Time series modeling method is a core component in traffic flow prediction, aiming to capture the dynamic evolution of traffic patterns. Early RNN-based approaches successfully modeled sequential features, but struggled with long-term dependencies (e.g., shifts from morning peak to midday off-peak periods) due to vanishing gradients. LSTMs addressed this limitation through gate mechanisms: the forget gate filters redundant information during low-flow periods, while the input and output gates preserve peak-flow features. For instance, the D-STN model [3] integrates convolutional operations with gating mechanisms, improving accuracy in modeling 12-h traffic flow trends on urban main roads. Faraz Malik Awan et al. [16] applied LSTM to multi-source data, integrating traffic flow and noise concentration, thereby reducing short-term prediction errors under rainy conditions. GRUs simplify the architecture while preserving essential functionality, making them more effective for short-term, high-frequency fluctuation scenarios. DeepTP [5] employs GRU to improve short-term prediction for 5-min interval data, while STGNN-FAM [6] uses bidirectional GRUs to capture morning and evening rush-hour patterns, enhancing accuracy during transition periods.

Beyond temporal dependency, complex spatial correlations within transportation networks are equally critical for traffic flow prediction. Early approaches introduced CNNs, which were originally designed for image processing to capture spatial patterns. However, the inherent non-Euclidean structure of transportation networks fundamentally limits grid-based representations, inducing topological distortion and connectivity loss. Consequently, graph-based modeling methods have become mainstream. GCNs capture non-Euclidean spatial dependencies by defining convolution operations directly on graph structures. DCRNN [17] employs diffusion convolution to model traffic flow propagation in directed graphs, enhancing spatial correlation representation. However, its reliance on fixed adjacency matrices fails to adapt to structural changes in dynamic traffic environments. To address this, T-GCN [18] integrates graph convolution for spatial feature extraction, whereas AGCRN [7] leverages adaptive graph learning to dynamically generate adjacency matrices. DSTGCN [19] constructs dynamic spatio-temporal graphs to characterize road network interactions, substantially improving accuracy. Despite these advances, modeling large-scale dynamic graphs remains computationally intensive.

Modeling exclusively either the temporal or spatial dimension cannot fully capture traffic flow complexity. Consequently, spatio-temporal fusion methods have emerged as a prominent research focus. Early approaches cascaded temporal and spatial modules sequentially, whereas advanced methods employ spatio-temporal graph convolutional networks to jointly model spatial adjacency and temporal dynamics. ST-CGCN [10] applies dynamic graph convolution for spatial feature extraction and LSTM for temporal dependencies, while leveraging complex graph structures to represent nonlinear interactions. STFGCN [11] introduces hierarchical graph convolution with adaptive fusion to model multi-level associations efficiently. The emergence of Transformers has significantly advanced traffic prediction. Transformer use multi-head self-attention to model global dependencies, enabling parallel long-sequence processing while overcoming the vanishing gradient and sequential computation limitations inherent to RNN. Unlike recursive architectures, Transformer excel in modeling non-stationary patterns and complex spatio-temporal dependencies. For instance, Bi-STAT [15] separates spatial and temporal Transformer modules, Traffic Transformer [13] integrates GCNs for better spatial perception, and GMAN [14] adopts graph-based multi-head attention to enhance dynamic pattern representation.

Although Transformer effectively capture global spatio-temporal dependencies, they model frequency-domain features inadequately. Multi-frequency information is essential: low-frequency components reflect long-term trends, whereas high-frequency components capture short-term fluctuations and anomalies. Traditional Transformer emphasize low-frequency trends but often diminish high-frequency signals [20]. In contrast, convolutional architectures such as TCNs better preserve high-frequency details [21]. To bridge this gap, Feng et al. [22] proposed a two-layer routing attention mechanism combining patch merging for multi-scale low-frequency extraction with MLPs for short-term high-frequency variation, achieving preliminary frequency-domain fusion. Nonetheless, fully integrating multi-frequency features while retaining spatio-temporal dependency modeling remains an urgent challenge.

