Exploring Heterogeneity in Car-Following Behaviors Based on Driver Visual Characteristics: Modeling and Calibration

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Introduction
Car-following models have long been a focal point in the feld of trafc fow theory.By modeling car-following behaviors, it becomes possible to quantify the longitudinal interactions between following vehicles (FVs, the vehicles located behind in the process of car-following, will receive the stimulus of the front car and produce a response) and leading vehicles (LVs, the leading vehicles in the process of car-following, which can bring certain stimulation to the FVs), thereby deciphering the operational characteristics of trafc fow and revealing the underlying mechanisms of micro-level driving behaviors.Since the inception of the car-following concept by Pipes [1], more than 70 years of development have transpired.Numerous carfollowing models have been proposed and gradually refned, with scholars like Dian-hei and Sheng [2] systematically categorizing and delineating these models from both trafc engineering and statistical physics perspectives.With the advent of big data and the rise of technologies such as machine learning and deep learning, various data-driven car-following model theories and trajectory prediction methods [3][4][5][6] have emerged.However, amidst the rapid theoretical progress of these models, their physical signifcance and interpretability have gradually waned, and attributes like driver characteristics and vehicle heterogeneity have been overlooked.Nevertheless, human-driven vehicles remain the primary actors in road trafc fow.Hence, drivers continue to be the most crucial element within road trafc components.Yao et al. [7] assessed patterns of individual emergence during the pandemic; Qu et al. [8] explored how ridership contributes to the planning and operation of urban and rural bus systems, showing that individual behavior rules can afect macro-trafc conditions.Tang et al. [9] introduced drivers' bounded rationality into the speed guidance model and demonstrated through simulation results that drivers' bounded rationality signifcantly impacts vehicle fuel consumption and emissions.Jin et al. [10] studied drivers' behavior of using mobile phones at intersections, and the results show that using mobile phones has a signifcant negative impact on driving behavior.Furthermore, Liao et al. [11] improved the traditional car-following model by taking into account drivers' driving habits, enhancing the model's safety and comfort.To better describe the impact of the driver's stochastic characteristics on car-following behaviors, Luo et al. [12] proposed a stochastic full velocity diference model (SFVDM) considering the stochastic variation of the desired velocity.Accurately comprehending the driving mechanisms of drivers during the driving process and establishing behavior models that are closer to real-world driving scenarios from a driver's perspective hold signifcant importance for a deeper understanding of driving behavior mechanisms and microtrafc simulation systems [13].
Conventional car-following models frequently assume homogeneity among both drivers and vehicles.However, in real-world scenarios, the presence of driver individuality, vehicle disparities, and even environmental distinctions such as weather and road conditions introduce heterogeneity into car-following behaviors.Tis heterogeneity is closely associated at a macroscopic level with phenomena including the reduction of road capacity, trafc congestion, trafc oscillations, and the emergence of stop-and-go waves [14,15].Ossen and Hoogendoorn [16] designated this form of heterogeneity as the divergences in car-following behavior exhibited between diverse drivers or distinct vehicle combinations operating within the same environmental context (i.e., identical road segments, comparable trafc conditions, and analogous weather conditions).
At the driver level, An et al. [17] introduced a delay parameter in reaction time to capture variations in responses among drivers with diferent levels of experience.Tey formulated the extended full velocity diference (FVD) model that takes driver heterogeneity into account.Subsequently, Cheng et al. [18] investigated the diferences in car-following characteristics among drivers with varying cultural backgrounds through virtual driving experiments.Pan and Guan [19] employed quantile regression to model driver heterogeneity at diferent quantiles.Makridis et al. [20] proposed a novel framework based on identifying driver characteristics through acceleration behavior, demonstrating driver heterogeneity in microsimulation scenarios.
At the vehicle level, Peeta et al. [21] pioneered categorizing diferent vehicle types into distinct car-following groups, examining diferences in car-following behavior between heavy vehicles and regular automobiles.Liu et al. [22] extended the intelligent driver model (IDM) by considering various vehicle combinations (C-C, C-T, T-C, and T-T, where C represents cars and T denotes trucks).Tey coupled the extended model with NGSIM dataset calibration to derive corresponding fundamental trafc diagrams.Raju et al. [23], utilizing data collected from two road sections in India, introduced "lateral separation" to combinations such as C-C, C-T, T-C, and T-T and recalibrated the Wiedemann model in Vissim software.
Existing studies predominantly focus on car-following behaviors between vehicles of diferent functional categories, considering combinations such as cars with trucks, buses, or heavy vehicles.Nevertheless, due to the limited representation of trucks and buses in actual collected data, the sample size often fails to adequately support their conclusions.Moreover, current research predominantly centers on heterogeneity in vehicle performance and driving behaviors among diferent functional vehicle types.However, there is limited investigation into the heterogeneity within the same functional category of vehicles.Furthermore, considering that the primary source of stimuli for drivers is visual input, the existing research that considers vehicle types still relies on traditional car-following variables, neglecting the investigation of the visual stimuli brought about by diferent vehicle types on drivers.
To address these issues, this study aims to characterize the infuence of heterogeneous vehicle types on carfollowing behaviors within the same functional category of vehicles from the perspective of drivers' visual characteristics.Te study utilizes the NGSIM dataset to extract all passenger cars, categorizes them into vehicle types, and obtains four types of car-following segments.To investigate vehicle-type heterogeneity in car-following, visual characteristics are introduced as variables and subjected to numerical simulation.Single-factor analysis of variance is employed to compare the diferences in car-following behavior performance between traditional car-following variables and visual characteristics.Finally, a drivers' visual angle (DVA) model incorporating visual characteristics is established, and its efectiveness is evaluated through comprehensive and type-specifc calibration, validation, and error sensitivity analysis.
Te contributions of this study can be summarized as follows.First, this study introduces the drivers' visual characteristic variables into the context of heterogeneous vehicle-type car-following models.Based on trajectory data, the visual angle and its rate of change are constructed to study vehicle-type heterogeneity from the perspective of drivers' visual characteristics, showcasing the efectiveness of visual characteristic variables in addressing heterogeneity in car-following scenarios.Second, an improved model is proposed based on visual characteristic variables.Trough comprehensive calibration and validation, as well as validation for four diferent combination types, the method is proven to signifcantly enhance model ftting performance.Additionally, the error sensitivity analysis demonstrates the model's robustness across various road conditions, vehicle combinations, and diferent error evaluation criteria.Finally, the statistical analysis of visual characteristic variables and model comparison substantiate that modeling from the perspective of drivers' visual characteristics is of vital 2 Journal of Advanced Transportation signifcance in enhancing model ftting performance and resolving the issue of heterogeneous car-following combination types.Tis study introduces novel avenues for investigating carfollowing behavioral heterogeneity.Te remainder of this paper is organized as follows.In Section 2, the preprocessing of trajectory data and the classifcation criteria of four heterogeneous car-following combination types are introduced, and the visual characteristic parameters are extracted for numerical simulation.Statistical diference analysis of heterogeneous car-following behaviors is introduced in Section 3. Section 4 elaborates the results of model calibration and verifcation and discusses the results.Te fnal section concludes the study.

