  for i to I do 
     4   Execute Procedure 3 Traffic condition prediction: 
    θ ← Get the best training parameter through training; 
5   Execute Procedure 4 Adaptive Matrix Generation: 
    Anew ← Get a new matrix through training; 
6   Execute Procedure 5 Multi-matrix Fusion: 
    A∗ ← Get the next iteration of input matrix through filtering and 
    fusion; 
  7  end 
 8  return A∗ 
_____________________________________________________________________________________    

DTS relationship matrix generation module

Traffic prediction is of great importance in intelligent transportation systems, but accurate road traffic prediction faces challenges due to the complex spatiotemporal dependencies in the traffic network. Existing methods have shortcomings in capturing spatial dependencies. They often consider only a single spatial relationship while ignoring crucial factors such as road characteristics and vehicle flow. This results in inaccurate descriptions of spatial relationships and difficulty in extracting comprehensive spatial dependencies, which affects the accuracy of traffic prediction (Shuman et al., 2013). In order to capture the spatial correlations in the traffic network more accurately and enhance the accuracy of road segment traffic prediction, multigraphs are employed in this module to conceptualize the complex traffic network. Three traffic network graphs are defined, namely Distance relationship, Transfer relationship, and Same-road relationship. Combining these three spatial relationships can better reflect the spatial connection between sensors and improve the accuracy of prediction results.

Distance relationship

A certain correlation is observed among sensors, with the strength of correlation being stronger among sensors that are relatively close to each other. To describe this relationships, the measure of distance among sensors is utilized. The distance between two sensors i and j in the road network, denoted as dist(i, j), is considered as the shortest distance when one or more paths exist between them. This is illustrated in Fig. 4A.

Definition 6 (Distance relationship matrix W_D). The distance relationship matrix W_D is defined by Laplace kernel function (Paclık et al., 2000), which is shown in Eq. (3). (3)${\displaystyle W_D\left(i,j\right)=\left\{\begin{array}{c}\hfill e^{\left(-\parallel dist\left(i,j\right)\parallel /{\theta }_1\right)},i\ne j\text{and}dist\left(i,j\right)>k\hfill \\ \hfill 0,\text{otherwise}\hfill \end{array}\right.}$ where θ₁ is a fixed parameter and k is a threshold. Although the Gaussian kernel function has become the standard for most distance modeling, in theory, a deep neural network can model any function according to the general approximation theorem. Additionally, since the provided experimental data includes the distance among sensors, the Laplacian kernel function is used to calculate W_D. By comparing the result calculated by Gaussian kernel function, it can be observed that the former ensures greater accuracy in the experimental outcome.

 
_____________________________________________________________________________________ 
 Algorithm 1: Procedure 1: DTS Relationship Matrix Generation 
    Input: original road network information R, hyper-parameters 
               k,θ1,θ2,p,q,Tsim, weight set W 
    Output: relationship matrix WD,WT,WS 
  1  Initialize WD,WT,WS,dist as zeros matrixs, M as a empty set; 
  2  // Method of generating WD 
  3  dist ← Get all distance values among sensors from R; 
  4  for dist(i,j) in dist do 
     5   if dist(i,j) > k then 
    6   WD ← e(−∥dist(i,j)∥/θ1); 
7   else 
    8   WD ← 0; 
9   end 
10  end 
11  // Method of generating WT 
12  NC ← Get all number of crossing sensors from R; 
13  G ← Create a graph based on the crossing sensors information from R; 
14  Enode ← Get vector representation of crossing sensors based on 
      Node2vec(G,p,q); 
15  for i in NC do 
     16   for j in NC do 
    17   WT(i,j) ← Enode(i) ⋅ Enode(j)/(∥Enode(i)∥∥Enode(j)∥); 
18   if WT(i,j) ≥ Tsim then 
19   WT(i,j) ← 1; 
20   else 
21   WT(i,j) ← 0; 
22   end 
23   end 
24  end 
25  // Method of generating WS 
26  WS ← Get the distance relationship matrix of each sensor on each road 
      segment; 
27  M ← Get a set of cluster center values by conducting cluster analysis 
      on WS; 
28  for ri,i in M, size(M) do 
     29   Gi(WS) ← e((WS−ri)2/2θ2 
2); 
30   Gall ← Gall + Gi(WS); 
31  end 
32  for ri,wi,i in M,W, size(M) do 
     33   fi(WS) ← Gi(WS)/Gall; 
34   WS ← WS + fi(WS) ⋅ wi; 
35  end 
36  return WD,WT,WS 
_____________________________________________________________________________________     

Adjacency matrix predefined module

The performance of the prediction module and the AMG-module significantly depends on the quality of input matrix’s initialization. Inaccurate or unreliable initial associations among sensors can lead to suboptimal optimization of the prediction module, subsequently compromising the overall performance of the entire model. Therefore, it is essential to ensure the accuracy and reliability of the initial input matrix to enhance the model’s predictive capability and overall performance.