Traffic flow prediction research has progressed from single-dimensional to multi-dimensional, multi-module collaborative modeling, employing methods such as RNNs, GCNs, STGCNs, and Transformers. These approaches have advanced spatio-temporal feature extraction, global dependency modeling, and dynamic structure learning. Nevertheless, persistent challenges impede the effective integration of multi-frequency components and the insufficient representation of dynamic relationships. These gaps necessitate an enhanced framework and form the theoretical foundation for the Bi-STAT+ model proposed in this study.

The Bi-STAT+ model builds upon the Encoder–Decoder architecture of Bi-STAT [15], optimizing critical components of spatio-temporal feature modeling in traffic flow prediction to address the limitations of existing methods.

As illustrated in Fig. 1, Bi-STAT+ consists of a Spatio-Temporal Adaptive Embedding (STAE) module, an encoder, a cross-attention module, and a decoder. STAE module enhances adaptability to spatio-temporal features and improves the efficiency of capturing complex spatio-temporal dependencies. Both encoder and decoder upgrade the original temporal and spatial Transformer structures to better address spatial heterogeneity and temporal multiscale characteristics in traffic flow prediction. The encoder contains multiple Spatio-Temporal Adaptive Transformer units, each of which integrates a Spatial Adaptive Agent Transformer (SAAT), a Frequency Domain Enhanced Temporal Adaptive Transformer (FDETAT), and an entanglement module for modeling spatio-temporal sequence interactions. The decoder adopts a dual-branch architecture that includes a prediction branch and a recall branch. The prediction branch performs similar functions to the encoder. Notably, the recall branch regularizes the model by learning historical traffic representations, omitting the DHM module to focus more on future traffic flow prediction.

Bi-STAT+ accepts road network spatial topology and traffic flow data as inputs, denoted by X. First, the STAE module learns efficient feature representations from the input, producing representative spatio-temporal features STAe. The encoder then jointly models spatial topology and temporal sequences, capturing complex dependencies in traffic flows to generate Xst. Based on this, the cross-attention module establishes an efficient information exchange between the encoder and decoder, comprising past–present and present–future cross-attention branches. The past–present branch outputs Xpp, enabling the decoder to fully exploit historical information and mitigate error accumulation. The present–future branch outputs Xpf, assisting in accurate future traffic prediction. Finally, the model is trained using a dual-branch structure comprising a prediction decoder and a recall decoder. The prediction decoder focuses on future traffic flow, outputting YP. The recall decoder reconstructs historical information to reduce overfitting, outputting YR. This design enhances robustness and predictive accuracy, ultimately enabling high-precision traffic flow forecasts.

Overall, compared with the original Bi-STAT, Bi-STAT+ replaces the original Temporal Adaptive Transformer with FDETAT, enhancing the modeling of multiscale temporal features. Instead of the original Spatial Adaptive Transformer, it employs SAAT to dynamically capture local and global interactions among road network nodes, improving the efficiency of spatial dependency modeling.

The STAE module takes temporal data and road network structural data as input, and generates a unified feature representation for the model. As shown in Fig. 2, the module is composed of spatial embedding, adaptive reconciliation, and temporal embedding. The implementation details are provided in Algorithm 1.

The spatial embedding branch captures spatial characteristics of the road network to model complex spatial dependencies among traffic sensors. A traffic network graph is constructed from actual road network data, where nodes denote traffic sensors and edges indicate road connectivity. Pairwise distances between all traffic sensors are then computed to better reflect actual traffic flow propagation paths. These distances are normalized using a Gaussian kernel function [23] to generate an adjacency matrix that captures spatial correlations between sensors. The node2vec method [24] maps graph nodes (traffic sensors) into low-dimensional feature representations for spatial embedding. These embeddings are then processed through two fully connected layers to align them with the target output dimension, yielding the spatial embedding Se.