Data Source and Trajectory Reconstruction.
To investigate the impact of vehicle type heterogeneity on driver behavior, this study utilizes the publicly available Next Generation Simulation (NGSIM) dataset [24] provided by the United States Federal Highway Administration.Trajectory data from two roadways, I-80 and US-101, are selected for analysis.Te dataset captures vehicle trajectories at a frequency of 10 Hz, encompassing dynamic vehicle motion information such as acceleration, velocity, and headway, as well as static vehicle attributes like width and length.Tese attributes are crucial for vehicle type analysis.To mitigate the infuence of high-occupancy vehicle (HOV) lanes and entrance/exit ramps, analysis is confned to vehicles on lanes 2 to 5 of the selected roadways.Te road confguration is illustrated in Figure 1.
Te raw trajectory data are acquired through video processing software.However, inherent anomalies and random noise in the data result in signifcant deviations between obtained trajectories and actual trajectories.Tus, prior to utilization, corrective actions are necessary to rectify outliers and smooth noise.In this study, the abnormal data points were corrected by threshold cleaning and spline interpolation, and the noise was smoothed by symmetric exponential moving average (sEMA) [25].Maczak et al. [26] conducted a comparative assessment of sEMA, locally weighted regression, Butterworth flters, Kalman flters, and multiple spline methods based on identical evaluation criteria.Ultimately, sEMA was determined to markedly minimize acceleration standard deviation and outlier counts.Tis method has since been widely adopted in subsequent analyses of NGSIM vehicle trajectory data [27,28].Te smoothing process is outlined in equations ( 1) and (2).
In equation (1), X(t k ) represents the ftted driving parameters of the vehicle at time t k , which includes position and driving speed.i denotes the sample point in the trajectory, dt is the sampling interval of 0.1 seconds, and m is the total length of the trajectory.In equation (2), D is the window width for boundary smoothing, and Δ is the window width for intermediate data smoothing.Tiemann et al. [25] conducted a comparative analysis of various window widths for displacement, velocity, and acceleration.Ultimately, they selected a displacement smoothing window T x of 0.5 seconds, a velocity smoothing window T v of 1.0 seconds, and an acceleration smoothing window T a of 5.0 seconds.
Te process involved selecting a random sample of vehicles from the I-80 and US-101 roadways.Te smoothing of vehicle speeds and accelerations is schematically depicted in Figure 2. Subsequently, the reconstructed trajectories from the I-80 and US-101 datasets were analyzed.Prior to reconstruction, approximately 12.4% of the acceleration values exceeded 10 ft/s 2 (approximately 3.048 m/s 2 ).However, following the reconstruction process, the accelerations stabilized within the range of ±3 m/s 2 .Moreover, the proportion of accelerations with magnitudes exceeding ±15 m/ s³ (referred to as jerk) decreased from 45.7% to 0%.Tis reduction underscores that the reconstructed trajectories align more closely with authentic driving scenarios.

