Hybrid signal decomposition and deep learning framework for vehicle–vehicle crash forecasting

Abstract

Road traffic crashes remain a significant concern for public safety and transport systems, and addressing their adverse effects forms a foundation for safety planning and policy development. This study presents a hierarchical hybrid framework that combines signal decomposition techniques, including Variational Mode Decomposition (VMD) and Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), with deep learning models: Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), Temporal Convolutional Network (TCN), and WaveNet. The framework uses daily vehicle–vehicle crash data from Yinzhou District, Ningbo City. Among all configurations, the VMD-GRU model produced the best results, with MAE = 2.960, RMSE = 3.750, and R² = 0.984, which reflects its ability to capture complex temporal structures. In contrast, the CEEMDAN-TCN model showed the weakest performance, with MAE = 14.559, RMSE = 19.481, and R² = 0.609. Furthermore, the Wilcoxon signed-rank test confirmed that the performance of VMD-GRU differs significantly from all other models at the 5% significance level. Residual analysis indicates that VMD-GRU maintains low prediction errors and aligns more closely with actual vehicle–vehicle crash values over time. This framework provides traffic authorities with a tool to identify shifts in crash patterns, make timely policy decisions, and allocate safety resources with greater precision.

Introduction

Road crash burden in China

Road traffic crashes remain a major concern in global public health. According to the Global Status Report on Road Safety 2018 by the World Health Organization, approximately 1.35 million fatalities occur each year due to road traffic crashes, while an additional 20 to 50 million individuals suffer non-fatal injuries^1,2. In China, the road transport sector has grown rapidly alongside economic expansion and a rise in travel demand³. A steady growth in the number of motor vehicles, licensed drivers, and total road length has been observed in recent years. Between the end of 2019 and the end of 2020, the number of civil cars increased by 19.64 million and private cars by 17.82 million compared to the previous year. During the same period, the total length of highways increased by 18,560 km, including 11,400 km of newly built expressways⁴. While infrastructure has advanced significantly, road traffic safety remains a critical concern. Although traffic accidents in China have declined in recent years, the country still reports the second highest number of annual road traffic deaths worldwide⁵.

Reliable prediction of such incidents aids the formulation of effective traffic safety strategies and allows for systematic assessment of policy interventions. Time series analysis has established itself as a fundamental approach in the empirical investigation of road traffic accidents. Its utility lies in the capacity to capture temporal dynamics, seasonal fluctuations, and structural variations inherent in the crash data. As road safety remains a central issue for policymakers and transport authorities, time series models prove valuable tools to reveal temporal patterns, forecast future accident counts, and evaluate the effect of regulatory or infrastructural measures. Researchers across the world apply and refine these models to better understand accident trends and assess policy outcomes. The following review is structured in two parts: the first examines studies based on classical statistical frameworks, while the second explores more recent developments that include machine learning and hybrid approaches.

Classical time series models for road traffic crash analysis

Classical models such as Seasonal Autoregressive Integrated Moving Average (SARIMA), Autoregressive Integrated Moving Average (ARIMA), and the Holt-Winters Exponential Smoothing method have maintained widespread use in road accident forecasting due to their structural simplicity and their capacity to capture seasonality and underlying trends. For instance, the SARIMAX model was applied to monthly accident data in Iran from 2016 to 2021 to examine the influence of average speed on both fatal and non-fatal crashes⁶. The findings indicated that SARIMAX outperformed traditional univariate models such as ARIMA and SARIMA. Similarly, SARIMA and Exponential Smoothing (ES) models were used to analyze road accident counts in the Khyber Pakhtunkhwa province of Pakistan from 2009 to 2020. The results revealed that the ES model performed better on the accident data over the SARIMA model⁷. The SARIMA model was applied in Japan to assess the impact of senior driver license renewal regulations on traffic fatalities over the 2005 to 2016 period⁸. In India, SARIMA model was applied to identify recurring accident peaks over the years 2001 to 2012⁹. In the United States, an autoregressive fractionally integrated moving average model with generalized autoregressive conditional homoscedasticity (ARFIMA-GARCH) was employed to assess road fatality data in Florida from 1975 to 2018. The approach was effective in capturing long-memory features and accommodating volatility in the time series¹⁰. In Malaysia, ARIMA, Poisson Generalized Linear Model (GLM), and Negative Binomial GLM were applied to examine gender-based differences in injury patterns using road crash data from 1975 to 2012¹¹.