The adaptive reconciliation branch does not process input data directly but provides a dynamically adjustable embedding mechanism to enhance the extraction of complex spatio-temporal features. It optimizes embeddings based on data variations by training a reconciliation matrix, improving adaptation to spatio-temporal dependencies in diverse scenarios. Specifically, a learnable parameter tensor of shape (time length, number of nodes, adaptive embedding dimension) is created as the reconciliation matrix and initialized using the Xavier method [25]. The reconciliation matrix is then transformed through two fully connected layers to adjust its dimension and map it to the target output dimension. Finally, the Mean module averages the feature tensors—output from the fully connected layers—over spatial dimensions (different locations) and temporal dimensions (historical, current, and future steps), yielding the adaptive reconciliation representation Ah with consistent dimensions.

The temporal embedding branch captures periodic patterns in traffic data to model recurring traffic flow behaviors. For each time step, two period embedding matrices are generated: day-of-week and time-of-day. These matrices represent weekly and daily periodic features, respectively. They are concatenated to form a comprehensive periodic feature matrix, integrating information across multiple time scales. The concatenated matrix is then fed into two fully connected layers. After a nonlinear transformation, the features are mapped to the target output dimension, producing the periodic embedding Te.

The spatial embedding Se, adaptive reconciliation Ah, and temporal embedding Te are summed to produce the fused spatio-temporal embedding STAe. This fusion enables Ah to dynamically adjust Se and Te, learning optimal feature representations from input context during training. This mechanism strengthens spatio-temporal embedding capability, thereby improving predictive performance.

Traffic flow data typically exhibit complex long-term trends alongside short-term fluctuations. However, the original Temporal Adaptive Transformer primarily models overall temporal dynamics, limiting its ability to capture features across multiple time scales. Moreover, traffic data often contain high-frequency noise (e.g., unexpected events, missing data), which distorts short-term trends and degrades overall prediction accuracy, thereby reducing model robustness and stability. To address these limitations, the FDETAT extends the original structure by integrating the Frequency-Enhanced Channel Attention Mechanism (FECAM) [26]. This integration strengthens the model’s ability to capture both long- and short-term traffic flow features via frequency-domain enhancement.

As shown in Fig. 1, the FDETAT comprises five key components: Multi-Head Attention, FECAM, Add & Norm, Transition Function, and DHM. The architecture takes the temporal feature Xtem as input and learns global dependencies through Multi-Head Attention, capturing correlations between different time steps. Subsequently, FECAM extracts both long- and short-term temporal features, enhancing the model’s perception of diverse temporal patterns. A residual connection followed by normalization transforms the temporal features into optimized representations, denoted as X^t. The Transition Function module then adjusts the feature space to maximize information flow, enabling efficient transformations between temporal features. Finally, the DHM component selectively halts or continues information delivery, prioritizing task-relevant data. This module improves information-processing efficiency and allows dynamic adjustment of computational steps based on task complexity. Ultimately, the optimized temporal features Xt are used in subsequent decoding stages to enhance task-specific performance.

As the core of FDETAT, FECAM processes traffic flow data in the frequency domain using the Discrete Cosine Transform (DCT) [27]. Through this transform, FECAM models temporal characteristics at multiple frequency scales, enhancing the model’s ability to capture complex spatio-temporal patterns. Low-frequency components reveal long-term trends and periodic patterns (e.g., day–night cycles, peak hours), whereas high-frequency components capture short-term fluctuations and localized changes (e.g., traffic accidents, missing data from sensor failures). FECAM compensates for the limitations of traditional multi-attention mechanisms in frequency-domain information extraction, thereby improving adaptability and robustness in complex traffic scenarios.

The overall architecture of FECAM is illustrated in Fig. 3. The input to FECAM is the traffic flow feature T, comprising time series data for N nodes, each of length L, representing traffic flow observations at discrete time points. Specifically, T is split along the node dimension into N subsequences {v0,v1,…,vN−1}, where each vi denotes the time series of a single traffic node. Each vi is transformed using the DCT to obtain its frequency representation Freqi. The DCT maps time-domain data to the frequency domain, enabling the model to capture cyclical patterns—information crucial for modeling periodic trends in traffic flow. The frequency representations {Freq0,Freq1,…,FreqN−1} from all nodes are stacked to form the complete frequency-domain tensor Freq, preserving frequency information across nodes. This tensor Freq passes through a fully connected layer to learn task-specific frequency-domain features, followed by normalization to produce an importance weight matrix that quantifies the contribution of each frequency component to prediction. The computed importance weights are applied element-wise to T, amplifying beneficial frequency components while suppressing irrelevant or noisy ones. The computational procedure of FECAM is formally defined in Eq. (1): (1)T′=T×(Norm(FC(Stack(DCT(Split(T))))))