Car-Following Segment Extraction and Classifcation.
Following the trajectory data reconstruction, car-following segments were further extracted with constraints on carfollowing gap, duration, and following vehicle (FV) speed, based on the studies by Liu et al. [22] and Higgs and Abbas [29].Te criteria for defning car-following behavior in this study are as follows.
① Te preceding vehicle's ID remains unchanged, ensuring that the vehicle consistently follows the LVs.② Te average speed of the FVs is ≥ 5 m/s to avoid uncertainties in car-following behavior during congested conditions.③ Te car-following gap is ≤ 120 m to ensure that the FVs operate under non-free-fow conditions.④ Te car-following duration is ≥ 30 s to ensure the stability of the car-following state.⑤ Te relative lateral displacement between the LVs and FVs is ≤ 1.5 m, ensuring that they remain in the same lane.Te car-following samples extracted based on these criteria are summarized in Table 1.
Segmentation of diferent car-following types requires vehicle classifcation.Based on the distribution characteristics of vehicle width on I-80 and US-101 roads, a critical vehicle width of 1.95 meters (corresponding to the 40th percentile for I-80 and the 50th percentile for US-101) was selected to diferentiate between small and large vehicle types.According to the vehicle types of the lead and following cars within car-following segments, these segments were categorized into four types: Small-Small (S-S), Small-Large (S-L), Large-Small (L-S), and Large-Large (L-L) carfollowing types.Te statistical results for each type of carfollowing segment are presented in Table 2.

Extraction of Driver's Visual Characteristics.
Conventional studies on car-following behavior often employ input variables such as following car velocity, relative velocity, and distance to obtain the following car's acceleration.However, psychological research suggests that drivers are unable to accurately perceive speed and distance information.Moreover, their judgments of the distance to the leading vehicles (LVs) are not based on these parameters.Car-following behavior fundamentally constitutes a driver's response to external trafc stimuli.Tese stimuli primarily originate from the LVs and directly impact the driver's visual perception.As the visual stimuli from the LVs change, drivers adopt various actions (such as maintaining a steady speed, accelerating, decelerating, or changing lanes) to achieve the desired following state.To characterize the visual stimuli perceived by drivers, considering both LVs' information and inter-vehicle distance, we introduce the concept of visual angle along with its rate of change, as depicted in Figure 3. Te calculation of these parameters is defned by equations ( 3) and ( 4): In equation ( 3), θ n (t) represents the visual angle of the FV's driver at time t, w n− 1 is the width of the LV, ∆x n (t) is the headway between the LV and the FV at time t (space headway), l n− 1 is the length of the LV, l 0,n− 1 is the distance from the rear of the LV to the front of the FV, and θ n ′ (t) is the change rate of the visual angle of the FV's driver at time t.Te sampling interval ∆t is 0.1 s.
By combining equations ( 3) and ( 4), visual angle and its rate of change sequences can be extracted for each carfollowing segment.To mitigate the impact of outliers, a twostep threshold cleaning method [29] is employed to cleanse the data.Firstly, the 98th percentile values of both variables are selected as the thresholds for the initial cleansing step, eliminating extreme outliers.Te postcleansing data