Therefore, this article does not use a single relationship matrix as the initial spatial relationship matrix, nor does it generate the initial matrix randomly. Instead, multiple matrices generated by the DTSRMG-module are used to construct the initial matrix, which includes different types of spatial dependencies. The initial input matrix A can be initialized by Eqs. (12)–(13). (12)${\displaystyle \text{i}.A_D={\tilde{D}}_D^{-\frac{1}{2}}{\tilde{W}}_D{\tilde{D}}_D^{-\frac{1}{2}}\text{ii}.A_T={\tilde{D}}_T^{-\frac{1}{2}}{\tilde{W}}_T{\tilde{D}}_T^{-\frac{1}{2}}\text{iii}.A_S={\tilde{D}}_S^{-\frac{1}{2}}{\tilde{W}}_S{\tilde{D}}_S^{-\frac{1}{2}}}$ (13)${\displaystyle A\left(i,j\right)=\frac{A_D\left(i,j\right)+A_T\left(i,j\right)+A_S\left(i,j\right)}{b\left[A_D\left(i,j\right)\right]+b\left[A_T\left(i,j\right)\right]+b\left[A_S\left(i,j\right)\right]}}$

where ${\tilde{W}}_k=W_k+I_N$, ${\tilde{D}}_k^{\left(i,j\right)}={\sum }_j^{}{\tilde{W}}_k^{\left(i,j\right)}$, k represents D, T and S. Equation (12) is a renormalization technique proposed by Kipf & Welling (2019), which ensures the comparability of different matrices. b[⋅] represents a binarization function, which is defined by Eq. (14). The specific process of initialization is shown in Algorithm 2 . (14)${\displaystyle b\left[x\right]=\left\{\begin{array}{ccc}\hfill 1\hfill & \hfill if\hfill & \hfill x\ne 0\hfill \\ \hfill 0\hfill & \hfill else\hfill \end{array}.\right.}$

 
__________________________________________________________________________________________________________ 
 
  Algorithm 2: Procedure 2: Generation of initial input matrix 
    Input: WD,WT,WS, a binaryzation function b[⋅] 
    Output: predefined adjacency matrix A∗ 
 1  Initialize W, Aset as a empty set, IN as a identity matrix of size 
      N × N, A as a zero matrix of size N × N; 
  2  Push WD,WT,WS into W; 
  3  for Wk in W do 
     4    ~ Wk ← Wk + IN; 
5   for i to N do 
    6    ~ D(i,j) 
k    ← Add all the columns corresponding to row i of  ~ Wk; 
7   end 
8   Ak ← ~ D− 1 
2 
k    ~ Wk ~ D− 1 
2 
k    ; 
9   Push Ak into Aset ; 
10  end 
11  AD,AT,AS ← Aset[0],Aset[1],Aset[2]; 
12  for i to N do 
     13   for j to N do 
    14   A∗(i,j) ← (AD(i,j) + AT(i,j) + AS(i,j))/(b[AD(i,j)] + 
b[AT(i,j)] + b[AS(i,j)]); 
15   end 
16  end 
17  return A∗ 
_____________________________________________________________________________________    

Prediction module based on improved loss function

Graph convolution-based spatio-temporal neural network is an approach for traffic prediction. It takes one or more adjacency matrices and historical time series data as input. The aim is to capture the spatio-temporal features that are hidden in the historical data, which is shown in Fig. 5A, The network consists of several spatio-temporal blocks typically. Each block captures the spatial dependencies of sensors in the road network after information aggregation using a GCN model. Simultaneously, a temporal attention module or GRUs is combined to capture long sequence dependencies and obtain temporal dependencies of traffic data.

For all the models used in this article, each layer of the GCN employs spectral graph convolution based on the Chebyshev polynomial approximation (ChebConv). As shown in Fig. 5B, it is able to capture the spatial dependencies, which can be expressed by Eq. (15). (15)${\displaystyle H^{\left(l\right)}=ChebConv\left(A^{\ast },H^{\left(l-1\right)},{\theta }^{\left(l\right)}\right)=\sigma \left(\sum _{k=0}^K{T}_k\left(A^{\ast }\right)H^{\left(l-1\right)}{\theta }^{\left(l\right)}\right)}$ where σ is the activation function, and T_k(x) represents the recursively defined Chebyshev polynomial given by T_k(x) = 2xT_k−1(x) − T_k−2(x), where T₀(x) = 1, T₁(x) = x. $A^{\ast }={\tilde{D}}^{-\frac{1}{2}}\tilde{A}{\tilde{D}}^{-\frac{1}{2}}$ is the normalized adjacency matrix, where $\tilde{A}=A+I$. $\tilde{D}$ is a diagonal matrix, ${\tilde{D}}_{ii}={\sum }_j^{}{\tilde{A}}_{ij}$. H^(l) represents the hidden features of layer l obtained after the convolution operation, and θ^(l) represents the learnable parameter corresponding to the l-th layer structure.