Furthermore, some studies have integrated exogenous variables into classical forecasting models. One investigation combined the Holt-Winters method with regression analysis to estimate crash frequencies across 31 provinces in Iran, incorporating macroeconomic and traffic-related indicators¹². Another study used SARIMA to examine how fuel prices influenced accident rates in England, Wales, and Scotland between 2005 and 2015¹³. Similarly, SARIMA was applied in Kurdistan province of Iran to evaluate accident trends under various policy measures and seasonal patterns from 2009 to 2015¹⁴.

Machine learning models for time series analysis of road traffic crash analysis

More recent research has adopted advanced methods such as Machine Learning (ML) and hybrid modeling approaches to better capture nonlinear relationships and improve the accuracy of road traffic crash forecasts. These methods are highly flexible in handling complex temporal dynamics and irregular crash patterns, which are often difficult to model using traditional statistical techniques. A study applied the Random Forest (RF) algorithm to estimate crash counts in Iran for the years 2016 to 2021 and compared its performance against the SARIMA model¹⁵. The findings indicated that RF yielded better results in short-term prediction accuracy. A hybrid model combining Elman Recurrent Neural Networks (ERNN) with SARIMA was developed to estimate road traffic data in China, which effectively captured both sequential dependencies and seasonal patterns¹⁶. Similarly, in USA, multiple modelling frameworks were assessed for crash data in Illinois, which included Periodic Autoregression (PAR), ARIMAX, neural networks, and Bayesian methods. Using data from 1975 to 2016, the study evaluated the influence of law enforcement strategies and climatic variables on crash outcomes¹⁷. A study on crash data from the City of Regina evaluated three forecasting models including Negative Binomial, SARIMAX, and Multi-Layer Perceptron (MLP). The Negative Binomial model demonstrated the highest average prediction accuracy. SARIMAX performed better for consistent crash patterns, while MLP handled more variable trends effectively¹⁸.

Rationale of the proposed research

Most of the existing studies have concentrated on monthly or annual aggregations, which tend to obscure abrupt shifts and short-term variability inherent in daily road traffic crash data. Therefore, this study presents a hierarchical framework that integrates Variational Mode Decomposition (VMD)^19,20,21 as well as Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN)^22,23,24 with advanced Deep Learning (DL) models, widely applied in road traffic analysis, including Long Short-Term Memory (LSTM)^25,26,27, Gated Recurrent Units (GRU)^28,29,30, Temporal Convolutional Networks (TCN)^31,32,33, and WaveNet^34,35. These decomposition methods extract intrinsic mode functions (IMFs) that capture distinct temporal patterns, from abrupt oscillations to long-term trends^36,37. DL models are independently trained on each IMF, and their individual forecasts are subsequently combined to produce the final prediction, which allows for better modeling of nonlinear and non-stationary crash patterns and reduces the impact of irregular noise in the original data. The proposed framework is illustrated in Fig. 1.

Fig. 1

Proposed Hybrid Signal decomposition-based DL framework for time series analysis of vehicle-vehicle crashes.

The remainder of the study is structured as follows: Sect. 2 provides the data and method, which include the crash records from Yinzhou District and the proposed hybrid forecasting framework, which applies signal decomposition techniques (VMD and CEEMDAN) and DL model (LSTM, GRU, TCN, and WaveNet). Section 3 illustrates the performance analysis results and Sect. 5 concludes the study with a summary of contributions and recommendation for future work.

Data and method

Crash data source

This study uses traffic crash data from Yinzhou District in Ningbo City. Yinzhou covers 1,346 km² and ranks first in economic output among all districts and counties in Ningbo. The dataset is based on records compiled by the traffic police of the Yinzhou Branch, Ningbo Public Security Bureau for the period from 10 April 2020 to 31 October 2021. Each record includes the date and time of the crash, geographic coordinates (longitude and latitude) of crash, crash type, cause of crash, as well as weather and environmental condition. Out of 137,931 total records, 134,394 contain valid GPS coordinates. The remaining 3,537 records are excluded due to missing or incorrect geographic information, which result from GPS signal loss or data entry issues. A spatial filter is applied to include only those records located within the official administrative boundary of Yinzhou District.