Numerical Analysis of Visual Characteristics.
To gain a deeper understanding of the performance of visual angle and its rate of change variables under diferent vehicle types, numerical simulations are conducted based on equations ( 3) and ( 4).A comparison is made between the visual angle variable and the traditional car-following gap in various vehicle types.Initially, equation ( 3) is substituted into equation ( 4) and further manipulated as follows.
where ∆x � l 1 − l 0 � ∆v • ∆t represents the change in space headway of the FVs at time ∆t, ∆v represents the relative velocity of vehicles, l 0 represents the current time's space headway, and l 1 represents the next time's space headway.
Based on the extracted car-following segment samples, the mean of l 1 is − 0.36 m/s, with a minimum of − 8.84 m/s and a maximum of 14.77 m/s, and hence it can be taken as Te numerical simulation of the visual angle variable is depicted in Figure 5.It is evident that as the space headway reduces, the visual angle gradually increases, with a larger increase observed when the headway is small.Tis suggests that drivers are more signifcantly infuenced by the LVs when the headway is tight.Additionally, for smaller headway, the visual angle increases notably with an increase in vehicle width.However, at greater distances, the diferences in visual angle among vehicles with diferent widths diminish, indicating that at longer distances, the stimuli from vehicles of varying widths remain relatively consistent, and drivers tend toward a state of free driving.
Concerning the visual angle rate of change variable, as indicated by equation ( 5), it varies with both space headway l 0 and ∆x. Figure 6 illustrates the distribution surfaces of the visual angle rate of change concerning ∆x under four vehicle width scenarios.Similarly, at longer headway, the visual angle rate of change tends to converge to a single plane and approaches zero for various vehicle widths.However, at smaller headway, signifcant diferences in the visual angle rate of change emerge among diferent vehicle widths.Larger vehicle widths correspond to larger visual angle rate of changes.In summary, visual angle and its rate of change, as visual characteristic variables of drivers, efectively refect the diversity in stimuli perception by drivers for diferent vehicle types at varying distances, aligning more closely with drivers' real-world car-following behaviors.

Analysis of Heterogeneous Car-Following Behaviors Based on Visual Characteristics
Te numerical simulation results presented earlier fnd validation in real-world driving situations.When following larger LVs, drivers often adopt more cautious driving behaviors, such as reducing vehicle speed or increasing space headway.Tis conservative response is attributed to the greater visual stimuli produced by larger vehicles, which also increases the psychological load on drivers.Hence, drivers tend to opt for safer driving strategies.In this section, real driving data will be utilized to compare the disparity between traditional carfollowing variables and visual characteristic variables across diferent car-following types.Furthermore, the signifcance of visual characteristic variables in modeling heterogeneous carfollowing behaviors will be analyzed.Journal of Advanced Transportation

Correlation Analysis of Car-Following Variables.
Firstly, the min-max normalization technique is employed to mitigate diferences stemming from varying scales among diferent features.Subsequently, partial correlation coefcients are calculated between diferent features using the method outlined in [30].Te correlation matrices of I-80 and US-101 roads are shown in Figures 7(a) and 7(b), respectively, where the horizontal and vertical axes denote following vehicle (FV) speed and acceleration, visual angle and its change rate, leading vehicle (LV) speed, space headway, and relative speed.According to [31], when the absolute value of a correlation coefcient is between 0 and 0.09, it is considered as having no or very weak correlation.A correlation coefcient between 0.1 and 0.3 is considered weak, 0.3 to 0.5 is considered moderate, and 0.5 to 1.0 is considered a strong correlation.Te analysis reveals a substantial correlation between visual angle and space headway, both of which exhibit strong correlation with FV speed (correlation coefcient: ±0.74 of I-80 and ±0.78 of US-101).
Similarly, the correlation between visual angle change rate and relative speed is noteworthy, exhibiting similar strong correlation with FV acceleration (correlation coefcients: − 0.57, 0.61 of I-80 and − 0.59, 0.64 of US-101).Consequently, visual angle and its change rate features can potentially replace traditional space headway and relative speed, rendering the analysis of car-following behavior from the perspective of driver visual characteristics a feasible approach.

