A Driving-Style-Adaptive Framework for Vehicle Trajectory Prediction

We are very grateful for the 3sDS's professional and meticulous comments. They have been instrumental in refining our presentation and strengthening the paper's scientific contributions.

S1: Add error bars to your results to demonstrate the stability of the method.

A: The evaluation metrics reported in Table 1 represent the mean ± standard deviation over five random seeds for the Argoverse (A) and nuScenes (N) datasets, and over three seeds for the Waymo (W) dataset.

Table 1. Multi-seed evaluation results on Argoverse (A), nuScenes (N), and Waymo (W). Metrics are reported as mean ± standard deviation.

S2: Integrate a summary of the limitations into the main text for a more conclusive understanding of the presented work.

(2) We will add the limitations in the main paper in the final version as follows:

Limitation We evaluate our method on prediction horizons up to 9 seconds (1 second of history and 8 seconds of future), which is the longest duration available in current open-access vehicle trajectory datasets. However, for significantly longer horizons (e.g., over one minute), direct long-term prediction may be unreliable and would likely require segment-wise modeling or hierarchical strategies. In addition, external factors such as strong conditions (e.g., traffic signals and regulatory constraints) and soft conditions (e.g., weather, which is often unlabeled in current datasets) can also affect trajectory prediction. Incorporating these contextual cues remains an important direction for future work.

S3: the authors may consider improving Figure 2 to illustrate the sequence of operations within the method in an improved manner.

A: We will revise Figure 2 by incorporating arrows to enhance clarity. On the left side, arrows will connect the highest-weighted experts in the histogram to their corresponding driving styles. On the right side, arrows will indicate the association between the predicted degrees and their respective polynomial bases.

Q1 & Significance/Originally Weaknesses: Have the authors considered incorporate the current situation in a more comprehensive (the current situation e.g.surroundings, weather, etc.) manner?

A: Driving styles reflect the behavioral characteristics of a driver [1-4], which typically remain stable over short temporal spans. Existing vehicle trajectory prediction datasets record trajectories of less than 10 seconds (Table 2). However, as you pointed out, how driving styles evolve in more comprehensive or long-term scenarios is indeed a compelling and valuable question. To address this issue, we plan to extend our work in future research as follows:

Trajectory slicing

For long-term vehicle trajectory prediction tasks (e.g., over 1 minute), we propose to segment the full trajectory into multiple shorter slices. Our DSA framework will be applied to each slice individually, and the final trajectory will be reconstructed by composing these predicted slices sequentially.

Fine-grained learning

The trajectory slices in above can be segmented with either equal or variable lengths. For segments exhibiting similar behavioral patterns, we may merge them into longer slices to reduce redundancy. Conversely, irregular or highly dynamic segments can be treated with greater attention, enabling fine-grained feature learning and more accurate predictions in critical scenarios.

Surrounding environment

Although existing datasets do not fully align contextual information (e.g., other traffic participants, weather conditions) with vehicle trajectories, we can incorporate available partial contexts and treat them as impact conditions for future feature learning. With the development of more comprehensive datasets, these contextual factors can be leveraged to enhance the accuracy and generalization capabilities of our framework.

Table 2: Comparison of existing vehicle trajectory datasets. "His" and "Pre" denote the durations of the historical and predicted trajectories, respectively, while "Total" refers to the overall recorded duration per vehicle.

[1] Yao X, et al. Driving heterogeneity identification using machine learning: A review and framework for analysis[J]. TRIP, 2025.

[2] Zheng X, Yang P, et al. Real-time driving style classification based on short-term observations[J]. IET Communications, 2022.

[3] Garrosa M, et al. Holistic vehicle instrumentation for assessing driver driving styles[J]. Sensors, 2021.

[4] Xu L, Hu J, et al. Establishing style-oriented driver models by imitating human driving behaviors[J]. IEEE TITS, 2015.

[5] https://www.cvlibs.net/datasets/kitti/.

[6] https://apolloscape.auto/.

[7] https://www.nuscenes.org/nuscenes.

[8] https://www.argoverse.org/.

[9] https://interaction-dataset.com/.

[10] https://levelxdata.com/ind-dataset/.

[11] https://levelxdata.com/round-dataset/.