The dataset categorizes crashes into six types and illustrated in Fig. 2 as crash density heatmap. The map were generated using Python with GeoPandas (version 1.0.1; https://geopandas.org) and Matplotlib (version 3.10.5; https://matplotlib.org). A total of 22,151 involve single vehicles, 21,406 involve a vehicle and a nonmotor vehicle, 74,216 involve two or more vehicles, 1,773 involve a vehicle and a pedestrian, 1,432 involve a nonmotor vehicle and a pedestrian, and 6,315 involve two nonmotor vehicles. This classification covers 127,293 records. It is pertinent to mention that the current study focuses on crashes involving two or more vehicles, which account for 74,216 cases, or approximately 58.3% of all classified records. These crashes are distributed across a wide range of crash location type and weather conditions as shown in Figs. 3 and 4. They frequently occur on road sections, at intersections, and within built-up areas. Weather conditions such as sunny, rainy, and cloudy days are most commonly associated with these crashes.

Fig. 2

Crash density heatmap in Yinzhou District, Ningbo City; (a) Vehicle-Vehicle crash heatmap; (b) Single Vehicle crash heatmap; (c) Vehicle–Pedestrian crash heatmap; (d) Vehicle–Nonmotor crash heatmap; (e) Nonmotor–Pedestrian crash heatmap; (f) Nonmotor–Nonmotor crash heatmap.

Fig. 3

Heatmap showing the distribution of accident types across different crash location types in Yinzhou District.

Fig. 4

Heatmap showing the distribution of crash types under different weather conditions in Yinzhou District.

Hybrid signal decomposition-based DL framework for vehicle-vehicle crash analysis

This study proposes a hybrid hierarchical forecasting framework, which combines signal decomposition methods including VMD and CEEMDAN with DL models such as LSTM, GRU, TCN, and WaveNet. The core rationale lies in utilization of decomposition strategy to extract distinct temporal patterns from the vehicle-vehicle crash time series data and modeling each IMFs with DL architectures. The framework is structured in 3 stages: (1) Time Series Signal Decomposition, (2) Component-wise modeling using DL models, (3) Reconstruction of the predicted series.

Time series signal decomposition

Let $X\left( t \right)$ represent the observed daily or monthly road traffic accident counts over time $t=1,2,...,T$, where T denotes the total number of observations in the dataset. Due to its non-stationary and nonlinear characteristics, directly modeling $X\left( t \right)$may obscure essential dynamics. To address this, two signal decomposition techniques including VMD and CEEMDAN are employed to extract informative sub-components. VMD decomposes the signal into K band-limited IMFs $\left\{ {{u_k}\left( t \right)_{{k=1}}^{K}} \right\}$, by solving a constrained variational problem. The objective is to minimize the total bandwidth of all modes under the constraint that their sum equals the original signal as shown in Eq. 1.

$$\mathop {\min }\limits_{\left\{ {{{{\mu }}_{\text{k}}}} \right\},\left\{ {{{{\omega }}_{{k}}}} \right\}} \left\{ {\sum\limits_{{{k}} = 1}^{\text{K}} {\left\| {{\partial _t}\left[ {\left( {\delta \left( {\text{t}} \right) + \frac{{\text{j}}}{{{{\pi t}}}}} \right) * {{{\mu }}_{\text{k}}}\left( {\text{t}} \right)} \right]{e^{ - {{j}}{{{\omega }}_{\text{k}}}{\text{t}}}}} \right\|_2^2} } \right\}{\text{ }}s.t{{. }} \sum\limits_{{\text{k = }}1}^{\text{K}} {{{{\mu }}_{\text{k}}}\left( {\text{t}} \right){\text{ = x}}\left( {\text{t}} \right)}$$

(1)

Here, ${\omega _k}$ is the center frequency of the k-th mode, $\delta \left( {\text{t}} \right)$ is the Dirac delta function, j is the imaginary unit, and $*$ denotes convolution. The solution yields K non-overlapping IMFs, each capture distinct temporal behavior.