Heterogeneous Car-Following Behavior Analysis.
To analyze the disparities in car-following behavior among heterogeneous vehicle combinations, it is necessary to extract stable car-following segments.Te extracted segments for analysis have a duration exceeding 30 seconds.Given the dynamic nature of car-following behavior, where drivers continuously adjust their actions in response to real-time stimuli from LVs, the duration of stable car-following segments is signifcant.For a comprehensive portrayal of micro-level driving behaviors, further small sample For each small sample segment, the mean following vehicle speed, mean headway distance (MHD), mean visual angle (MA), mean relative speed, and mean acceleration are extracted as corresponding car-following features.Tese features serve as the foundation for analyzing heterogeneous car-following behaviors across diferent vehicle types.
To elucidate the disparities in car-following behavior types across various driving conditions, the average headway distance (MHD) and mean visual angle (MA) for each carfollowing type are examined within distinct car-following vehicle speed ranges, as indicated in Table 3. Te values within parentheses indicate the growth rate of the carfollowing features when transitioning from a small car leading to a large car [28].Te analysis reveals that with increasing car-following vehicle speed, the headway distance signifcantly increases while the visual angle decreases  notably.Tis suggests that at higher speeds, drivers tend to maintain a safer driving state, resulting in a larger following distance or reduced visual stimulation.Furthermore, across diferent speeds, the shift from a small car leading to a large car is associated with a respective 7.53% increase (S-S to S-L) and 7.37% increase (L-S to L-L) in average headway distance.In contrast, the visual angle exhibits a more substantial increase of 22.32% (S-S to S-L) and 29.17% (L-S to L-L).Tis signifcant increase is attributed to the ability of the visual angle to refect drivers' sensitivity to the stimulus of the LV size.Te visual angle variable efectively captures this sensitivity from both physiological and psychological perspectives, highlighting its crucial role in describing carfollowing behavior.Te same conclusions are drawn from the analysis of the US-101 road.
Subsequently, using car-following type as a categorical variable, one-way ANOVA is conducted on the normalized data to explore the diferences in headway distance and visual angle distributions across car-following types.Tables 4 and 5 illustrate that both the mean gap distance (MGD) and mean angle (MA) variables exhibit statistically signifcant diferences across the four car-following types.Subsequent quantifcation using η 2 (partial eta-squared) and Cohen's f values further afrms this diference.For both the I-80 and US-101 roads, signifcant diferences are identifed in MGD and MA among diferent car-following types.When following larger vehicles, both MGD and MA signifcantly increase.Te quantifcation analysis reveals that the differences in MGD and MA among various car-following types are 0.5% and 3.6%, respectively.Corresponding Cohen's f values are 0.074 and 0.192, signifying that the visual angle variable efectively captures the diferences among diferent car-following types.Tis variable reveals the physiological and psychological stress experienced by drivers when facing vehicles of diferent sizes.

Constructing Car-Following Models Based on Visual
Characteristics.Calibration results from existing highway data models reveal that the FVD (full velocity diference) model outperforms other car-following models such as the GHR model and the Gipps model, demonstrating advantages including higher calibration accuracy, fewer parameters with clear physical signifcance, and robustness [32].To comprehensively compare the diferential modeling efects of visual characteristic variables and traditional variables in car-following behavior, this study selects the FVD model based on headway distance and the DVA (drivers' visual angle) model based on visual angle for calibration and validation.Te FVD model is represented by equations ( 6) and (7).
where a n (t) denotes the acceleration of the following vehicle at time t, V[∆x n (t)] is the driver's desired speed function based on headway distance, and α, λ, V 1 , V 2 , c 1 , and c 2 are the model parameters.

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Journal of Advanced Transportation +29.17 Note that the Mann-Whitney U test was employed, where * * * , * * , and * denote signifcance levels of 1%, 5%, and 10%, respectively.Values within parentheses signify the growth rate of car-following features when transitioning from a small car leading to a large car within the same car-following vehicle type.Cases with signifcance levels exceeding 5% were disregarded.
To evaluate the performance of visual angle and its rate of change variables on heterogeneous vehicle types, the visual angle and its rate of change variables extracted are incorporated into the improved FVD model, creating the DVA model.Tis model has been validated through stability analysis and numerical simulation [33].Te specifc form of the model is presented in equations ( 8) and (9).
where V[θ n (t)] represents the driver's desired speed function based on visual angle.