For the training of the previous prediction modules, the focus is usually on designing a loss function to enhance the accuracy of predicting the value of individual sensors, rather than prioritizing the accurate prediction of traffic conditions on the road segment where the sensor is located. This approach may result in less accurate predictions. This article aims to predict the future traffic conditions of each road segment. Instead of calculating the loss function between the real value Y and the predicted value Pred of a single sensor directly, some improvements should be made to ensure that the designed loss function optimizes the model’s performance in predicting road traffic conditions.

Firstly, a matrix W_r of size N × S should be generated. This matrix is obtained by considering the distribution of sensors in the road segments. The number of sensors in the network is denoted as N, while the number of road segments is represented by S. The matrix W_r can be defined by Eq. (16). (16)${\displaystyle W_r\left(i,j\right)=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill V_{i}locateson{S}_j\hfill \\ \hfill 0\hfill & \hfill otherwise\hfill \end{array}\right.}$ where V_i ∈ N represents the ith sensor, and S_j ∈ S represents the jth road. If the ith sensor locates on the jth road, the corresponding matrix value is set to 1, otherwise it is 0. This grouping allows sensors to be categorized according to the road they belong to.

Secondly, the ground truth Y and predicted value Pred are processed. As shown in Eqs. (17)–(18), the actual value Y_road of each road segment and the predicted value Pred_road are obtained. (17)${\displaystyle Y_{road}=Y\cdot W_r}$ (18)${\displaystyle Pre{d}_{road}=Pred\cdot W_r.}$

An array C of length S is calculated to record the number of sensors for each road segment. Additional processing is conducted on Y_road and Pred_road. The average real value Y_road and the average predicted value Pred_road are obtained by Eqs. (19)–(20). (19)${\displaystyle Y_{road}=Y_{road}/C}$ (20)${\displaystyle Pre{d}_{road}=Pre{d}_{road}/C.}$

Finally, the minimization of the L1 loss between the predicted value of the road segments and the real value is selected as the training goal of the prediction module, which is shown in Eq. (21). (21)${\displaystyle L_p\left(Y_{road},Pre{d}_{road}\right)=\left|Y_{road}-Pre{d}_{road}\right|.}$

Through such a loss function, after several iterations, correlations that are favorable for road segment traffic prediction are highlighted, while weaker correlations are erased gradually. The prediction modules of the experiments are all implemented based on several GCNs models with good prediction effects. The specific process is shown in Algorithm 3 .

 
_____________________________________________________________________________________ 
 
 Algorithm 3: Procedure 3: Traffic condition prediction 
    Input: time series data of traffic feature X, target data Y , normalized 
               adjacent matrix A∗, sensor distribution matrix Wr, array C, 
               GCN-based spatiotemporal neural networks for traffic 
               prediction STnet(⋅), number of epoch Nepoch. 
    Output: well-trained parameter θ of the prediction module 
  1  for i to Nepoch do 
     2   pred ← STnet(X,A∗,θ) ; 
3   Yroad ← Y ⋅ Wr/C ; 
4   predroad ← pred ⋅ Wr/C ; 
5   Compute Lp(predroad,Yroad) ; 
6   Update model parameters θ through Adam; 
  7  end 
 8  return θ 
_____________________________________________________________________________________    

Adaptive matrix generation module

In the context of traffic prediction, accurately representing the spatial dependencies among sensors is crucial for obtaining accurate predictions. However, traditional end-to-end training methods often face challenges in determining the training direction of learnable parameters, resulting in unclear information from the generated graph and limited improvements in prediction performance.

To address these limitations, the AMG-module is introduced. It can generate matrices based on the DTSRMG-module, which provides initial spatial dependencies. By utilizing the training results of the prediction network and incorporating prior knowledge in the iterative process, the AMG-module can continuously generate new spatial dependency matrices. This allows the matrices to better capture and represent the stronger dependencies among sensors, thereby improving the accuracy of the model’s predictions.

