2.1. Analysis of Factors Influencing the Operating Speed of Ramps

Recognizing that the operating speed of vehicles is primarily influenced by road alignment conditions, most researchers utilize road geometry to forecast the operating speed, such as slope, curvature, and other factors [15, 16]. Jafarov et al. [17] conducted an analysis of the operating speeds of vehicles on 18 ramps within 10 highway interchanges in Moscow. They formulated a model for operating speed and acceleration, utilizing the ramp slope as the sole input variable. They employed a linear regression methodology to predict operating speed. In a separate investigation, Xie et al. [18] scrutinized the driving performance of vehicles on mountain grade separation ramps and introduced a speed prediction method that takes into account curvature, curve length, and cumulative steering angle. Gong et al. [19] identified that curve factors, specifically curve radius and length, predominantly influence the driver’s speed choice. Hashim et al. [20] established a speed profile model for two-lane rural roads by considering the geometric design of horizontal and vertical alignments. Paolo et al. [21] developed an operating speed model for low-capacity roads, incorporating factors such as relative sight distance, radius, and curvature change rate. Hence, based on the literature discussed earlier, this study develops models to predict vehicle operating speeds on interchange ramps. It considers factors such as slope gradient, curvature, and the length of vehicle entrance to a ramp.

Additionally, the velocity of vehicles on ramps is influenced by spatial correlation. Spatial correlation describes the behavior of a vehicle’s speed at a specific point during its journey. This speed is influenced not only by the immediate geometric characteristics of the road at that point—such as curvature, slope, and surface conditions—but also by the state of the trajectory at preceding locations (i.e., adjacent points along the vehicle’s trajectory) and the cumulative effect of the vehicle’s speed at these locations. This correlation highlights the continuity and dynamism of driving behavior, indicating that the vehicle’s motion is a continuous process that evolves over time and space. Consequently, the current speed state is a cumulative response to past road conditions and speed states. While the immediate geometric characteristics of the road have been considered in previous studies, spatial correlation has often been overlooked. Recognizing spatial correlation is pivotal because, for interchange ramps, speed fluctuations frequently manifest in closely observed spaces, where the operating speed of a vehicle at a particular trajectory point heavily relies on the operating speed of the preceding trajectory point and alterations in road geometric conditions [22, 23]. Therefore, in our study, the GLMS and the DNNS introduce intermediate speed parameters (representing vehicle speeds at preceding trajectory points), intermediate curvature parameters (indicating road curvature at preceding points), and intermediate slope parameters (reflecting road slope at preceding points) to consider the spatial correlation of vehicle travel speed and enhance the accuracy of predicting vehicle speeds on ramps.

2.2. Operating Speed Prediction Methods

Recently, various models have been employed for speed prediction, encompassing linear regression [24], GLM, random effect models [25, 26], and neural network models [27]. Among these models, the multiple linear regression model stands out as the most commonly employed. For instance, Yao et al. [28] utilized the linear regression method to formulate a speed prediction model for super long tunnels on highways, considering tunnel geometric parameters. Banihashemi et al. [29] applied the linear regression method to establish a speed model for rural two-lane highways. Thiessen et al. [30] used a GLM to predict speeds at the midpoint of straight segment roads, leveraging panel data from straight segments in Canada. Maji et al. [31] crafted three operating speed prediction models through stepwise multiple linear regression analysis.

The random effects model primarily explains the random variations among individuals. However, it necessitates a well-structured dataset and adequate data to achieve reliable estimation results. Bassani et al. [32] established a random effects model using environmental parameters related to road geometry and lighting conditions to predict the average speed of vehicles. Their model assumes that the operating speeds at different points during vehicle travel are independent of each other. However, both the GLM and the random effects model merely account for the impact of road geometry on vehicle speed, failing to consider the influence of spatial correlation on travel speed.