Driver Reaction Time Calibration.
During car-following processes, individual drivers exhibit variations in their reaction times [34].In this study, the two-related sequences coefcient method [35] is employed to calibrate driver reaction times at the individual level.Based on prior research, the relative speed and acceleration of the preceding and following vehicles are used for calculation.Te specifc procedure involves predefning a series of reaction time values at intervals of 0.2 seconds within the range of 0-2 seconds.For each reaction time value, the correlation coefcient between relative speed and acceleration is computed.Te reaction time corresponding to the maximum correlation coefcient is selected as the calibrated reaction time for that driver.Te distribution of the maximum correlation coefcients for each driver's two sequences is shown in Figure 9(a).Te two sequences exhibit a high correlation, with reaction times primarily falling within the range of 1.0-1.6 seconds.
After obtaining the calibrated reaction times for each driver, the distribution of reaction times for diferent-sized vehicles is compared (Figure 9(b)).Te distribution curve for larger vehicles shifts toward the lower right corner, and the percentage of vehicles with reaction times exceeding 1.4 seconds or less than 1.8 seconds is higher for larger vehicles compared to smaller ones.Tis suggests that the distribution of reaction times for larger vehicles is more dispersed.Statistical analysis conducted on diferent-sized vehicles from the two roadways reveals that larger vehicles have a 0.458-meter increase in width, representing a 26.4% rise.Furthermore, the average reaction time increases by 0.33 seconds, indicating a 2.8% increment.Specifcally, the average reaction time for larger vehicles is 1.212 seconds, while it is 1.178 seconds for smaller vehicles.Te standard deviation also increases by 0.434 seconds.Tis can be attributed to the inherent characteristics of larger vehicles, including their acceleration, deceleration capabilities, and inertia.

Error Index Selection and Improvement.
To assess the disparity between model calibration results and actual outcomes, it is essential to establish appropriate error evaluation metrics and criteria.In past car-following model calibrations, parameters like car-following speed or headway distance have been commonly employed as evaluation   [36][37][38][39].In this study, to contrast the modeling efcacy of headway distance and visual angle, both carfollowing speed and headway distance variables are adopted as evaluation indicators.Furthermore, the changes in evaluation outcomes under diferent weightings are analyzed.
As the error indicators, car-following speed and headway distance are chosen.Te mean absolute relative error (MARE) is employed as the evaluation function to compare the goodness of ft of the models.Te objective function is defned as follows.
where MARE(v, ∆x) represents the comprehensive average percentage error of car-following speed and headway distance and MARE(v) and MARE(∆x), respectively, denote the average percentage errors of car-following speed and headway distance.T represents the number of data points in the car-following segment.v real i and ∆x real i represent the actual speed and actual headway distance of the following car at time i, while v pre i and ∆x pre i denote the model-predicted car-following speed and headway distance at the same time.w 1 and w 2 are the weighting coefcients for the relative speed error and headway distance error, respectively, both initially set to 0.5 during the initial calibration.

Calibration and Validation of the Overall Samples.
Five hundred car-following segments were selected randomly from both I-80 and US-101 highways for calibration and validation using a genetic algorithm combined with a 5fold cross-validation approach [39].Among these, 400 segments were designated for calibration, while the remaining 100 segments were reserved for validation.Te model parameter calibration outcomes are detailed in Table 6.Te results indicate that the DVA model exhibited calibration errors below 0.5 for both roadways.Conversely, the FVD model displayed calibration errors nearing 0.8, marking an increase of 51.93% and 42.22% for the I-80 and US-101 segments, respectively.Furthermore, the standard deviation of the FVD model's calibration results also exhibited signifcant augmentation.
Te parameter means obtained after calibration were employed as model parameters for validation on the reserved dataset.Te validation results are presented in Table 7, while the cumulative distribution of calibration and validation errors is depicted in Figure 10.Te I-80 road's validation error decreased from 1.409 to 0.611, resulting in a precision improvement of 56.61%.Similarly, the US-101 road's validation error decreased from 1.425 to 0.780, leading to a 45.26% enhancement in precision.Te combined improvement across the two roadways was 50.94%.Te calibration and validation outcomes across both roadways underscore the superior precision of the DVA model in comparison to the FVD model.
Further investigation into the relationship between error and sample duration is illustrated by the error distribution against the car-following duration, as depicted in Figure 11.Te linear ftting of the two models' errors with respect to sample duration shows that both models' errors increase in tandem with sample duration.Tis indicates that both models are comparably infuenced by the sample duration.
Additionally, an examination of the FVD model's ftting errors revealed a denser distribution within the 0-1 range, with greater dispersion in ftting errors beyond 1.Consequently, the ftting line shifts upward, indicating higher ftting errors.To elucidate the origin of this discrete error, FVD model calibration results with errors exceeding 1.0 were extracted.Upon categorizing these errors, it was observed that among the two roadways, the L-L type samples accounted for 84 instances, constituting 50.91% of the total.Te remaining types were distributed as follows: L-S: 34 instances, 20.61%; S-L: 27 instances, 16.36%; and S-S: 20 instances, 12.12%.Tis highlights that the increase in FVD model errors primarily stems from the L-L type, signifying that optimal ftting outcomes are achieved when both the lead and following vehicles are small cars.Conversely, as the lead or following vehicle transitions to a large car, the ftting  performance diminishes, reaching its poorest state when both are large cars, thus demonstrating a signifcant degree of error dispersion.