CEEMDAN is an extension of EMD that mitigates mode mixing by injecting white noise across multiple ensembles. The signal is decomposed iteratively into IMFs $\left\{ {IM{F_k}\left( t \right)_{{k=1}}^{K}} \right\}$ and a residual ${r_K}\left( t \right)$. The procedure begins by computing the first IMF as shown in Eq. 2.

$$W = \sum\limits_{i = 1}^n {\operatorname{sgn} \left( {{d_i}} \right)} \cdot {R_i}$$

(2)

Where, $\epsilon_{n}^{{\left( 1 \right)}}\left( t \right)$is white Gaussian noise and ${\text{E}}\left( \cdot \right)$ denotes the EMD operator.

The residual is calculated by using Eq. 3 and subsequent IMFs are computed as Eq. 4. The final reconstruction is then obtained by using Eq. 5.

$${r_1}\left( t \right) = X\left( t \right) - IM{F_1}\left( t \right)$$

(3)

$$IM{F_k}\left( t \right)=\frac{1}{N}\sum\limits_{{n=1}}^{N} {E\left( {{r_{k - 1}}\left( t \right)+\varepsilon _{n}^{{\left( k \right)}}\left( t \right)} \right)} ,{\text{ }}k=2,...,K$$

(4)

$$X\left( t \right)=\sum\limits_{{k=1}}^{K} {IM{F_k}\left( t \right)+{r_K}\left( t \right)}$$

(5)

Component-wise DL modeling

Each decomposed component ${C_k}\left( t \right)$(from either VMD or CEEMDAN) is modeled independently using a deep learning model ${F_k}$ to learn its temporal dynamics. The objective is to forecast the next value ${\hat {C}_k}\left( {t+1} \right)$as shown in Eq. 6.

$${\hat {C}_k}\left( {t+1} \right)={F_k}\left( {{C_k}\left( t \right),{C_k}\left( {t - 1} \right),...,{C_k}\left( {t - p+1} \right)} \right)$$

(6)

Where, p is the look-back window size. Four deep learning models are applied including LSTM, GRU, TCN, and WaveNet. Each DL model is described below.

• a) Long Short-Term Memory (LSTM).

LSTM networks control information flow through three gates: forget, input, and output, along with a memory cell. At each time step t, the forget gate decides how much past memory to retain as illustrated in Eq. 7.

$${f_t}=\sigma \left( {{W_f}{x_t}+{U_f}{h_{t - 1}}+{b_f}} \right)$$

(7)

The input gate and candidate memory prepare new information, which is provided in Eq. 8.

$${i_t}=\sigma \left( {{W_i}{x_t}+{U_i}{h_{t - 1}}+{b_i}} \right),{\text{ }}{\tilde {c}_t}=\tanh \left( {{W_c}{x_t}+{U_c}{h_{t - 1}}+{b_c}} \right)$$

(8)

The cell state is updated as shown by Eq. 9 and the output gate and hidden state are given by Eq. 10

$${c_t}={f_t} \odot {c_{t - 1}}+{i_t} \odot {\tilde {c}_t}$$

(9)

$${o_t}=\sigma \left( {{W_o}{x_t}+{U_o}{h_{t - 1}}+{b_o}} \right),{\text{ }}{h_t}={o_t} \odot \tanh \left( {{c_t}} \right)$$

(10)

These computations allow the LSTM to retain both recent and distant patterns, which are relevant to vehicle-vehicle crash sequences.

• b) Gated Recurrent Unit (GRU).

The GRU models simplifies LSTM by combining the forget and input gates into an update gate as shown by Eq. 11. A candidate hidden state in case of GRU is then computed as Eq. 12 and subsequently, the final hidden state is given by Eq. 13. This compact gating structure efficiently learns temporal patterns in each IMF of the vehicle-vehicle crash signal.

$${z_t}=\sigma \left( {{W_z}{x_t}+{U_z}{h_{t - 1}}+{b_x}} \right),{\text{ }}{r_t}=\sigma \left( {{W_r}{x_t}+{U_r}{h_{t - 1}}+{b_r}} \right)$$

(11)

$${\tilde {h}_t}=\tanh \left( {{W_h}{x_t}+{U_h}\left( {{r_t} \odot {h_{t - 1}}} \right)+{b_h}} \right)$$

(12)

$${h_t}=\left( {1 - {z_t}} \right) \odot {h_{t - 1}}+{z_t} \odot {\tilde {h}_t}$$

(13)

• c) Temporal Convolutional Network (TCN).

