Air Quality Index Forecasting Based on Quadratic Decomposition and Transformer-BiLSTM—A Case Study of Beijing

Abstract

Accurate Air Quality Index (AQI) forecasting is crucial for environmental pollution control. However, the strong nonlinearity and pronounced non-stationarity of AQI time series limit the precision of single-model predictions. This paper therefore proposes an efficient new AQI forecasting model. First, the raw AQI sequence is decomposed using Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN). This is combined with Sample Entropy (SE) and K-means clustering to reconstruct high-, medium-, and low-frequency sub-sequences. For the high-frequency component, a second decomposition is performed using Variational Mode Decomposition (VMD) optimised by the Crested Porcupine Optimizer (CPO). This forms the basis for constructing a hybrid forecasting model: the CEEMDAN–SE–CPO–VMD–Transformer-BiLSTM model. Finally, the prediction error is corrected via Least Squares Support Vector Machine (LSSVM). Empirical analysis based on AQI data of Beijing in summer 2023 demonstrates that this model significantly outperforms traditional models and single-decomposition models in terms of MAE, RMSE, MAPE, and R² metrics. Cross-seasonal experiments further confirm its excellent predictive performance and robustness across the spring, autumn, and winter. This model provides a new, efficient, and reliable approach for AQI forecasting.

1. Introduction

In recent years, with the rapid advancement of China’s industrialisation and the continuous expansion of its population, air pollution has become an increasingly severe issue, exerting significant negative impacts on public health and socio-economic development [1,2]. Accurate air quality forecasting facilitates effective prevention, control, and mitigation of atmospheric pollution [3]. While individual pollutant concentrations can be used to assess air quality, the public and policymakers struggle to interpret these data to determine current air quality conditions. The Air Quality Index (AQI), however, simplifies pollutant concentrations—including PM_2.5, PM₁₀, CO, SO₂, O₃, and NO₂—into a single numerical value, providing a more intuitive reflection of overall air quality.

The atmospheric system is inherently dynamic, non-linear, and non-stationary, significantly increasing the complexity of AQI forecasting. Consequently, developing an appropriate forecasting framework is crucial [4]. Existing AQI forecasting techniques are primarily categorised into four types [5]: physical methods, statistical methods, artificial intelligence methods, and hybrid models. Physical models, based on physicochemical principles, are commonly used to simulate the dispersion and transport of atmospheric pollutants. However, the primary drawbacks of this approach lie in their high computational complexity, substantial time requirements, and limited predictive accuracy [6]. Traditional statistical methods, such as the Autoregressive Integrated Moving Average (ARIMA) model [7] and Grey Model [8], can infer short-term trends using historical sequences but struggle to capture the pronounced non-linear and non-stationary characteristics of AQI data [9]. Artificial intelligence methods, particularly deep learning approaches, such as convolutional neural networks (CNN), recurrent neural networks (RNN), long short-term memory networks (LSTM), gated recurrent units (GRU) [10], and the recently emerging Transformer model, have been widely applied in air quality forecasting due to their robust non-linear fitting capabilities and ability to learn temporal features [11]. For instance, Ansari A et al. [12] conducted a study in Azamgarh, India, using 8760 hourly air quality measurements (PM_2.5, PM₁₀, NO₂, SO₂) and meteorological data from July 2022 to June 2023, employing multiple statistical methods to demonstrate significant pollutant variations across different timescales. By comparing six deep learning models including Transformers and LSTMs, they that found feedforward neural networks (FNNs) achieved the optimal hourly AQI prediction performance (MAE 2.89, RMSE 4.99, R² = 0.9971), with PM_2.5, NO₂, and SO₂ exerting the strongest influence on their predictions. However, single-model approaches remain significantly constrained in handling the multi-scale characteristics and complex spatiotemporal dependencies inherent in air quality time series [13].

Presently, to overcome the limitations of single models in terms of accuracy and adaptability, deep learning-based hybrid models have emerged as the mainstream approach for AQI forecasting [14]. Nguyen A T et al. [15] proposed a hybrid deep learning model that integrates Attention Convolutional Neural Networks, ARIMA, LSTM enhanced by Quantum Particle Swarm Optimisation, and XGBoost. Utilising Seoul air quality data from 2021–2022, they achieved AQI prediction through a two-stage process (ARIMA fitting for linear components + hybrid model processing for non-linear components). Results demonstrated significant improvements over traditional models in terms of metrics such as MSE, MAE, and R², with superior performance at both city and site levels. Qian S et al. [16] proposed a hybrid deep learning model integrating XGBoost feature selection, Gaussian data augmentation, an improved manta ray foraging optimisation algorithm, and TCN-GRU. By synergistically enhancing spatio-temporal feature extraction and model robustness across multiple components, it significantly reduced prediction errors compared to baseline models in AQI forecasting for four cities, providing reliable support for air quality early warning systems.