Calibration and Validation by Diferent Car-Following
Types.To compare the performance of the two models under diferent car-following scenarios, a subset of carfollowing samples was selected from various car-following types on the I-80 and US-101 highways.Specifcally, 300 samples with a car-following duration ≤60 s were randomly chosen.Among these, 200 samples were designated for calibration, leaving 100 for validation.A 3-fold crossvalidation method was employed.Te model calibration and validation procedures outlined in Section 4.4 were repeated for each of the four car-following types, yielding parameter calibration results as presented in Table 8.Te analysis of the results indicates notable disparities between the DVA and FVD models in terms of sensitivity coefcients α and λ.In the former, λ is signifcantly larger, while α is signifcantly smaller, compared to the latter.Moreover, both are comparatively smaller in the FVD model, signifying lower sensitivity to relative velocity and distance, as well as a reduced capacity for diferentiation between the two.Te heightened sensitivity of the DVA model to changes in visual angle is attributed to the congested trafc conditions on both roadways.Given the prevalent low speeds of drivers, adhering to the expected velocity is challenging, making the direct stimulus from the LVs a prominent infuencer on the car-following behaviors.
Furthermore, a comparison of the calibration errors for diferent car-following types between the two models is illustrated in Figure 12.Across both roadways, the DVA model exhibited a signifcant improvement in the mean calibration error in comparison to the FVD model.Additionally, the standard deviation of errors for the DVA model noticeably decreased, indicating a more concentrated error distribution.Tis reduction in error variability underscores the higher stability of the model calibration process.Furthermore, the DVA model displayed a more uniform error distribution across all four car-following types.
Using the optimal parameter means from the calibration results as model parameters, the validation process was conducted on a set of 100 samples from the validation set.Te validation results are presented in Table 9. Te outcomes reveal signifcant accuracy improvements across all four carfollowing types on the I-80 highway, with the highest increase reaching 62.0% for the L-S type.Te remaining types exhibited accuracy enhancements exceeding 50%.On the US-101 highway, accuracy improvements varied considerably among diferent types.For the S-L and L-L types, the model's accuracy increased by 35.8% and 32.3%, respectively.In contrast, for the S-S and L-S types, the model's accuracy is improved by 46.1% and 44.1%, respectively.Tis indicates that the ftting accuracy of the DVA model is more signifcantly enhanced when following a small LV.As illustrated in Figure 13, the DVA model incorporating the visual angle variable demonstrated substantial improvement in ftting efectiveness under various car-following types, showcasing its adaptability and stability across diferent types of car-following combinations.