Firstly, an initial relationship matrix is generated according to the cosine similarity by Eqs. (22)–(23). (22)${\displaystyle sim\left(i,j\right)=\frac{v_i\cdot v_j}{∥v_i∥∥v_j∥}}$ (23)${\displaystyle A_{ij}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill ifsim\left(i,j\right)>0\hfill \\ \hfill 0,\hfill & \hfill otherwise\hfill \end{array}\right.}$ where v_i, v_j are the eigenvectors of sensors i and j, respectively. $∥\cdot ∥$ represents the modulus of the vector. A_ij represents the degree of correlation between sensors i and j. Then a learnable matrix A₁ is introduced and combined with the initial relationship matrix A by Eq. (24) to obtain the initial learnable matrix M_init. (24)${\displaystyle M_{init}=ReLU\left(A+A_1\right).}$ A₁ ∈ ℝ^N×N are learnable parameters. In order to enhance the sparsity of M_init, the activation function ReLU can set the diagonal position of the matrix and the other half positions to 0.

Secondly, an attention mechanism is used to fuse the old matrices with the newly generated spatial dependency matrix. Then, an attention weight α_ij between each pair of nodes can be obtained by Eq. (25). (25)${\displaystyle {\alpha }_{ij}=softmax\left(LeakyReLU\left(W\cdot \left[v_i\left|\right|v_j\right]\right)\right)}$ where W is a learnable weight matrix, v_i and v_j are eigenvectors of nodes i and j respectively, $\left|\right|$ represents vector concatenation operation, LeakyReLU is a linear rectification function with leakage, and the softmax function is used to normalize attention weights. The fusion matrix M_fs is obtained according to the attention weights α_ij by Eq. (26). (26)${\displaystyle M_{fs}\left(i,j\right)={\alpha }_{ij}\cdot M_{init}+\left(1-{\alpha }_{ij}\right)\cdot M_{old}}$ where M_old is the old spatial dependency matrix.

Finally, to facilitate calculation, the fused matrix is adjusted. In this process, certain elements with relatively small values are filtered out, and control is exerted through the utilization of a hyperparameter σ. Furthermore, by employing the renormalization technique (Kipf & Welling, 2019), a newly generated matrix M_new is obtained by Eqs. (27)–(28). (27)${\displaystyle M_{fs}^{{}^{\prime }}=ReLU\left(D_{fs}^{-\frac{1}{2}}{M}_{fs}{D}_{fs}^{-\frac{1}{2}}-\sigma \right)}$ (28)${\displaystyle M_{new}={\left(D_{fs}^{{}^{\prime }}\right)}^{-\frac{1}{2}}{M}_{fs}^{{}^{\prime }}{\left(D_{fs}^{{}^{\prime }}\right)}^{-\frac{1}{2}}}$ where D_fs, $D_{fs}^{{}^{\prime }}$ are diagonal matrices, $D_{fs}^{\left(i,j\right)}={\sum }_{j=1}^N{M}_{fs}^{\left(i,j\right)},{\left(D_{fs}^{{}^{\prime }}\right)}^{\left(i,j\right)}={\sum }_{j=1}^N{\left(M_{fs}^{{}^{\prime }}\right)}^{\left(i,j\right)}.\sigma \in \left(0,1\right)$ is a custom threshold, the specific process is shown in Algorithm 4 .

 
________________________________________________________________________________________ 
 
  Algorithm 4: Procedure 4: Adaptive Matrix Generation 
    Input: time series data of traffic feature X, target data Y , normalized 
               adjacent matrix A∗, sensor distribution matrixa Wr, array C, 
               GCN-based spatiotemporal neural networks for traffic 
               prediction STnet(⋅), the set of eigenvector of each node v, the 
               parameters A1 ∈ RN, W, σ, number of epoch Nepoch 
     Output: adaptive generator matrix Mnew 
  1  Mold ← A∗; 
  2  for i to Nepoch do 
     3   Mnew ← Generate a new matrix G(Mold,v,A1,W,σ) ; 
4   pred ← STnet(X,Mnew,θ) ; 
5   Yroad ← Y ⋅ Wr/C ; 
6   predroad ← pred ⋅ Wr/C ; 
7   Compute Lp(predroad,Yroad) ; 
8   Fix θ, update model parameters A1,W,σ through Adam ; 
  9  end 
10  return Mnew 
_____________________________________________________________________________________    

Multi-matrix fusion module

After the AMG-module, a new adjacency matrix is generated. This adjacency matrix is then input into the MF-module. For the N_m matrices in this module, they are fused to generate the input matrix A^∗ for the next iteration of the prediction module. The main task of this module is to weight the importance of different subgraphs through the prediction loss of the prediction model. By doing so, it can identify the most relevant and useful subgraphs for the prediction task. Subsequently, these weights are used to fuse different subgraphs to obtain the input A^∗ for the next iteration of training. This ensures that the model can make full use of the information from different subgraphs, enhancing the accuracy and reliability of the prediction.