With technological advancements, numerous scholars have incorporated signal decomposition techniques such as the Extended Empirical Mode Decomposition (EEMD) into AQI forecasting. By decomposing AQI data into frequency-based components through signal decomposition, these methods better capture multi-scale characteristics and enhance prediction accuracy [17]. Wang K et al. [18] developed an integrated model (IAMSSA-VMD-SSA-LSTM) for AQI forecasting. This combines an improved adaptive variant sparrow search algorithm (IAMSSA) to optimised variational mode decomposition (VMD) parameters with a sparrow search algorithm (SSA) to optimise LSTM. After decomposing the non-linear, non-stationary AQI sequence into multiple intrinsic modal functions (IMFs) and residuals (RES), they modelled these components separately. Across data from Chengdu, Guangzhou, and Shenyang, the model achieved MAE, RMSE, MAPE, and R² values of 3.692, 4.909, 6.241, and 0.981, respectively, outperforming comparison models such as LSTM and SSA-LSTM. Qian S et al. [19] proposed the STL–Metis–MHBA–TC hybrid model for AQI forecasting. This approach processes raw sequences via seasonal-trend decomposition (STL), employs the Metis algorithm to optimise hyperparameters of TimesNet and Crossformer for component prediction, and then fuses results using an enhanced honey badger algorithm (MHBA). This significantly enhances forecasting accuracy compared to Transformer-based models. Owing to AQI’s complex characteristics, including strong non-stationarity, some single-stage decomposition methods fail to effectively address high-frequency disturbances, and often retain noise interference. Others may yield suboptimal decomposition outcomes due to heavy reliance on parameters [20]. Consequently, researchers have explored two-stage decomposition strategies, combining the strengths of both approaches: initial decomposition followed by refined processing of complex components to enhance feature extraction quality [21]. Tang C et al. [22] utilised AQI data from the Beijing–Tianjin–Hebei region to propose two foundational AQI prediction models: one based on Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) with GRU, and another hybrid model incorporating VMD and Sample Entropy (SE) optimisation. The improved model achieved a maximum R² of 0.984 and a mean absolute error (MAE) of 2.476 across multiple regions, validating its reliability and applicability.

Consequently, this paper proposes a hybrid prediction model integrating CEEMDAN, SE, VMD, and Transformer–BiLSTM. The Crested Porcupine Optimizer (CPO) algorithm is employed to optimised VMD parameters, enhancing decomposition accuracy. Its performance is validated using AQI data of Beijing. Experimental results demonstrate that the CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM framework significantly outperforms traditional models in both prediction accuracy and efficiency. To further enhance prediction precision, LSSVM is introduced for error correction, and this error correction step provides crucial reference for environmental management.

The innovations and primary contributions of this study are as follows:

(1)

Proposal of a novel hybrid prediction framework: We designed the CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM hybrid prediction model. CEEMDAN decomposition enhances data utilisation, SE optimises signal reconstruction, the CPO algorithm adaptively optimises VMD parameters to enhance decomposition accuracy, Transformer–BiLSTM collaboratively captures long- and short-term temporal features, and LSSVM error correction further improves prediction accuracy. Empirical analysis demonstrates the framework’s superior performance in AQI forecasting, significantly outperforming traditional and single-decomposition models.

(2)

Validating of the hybrid framework’s superiority and robustness: Cross-season comparison experiments using AQI data of Beijing across spring, summer, autumn, and winter 2023 data confirmed the proposed model’s superiority across metrics including MAE, RMSE, MAPE, and R², alongside its universality and stability across seasonal scenarios. LSSVM error correction substantially optimised prediction outcomes.

(3)

Optimisation of VMD parameter selection: Addressing the subjective nature of selecting VMD parameters (mode number K and penalty factor $\alpha$), we introduce CPO for adaptive optimisation. This enhanced the accuracy of quadratic decomposition, providing high-quality inputs for Transformer–BiLSTM forecasting and LSSVM error correction, and thus ensuring model robustness and prediction reliability.