Sensitivity Analysis of the Errors.
With the multitude of existing car-following models, a unifed evaluation standard for model performance remains lacking.To investigate the performance of both models under diferent evaluation criteria, by assigning diferent weights to the space headway and speed, we improve the traditional error functions seeing in equations (11) and (12).A series of values are set for w 1 and w 2 , and 400 samples are randomly selected from I-80 road and US-101 road for calibration.Te calibration results are illustrated in Figure 14.It is evident from these results that when w 1 is equal to 0 (at this moment, w 2 equals 1), considering only the headway as the error indicator, both model errors for all four scenarios reach their maximum.As w 1 increases, the errors for both models gradually decrease.When w 1 equals 1 (w 2 equals 0), with consideration solely given to the following vehicle's speed, the error reaches its minimum.
Further comparison reveals that the FVD model exhibits substantial discrepancies in ftting results between the two road types, while the DVA model's performance remains similar across both road types.Tis suggests that the DVA model displays higher adaptability under varying road conditions.As w 1 increases, the error of DVA model decreases slowly, indicating its overall stability, while the FVD model demonstrates a more pronounced decline.Tis signifes that the DVA model boasts greater robustness against diferent error indicators, resulting in a more consistent model performance.To delve into this phenomenon, a comparison of errors for diferent indicator weights and combinations is conducted, as illustrated in Figure 15.Tis analysis reveals that under varying weights, the FVD model shows signifcant diferences in error outcomes among     Journal of Advanced Transportation

Conclusions
Tis study is based on heterogeneous car-following segments extracted from the NGSIM dataset.Visual characteristic variables are extracted for numerical simulations and compared with traditional car-following variables to investigate the diferences in heterogeneous car-following behaviors.Statistical analysis reveals substantial variability in driver behavior within heterogeneous car-following scenarios.Furthermore, the use of visual characteristic variables efectively refects the visual stimuli experienced by drivers when following larger vehicles.In contrast to traditional distance variables, these visual stimuli exhibit more pronounced diferences across diferent car-following types, emphasizing the signifcance of incorporating driver visual characteristics in the study of heterogeneous carfollowing behaviors.
To evaluate the merits of modeling driver visual characteristics in comparison with traditional car-following variables, both an improved DVA model and an FVD model were calibrated and validated.Te results demonstrate that the enhanced DVA model signifcantly outperforms the FVD model.
Te calibration results for diferent car types and the sensitivity analysis of errors reveal that the DVA model, based on driver visual characteristics, exhibits high adaptability and stability across diverse road conditions, vehicle types, and various error metric weights.Tis indicates the model's potential for broader application and implementation.Terefore, investigating micro-driving behaviors from the driver's perspective, analyzing physiological and psychological characteristics during driving, refning carfollowing modeling theories, and addressing the challenges of heterogeneous car-following are of paramount importance.
It should be noted that this study solely focuses on improving the input variables of the FVD model, which yielded signifcant improvements.However, the potential infuence of model structure on diferent variables cannot be ruled out.Further experimentation is needed for other commonly used models such as the Gipps model and the Wiedemann model.Additionally, the NGSIM dataset features high trafc fow on both roadways, typically involving car-following distances below 50 meters.Drivers are subjected to substantial visual stimuli in such scenarios.As carfollowing distances increase further, driver stimuli tend to diminish.Analyzing the changing characteristics of driver

Figure 3 :
Figure 3: Schematic diagram of driver visual angle calculation.

Figure 4 :
Figure 4: Comparison of distribution before and after data cleaning.(a) Visual angle before cleaning.(b) Visual angle change rate before cleaning.(c) Visual angle after cleaning.(d) Visual angle change rate after cleaning.

Figure 5 :
Figure 5: Numerical simulation of visual angle under diferent vehicle widths.

Figure 6 :
Figure 6: Distribution characteristics of visual angle change rate.(a) Numerical simulation of visual angle change rate under diferent vehicle widths.(b) Evolution characteristics of visual angle change rate under diferent vehicle widths.

Figure 9 :
Figure 9: Correlation coefcient and reaction time distribution.(a) Distribution of correlation coefcients.(b) Distribution of response time.

Figure 10 :
Figure 10: Cumulative distribution of the overall calibration and validation errors.(a) Cumulative distribution of calibration errors.(b) Cumulative distribution of validation errors.

Figure 12 :
Figure 12: Calibration results of diferent car-following combinations.(a) Distribution of calibration error at I-80.(b) Distribution of calibration error at US-101.

Table 1 :
Sample statistics of efective following fragments.

Table 2 :
Sample statistics of heterogeneous car-following segments.

Table 3 :
Statistics of average headway and visual angle under diferent following models on I-80 road.

Table 5 :
Results of one-way analysis of variance of following behavior of heterogeneous vehicle on I-80 road.

Table 6 :
Overall calibration results of model.

Table 7 :
Overall validation results of the model.

Table 8 :
Calibration results of heterogeneous car-following model.

Table 9 :
Verifcation results of heterogeneous vehicle-following model.