Firstly, the prediction module is utilized to calculate the corresponding prediction loss for all matrices in the matrix set A, which is shown in Eq. (29). (29)${\displaystyle L_k=L_p\left(STnet\left(X,A_k,\theta \right)\cdot W_r/C,Y_{road}\right)}$ where STnet(⋅) is the prediction network. θ is the parameter that makes the best prediction currently. k refers to the k-th submatrix, W_r and C are the matrix and array mentioned in Prediction module. The predicted and actual values of the road are calculated to obtain the predicted loss of the k-th submatrix. These prediction loss values are combined into a vector $\overrightarrow{l}={\left(L_1,L_2,\dots ,L_m\right)}^T$, and then the maximum value is take as $L_{max}=max\left(\overrightarrow{l}\right)$. Then, it is necessary to judge whether the current matrix number m is greater than the maximum capacity N_max. If it exceeds, it is necessary to remove the matrix with the largest predicted loss value in $\overrightarrow{l}$, which is shown in Eq. (30). Additionally, the maximum value L_max is updated. (30)${\displaystyle \overrightarrow{l}=remove\left(L_{max}\right).}$

 
__________________________________________________________________________________________________________ 
 
  Algorithm 5: Procedure 5: Multi-matrix Fusion 
    Input: time series data of traffic feature X, target data Y , matrix set 
               A, optimized parameter θ, the capacity Nmax of A 
    Output: fusion matrix A∗ 
 1  m ← Get the number of items of A ; 
  2  for i to m do 
     3   Lk ← Calculate the predicted loss value for each submatrix ; 
4   push Lk into ~ l; 
  5  end 
 6  if m > Nmax then 
     7   Remove the maximum predicted loss value from ~ l; 
8   Remove the corresponding matrix from A ; 
9   m ← Get the number of items of A ; 
10  end 
11  Lmean ← mean(~l ) ; 
12  w ← g(~l− Lmean) ; 
13  for i to m do 
     14   A∗ ← A∗ + wiAi ; 
15  end 
16  return A∗ 
_____________________________________________________________________________________    

Secondly, the weight vector $\overrightarrow{w}$ of all matrices remaining in A is calculated by Eq. (31). (31)${\displaystyle \overrightarrow{w}=g\left(\overrightarrow{l}-mean\left(\overrightarrow{l}\right)\right)}$ where $mean\left(\overrightarrow{l}\right)$ represents the average of the prediction loss of all matrices. g represents the normalization function, which is defined by Eq. (32). (32)${\displaystyle g\left(\overrightarrow{x}\right)=\left|\frac{e^{\overrightarrow{x}}}{\sum _{i=1}^m{e}^{\overrightarrow{x}}}.\right|}$

Finally, the input matrix of the next iteration can be obtained through the MF-module, which is shown in Eq. (33). (33)${\displaystyle A^{\ast }=\sum _{k=1}^m{w}_k{A}_k}$

The same normalization is performed on A^∗ with Eqs. (12)–(13). The specific process of MF-module is shown in Algorithm 5 .

Datasets

In order to verify the performance of the model, experiments are conducted on two public datasets, METR-LA and PEMS-BAY. The detailed statistics of the two datasets are shown in Table 2.

METR-LA: This traffic dataset contains traffic information collected from loop detectors in the highway of Los Angeles County (Gehrke et al., 2014). Since there are a large number of missing values in this dataset and to facilitate the experiment, traffic speed statistics data from March 1, 2012, to April 30, 2012, encompassing 207 sensors along Los Angeles County highways is utilized to mitigate the impact of missing values on experimental results. Moreover, missing values are handled.

PEMS-BAY: This traffic dataset is collected by California Transportation Agencies (CalTrans) Performance Measurement System (PeMS). We selected six months of traffic speed statistics from January 1, 2017 to June 30, 2017, covering 325 sensors in the Bay Area.

The same data pre-processing procedures are adopted as Li et al. (2018). Observations from the sensors are aggregated into 5-minute windows. The original spatial dependencies such as road distance and direction are used as the input of the DTSRMG-module to generate corresponding relationship matrices. The input data are normalized using Z-score. Both datasets are split by time sequence with 70%, 15% and 15% for training, validation and testing, respectively.

Baseline

Three predictive neural networks based on GCN with good performance are employed as prediction models for experiments. The structures of these three GCN-based frameworks are shown in Fig. 6.