2. Materials and Methods

2.1. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) [23] represents an enhanced signal decomposition method derived from Empirical Mode Decomposition (EMD). By repeatedly introducing adaptive white noise during decomposition, it significantly alleviates the modal aliasing inherent in EMD. Meanwhile, its decomposition process exhibits excellent completeness, enabling precise reconstruction of the original signal. The specific steps of this method are as follows:

First, the AQI signal is decomposed into K zero-mean white noise components:

$$x_i(t)=x(t)+\epsilon {\delta }_i(t)$$

(1)

where $\epsilon$ represents a Gaussian white noise weighting factor; ${\delta }_i(t)$ denotes the $i$-th generated white noise component

Subsequently, perform EMD is performed on $x_i(t)$, treating the first decomposed modal component as the first modal component obtained from CEEMDAN decomposition:

$$IM{F}_1(t)={\displaystyle \frac{1}{K}}\sum _{i=1}^{K}IM{F}_1^i(t)$$

(2)

$$r_1(t)=x(t)-IM{F}_1(t)$$

(3)

Among these, $IM{F}_1(t)$ represents the first modal component of this decomposition; $r_1(t)$ denotes the residual.

After adding specific adaptive noise to the residuals obtained at the $j$-th stage of the decomposition, the EMD continues:

$$IM{F}_j(t)={\displaystyle \frac{1}{K}}\sum _{i=1}^K{E}_1\left\{r_{j-1}(t)+{\epsilon }_{j-1}{E}_{j-1}[{\delta }_i(t)]\right\}$$

(4)

$$r_j(t)=r_{j-1}(t)-IM{F}_j(t)$$

(5)

where $IM{F}_j(t)$ denotes the $j$-th modal component of this decomposition.

Finally, if the residual signal obtained from the $K$-th decomposition is a monotonic signal, the iteration stops and the CEEMDAN algorithm concludes.

2.2. Sample Entropy

To address the issue of the generation of numerous components after decomposition whose practical significance is difficult to determine, Richman and Moorman proposed Sample Entropy (SE) [24]. SE is a non-linear dynamic method for assessing data complexity: the simpler the time series, the smaller the SE value; conversely, the larger the SE value. SE computation consists of three steps: first, determining the subsequence length and similarity threshold; second, constructing the subsequences; and finally, calculating similarity and probability. In our experiments, we employed the SE algorithm to compute the entropy for each intrinsic mode function (IMF). Based on these entropy values, we assessed the randomness of each IMF and them as the basis for merging and reconstructing components, to generate three types of components: low-frequency, high-frequency, and trend components. To further optimise component classification, we employed the K-means clustering algorithm to categorise components into high-frequency, medium-frequency, and low-frequency groups. This approach reduces the number of components and improves computational efficiency.

2.3. Crested Porcupine Optimizer

The Crested Porcupine Optimizer (CPO) [25] is a novel bio-inspired meta-heuristic optimisation algorithm. Its core lies in constructing an optimisation framework by simulating four typical defence behaviours of crested porcupines: visual, acoustic defence, olfactory defence, and physical attack. These behaviours are hierarchically designed to balance exploration and exploitation: the first two strategies (visual and acoustic) facilitate global search, while the latter two (olfactory and physical) enhance local refinement. This design endows CPO with three critical advantages:

Strong global exploration: Effectively avoids local optima by mimicking the porcupine’s threat assessment mechanisms;

Rapid convergence: Achieves fast optimisation through cyclic population reduction, which dynamically adjusts search intensity;

Low computational cost: Reduces traversal time in complex parameter spaces via adaptive defence strategy switching.

The detailed mathematical formulations of CPO, including its position update rules and cyclic population reduction technique, are given in Reference [26]. This algorithm has been validated in various engineering applications for its robustness and efficiency in solving high-dimensional optimisation problems. The specific workflow is illustrated in Figure 1.

2.4. Variational Mode Decomposition

Variational Mode Decomposition (VMD) represents a novel approach to decomposing complex signals [27]. Unlike EMD and its variants, the VMD algorithm has rigorous mathematical derivation and strong noise robustness, thereby ensuring decomposition accuracy and stability. Consequently, VMD effectively resolves the modal aliasing issue inherent in EMD [28]. It can adaptively optimise the decomposition of non-stationary and highly complex signals into several relatively stable sub-modes. The specific steps are detailed in the Reference [29].

2.5. Transformer–BiLSTM

AQI forecasting necessitates processing multidimensional time-series data. Traditional RNNs struggle to model long-term sequence dependencies, while Transformers excel at capturing global correlations but lack sufficient short-term dynamic capture capability. Hence, this paper proposes the Transformer–BiLSTM model, which synergistically learns to extract long- and short-term features through self-attention mechanisms and BiLSTMs, thereby enhancing the precision and robustness of AQI predictions. The model structure and principles are as follows:

(1)

Position Encoding: Imbuing temporal information into the model

The Transformer first applies positional encoding to the input AQI time-series features, embedding temporal sequence information into the model. The formula are:

$$P_E\left(pos,2i\right)=\sin \left({\displaystyle \frac{pos}{{10000}^{\frac{2i}{d}}}}\right)$$

(6)

$$P_E\left(pos,2i+1\right)=\cos \left({\displaystyle \frac{pos}{{10000}^{\frac{2i}{d}}}}\right)$$

(7)

where $pos$ denotes the temporal step position and $d$ represents the embedding dimension. Position encoding enables the model to capture the temporal sequence of AQI features, preventing the loss of temporal information inherent in the Transformer’s “unordered list” processing approach.