Neural network models excel in managing intricate nonlinear relationships, showcasing robust learning and predictive capabilities, and autonomously discerning patterns within data [33]. Wu et al. [34] formulated a short-term prediction model for lane driving speed within extra-long tunnels, employing the temporal Transformer framework. The constructed speed prediction model achieves a commendable accuracy of 97.82%. In a previous study, Wu et al. [35] devised a speed prediction model based on a gray neural network, amalgamating the advantages of the low data requirements inherent in the gray prediction model and adaptive capabilities of neural networks. However, despite the effectiveness of these models in capturing the temporal dependencies of speed sequences in time-series prediction, they fail to simultaneously account for road alignment conditions and spatial correlations—factors that are crucial for accurately forecasting travel speed.

To investigate the impact of road alignment conditions and spatial correlation on driving speed in urban interchange ramps, this study employs the GLM as a baseline. It integrates the Exponential Moving Average (EMA) method to account for spatial correlation in vehicle operations. Building on this, the study enhances the GLM and neural network models to incorporate spatial effects on vehicle speed, enabling accurate predictions of speed fluctuations across the entire ramp.

3.1. Natural Driving Experiment

We conducted natural driving experiments in Shanghai (SNDD) and selected time periods during which vehicles were relatively less influenced by surrounding traffic conditions for data collection. The experimental vehicles adopt the small car design specified in the Chinese standard “Technical Standards for Highway Engineering” (JTG B01-2014), with a wheelbase of ≤ 7 m and a specific power of > 15 kW/ton. These vehicles are outfitted with onboard Data Acquisition System (DAS) which comprise devices such as Doppler radar, triaxial accelerometers, and Global Positioning Systems (GPS), event button, four synchronized high-definition cameras, and data integration recorder all operating at a sampling frequency of 10 Hz.

Drivers navigate their vehicles continuously to collect data in real conditions according to their customary driving behavior and patterns. The DAS integrated within the test vehicle, diligently captures and records driver behavior metrics, commencing approximately 90 s after the ignition of the vehicle’s engine and persisting until its subsequent shutdown. Consequently, every datum recorded by the computerized system, inclusive of video footage and GPS data, can be construed as a comprehensive portrayal of a driver’s travel behavior.

From the data obtained from these experiments, we have selected trajectory data segments of vehicles navigating within urban expressway interchange ramps for our research [36]. Specific trajectory data tailored to its research objectives and subjects are employed, encompassing the following: (1) GPS data capturing vehicle speed within the mainline area of urban expressway interchange ramps. This data concentrates on sections unaffected by merging and diverging zones. (2) Video data of vehicles driving within the urban expressway interchange ramp area. The videos feature forward-facing views, facilitating the evaluation of road traffic conditions during vehicle movement.

3.2. Interchange Ramp Data Extraction

The DAS in our experiments captures all the characteristic parameters of the experimental vehicles from startup to shut down and consolidates them into a unified CSV file. The data file aligns with a consistent timeline shared with the corresponding video data. The current dataset comprises 9207 CSV files along with their corresponding video records. To acquire trajectory data of vehicles on ramps, data cleaning and extraction of urban expressway interchange ramp trajectory data are undertaken as following: (1) Given the study’s emphasis on travel records on urban expressways, where the process of entering and exiting these expressways typically involves longer driving times, CSV files smaller than 1000 KB are excluded, resulting in 3556 remaining records. (2) The latitude and longitude data in the CSV files are imported into ArcMap software to identify concentrated travel trajectories around urban expressway interchanges. (3) Given the significant speed differences between the ramp merging/diverging areas and the mainline sections, this study focuses on the region between the noses of the ramp diverging triangle and the merging triangle, excluding the acceleration and deceleration lanes. Trajectory data from this area are extracted using the K-Nearest Neighbors (KNN) algorithm, with the number of neighbors set to 1 to identify the nearest point to the noses of both the diverging and merging areas. Based on these points, the trajectory data between them are retrieved from the CSV file. Finally, the accuracy and validity of the extracted trajectory data are evaluated by comparing it with actual observed video and trajectory data. The final dataset comprises a total of 974 interchange ramp trajectory data. The extracted interchange ramps encompass three types: ① direct ramps, ② loop ramps, and ③ indirect ramps, as shown in Figure 1. There are 600, 115, and 259 trajectory data for the three types of interchange ramps.