• ASTGCN (Guo et al., 2019): Attention Based Spatio-Temporal Graph Convolutional Network, which combines graph convolutions with spatial attention to capture spatial patterns, and leverages standard convolutions and temporal attention to extract temporal features.

• TGCN (Zhao et al., 2020): Temporal Graph Convolutional Network model, which utilizes recurrent models to extract temporal features and graph convolutions to capture spatial dependencies, respectively.

• DCRNN (Li et al., 2018): Diffused Convolutional Recurrent Neural Network, which replaces matrix multiplication in recurrent models (i.e., GRU and LSTM) with graph convolutions, and extracts spatial and temporal features in an encoder–decoder manner simultaneously.

The above three models also serve as baseline models, along with the following ones, including both classic methods and state-of-the-art approaches:

• HA: Historical average, which uses the average of historical traffic flow data to complete the task.

• STGCN (Yu, Yin & Zhu, 2017): Spatio-Temporal Graph Convolutional Network, which combines 1D convolution and graph convolution.

• Graph WaveNet (Wu et al., 2019): A convolutional network architecture, which introduces adaptive graphs to capture hidden spatial dependencies and uses dilated convolutions to capture temporal dependencies.

• DCRNN+AdaGL (Zhang et al., 2022): An Adaptive Graph Learning Algorithm for Traffic Prediction Based on Spatiotemporal Neural Networks, which combines the proposed adaptive graph learning module with a DCRNN to find the adjacency matrix relations that makes traffic prediction work well.

Evaluation metrics

Three common metrics for traffic prediction are employed to evaluate the performance of different models, including MAE, RMSE and MAPE. Their definitions are shown in Eqs. (34)–(36).

• MAE: Mean absolute error measures the average magnitude of errors in the predictions without considering their direction. This makes it a straightforward metric for evaluating the accuracy of predictions in real-world applications like traffic prediction, where absolute deviations matter. (34)${\displaystyle \mathrm{MAE}\left(\mathrm{Y},\widehat{\mathrm{Y}}\right)=\frac{1}{\left|\Omega \right|}\sum _{i\in \Omega }^{}|y_i-{\widehat{y}}_i|}$

• RMSE: Root mean square error is particularly sensitive to larger errors, which can be critical in traffic forecasting scenarios where large deviations from actual values could have significant real-world consequences (e.g., incorrect predictions during peak traffic hours). (35)${\displaystyle \mathrm{RMSE}\left(\mathrm{Y},\widehat{\mathrm{Y}}\right)=\sqrt{\frac{1}{\left|\Omega \right|}\sum _{i\in \Omega }^{}{\left(y_i-{\widehat{y}}_i\right)}^2}.}$

• MAPE: Mean absolute percentage error provides a normalized error measurement, making it useful when comparing prediction performance across datasets with different scales (e.g., high-traffic vs. low-traffic areas). (36)${\displaystyle \mathrm{MAPE}\left(\mathrm{Y},\widehat{\mathrm{Y}}\right)=\frac{1}{\left|\Omega \right|}\sum _{i\in \Omega }^{}\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|\times 100\text{\%}.}$

Experiment settings

In all experiments, the traffic speed is predicted over the next hour using the traffic speed from the previous hour, hence T = 12. The parameters θ, k, and T_sim used in generating the three relationship matrices W_D, W_T, W_S are adjusted according to the scale of the data. For the prediction module, the number of training epochs N_epoch is set to approximately 10 based on the convergence rate of the prediction module. In the AMG-module, the dimensions of A₁ and A₂ are configured as 64, and σ is selected from the range [0,1]. Regarding the MF-module, the size of the matrix set A is set to 3.

All experiments are executed under a platform with NVIDIA GeForce RTX3080-10GB graphics card. For all deep learning based models, the training process is implemented in Python with Pytorch 1.8.0. Adam (Wang, Xiao & Cao, 2022) is utilized as the optimization method with a learning rate of 0.001. Additionally, an early stopping strategy is employed to determine whether the stopping criterion is met. Furthermore, the optimal parameters are determined based on the performance on the validation dataset.