(2)

Transformer Self-Attention: Capturing Global Dependencies

Self-attention is the core of the Transformer. The input to self-attention is represented by matrix X. Via linear transformations of the query matrix (Q), key matrix (K), and value matrix (V), it calculates the correlation weights between AQI features across different time steps (e.g., the correlation between pollutant concentrations at the current time and those at historical times, or with historical AQI values).

$$\left\{\begin{array}{c}Q=X{W}^Q\\ K=X{W}^K\\ V=X{W}^V\end{array}\right.$$

(8)

$$A\left(Q,K,V\right)=Soft \max \left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)V$$

(9)

where $A\left(Q,K,V\right)$ represents the attention score, $X$ denotes the input matrix, and $W^Q,$$W^K,$$W^V$ are the trainable matrices within the model; $Q{K}^T$ characterises the similarity between queries and keys (i.e., feature relevance); $\sqrt{d_k}$ is a scaling factor to prevent excessive inner products; Multi-Head Attention further maps features to multiple subspaces, capturing multi-scale global dependencies. The formula is:

$$on\left(Q,K,V\right)=concat\left(hea{d}_1,hea{d}_2,\dots ,hea{d}_h\right)W_{ei}$$

(10)

where $hea{d}_h$ denotes the output of the $h$-th head, and $W_{ei}$ represents the linear transformation matrix. Through multi-head attention, the model can simultaneously learn long-range correlations within AQI sequences.

(3)

BiLSTM Decoder: Learning Local Temporal Dependencies

The global feature vector processed by the Transformer serves as input to the BiLSTM, which comprises a forward LSTM and a backward LSTM: the forward LSTM processes the sequence chronologically to learn forward temporal relationships; the backward LSTM processes the sequence in reverse order to learn backward temporal relationships.

The core of LSTM lies in its gating mechanism ($f_t,$$i_t,$$o_t$). The forward computation process of LSTM is as follows:

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}{f}_t=\sigma \left(U_f\cdot h_{t-1}+W_f\cdot x_t+b_f\right)\\ i_t=\sigma \left(U_i\cdot h_{t-1}+W_i\cdot x_t+b_i\right)\end{array}\\ o_t=\sigma \left(U_o\cdot h_{t-1}+W_o\cdot x_t+b_o\right)\end{array}\\ {\tilde{c}}_t=tanh\left(U_c\cdot h_{t-1}+W_c\cdot x_t+b_c\right)\\ c_t=f_t\odot c_{t-1}+i_t\odot {\tilde{c}}_t\\ h_t=o_t\odot tanh\left(c_t\right)\end{array}\right.$$

(11)

where $x_t$ represents the AQI feature sequence (incorporating contextual information from pollutant concentrations, meteorological factors, and historical AQI data). The hidden state $c_t$ captures short-term AQI dynamics. Through bidirectional processing, BiLSTM can simultaneously learn “past → present” and “present → past” relationships, enhancing its modelling capability for short-term fluctuations.

(4)

Output Layer: AQI Prediction

The BiLSTM output is mapped to a single-dimensional prediction value via a fully connected layer, and model training is completed using regression loss. Ultimately, the model outputs the predicted AQI value for the next time step.

The Transformer–BiLSTM model provides a more accurate and robust approach for AQI forecasting through synergistic learning of “global correlations + local dynamics”.

2.6. Prediction Framework

To address the non-stationarity and complex temporal characteristics in AQI forecasting, this paper proposes an integrated AQI prediction workflow combining a two-stage modal decomposition method based on CEEMDAN-SE-CPO-VMD with an ensemble Transformer–BiLSTM model. This workflow comprises three stages: preprocessing of raw AQI data, signal processing via secondary modal decomposition based on CEEMDAN-SE-CPO-VMD, and training and testing of the integrated Transformer–BiLSTM model. The prediction steps are as follows:

(1)

Data Preprocessing

Collect historical AQI data and feature vectors (pollutant concentrations such as PM2.5, PM10, NO2, SO2, CO, O3, and meteorological factors including wind speed, wind direction, temperature, and humidity). Preprocess the raw data to construct standardised input sequences, providing a high-quality data foundation for subsequent decomposition and modelling.