These data serve as the foundation for subsequent analysis and prediction. Geometric parameters and types of interchange ramps are delineated in Table 1. Furthermore, the extracted interchange ramps in this study are all single-lane ramps. The road widening caused by the horizontal curve is not significant due to its location on the ramp. Moreover, the lighting conditions remain consistent along the entire road for each interchange ramp.

4.1. Overall Speed Characteristics

Based on the above analysis results, the subsequent models will factor in the influence of ramp curvature, the square of the slope, ramp length, and the interaction term between ramp curvature and length on operating speed.

4.2. Analysis of Speed Variation Trends

This section investigates the fluctuations of vehicle speeds at different positions on the same ramp, employing the K-means method for trajectory clustering to analyze vehicle speed trajectories. Initially, all trajectories on each ramp are aligned based on the ramp entry position. Subsequently, the average velocity for each ramp is calculated to derive the velocity-position trajectory across different ramps. Finally, the Dynamic Time Warping (DTW) algorithm is used to measure the trajectory similarity, which can adjust for time compression and expansion to mitigate the influence of segment length. The Euclidean distances of geometric shape, velocity, and direction are calculated separately and assigned weights of 0.69, 0.23, and 0.08 for aggregation based on entropy weight method. Here, geometric shape refers to the curvature and length of the ramp, velocity refers to the average speed of vehicles traveling at different trajectory points on the ramp, and direction refers to the slope of the ramp. Figure 4 displays the clustering outcomes, which illustrate the trajectory distribution across three distinct clusters, differentiated by red, blue, and green, respectively.

Based on the clustering results, it is observed that operating speed curves of ramps of different lengths exhibit different oscillation patterns. Subsequently, to delve deeper into the driving characteristics of vehicles on various ramps, an examination of operating speed on ramps of different lengths is undertaken, as depicted in Figure 5. Speed curves of distinct colors denote varying lengths of the ramp: (1) When the ramp length is less than 400 m, the operating speed exhibits a clear decreasing trend followed by an increasing trend, with little or no prolonged periods of stable speed driving. (2) When the ramp length is between 400 and 700 m, the operating speed is unstable, showing more fluctuations. (3) When the ramp length exceeds 700 m, vehicles tend to maintain a relatively stable speed during the middle section of the ramp.

5.1. GLM

5.2. GLM With Spatial Correlation Consideration

Given the complex linear combinations of urban expressway ramps, a vehicle’s speed at a specific point is consistently influenced by the states of adjacent trajectory points, including road curvature, speed, and gradient. To address this, we developed a GLM that incorporates spatial correlation using an EMA framework. This method, widely recognized for its effectiveness in capturing trends and fluctuations in time-series data, is equally adept at handling spatially correlated data. In our model, the EMA structure adjusts for distance-based weighting, reflecting the varying impact of trajectory points at different distances on the current speed.

5.3. DNNS

The DNN can discern higher-level and more abstract features, enhancing the model’s ability to generalize and achieve more precise predictions of operating speed curves. Consequently, we introduce the DNNS to predict the short-term operating speed of small passenger cars. This model aims to capture the speed fluctuations of vehicles on urban expressway interchange ramps comprehensively and provide a detailed overview of the speed distribution of vehicles throughout the interchange ramp.

The DNNS takes inputs such as y_i,k−1, H_i,k, x_i,k, $$, and x_i,k · h_i,k, with v_i,k−1, as the corresponding output. The meanings of these parameters correspond to those in the “Generalized Linear Model Considering Spatial Correlation.”

6.1. Model Calibration

Parameter calibration for both the GLM and the GLMS is executed using the R software. The Markov Chain Monte Carlo (MCMC) sampling method is employed for parameter estimation. The mean values of the estimated model parameters are presented in Table 3, while Table 4 displays the 95% confidence intervals.