The TCN model sequential data using dilated causal convolutions, allowing for efficient learning of long-range dependencies without recurrence. For a given input sequence $\left\{ {{x_t}} \right\}_{{t=1}}^{T}$, the output of a dilated convolution at time t is defined as Eq. 14.

$${y_t}=\sum\limits_{{i=0}}^{{k - 1}} {{\omega _i} \cdot {x_{t - d.i}}}$$

(14)

where, k denotes the filter size, d is the dilation factor that determines the spacing between input elements, $\omega$are the learnable convolutional weights, and ${x_{t - d.i}}$ represents the past inputs sampled at intervals of d time steps.

The dilation ${d_l} = {2^{l - 1}}$ controls the receptive field, which expands exponentially with depth. For a stack of L layers with exponentially increasing dilations ${d_l}={2^{l - 1}}$, the receptive field is given by Eq. 15.

$$R=1+\left( {k - 1} \right)\sum\limits_{{l=1}}^{L} {{2^{l - 1}}=} 1+\left( {k - 1} \right)\left( {{2^L} - 1} \right)$$

(15)

This structure allows the TCN to capture temporal dependencies over long sequences while requiring relatively few layers. To preserve causality, zero-padding is applied so that the output ${y_t}$ depends only on input values ${x_{t^{\prime}}}$ for $t^{\prime} \leqslant t$. In this framework, each IMF ${C_k}\left( t \right)$ is modeled using an independent TCN, which maps the input subsequent $\left\{ {{C_k}\left( {t - p+1} \right),...,{C_k}\left( t \right)} \right\}$ to the forecasted value ${\hat {C}_k}\left( {t+1} \right)$.

• d) WaveNet.

WaveNet is a deep autoregressive neural architecture, which models sequential data through stacked dilated causal convolutions and gated activation units. This structure provides high-resolution temporal representation while preserving causality. The core computation at each time step t relies on a gated activation function, which is defined as Eq. 16.

$${z_t}=\tanh \left( {{W_f} * x} \right) \odot \sigma \left( {{W_g} * x} \right)$$

(16)

Where, ${W_f}$ and ${W_g}$ are convolution filters applied to the input sequence x, $\sigma \left( \cdot \right)$denotes the sigmoid activation function, and$\odot$represents element-wise multiplication.

The hyperbolic tangent captures the nonlinear transformation of the input, while the sigmoid gate regulates the signal flow through the network. Causal convolution ensures that the output at t is conditioned only on the current and past inputs, thereby respecting the autoregressive nature of time series forecasting. Dilations are applied to increase the receptive field exponentially across layers, which allows the model to learn both short- and long-range temporal structures efficiently. In our proposed framework, each decomposed component ${C_k}\left( t \right)$is passed through a separate WaveNet model. Given an input subsequence $\left\{ {{C_k}\left( {t - p+1} \right),...,{C_k}\left( t \right)} \right\}$, the model produces a forecast ${\hat {C}_k}\left( {t+1} \right)$, which is later aggregated with forecasts from other components to reconstruct the original time series.

Reconstruction of the predicted series

After forecasting each IMF independently using its assigned model, the outputs are aggregated to reconstruct the final prediction of the original vehicle-vehicle crash time series as shown in Eq. 17

$$\hat {X}\left( {t+1} \right)=\sum\limits_{{k=1}}^{K} {{{\hat {C}}_k}\left( {t+1} \right)}$$

(17)

This process combines short-term variations and long-term patterns learned separately from each mode into a single, coherent forecast.

Performance evaluation

This section evaluates the accuracy of the proposed forecasting framework using standard error metrics and a post-hoc statistical test to facilitate model comparison. The evaluation incorporates both numerical accuracy measures and statistical significance testing to determine the relative performance of the models.