(2)

CEEMDAN-SE-CPO-VMD Second-Order Modal Decomposition

The CEEMDAN algorithm to decompose the original non-stationary AQI time series into multiple IMF components. Calculate the SE for each IMF component to quantify sequence complexity. Using the SE values of each IMF as feature vectors, apply the K-means clustering algorithm to partition the IMF components into three clusters (high-, medium-, and low-frequency Co-IMF). The cluster with the highest complexity (Co-IMF1) is selected and further decomposed into finer-grained sub-components (VMD-IMF) using VMD to uncover micro-oscillation patterns. To avoid arbitrary parameter selection in VMD, CPO is employed to optimise the modal number K and penalty factor $\alpha$.

(3)

Ensemble Transformer–BiLSTM Prediction and Result Reconstruction

The sub-components obtained from VMD and the remaining clusters from CEEMDAN decomposition are fed into a Transformer–BiLSTM hybrid prediction model for training and forecasting. The Transformer captures global correlations within the AQI sequence (e.g., pollutant concentration trends across days) via self-attention mechanisms, while the BiLSTM models local temporal dependencies (e.g., short-term pollutant fluctuations). Finally, the predicted values of all subcomponents (VMD-decomposed subcomponents and remaining Co-IMFs) are linearly superimposed to reconstruct the final prediction of the original AQI time series.

This approach effectively captures both short-term and long-term dynamic characteristics of AQI sequences through integrated modelling using secondary modal decomposition and Transformer–BiLSTM, thereby enhancing prediction accuracy and robustness. The prediction model workflow is illustrated in Figure 2.

3. Date Sources and Processing

3.1. Data Sources and Preprocessing

Beijing, situated in the northern part of the North China Plain, experiences a temperate monsoon climate with distinct seasonal characteristics. Summers are hot and rainy, while winters are cold and dry. Daily temperature variations are moderate, and the region enjoys abundant sunshine. Precipitation is predominantly concentrated in summer, and spring is prone to sandstorms. Beijing’s industrial structure centres on the tertiary sector, prioritising modern services, high-tech industries, and cultural and creative industries. Heavy industry accounts for a relatively low proportion, though certain areas still host energy processing and high-end manufacturing sectors. Significant disparities exist in pollutant emissions between urban and suburban zones. Regarding pollution sources, Beijing’s atmospheric environment is primarily influenced by two factors: natural factors, notably spring sandstorms and surface dust pollution; and anthropogenic factors, chiefly motor vehicle exhaust emissions, industrial pollutant discharges, and energy consumption during the winter heating season.

This study uses hourly AQI, pollutant, and meteorological data from Beijing spanning 1 June to 31 August 2023, comprising 2208 data points. Pollutants include PM_2.5, PM₁₀, SO₂, NO₂, O₃, and CO, while meteorological factors encompass air temperature, humidity, atmospheric pressure, and wind speed. AQI and pollutant data were sourced from the National Urban Air Quality Real-time Publishing Platform of the China National Environmental Monitoring Centre (CNEMC) [Online]. Available: https://air.cnemc.cn:18007/ (accessed on 15 October 2025), while meteorological data were sourced from the National Meteorological Information Centre (NMIC) [Online]. Available: http://data.cma.cn/ (accessed on 15 October 2025).

AQI data are presented in Figure 3.

The descriptive statistics presented in

Table 1. Descriptive Statistics of AQI, Pollutants, and Meteorological Parameters.

Variable	Mean	Std. Dev.	Min	25%	50%	75%	Max	Skewness
AQI	48.84	25.44	7	31.0	43.0	59.0	150	1.299
PM2.5 (${\mathsf{\mu }\mathrm{g}/\mathrm{m}}^3$)	20.84	13.84	1	11.0	18.0	28.0	83	1.269
PM10 (${\mathsf{\mu }\mathrm{g}/\mathrm{m}}^3$)	39.86	22.66	3	22.0	37.0	53.0	140	0.901
SO2 (${\mathsf{\mu }\mathrm{g}/\mathrm{m}}^3$)	2.78	0.73	2	2.0	3.0	3.0	18	5.287
NO2 (${\mathsf{\mu }\mathrm{g}/\mathrm{m}}^3$)	16.57	8.27	4	10.0	15.0	21.8	53	1.007
O3 (${\mathsf{\mu }\mathrm{g}/\mathrm{m}}^3$)	101.90	54.90	8	61.0	93.0	131.8	299	0.836
CO (${\mathrm{mg}/\mathrm{m}}^3$)	0.45	0.17	0	0.3	0.4	0.6	1	0.577
Temperature (°C)	28.17	4.82	17	24.8	27.6	31.2	42	0.425
Humidity (%)	60.44	25.08	9	39.0	63.0	83.0	100	−0.257
Pressure ($\mathrm{hPa}$)	1001.84	4.62	986	999.0	1002.1	1005.3	1011	−0.707
Wind Speed ($\mathrm{m}/\mathrm{s}$)	2.24	1.24	0	1.4	2.0	2.8	7	1.028

offer a comprehensive overview of the dataset and reveal the dominant influence of specific environmental and meteorological parameters on the AQI during the summer of 2023 in Beijing.