[12] https://levelxdata.com/highd-dataset/.

[13] https://waymo.com/open/.

Q2: Have the authors considered scaling the proposed method by identify and incorporate driving styles in an automatic manner rather than predefining the driving styles?

A: We consider adopting (1) semi-supervised and (2) unsupervised method for automatic categorization, and (3) evaluating predefined categories under multiple values beyond our proposed three-category setting.

(1) Semi-supervised approach. Based on a set of partially predefined driving styles, we define a similarity threshold $c_{\text{style}}$ for each style. Trajectories with similarity scores below the corresponding $c_{\text{style}}$ are assigned to that predefined style. In contrast, those exceeding the threshold are categorized as newly discovered or blended styles, enabling further refinement and enrichment of the driving style taxonomy.

(2) Unsupervised approach. We extract trajectory features such as speed, acceleration and other motion attributes. Then, we employ deep clustering-based methods, such as SharPer [14] and SMYRF [15] to automatically partition the driving styles into distinct categories. The optimal number of categories is determined using standard clustering evaluation metrics [16].

(3) We evaluate multiple $k$-category settings using K-means clustering, and report the corresponding metrics (log-normalized except for Silhouette) in Table 3.

Table 3. Clustering evaluation results on Argoverse (A) and nuScenes (N). A value of “1” indicates the best performance among all settings.

Our three-category configuration yields the highest number of best scores across the metrics, supporting the validity and rationality of our chosen driving style taxonomy.

Note: For all metrics listed below, larger values indicate better clustering performance.

Note: For all metrics listed below, larger values indicate better clustering performance.

-$\Delta$WSS (Within-Cluster Sum of Squares): Measures the improvement in intra-cluster compactness. A higher value suggests tighter grouping of samples within clusters after clustering.

-Silhouette: Reflects both the cohesion within a cluster and the separation between clusters. A higher silhouette score indicates that samples are well matched to their own cluster and poorly matched to neighboring clusters.

-$\Delta$CHI (Calinski–Harabasz Index): Captures the variation in inter-cluster separability and intra-cluster compactness. Higher values indicate better-defined and more distinct clusters.

[14] Amiri M J, Agrawal D, El Abbadi A. Sharper: Sharding permissioned blockchains over network clusters[C]//sigmod. 2021.

[15] Daras G, Kitaev N, Odena A, et al. Smyrf-efficient attention using asymmetric clustering[J]. NIPS, 2020.

[16] Vysala A, Gomes J. Evaluating and validating cluster results[J]. arXiv, 2020.

Small Formatting Corrections:

-Remove the highlight on $\psi \left( x \right)$, the extra comma (Line 186), and the extra space (Line 77).

-Figure 2: Correct to …while the arrow "indicates" the direction (history or future)…

We greatly appreciate the Hx7U's detailed and constructive comments. These would help us better position our work.

Question 1: Clarify the number of training samples corresponding to each motion type in Table 5 Appendix B.2.

A: We present the rate of each motion class in Table 1, where the motion definitions follow the ClassifyTrack class schema of the Waymo dataset*.

Table 1: The b-minFDE values and motion occurrence rates on the nuScenes dataset. The best results are highlighted.

The weighted average b-minFDE is 2.719 for MTR and 2.627 for our framework, indicating that our method improves prediction accuracy by a margin of 0.092.

*Waymo: https://github.com/waymo-research/waymo-open-dataset.

Q2: Are the reported results based on the average of multiple runs, or do they reflect a single best run?

A2: The result are the average value under five (Argoverse, nuScenes) and three (Waymo) seeds. The whole results are shown in Table 2.

Table 2. Multi seeds results in Argoverse (A), nuScenes (N) and Waymo (W). The metrics are listed by average ± standard.

Weakness 1: Several typos.

A: We will modify it in final version.

W2: The paper lacks a visual analysis showing whether the system accurately (1) distinguishes between different driving styles and whether the (2) expert weights appropriately reflect those behaviors.

A: (1) Based on the characteristics of the three driving styles [1], we compute statistical measures for the datasets (Table 3). Specifically, we report the mean and variance (var) for key motion indicators, including velocity ($v$), acceleration ($a$), space headway ($S_{h}$), and time headway ($T_{h}$).