Performance analysis

Table 3 presents the performance comparison of different baselines and DTS-AdapSTNet on the two datasets for road speed prediction in the next 15 min, 30 min and 60 min, respectively. The prediction networks, which serve as the foundations for DTS-AdapSTNet, are denoted within brackets (e.g., DTS-AdapSTNet (ASTGCN) refers to using ASTGCN as a prediction network). The results demonstrate that DTS-AdapSTNet achieves excellent results in all metrics across the prediction range. Superior performance is observed for DTS-AdapSTNet based on three different GCN prediction networks compared to the other baselines, for both the METR-LA and PEMS-BAY datasets. Particularly, DTS-AdapSTNet based on DCRNN exhibits the best performance. When compared with the traditional model HA for a 15-minute prediction on both datasets, it is found that MAE, RMSE, and MAPE are reduced by 43%, 39%, and 54%, respectively. This demonstrates the significance of considering spatial dependencies among sensors. Other spatio-temporal GCN models are also compared, revealing an average reduction in prediction error for each metric by 10%, 6% and 13%, respectively. This reduction can be attributed to the accuracy of the learned adjacency matrix, which has a relatively large impact on prediction results. It is evident from the results presented in Table 3 that as the prediction window increases, the accuracy of prediction decreases for each model. However, DTS-AdapSTNet outperforms other models consistently, with a more noticeable improvement.

For the MAE, when the prediction window is 15 min, the DTS-AdapSTNet (DCRNN) model exhibits a 24% lower error compared to the ST-GCN model. With a prediction window of 30 min, the error is reduced by 26%. Furthermore, with a prediction window of 60 min, the error is reduced by 32%. This showcases the increasing complexity of the traffic network and the growing difficulty in prediction as the prediction range expands. However, it is noteworthy that DTS-AdapSTNet is able to outperform in long-term forecasting as well, highlighting the stability of the method employed in this study.

When compared with the AdapGL+DCRNN model, the method in this article has different prediction focuses and specific implementation methods. In different GCN prediction networks, the performance of the DTS-AdapSTNet model has been improved. The most obvious improvement is observed in DTS-AdapSTNet (DCRNN). Particularly noteworthy is the significant decline observed in all three metrics for the METR-LA dataset. Similarly, for the PEMS-BAY dataset, DTS-AdapSTNet continues to remain competitive or even show further enhancements compared to the AdapGL+DCRNN model.

In order to provide a more intuitive comparison of the performance of DTS-AdapSTNet and other models with superior performance under different prediction windows, a line chart illustrating the changes in each evaluation metric corresponding to different models under various prediction windows is depicted in Fig. 7. It can be observed that DTS-AdapSTNet based on different GCNs demonstrates relatively good performance in each prediction range on the two datasets. Specifically, on the METR-LA dataset, as the prediction window increases, the MAE, RMSE and MAPE of the three models proposed in this article increase among 0.32−2.15, 1.00−5.26 and 1.07−7.60, respectively. The corresponding increases of the other three baseline models are 0.32−2.21, 0.98−5.45 and 1.07−7.63, respectively. It can be observed that the error rise of the DTS-AdapSTNet model changes less as the prediction window increases, illustrating the effectiveness of the proposed model in long-term prediction further.

In order to compare the prediction effectiveness of DTS-AdapSTNet with baselines, the comparison between the prediction curves of different models for the speed of the same road on the same day and the ground truth is displayed intuitively in Fig. 8A. It can be observed that during periods of drastic speed fluctuations (i.e., 6:00-12:00), the prediction curve of Graph WaveNet does not fit well with the curve of the ground truth, failing to capture abrupt changes accurately. The curves of AdapGL+DCRNN and DTS-AdapSTNet (DCRNN) align more closely with the actual change shape of the ground truth, depicting the beginning and end of peak hours accurately. However, a considerable deviation from the ground truth is noticeable with AdapGL+DCRNN during the 6:00-7:00 period in the morning, and a significant deviation from the ground truth speed value is observed at 9:00 as well. In contrast, the curve of DTS-AdapSTNet (DCRNN) aligns with the curve of the ground truth better. Similar conclusions can be drawn from the analysis of Fig. 8B. This demonstrates that although DTS-AdapSTNet and AdapGL+DCRNN have a similar structure, the method designed by DTS-AdapSTNet is more conducive to extracting potential spatial relationships between sensors within each road segment. It exhibits a stronger modeling ability for complex and changeable traffic conditions.

Ablation experiment

To verify the effectiveness of the main modules proposed in this article, ablation studies are conducted on METR-LA and PEMS-BAY datasets.

Effect of DTSRMG-module and AMG-module

In order to verify the influence of the DTSRMG-module and AMG-module on the experimental results, three variants of each model are tested on two datasets. The results displayed in Fig. 9 indicate that whether it is the METR-LA dataset or the PEMS-BAY dataset, the performance of each model significantly deteriorates when one of the two modules is removed, particularly evident in the METR-LA dataset. As depicted in Figs. 9A–9C, when the DTSRMG-module is eliminated, there is an average increase of 8.9%, 6.1%, and 10.7% in the MAE, RMSE, and MAPE for each model, respectively.