Prediction accuracy metrics

To assess the performance of the reconstructed signal ${\hat {y}_t}$, the Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), and Coefficient of Determination (R²) are used as evaluation metrics. These are defined as follows in Eqs. 18–20

$$MAE=\sum\limits_{{t=1}}^{N} {\left| {\frac{{{y_t} - {{\hat {y}}_t}}}{{{y_t}}}} \right|}$$

(18)

$$RMSE=\sqrt {{{\sum\limits_{{t=1}}^{N} {\left( {{y_t} - {{\hat {y}}_t}} \right)} }^2}}$$

(19)

$${R^2}=1 - \frac{{{{\sum\limits_{{t=1}}^{N} {\left( {{y_t} - {{\hat {y}}_t}} \right)} }^2}}}{{{{\sum\limits_{{t=1}}^{N} {\left( {{y_t} - \bar {y}} \right)} }^2}}}$$

(20)

Where; ${y_t}$ is the actual value at time t, ${\hat {y}_t}$ is the reconstructed predicted value at time t, ${\hat {y}_t}$is the mean of actual values, and N is the total number of time steps in the evaluation period.

Post-Hoc statistical test

While standard metrics provide useful insights, they may not sufficiently distinguish between competitive models when performance values are close. To validate the superiority of the proposed framework over alternatives, this study applies the Wilcoxon signed-rank test³⁸, a non-parametric method for assessing paired differences. Let ${d_i}={x_i} - {y_i}$ denote the difference in prediction errors between two models across n test points. The test ranks the absolute differences $\left| {{d_i}} \right|$, ignoring zeros, and assigns the original sign to each rank. The test statistic W is computed as Eq. 21.

$$W = \sum\limits_{i = 1}^n {\operatorname{sgn} \left( {{d_i}} \right)} \cdot {R_i}$$

(21)

Where; $\operatorname{sgn} \left( {{d_i}} \right) \in \left\{ { - 1,0,+1} \right\}$ is the sign function, and ${R_i}$ is the rank of $\left| {{d_i}} \right|$.

The hypothesis formulation is as follows:

• H₀: here is no significant difference in performance between the best performing model and the model it is being compared with.

• H₁: A significant difference exists between the best performing model and the compared model.

At a significance level of α = 0.05, the test determines whether to reject H₀ based on the p-value or critical value comparison.

Results and discussion

Stationarity evaluation and trend analysis

The original time series of vehicle-vehicle crashes is shown in Fig. 5. The Augmented Dickey-Fuller (ADF) test was applied to the monthly crash data to evaluate its temporal properties. The test returned an ADF statistic of − 1.982 and a p-value of 0.290, as reported in Table 1. These values confirm that the series is not stationary and retains a unit root, which reflects the presence of long-term patterns or structural changes in crash occurrences over time. This pattern matches the overall shape of the time series and the gradual decline in average crash counts observed in yearly summaries.

Fig. 5

Daily counts of vehicle–vehicle crashes in Yinzhou District, Ningbo, from April 2020 to October 2021.

Table 1 Results of the ADF test for stationarity of the vehicle-vehicle crash time series.

The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots in Fig. 6 show a gradual decline, which confirms the non-stationary nature of the vehicle-vehicle crash time series. Instead of relying on conventional differencing to impose stationarity, the proposed framework introduces a hybrid structure that combines signal decomposition with deep learning. This design extracts relevant temporal patterns directly from the raw data and avoids strict assumptions about mean or variance stability. Further decomposition through Variational Mode Decomposition (VMD) and Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), shown in Figs. 7 and 8, respectively, yields several IMFs, each of which isolates a unique component of the series. These IMFs show short-term variation, seasonal structure, and long-term evolution, which allows the model to interpret the data across multiple temporal levels.

Fig. 6

ACF and PACF plots for daily vehicle–vehicle crash counts in Yinzhou District.

Fig. 7

VMD of daily vehicle–vehicle crash counts. CEEMDAN of daily vehicle–vehicle crash counts.