Table 3. Statistical characteristics of driving styles in the Argoverse (A-Metric) and nuScenes (N-Metric) datasets.

Note: $S_h$: Spacing provides the distance between the front centre of a vehicle to the front center of the preceding vehicle.

$T_h$: The time to travel from the front center of a vehicle to the front center of the preceding vehicle.

[1] Zheng X, et al. Real‐time driving style classification based on short‐term observations[J]. IET Communications, 2022.

(2) To reflect the relationship between expert weights and driver behaviors (styles), we compute the correct matching rate, defined as the proportion of cases where the highest expert weight (Top-1) corresponds to the expected polynomial $p_n$ under each driving style. The results are reported in Table 4, where the first letter denotes each driving style.

Table 4. The matching relationship between different driving styles and their corresponding $p_n$. The percentage indicates the rate at which each $p_n$ has the highest weight within each driving style.

W3: (1) Training time or provide key implementation details.

A: (1) The average training time per epoch (in minutes:seconds) across the three datasets is reported in Table 3. The second column indicates the number of GPUs used and their specific model (NVIDIA).

Table 3. Average training time per epoch for each dataset.

Key implementation details: The model is implemented using PyTorch. We employ single-layer KANs with three expert networks (TopK, K=3), and polynomial degrees ranging from 1 to 10. The initial weights for the distance loss and MoE-$K$ loss are both set to 0.5. Further implementation details will be provided in the final version.

W3: The (2) evaluation metrics and (3) dataset characteristics (e.g., size, preprocessing).

A(2): We will include the mathematical definitions in final version. Specifically,

-- Average / Final Displacement Error (ADE / FDE):

ADE $=\frac{1}{T} \sum\nolimits^{T}_{t=1} \| \widetilde{p_{t}} -p_{t}\|_{L^{2}}$ , FDE $=\| \widetilde{p_{T}} -p_{T}\|_{L^{2}} .$

Where $T$ represents the prediction length. For $\widetilde{p_{t}}$ and $p$ are the predicted and ground truth $(x,y)$ positions respectively.

-- Missing Rate (MR): The ratio of whether the Euclidean distance between the $\widetilde{p_{t}}$ and $p$ at the FDE for all predictions is larger than 2m.

-- Brier-minFDE ($b$-minFDE): $\text{bminFDE} = \text{minFDE} + (1 - p)^2$, where $p$ is the probability of probability of the best forecasting trajectory with minimum endpoint error (minFDE).

(3) We preprocess the three datasets using the official pip packages provided by their respective baselines. The characteristics of each dataset are summarized in Table 4. The preprocessing procedures are described below.

Table 4. Characteristics of the evaluation datasets.

[2] https://www.argoverse.org/index.html.

[3] https://www.nuscenes.org/nuscenes.

[4] https://waymo.com/open/.

[5] https://github.com/waymo-research/waymo-open-dataset.

Preprocessing is based on the corresponding official libraries. First, extract trajectory features (position, velocity, acceleration) and corresponding map elements (e.g., lane lines, traffic lights). Then, align them by timestep to ensure data integrity and output.

W4: Significance tests to assess whether the performance improvements are statistically meaningful.

A: To verify the robustness of our improvements over the baseline, we conduct two analyses: (1) comparison against existing methods and (2) self-evaluation under multiple random seeds.

(1) We perform $t$-tests against Context-Aware [6], the most recent baseline method with publicly available code. For fairness, we use the metrics reported in their original paper. The results are presented in Table 5.

Table 5. Multi-seed results on the nuScenes dataset. "No." indicates the seed value. Columns 2–5 show baseline [6] results, and the last four columns report our results.

All four metrics yield $p$-values less than 0.01, indicating that our method significantly outperforms the baseline.

(2) Meanwhile, the evaluation metrics presented in Table 6 are averaged over five random seeds for Argoverse and nuScenes and three seeds for Waymo.

Table 6. Multi-seed results on Argoverse (A), nuScenes (N), and Waymo (W). Metrics are reported as mean ± standard deviation.

[6] Pei Xu, et al. Context-aware timewise vaes for real-time vehicle trajectory prediction. IEEE Robotics and Automation Letters, 2023.