When the AMG-module is removed, there is an average increase of 6.8%, 6.2%, and 8.8% in the corresponding three metrics, respectively. This demonstrates that removing the DTSRMG-module has a larger negative impact on the performance of most models compared to removing the AMG-module. This highlights the vital role of the DTSRMG-module in the predefined relationship matrix. Without the DTSRMG-module, the effectiveness of the AMG-module cannot be realized fully. Similar findings are observed in the analysis of the PEMS-BAY dataset, which are shown in Figs. 9D–9F.

Effect of generator matrix

The effectiveness of the generated matrix can be observed more intuitively by utilizing DTS-AdapSTNet (DCRNN) for experiments on the METR-LA dataset, where 40 sensors in the dataset are selected. The heat map shown in Fig. 10 depicts the distance relationship matrix, initialization matrix and generation matrix, respectively. Consequently, it is evident from the previous experiments that the experimental prediction results can be improved significantly through the combination of the generation matrix and the DTSRMG-module.

The position distributions of the sensors 160, 164, 161 are shown in Fig. 10E. Firstly, the distance relationship is shown as the small squares 1 and 2 in Fig. 10A, where square 1 represents the relationship between sensors 160 and 164, and square 2 represents the relationship between sensors 164 and 161. It is evident that the distance between them is related closely. Secondly, after the initialization matrix is generated by the AMP-module, the spatial relationships among them are shown in Fig. 10B. The color of squares 1 and 2 becomes lighter, indicating that the relationships among sensors at this stage not only consider the distance factor but also take into account other factors. Finally, after the AMG-module, the spatial relationships among the three sensors are shown in Fig. 10C. It can be seen that the color of square 1 is darker, indicating that after the model training, it is believed that the relationship between sensors 160 and 164 is closer.

As depicted in Fig. 10E, several observations can be made. Firstly, sensor 164 is situated downstream of sensor 160, indicating a strong correlation between the two sensors after model training. Secondly, there is a junction in the vicinity of sensor 164 where traffic can merge, and another junction near sensor 161 where traffic can flow out. Therefore, sensors 164 and 161 cannot be classified simply as part of the same segment. After model training, the color of square 2 becomes lighter, indicating a weakening of the spatial relationship between sensors 164 and 161. Finally, although all three sensors are located on the same road and are relatively close in distance, effective capturing of potential spatial relationships among them after training can enhance beneficial dependence relationships while weakening unfavorable ones. Consequently, new spatial relationships conducive to road speed prediction are obtained. It is evident that the proposed model demonstrates effectiveness in learning the spatial relationships among sensors on the same road, and endeavors to represent real road conditions accurately.

Regarding the spatial relationships among different roads, the proposed model also describes them through the spatial relationships of sensors located on different roads. Firstly, the two sensors located on the road 62 and the road 85 show a weak correlation in both the distance relationship matrix and the initialization relationship matrix, which are shown by square 3 in Figs. 10A and 10B. Secondly, after learning by the proposed model, it can be seen in the generated graph that the spatial relationship between the two sensors enhancs, which is shown by square 3 in Fig. 10C. This indicates that the two roads where the two sensors are located have a high similarity in traffic changes. Finally, to verify this, the real speed variation of the two roads in a day is plotted. As shown in Fig. 10D, it can be seen that road 62 and road 85 have almost identical speed curves. This further demonstrates the importance of spatial relationships among the sensors represented by the generated graph in road network speed prediction.

In summary, the AMG-module is capable of effectively learning spatial relationships among sensors located on the same road as well as among sensors located on different roads. These spatial relationships contribute to making predictions more accurate.

Improvement of the loss function

In order to investigate the contribution of the proposed loss function to the experimental outcomes, experiments are conducted on the METR-LA dataset using the DTS-AdapSTNet (ASTGCN) model. All other conditions remain unchanged, with only the loss function in the training process being modified. One version of the loss function is the general one, while the other is the optimized loss function proposed in this article. A comparison of the prediction results is performed, and the findings are presented in Table 4. It can be seen that after the proposed optimized loss function is used, the prediction performance of the model improves by 10.9%, 7.2% and 17.9% across all metrics, demonstrating the effectiveness of the proposed loss function.

Furthermore, in order to compare the impact of different loss functions on prediction results more intuitively, the alteration in speed for a specific road throughout a day in the test data is depicted in Fig. 11. Upon training with the proposed loss function, the resulting prediction curve aligns more closely with the actual data. Instead, when not utilizing the proposed loss function, the error is increased, illustrating the effectiveness and improvement introduced by the proposed loss function for road speed prediction.