The average AQI was only 48.84, with a maximum of 150, indicating air quality was predominantly excellent or good and no heavy pollution episodes occurred. Among the six criteria pollutants, ozone (O₃) exhibited the highest mean concentration (101.90 μg/m³) and by far the greatest variability (Std. Dev = 54.90 μg/m³), while primary pollutants such as PM_2.5 (mean: 20.84 μg/m³), PM₁₀ (mean: 39.86 μg/m³), NO₂ (mean: 16.57 μg/m³), and CO (mean: 0.45 mg/m³) remained at remarkably low levels. This pattern suggests that secondary photochemical processes, rather than direct emissions, were the primary drivers of AQI variations during this period.

Temperature emerged as a key meteorological driver, with a mean of 28.17 °C and moderate variability (Std. Dev = 4.82 °C), creating favourable conditions for intense solar radiation and the rapid photochemical formation of ozone from precursor pollutants. Relative humidity averaged 60.44% with marked dispersion (Std. Dev = 25.08%), reflecting frequent convective rainfall events, which effectively scavenged particulate matter through wet deposition, thereby suppressing PM concentrations. Wind speed was relatively low (mean: 2.24 m/s, Std. Dev = 1.24 m/s), indicating generally calm conditions, which limited long-range pollutant transport and favoured the local accumulation of photochemically produced pollutants. Atmospheric pressure showed minor fluctuations (Std. Dev = 4.62 hPa), consistent with the stable dominance of the East Asian summer monsoon and the absence of strong subsidence inversions characteristic of colder seasons.

Taken together, the interplay of high temperatures, moderate-to-high humidity coupled with frequent precipitation, low wind speeds, and stable pressure patterns created a meteorological regime in which temperature-driven ozone production overwhelmingly dominated AQI variability, while particulate matter was efficiently removed by wet deposition. This contextual understanding of the dataset, alongside the specific roles played by these environmental and meteorological parameters, lays a solid foundation for subsequent correlation analysis and modelling stages.

During data preprocessing, missing data caused by equipment failure were first addressed using mean imputation to ensure data integrity and minimise its impact on the model. Subsequently, outliers arising from equipment malfunctions, transmission errors, or extreme weather were detected using the Z-score method. Given the limited sample size, outliers with |Z| > 3 were corrected via threshold-based correction. Next, to eliminate the impact of differing variable dimensions on analysis and accelerate the convergence of machine learning algorithms, Min-Max normalisation was applied to scale the data to the [0, 1] range. Finally, the 2208 empirical data points were divided into a training set (1766 records) and a test set (442 records) in an 8:2 ratio in chronological order.

3.2. Feature Selection

AQI is influenced by multiple factors. with influencing factors as model inputs risks increasing computational workload and data redundancy, making feature selection crucial. Correlation analysis among features enables the identification of optimal subsets from numerous features, eliminating non-critical factors and significantly enhancing model training efficiency. Given the pronounced influence of weather conditions on air quality, Pearson’s correlation coefficient was employed to investigate the relationship between concentrations of six primary pollutants and meteorological variables. This coefficient measures the strength of data association, calculated as the covariance divided by the product of their respective standard deviations. Results range from −1 to 1, with values closer to 1 indicating a stronger positive correlation between the factor and AQI. Using the concentrations of six AQI-influencing pollutants as baseline data, combined with four categories of meteorological data, the aforementioned method was applied the aforementioned method after screening and preprocessing. This ultimately yielded a Pearson correlation coefficient heatmap (Figure 4), revealing the interaction mechanisms between pollutants and meteorological factors while enabling the selection of relevant characteristics.