Performance assessment

The vehicle–vehicle crash dataset contains 74,207 records collected between 01/04/2020 and 31/10/2021. The experiments were conducted on a workstation with Intel i7 CPU, 32 GB RAM, and NVIDIA RTX 3080 GPU. For model development, the data were split into 70% for training (51,945 samples) and 30% for testing (22,262 samples). Each decomposed component from the original vehicle-vehicle crash time series was independently modeled using DL architectures. The VMD was carried out with the following parameter settings: the balancing parameter α = 100, time-step constraint τ = 0, number of IMFs, K = 4, DC component inclusion DC = 0, initialization method init = 1, and convergence tolerance tol = 1 × 10⁻⁷. Each IMF captured a distinct temporal frequency component from the centered vehicle-vehicle crash series. The resulting decomposition isolated short-, medium-, and long-term variations, which allowed individual modeling of the temporal behaviors embedded in the original series. In comparison, the CEEMDAN method yielded seven IMFs, which captured various oscillatory modes embedded in the same signal.

Each IMF was subsequently modeled using DL architectures including LSTM, GRU, TCN, and WaveNet. For every IMF–model pair, hyperparameter tuning was conducted using Bayesian optimization^39,40. This approach efficiently searched for optimal configurations of model-specific parameters, such as the number of hidden units, learning rate, batch size, number of layers, and convolutional kernel size⁴¹. The use of specific hyperparameter tuning allowed each network to capture the distinctive dynamics of its corresponding IMF with precision⁴². The final output was reconstructed by aggregating the predictions of all IMF-specific models.

Table 2 presents the post-reconstruction performance of each hybrid configuration combining VMD and CEEMDAN with LSTM, GRU, TCN and WaveNet. The models were evaluated using three metrics including MAE, RMSE and R². Among all the models, the VMD-GRU combination achieved the best overall results, with an MAE of 2.960, RMSE of 3.750, and an exceptionally high R² value of 0.984. This indicates that the GRU network was able to capture the decomposed temporal patterns with minimal residual error while explaining nearly all the variance in the original time series. Following closely was the VMD-WaveNet model, which also demonstrated strong performance with an MAE of 5.142, RMSE of 6.179, and R² of 0.957. This model also showed a strong ability to generalize complex temporal dynamics. The VMD-LSTM model produced moderately good performance (MAE = 5.760, RMSE = 7.641, R² = 0.934), which illustrates its recurrent structure was effective in learning the patterns embedded in the decomposed IMFs. However, the VMD-TCN model showed the weakest performance among the VMD-based group, with a higher MAE of 9.847, RMSE of 12.638, and a lower R² of 0.820, which indicated less reliable reconstruction.

On the other hand, models built on CEEMDAN decomposition show lower predictive accuracy compared to those based on VMD. Among the CEEMDAN group, the GRU model yielded the strongest performance, with an MAE of 11.766, RMSE of 14.910, and R² of 0.771. The WaveNet and LSTM models followed with slightly lower scores. However, none of the CEEMDAN-based models reached the accuracy levels achieved by the VMD-based models. The TCN model recorded the weakest outcomes, with the highest MAE and RMSE (14.559 and 19.481) and the lowest R² of 0.609, which reflects a significant gap in prediction precision.

Table 2 Performance measures of signal decomposition-based DL models.

To complement the quantitative evaluation of model performance, a series of plots illustrate the prediction behavior of the best model (VMD-GRU), across each IMF as shown in Fig. 8. These plots compare actual and predicted values for IMF 1 through IMF 4, along with the final reconstructed series, which results from the sum of all IMF-level outputs. Figure 8a presents the prediction for IMF 1, which reflects the dominant low-frequency component of the signal. The model follows the actual trend closely, especially across sections where sharp shifts appear. In IMF 2 (Fig. 8b), the GRU structure tracks moderate oscillations with precision, capturing the pattern which appears more irregular. IMF 3 and IMF 4 (Figs. 8c-d) relate to higher-frequency components and residuals, which add complexity. Despite the added complexity, the predicted values show close approximation to the observed series. The last plot (Fig. 8e) shows the final reconstruction, which aligns well with the original crash counts.

Fig. 8

Actual and predicted values for the best VMD-GRU model; (a) IMF 1, (b) IMF 2, (c) IMF 3, (d) IMF 4, (e) Final reconstructed series.