We sincerely appreciate the 5qot's insightful comprehensively consider multiple aspects of our vehicle trajectory work.

Question1: How are multi-modal predictions generated during experiments? Are multiple hypotheses sampled, or is a single deterministic output used for evaluation?

A: Generating multi-modal predictions is equivalent to producing $k$ distinct trajectory hypotheses. This is reflected by the subscript $k$ in the evaluation metrics. By independently repeating the stochastic generation process $k$ times, the model outputs $k$ diverse trajectories, from which the best prediction is selected based on the evaluation criterion.

Q2: Is the polynomial degree fixed or adaptively updated during training? How is the chosen degree used during inference?

A: The degree is adaptive during training but fixed during inference. In the training phase, our framework learns the optimal degree for each expert (corresponding to a specific driving style). During inference, these learned degrees are used in combination to obtain appropriate expert weights.

Q3 & Weaknesses 2: (1) Despite the focus on modeling driving styles, whether specific polynomial operators consistently correspond to particular driving styles in practice, missing an opportunity to demonstrate the model's key advantage.

A: We calculate the cosine similarity between the trajectory corresponding to the highest-weighted polynomial $p_n$ and the predefined driving style standards [1-4], which include Conservative (Con), Aggressive (Agg) and Normal (Nor) categories. The results on the nuScenes dataset are presented below:

Table 1. Cosine similarity reflecting consistent matching between polynomial based predictions and driving style standards on the nuScenes dataset.

[1] Yao X, Calvert S C, Hoogendoorn S P. Driving heterogeneity identification using machine learning: A review and framework for analysis[J]. Transportation Research Interdisciplinary Perspectives, 2025.

[2] Zheng X, Yang P, Duan D, et al. Real-time driving style classification based on short‐term observations[J]. IET Communications, 2022.

[3] Garrosa M, Olmeda E, Fuentes del Toro S, et al. Holistic vehicle instrumentation for assessing driver driving styles[J]. Sensors, 2021.

[4] Xu L, Hu J, Jiang H, et al. Establishing style-oriented driver models by imitating human driving behaviors[J]. IEEE Transactions on Intelligent Transportation Systems, 2015.

Q3: (2) Furthermore, it would be valuable to explore whether increasing the diversity of basis operators (e.g., adding more types of polynomials) leads to better performance or simply increases model complexity without clear gains.

A: We incorporate additional KAN variants: TaylorKAN [5], JacobiKAN [6], FourierKAN [7] and WaveKAN [8] into our KANlayer named "multi". The performance of these basis function combinations under the TopK=3 MoE setting is evaluated on the nuScenes dataset. The corresponding results are summarized in Table 2.

Table 2. Comparison of multiple basis operators with our proposed method on the nuScenes dataset under the MoE TopK=3 setting.

From Table 2, although the values of FDE$_1$ and FDE$_5$ show only marginal increases (at the thousandth level), all other metrics exhibit noticeable performance degradation compared to our proposed method. Due to the larger pool of experts, some are selected less frequently during training resulting in undertraining. This imbalance further leads to overfitting in the gating network, ultimately compromising the generalization ability of the model.

[5] https://github.com/Muyuzhierchengse/TaylorKAN.

[6] https://github.com/SpaceLearner/JacobiKAN.

[7] https://github.com/GistNoesis/FourierKAN.

[8] https://github.com/cconut/WaveKANet.

Weaknesses 1: The model's interpretability is limited, as the contribution of each polynomial expert is not visualized or analyzed across scenarios, making it unclear if driving styles are explicitly learned or used during inference.

A: We compute the proportions of each polynomial expert p_n being selected as the Top-1 weight as shown in Table 3. Here, $p_n$ consists of the Bernstein $B_n$, Chebyshev $T_n$, and Legendre $L_n$ polynomials.

Table 3. Top-1 selection proportions for each polynomial expert.

These results indicate that all experts are utilized, with none having a selection rate of zero.

W3: The justification for using Legendre polynomials to represent "normal" driving style is vague and overlaps with the conservative case, weakening the conceptual distinction between the styles.