As shown in Figure 4, the correlation coefficient between SO₂ and AQI is merely 0.117, while that for NO₂ is 0.065, indicating extremely weak linear correlations with AQI. The correlation coefficient between humidity and AQI was −0.361, between atmospheric pressure and AQI was −0.100, and between wind speed and AQI was 0.149. These meteorological variables also exhibited relatively weak correlations with AQI, indicating limited direct explanatory power for AQI. Conversely, PM_2.5 exhibits a moderate positive correlation with AQI (correlation coefficient 0.565), while PM₁₀ shows a strong positive correlation (0.662). O₃ demonstrates the strongest positive correlation with AQI (0.751), and CO exhibits a moderate positive correlation (0.380). Air temperature also exhibits a moderate positive correlation with AQI (0.585). Notably, while PM_2.5 and PM₁₀ demonstrate extremely strong collinearity (correlation coefficient 0.907), their influence on AQI differs due to their distinct particle size dimensions (as pollutants of different diameters). O₃ and CO show weaker collinearity with other pollutants, providing independent types of pollution information. Although air temperature exhibits strong collinearity with O₃ (correlation coefficient 0.778), as a meteorological variable it maintains a close association with AQI through unique mechanisms. Consequently, PM_2.5, PM₁₀, O₃, CO, and air temperature were selected as predictive features for AQI.

4. Results and Discussion

The experiments in this paper were conducted in MATLAB 2023b. The computer environment configuration is provided in

Table 2. Computer Environment Configuration.

Experimental Environment	Specific Settings
GPU	NVIDIA GeForce RTX 4060 Laptop GPU/Memory: 8 GB
CPU	AMD Ryzen 7 7735H with Radeon Graphics
Memory	6 GB 4800 MT/s
Default hard disc	512 GB SSD
Operating System	Windows 11

.

4.1. Evaluation Metrics

This paper employs Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), and Coefficient of Determination (R²) as evaluation metrics for assessing the performance of each model in predicting AQI. Their respective formulas are as follows:

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n|y_i-{\widehat{y}}_i|}$$

(12)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}$$

(13)

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}\times 100\%$$

(14)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}$$

(15)

where $n$ denotes the number of samples; $y_i$ represents the true value of the $i$-th sample; ${\widehat{y}}_i$ signifies the predicted value of the $i$-th sample; and $\overline{y}$ indicates the mean of the true values.

4.2. Parameter Configuration

The parameter configurations were determined based on multiple experimental validations and reviews of relevant literature to ensure the rationality and effectiveness of model performance (

Table 3. Model Parameter Settings.

Model Name	Parameter Name	Parameter Settings
CEEMDAN-SE-CPO-VMD	CEEMDAN Standard Deviation	0.2
	CEEMDAN Number of Ensembles	500
	CEEMDAN Maximum Number of Iteration	5000
	SE dim	2
	SE r	0.2
	CPO α Optimisation Range	[100, 2000]
	CPO k Optimisation Range	[2, 10]
	CPO Initial Population Size	30
	Maximum Number of Iterations	50
	Convergence Rate	0.2
Transformer–BiLSTM	Number of Multi-Head Attention Heads	4
	Key/Value Number of Channels	128
	Attention Mask	Causal
	Number of Hidden Units in BiLSTM Layer	50
	Positional Encoding Embedding Dimension	256

).

4.3. CEEMDAN-SE-CPO-VMD Data Decomposition

First, the AQI dataset was decomposed using the CEEMDAN algorithm. The decomposition results of the AQI dataset following CEEMDAN are shown in Figure 5.

As shown in Figure 5 following CEEMDAN decomposition, the original AQI time series is decomposed into 11 IMF components sorted by frequency from high to low. Among these, the high-frequency components exhibit higher noise content and more complex time-domain structures, making it challenging for models to adequately capture fluctuation characteristics across various time scales. In light of this, we first calculate the SE for each IMF component to quantify its complexity. The SE results for each IMF are presented in

Table 4. SE Values for Each IMF Component.

IMF Component	SE	IMF Component	SE
IMF1	1.06112	IMF7	0.20416
IMF2	1.07190	IMF8	0.10191
IMF3	0.84298	IMF9	0.04570
IMF4	0.51092	IMF10	0.01462
IMF5	0.51110	IMF11	0.00360
IMF6	0.44369

.

Based on the SE calculation results in Table 4, the 11 IMF components were fed into the K-means algorithm with the number of clusters set to 3. The three modal clusters obtained from clustering were designated as Co-IMF1, Co-IMF2, and Co-IMF3: the high-frequency group Co-IMF1 comprises IMF1, IMF2, and IMF3; the mid-frequency group Co-IMF2 comprises IMF4, IMF5, and IMF6; and the low-frequency group Co-IMF3 consists of IMF7–IMF11. After completing K-means clustering, for the high-frequency component Co-IMF1 obtained from clustering, the VMD algorithm was further employed to perform in-depth secondary modal decomposition on it. The CPO algorithm was used to optimise the parameters of VMD (K and $\alpha$), yielding K = 6 and $\alpha$ = 110.08 after CPO optimisation. The secondary decomposition via CEEMDAN-SE-CPO-VMD is shown in Figure 6.