By incorporating attention in the frequency domain, FECAM focuses on critical frequency components in time-series data, thereby improving traffic flow prediction and enhancing the capture of both periodic and trending patterns.

Spatial relationships in traffic flow data typically show strong localization, where traffic in a given road segment is mainly influenced by its neighboring segments, and correlations diminish as spatial distance increases. In traditional spatial adaptive transformers, the multi-head attention mechanism assigns weights to all nodes to capture global relationships. However, this global allocation can cause weight dispersion, reducing the ability to effectively capture critical local interactions.

As illustrated in Fig. 1, the SAAT model replaces conventional multi-head attention with the Agent Attention mechanism [28]. The module receives the spatial feature matrix Xspa as input, applies Agent Attention to adaptively learn the importance of different regions, and dynamically reallocates computational resources, thereby improving its ability to capture both local and global dependencies. After residual concatenation and normalization, the spatial features are transformed into optimized representations X^s. A transition module then adjusts the feature representation space to optimize information transfer and ensure effective mapping between spatial dimensions. Finally, the DHM component selectively halts or continues information transfer, enabling the model to dynamically focus on key features relevant to the current task. The resulting optimized spatial feature Xs serves as the final representation for subsequent decoding stages, enhancing performance, particularly in complex spatio-temporal dependency scenarios.

In Agent attention, the traditional attention triplet (Q,K,V) is extended to a quadruplet (Q,A,K,V), introducing an additional agent vector A for aggregating information from K and V, and then transferring to Q to model global information. The overall structure of Agent Attention comprises two standard Softmax Attention operations and is mathematically equivalent to a generalized linear attention mechanism. In this way, Agent attention seamlessly integrates high-performance Softmax Attention with efficient linear attention, maintaining sensitivity to local influences while preserving a global understanding of the entire transportation network.

Fig. 4 depicts the detailed workflow of the Agent attention module. The process begins with a linear transformation of the input traffic network adjacency matrix S, generating the query (Q), key (K), and value (V) matrices for subsequent attention computation. The query matrix Q then undergoes pooling to extract Agent Tokens (A), which aggregate features from multiple sensor nodes. In the first stage, Agent Features are computed as shown in Eq. (2). The Agent Tokens A serve as new queries in a Softmax Attention operation with the key matrix K, producing Agent Features Oagent that enhance the model’s perception of critical traffic patterns. (2)Oagent=softmax(AKTd+B1)⋅V

Next, perform a Softmax Attention operation taking Oagent as value and A as key, and keeping the original Q as query, which strengthens global feature interactions and yields the intermediate feature O, as Eq. (3) shows. (3)O=softmax(Q(A)Td+B2)⋅Oagent

The final output Y directly fused the intermediate feature O and local feature which is processed by Depthwise Separable Convolution (DWC), as shown in Eq. (4). (4)Y=O+DWC(V)

This workflow combines a two-stage Softmax Attention mechanism with a local fusion stage via deep convolution, enabling the model to capture both global context and local traffic dynamics.

This study employed four real-world traffic flow datasets for model validation, comprising two public and two private datasets. The public datasets are the widely used traffic flow prediction benchmarks PeMS04 and PeMS08 [29]. The private datasets, obtained from traffic surveillance systems in Chengdu and Baotou, China, encompass diverse traffic scenarios and complex environmental conditions, thereby enhancing the comprehensiveness and generalizability of model validation. Detailed dataset descriptions are provided in Table 1, while Fig. 5a,b illustrates the spatial distribution of sensors in the two private datasets.