A: We justify the use of Legendre polynomials $L_n$ to represent the Normal driving style from two perspectives: i) empirical evidence and 2) mathematical proposition analysis.

Experimental Results

We conduct ablation experiments by replacing the $L_n$, originally used for Normal (Nor) drivers with Hermite polynomials $H_n$, and replacing the Bernstein polynomials $B_n$ (for Conservative drivers) with Charlier polynomials $C_n$. The performance comparison is reported in Table 4 (Argoverse, A-Method) and Table 5 (nuScenes, N-Method), respectively.

Table 4. Replacement results on the Argoverse dataset (best results in bold).

Table 5. Replacement results on the nuScenes dataset (best results in bold).

Across the two datasets, our original configuration achieves 9 out of 10 best results. The only exception is a marginal 0.06 gap in ADE$_{10}$ on the nuScenes dataset. These results strongly support the validity of the proposed matching between driving styles and polynomial basis. Specifically, using $L_n$ for Normal drivers and $B_n$ for Conservative drivers.

Mathematical Proposition analysis

Conservative drivers typically maintain the lowest average speed among all driving styles and exhibit minimal behavioral variation. Their trajectories are characterized by smoothness and stability, with few abrupt changes. The uniform convergence property of $B_n$ ensures that the approximation error diminishes evenly across the interval, making them well-suited for modeling such stable, predictable trajectories.

In contrast, normal drivers exhibit more regular but moderate changes in behavior. Their trajectories are generally smooth with consistent fluctuations, which are neither overly abrupt nor completely static. The $L_n$ known for their orthogonality and optimal approximation under the $L^2$ norm, are particularly effective for capturing such patterns. Their ability to represent smooth variations with computational efficiency makes them a suitable choice for approximating the trajectories of Normal drivers.

Dear Reviewer 5qot,

You mentioned key concerns on the interpretability, evaluation, etc. Please check the rebuttal and provide feedback. Thank you.

Best,

AC

We thank the reviewer ejCn for raising thought-provoking questions, which explanations and significantly improve the quality of the paper.

Weaknesses & Question 1: The precise architecture of the Expert networks? How are the assigned polynomial basis functions integrated?

A: The combined feature $z$ is integrated by computing the weighted sum of the expert outputs $E$, using the gating network $G$ as weights.

In Polynomial Combination algorithm (Expert networks), each expert $E_j,(j=1,2,3)$ is implemented as a KAN-based module corresponding to a specific polynomial basis (Bernstein, Chebyshev and Legendre), each associated with a distinct driving style..

For any input vehicle trajectory $X_i (i=1,\cdots,N)$, we first compute its score with respect to each expert using the following equation:

$$H_{j}\leftarrow (X\cdot W_{g})_{i}+\mathsf{N} (0,1)\cdot \mathrm{Softplus} \left[ \left( X\cdot W_{n}\right) \right]_{i},$$

where $W_g$ and $W_n$ are trainable weight matrices (i.e., implemented as nn.Linear layers), and $\mathcal{N} (0,1)$ denotes standard Gaussian noise.

Next, we compute the weight for each expert via the Softmax function:

$G_{j}= \text{Softmax}(H)$, where $H=\left( H_{1},H_{2},H_{3}\right)$.

Finally, the combined feature representation $z^{\text{Com}}_i$ for vehicle $i$ in the DSA Polynomial Combination block is computed as a weighted sum over the experts:

$$z^{\text{Com} }_{i}=\sum^{3}_{j=1} G_{j}\left( X_{i}\right) \cdot E_{j}\left( X_{i}\right) .$$

2: (1)The CAN taxonomy ignores continuous/style-blending behaviors (e.g., moderately aggressive) and relies on heuristic categorization.

A: (1) We use three typical driving style categories to represent two behavioral extremes (Aggressive, Conservative) and an intermediate type (normal), following the convention established in prior works [1–4]. For more nuanced behaviors that fall between these categories (e.g., moderately aggressive), the weights of the three polynomial experts in the MoE-TopK network can be adaptively adjusted.

[1] Yao X, Calvert S C, Hoogendoorn S P. Driving heterogeneity identification using machine learning: A review and framework for analysis[J]. Transportation Research Interdisciplinary Perspectives, 2025.