Post-hoc statistical significance test

To investigate whether the VMD-GRU model performs significantly differently from each of the other models, the Wilcoxon signed-rank test was applied. This non-parametric test was based on repeated experimental results, in which VMD-GRU was compared with each alternative model individually. The analysis aimed to assess whether the observed differences were statistically meaningful. The results indicate that VMD-GRU shows distinct behavior compared to the other forecasting models evaluated as shown in Table 3.

Table 3 Results of the Wilcoxon signed-rank test comparing VMD-GRU with other forecasting models (α = 0.05).

Prediction error and residual diagnostics

Figure 9 presents prediction error distributions which show how the predicted vehicle-vehicle crash counts correspond to the actual values across all tested models. Each subplot includes an identity line which marks perfect prediction and a best-fit regression line which reflects the trend of the model output. Models based on VMD decomposition produce plots where the predicted values align closely with the identity line. This indicates that those models capture the structure of the vehicle-vehicle crash series with higher accuracy. In contrast, models based on CEEMDAN decomposition show wider scatter around the identity line.

Fig. 9

Prediction Error plots; (a) VMD-LSTM; (b) VMD-GRU; (c) VMD-TCN; (d) VMD-WaveNet; (e) CEEMDAN-LSTM; (f) CEEMDAN-GRU; (g) CEEMDAN-TCN; (h) CEEMDAN-WaveNet.

Furthermore, the residual plots in Fig. 10 show the difference between actual and predicted vehicle-vehicle crash values for each model. These plots indicate how well each model captures the fluctuations in the observed data. VMD-based models, such as VMD-GRU and VMD-WaveNet, produce residuals which stay closer to zero. In contrast, CEEMDAN-based models yield wider residual spreads, with larger and more frequent deviations from the zero baseline. Especially, the CEEMDAN-TCN and CEEMDAN-LSTM models show more irregular and scattered residuals. The narrower spread and more stable alignment of VMD-based model errors point to stronger predictive alignment with the actual series.

Fig. 10

: Residual plots; (a) VMD-LSTM; (b) VMD-GRU; (c) VMD-TCN; (d) VMD-WaveNet; (e) CEEMDAN-LSTM; (f) CEEMDAN-GRU; (g) CEEMDAN-TCN; (h) CEEMDAN-WaveNet.

Conclusions and recommendations

This research develops a hybrid signal decomposition–driven deep learning framework for predicting daily vehicle–vehicle crashes in the Yinzhou District of Ningbo. The framework integrates VMD and CEEMDAN with four neural architectures: LSTM, GRU, TCN, and WaveNet. The crash time series is first decomposed into intrinsic mode functions (IMFs), which are then modeled individually. Among all model configurations, the VMD–GRU approach achieved the best accuracy, recording the lowest prediction errors (MAE = 2.960, RMSE = 3.750) and the highest explanatory power (R² = 0.984). Results from the Wilcoxon signed-rank test further verified that VMD–GRU significantly outperformed the alternative models at the 5% level. The proposed framework, based on component-wise learning, captures the temporal evolution of crash patterns with high fidelity to observed trends.

Limitations of the study

This study is limited to vehicle–vehicle crashes, which excludes crashes involving pedestrians or nonmotor vehicles. The model does not include external influencing variables such as traffic volume, road features, or enforcement activity, which could provide additional insights. Furthermore, the dataset is specific to Yinzhou District, which may reduce the broader applicability of the results without further validation.

Future recommendations

Future studies may broaden the scope of analysis by incorporating additional crash categories, such as single-vehicle, vehicle–pedestrian, and non-motor vehicle–related crashes, which would provide a more comprehensive understanding of overall road safety dynamics. Beyond crash types, the integration of influential external factors including road geometry, traffic flow intensity, enforcement activities, land use characteristics, socio-economic indicators, and weather severity could substantially enhance the explanatory capacity and predictive reliability of the framework. Expanding the application of the proposed approach to other districts and cities with diverse demographic, infrastructural, and policy contexts would also help assess its adaptability and general usefulness. Furthermore, an important extension lies in the incorporation of real-time traffic and environmental data streams, which could enable the framework to function as a proactive monitoring and early warning tool. Such real-time forecasting would support local authorities in implementing dynamic safety interventions, optimizing resource allocation, and improving incident response strategies, thereby strengthening the practical value of the framework for road safety management.