During data preprocessing, a comprehensive data cleaning and processing scheme was developed. First, based on integrity analysis, the original monitoring data were screened. Data from monitoring points with high missing rates, along with their corresponding periods, were removed to ensure reliability. Second, for the filtered dataset, missing values were filled using linear interpolation. This method preserves temporal continuity and ensures spatial consistency. To ensure experimental rigor, a stratified sampling strategy was applied, dividing all datasets into training, validation, and test sets in a 7:1:2 ratio based on temporal order (see Table 1 for details). The division strictly followed temporal continuity, ensuring no overlap in time between subsets. This approach effectively prevented data leakage and provided a robust foundation for subsequent model training and evaluation.

To thoroughly assess the predictive performance of the Bi-STAT+ model, 14 representative benchmark models were selected for comparative experiments. These comprise one traditional statistical model, HA (1997) [30]; five deep learning models based on Graph Neural Networks (GNNs)—ASTGCN (2019) [8], AGCRN (2020) [7], STFGNN (2021) [31], DSTAGNN (2022) [32], and ST-CGCN (2023) [10]; and eight deep learning models based on the Transformer architecture—GMAN (2020) [14], ASTGNN (2021) [33], Traffic Transformer (2022) [13], Bi-STAT (2022) [15], MFE-STL (2024) [34], STFGCN (2024) [11], GAMAN (2025) [35], and FDGT (2025) [36].

The experimental platform runs on Ubuntu 20.04 LTS, with model implementation based on the PyTorch 2.1.1 deep learning framework. The hardware configuration comprises an Intel® Xeon® E5-2667 v3 processor (3.20 GHz), 24 GB RAM, and an NVIDIA GeForce RTX 3090 GPU (CUDA 12.2).

Following experimental validation and parameter tuning, the model training parameters were configured as follows. The batch size was set to 4 for the PeMS04 and PeMS08 datasets, and to 2 for the larger Baotou and Chengdu datasets. The encoder–decoder depth was set to two layers for all datasets, except for PeMS04, where a single layer was used due to its data characteristics. The Adam optimizer was employed with an initial learning rate of 0.001, combined with a ReduceLROnPlateau scheduler for dynamic adjustment. Training was conducted for 100 epochs, with early stopping triggered if validation performance failed to improve for 10 consecutive epochs. The DHM penalty term weight and recall decoder weight were both set to 0.001, and the reconciliation matrix dimension was fixed at 80.

To assess model prediction performance, three widely used evaluation metrics are employed: mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). All three metrics quantify the deviation between predicted and actual values, with smaller values indicating higher prediction accuracy and larger values indicating greater prediction error. The formulas for each metric are provided in Eqs. (5)–(7). (5)MAE=1n∑i=1n|yi−y^i| (6)RMSE=1n∑i=1n(yi−y^i)2 (7)MAPE=1n∑i=1n|yi−y^iyi|×100%

This study conducts ablation experiments on four datasets to evaluate the contribution of each core innovation in the Bi-STAT+ model to traffic flow prediction. Four variant models are designed to analyze the specific effects of individual components.

Variant Model 1: Removes the FECAM module and replaces the Agent Attention Mechanism with multi-head attention to evaluate the combined contribution of FECAM and Agent Attention to overall performance.

Variant Model 2: Removes the FECAM module from FDETAT, which reduces the model’s ability to capture temporal contextual information.

Variant Model 3: Replaces the Agent Attention Mechanism in SAAT with multi-head attention, thereby reducing the model’s ability to adaptively allocate attention across nodes.

Variant Model 4: Removes the Adaptive Reconciliation Embedding (AAE) module, reducing the model’s ability to embed dynamic temporal and spatial information.

Table 8 summarizes the performance of Bi-STAT+ and its variants across the four datasets, followed by the corresponding analysis.

Contribution of the FECAM Module: Removing the FECAM module led to a notable performance decline, particularly on the Baotou dataset, where the MAE rose from 49.79 to 54.80. This demonstrates that the FECAM module leverages frequency-domain analysis to jointly model high-frequency details and low-frequency trends, thereby improving the representation of multi-scale temporal features.