[2] Zheng X, Yang P, Duan D, et al. Real-time driving style classification based on short‐term observations[J]. IET Communications, 2022.

[3] Garrosa M, Olmeda E, Fuentes del Toro S, et al. Holistic vehicle instrumentation for assessing driver driving styles[J]. Sensors, 2021.

[4] Xu L, Hu J, Jiang H, et al. Establishing style-oriented driver models by imitating human driving behaviors[J]. IEEE Transactions on Intelligent Transportation Systems, 2015.

2: (2) Real driving styles are context-dependent, but DSA uses fixed polynomial mappings.

A: (2) We justify the choice of "fixed polynomial" types using both mathematical theory (theoretical) and empirical experiment results.

Theoretical Justification

We establish a theoretical matching between the three driving styles and corresponding polynomial bases based on the behavioral characteristics of each style:

-Conservative drivers tend to maintain the lowest average speed and exhibit minimal behavioral changes. Their trajectories are smooth and stable with few abrupt accelerations or decelerations. The uniform convergence property of Bernstein polynomials ($B_n$) ensures that the approximation error decreases consistently across the interval, making them well-suited for approximating conservative trajectories.

-Aggressive drivers, in contrast, display higher speeds and more abrupt behavioral changes, resulting in trajectories that are less smooth and more prone to sudden shifts. Chebyshev polynomials ($T_n$) which minimize the maximum approximation error, are particularly effective in handling sharp variations and limiting the influence of extreme points.

-Normal drivers represent a balance between conservative and aggressive styles. Their trajectories tend to maintain moderate speeds and accelerations with regular but smooth behavioral changes. Legendre polynomials ($L_n$) known for their orthogonality and optimal approximation under the $L^2$ norm, are computationally efficient and capable of capturing continuous and moderately fluctuating trajectory patterns.

Experiment Results

We replace each original polynomial in ours with alternative types, including Charlier ($C_n$), Hermite ($H_n$) and Second-order ($S_n$) polynomials. The best-performing results (in bold) are highlighted in Table1.

Table 1. Evaluate different p_n in the DSA framework on the Argoverse dataset. The "Replace" column indicates the original → replacement mapping and the corresponding driving style.

From Table 1, our DSA framework yields the largest improvements: 27.8%, 43.6% and 37.3% in key metrics when using the original polynomial assignments. Although the combination $B_n+T_n+H_n$ results in a slightly lower FDE$_1$ (from 2.81 to 2.85, a 1.4% difference), the original DSA configuration still ranks second for FDE$_1$, confirming its overall robustness and effectiveness.

3: Evaluations focus on (1) highway-like scenarios (nuScenes/Waymo). Performance in unstructured environments (e.g., (2) pedestrians, intersections) is unverified.

A: (1) We evaluate our framework on three real-world datasets that span both urban and suburban environments: (i) Boston and Singapore (nuScenes [5]), (ii) Miami and Pittsburgh (Argoverse [6]) and (iii) Phoenix, AZ, Kirkland, and other cities (Waymo [7]). The maps provided in these datasets include diverse road geometries, including standard 3-way and 4-way intersections, as well as T-intersections.

[5] https://www.nuscenes.org/nuscenes.

[6] https://www.argoverse.org/av1.html#overview-link.

[7] https://waymo.com/blog/2019/08/waymo-open-dataset-sharing-our-self.

(2) Our current work focuses on vehicle trajectory prediction. Thank you for your valuable suggestion, we plan to extend our framework to predict the trajectories of other traffic participants, such as pedestrians and cyclists in future work.

4: Degree Adaptation is potentially misleading. The method describes using SMAC3 to solve an optimization problem for the degree $n$ based on validation set performance. This sounds like an offline hyperparameter tuning step that finds a single optimal degree for each driving style, which is then fixed. If the degree is tuned offline, calling it "adaptation" is an overstatement. If it is adapted dynamically for each sample during inference, the mechanism and its computational overhead are not explained at all.

A: We will revise Section 3.3.2 to be titled "Degree Adaptation". During the training process, our framework learns the optimal polynomial degree for each expert (corresponding to a driving style). During inference, the learned degree is used to determine suitable expert weights, analogous to how attention weights are computed in Transformer architectures.