Table 4 indicates that the 95% confidence intervals of GLMS do not encompass zero, signifying the significance of the parameters at a higher significance level. Conversely, the confidence intervals of GLM include zero in fewer instances, suggesting that some parameters of this model lack significance at a higher significance level. GLM assumes independence among operating speeds at different trajectory points during vehicle travel, neglecting the inherent correlation in operating speeds as time series. The significance levels derived from the MCMC sampling estimation underscore the limitations of GLM. In contrast, the GLMS proposed in this study acknowledges and incorporates the spatial correlation in operating speeds, resulting in improved significance in parameter estimation from MCMC sampling.

Furthermore, an orthogonal experiment was designed to determine the optimal network structure, and the considered factors along with their respective values are detailed in Table 5. After a series of cross-validation processes, the hyperparameters for DNNS are selected as outlined in Table 6. The loss curve of the DNNS model training, depicted in Figure 6, illustrates the convergence of the value infinitesimally close to the x-axis and the loss value shows 0.0000001, affirming the model’s convergence post-training.

6.2. Model Validation

The established models are employed to predict the operating speeds of 10 additional ramps that were not involved in parameter calibration. These predicted results are then compared with the actual operating speed data obtained from natural driving experiments. For evaluation purposes, R², MAPE, and MSSE are calculated and presented in Table 7, which shows the maximum MAPE for GLMS to be 10.80%, and the maximum MAPE for GLM to be 24.12%. This highlights the significance of considering the dependence of operating speeds between vehicle trajectory points to achieve accurate speed prediction on urban expressway interchange ramps. On the other hand, the maximum MAPE for DNNS is 9.14%, indicating that DNNS demonstrates higher accuracy in predicting operating speed compared to GLM and GLMS for these 10 ramps. In most cases, the predictive accuracy of DNNS is approximately 95%, as revealed by the MAPE indicator. Moreover, the average R² and MSSE for DNNS are 0.81 and 0.92, respectively. GLM, GLMS and DDNS had chi-square test significances (Sig.) greater than 0.05, indicating acceptance of the null hypothesis, suggesting that there is no significant difference between the predicted operating speeds and the actual measurements. These values indicate that DNNS successfully captures the relationship between operating speed and the geometric characteristics of urban expressway interchange ramps, effectively reflecting the speed fluctuations of vehicles on the ramp.

The velocity fluctuation curve predicted by the established models is depicted in Figure 7, where the results from the Transformer model are also included. The Transformer model has recently garnered attention and achieved some success in time series prediction. As illustrated in Figure 7, in comparison to the GLM, the GLMS, and the Transformer model, the DNNS model exhibits higher prediction accuracy and better reflects the fluctuation trend of vehicle speed on the ramp.

Overall, the estimation results of the DNNS showcase superior robustness and generalization performance in accurately predicting the short-term operating speed for compact passenger vehicles on urban expressway interchange ramps. In other words, both the DNNS and the GLMS exhibit certain advantages over the GLM. Recognizing the dependence of operating speeds between vehicle trajectory points is essential for accurate speed prediction in practical scenarios. The DNNS demonstrates distinct advantages over the GLM and the GLMS, with the DNN significantly outperforming the GLM in capturing the speed fluctuations of vehicles on ramps.

6.3. Model Application

The developed model serves as a valuable tool for assessing road safety, specifically in reviewing the speed uniformity of vehicles on ramps. In cases where significant fluctuations in vehicle speed occur, it is advisable to devise precise and targeted speed control measures.

As depicted in Figure 7, the DNNS model adeptly predicts the short-term operating speed of compact passenger vehicles traversing an interchange ramp. The model’s calculations indicate that the speed of vehicles entering the ramp is 80 km/h. However, within a distance range of 200–400 m along the ramp, the operating speed decreases to approximately 60 km/h and stabilizes at this level. Subsequently, as vehicles cover a distance of 600–800 m along the ramp, the operating speed exhibits an upward trend. The resulting operating speed curve enables road managers to accurately anticipate speed fluctuations on ramps and develop more precise speed control measures based on both the position and magnitude of speed variations [42].