In real-world traffic flow data collection, sensor measurements are frequently affected by factors such as poor contact and natural aging, resulting in noise interference. Environmental conditions and limited sensor lifespans make it difficult to fully eliminate noise caused by aging. Such interference can significantly reduce the accuracy of traffic flow prediction models.

To evaluate the robustness of Bi-STAT+ in noisy environments, we compare its performance with several benchmark models on the Baotou dataset. Data loss is simulated by randomly removing 20%, 40%, and 60% of the records. Gaussian noise with a mean of 10 and a standard deviation of 500 is then added at the same proportions to mimic real-world uncertainties, including measurement errors and outliers. Tables 9 and 10 summarize the prediction performance of Bi-STAT+ and the benchmark models under different levels of missing data and noise.

The results show that Bi-STAT+ exhibits greater robustness than the benchmark models under missing data and noise conditions. With 20% missing data and noise, the MAE of Bi-STAT+ rises only slightly, while other graph-based models suffer more substantial degradation. Even as missing data and noise levels reach 40% and 60%, Bi-STAT+ maintains relatively stable accuracy, with only moderate increases in MAE. Under severe noise interference, it effectively suppresses the impact of Gaussian noise, demonstrating strong noise tolerance.

As shown in Section 4.5, prediction errors on private datasets are notably higher than on public datasets, mainly due to differences in data collection frequency. The PEMS dataset employs a 5-min sampling interval, which yields strong correlations between consecutive time steps. Consequently, a 12-step prediction covers only the next hour, thereby facilitating the capture of short-term dynamics. In contrast, private datasets are sampled hourly, greatly weakening temporal correlations. A 12-step prediction must then span 12 h of traffic flow changes, thereby amplifying cumulative errors and increasing the difficulty of the prediction.

To examine the effect of sampling frequency on prediction accuracy, we performed resampling experiments on the Baotou and Chengdu datasets. The original traffic data were resampled to 30-min, 15, 10, and 5-min intervals. Two preprocessing steps were applied: (1) linear interpolation filled in missing time steps during resampling; and (2) a moving-average method smoothed the prediction values for the final time step. Tables 11 and 12 present the prediction errors of Bi-STAT+ and the benchmark models under different sampling frequencies.

Experimental results show that as the sampling interval decreases from 1 h to 5 min, the prediction errors of both Bi-STAT+ and the benchmark model decline markedly. This supports our hypothesis that longer intervals increase prediction uncertainty and error. Shorter intervals strengthen temporal correlations, allowing the model to capture traffic flow dynamics more accurately and thereby reduce prediction errors.

Using the 5-min sampling frequency traffic flow dataset from Baotou City, two sensor nodes with distinct location characteristics—No. 98 (residential area) and No. 205 (commercial area)—were selected for analysis, with their spatial positions shown in Fig. 8a,b. Traffic flow prediction performance was evaluated for different forecast horizons (15, 30, and 60 min) on 07 July 2024 (Sunday). As shown in Fig. 9a–f, the Bi-STAT+ predictions (red dashed line) closely align with the actual traffic flow (blue solid line) over time. Across both short-term (15 min) and long-term (60 min) forecasts, Bi-STAT+ maintains high accuracy and stability across different nodes. Notably, during peak traffic periods, it captures rapid fluctuations in traffic speed and accurately reflects dynamic changes in traffic volume. Moreover, performance variations between nodes in different locations suggest that Bi-STAT+ offers advantages in handling geospatial heterogeneity.

To validate the real-time capability of Bi-STAT+, we measured its inference speed on traffic networks from Baotou and Chengdu under identical experimental settings (Section 4.3). Results show that Bi-STAT+ processes full-network predictions in 0.97 s (Baotou) and 3.88 s (Chengdu), while the baseline Bi-STAT requires 0.15 and 1.52 s, respectively. Although Bi-STAT+ incurs higher computational costs due to its enhanced architecture, both models operate well within the 5 s real-time threshold for traffic management systems. This efficiency ensures practical deployment in smart city applications without sacrificing prediction quality.