The secondary decomposition via CEEMDAN-SE-CPO-VMD not only reduced data complexity but also extracted key features, effectively capturing both short-term fluctuations and long-term trends in AQI. The data after secondary decomposition was then fed into the Transformer–BiLSTM model for prediction.

4.4. CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM Prediction Results

4.4.1. Comparison of Different Models

To validate the superiority of the CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM model, four reference models (BiLSTM, Transformer, Transformer–BiLSTM, and CEEMDAN–SE–Transformer–BiLSTM) are selected for the comparison of experimental results, as follows:

A comprehensive analysis of Figure 7 and Figure 8 reveals that the proposed CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM model demonstrates outstanding performance in AQI forecasting. The prediction fit plot demonstrates that the model’s forecast curve aligns most closely with the true value curve, accurately tracking AQI trends across varying fluctuation ranges. The percentage error plot further indicates that this model exhibits the narrowest error band with the most concentrated distribution, significantly outperforming other comparison models. Combined with the quantitative metrics in

Table 5. Evaluation Metrics for Each Model.

Model	MAE	MAPE	RMSE	R2
BiLSTM	2.6875	6.8627%	3.8892	0.9427
Transformer	2.6463	6.8830%	3.7046	0.9480
Transformer–BiLSTM	2.4445	6.2439%	3.5867	0.9512
CEEMDAN–SE–Transformer–BiLSTM	1.3737	3.5598%	2.0378	0.9843
CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM	0.7963	2.0808%	1.0544	0.9958

, this model achieved the lowest values for MAE (0.7963), RMSE (1.0544), and MAPE (2.0808%), alongside the highest R² (0.9958). This indicates not only high fitting accuracy but also robust predictive stability.

Figure 9 presents the Taylor diagram of the prediction results of all models, intuitively illustrating the agreement between the model predictions and the actual data. The CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM model exhibits the smallest angle with the observation point and the highest correlation coefficient with the observed values, while its standard deviation is close to that of the observed values—indicating that the output range of this model is in good agreement with the observed data. Although the standard deviations of other models are also relatively close to those of the observed values, their correlation coefficients with the observed values are lower than that of the CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM model.

The results demonstrate that incorporating preprocessing methods—including CEEMDAN decomposition, SE-based clustering, and CPO-optimised VMD—effectively enhances the model’s ability to capture complex temporal features and non-linear patterns within AQI sequences, thereby significantly improving predictive performance.

4.4.2. Error Correction Strategy

Existing research confirms that error correction strategies markedly enhance model predictive performance [30,31]. Inspired by this body of research, this study employs the Least Squares Support Vector Machine (LSSVM) [32]—a model characterised by its simple structure and compact parameter set, to predict the error sequence generated from the combined model’s prediction results, thereby achieving error correction. The final corrected prediction is obtained by integrating the error prediction results with the initial prediction results of the combined model. The specific experimental steps are illustrated in Figure 10 (see also

Table 6. Evaluation Results After Error Correction.

Model	MAE	MAPE	RMSE	R2
CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM	0.7963	2.0808%	1.0544	0.9958
CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM–LSSVM	0.7102	1.8543%	0.9154	0.9968

).

Following the incorporation of the LSSVM error correction strategy into the CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM model, all prediction accuracy metrics were further enhanced. For the error-corrected model, MAE was reduced from 0.7963 to 0.7102, RMSE from 1.0544 to 0.9154, and MAPE from 2.0808% to 1.8543%, while R² was increased from 0.9958 to 0.9968. This indicates that the LSSVM-based error correction method effectively identified and rectified the systematic bias in the original combined model. By predicting and integrating the error sequence, the final output results were further optimised, validating the effectiveness of the error correction strategy in enhancing AQI prediction accuracy.

4.4.3. Seasonal Validation

AQI values across different seasons within the same city exhibit significant spatio-temporal variations influenced by meteorological conditions and emission source intensities, with their fluctuation characteristics directly challenging the model’s environmental adaptability. Having previously validated the baseline model performance using Beijing’s summer 2023 data (June–August), we supplemented the validation with seasonal data from Beijing’s spring 2023 (March–May), autumn (September–November), and winter (December 2023–February 2024). The supplementary data included AQI values alongside corresponding pollutant concentrations and meteorological factors. The cross-seasonal validation results are presented in

Table 7. Evaluation Metrics for Each Model Across Different Seasons.

Season	Model	MAE	MAPE	RMSE	R2
Spring	BiLSTM	6.4208	7.7047%	14.3550	0.8951
	Transformer	5.7344	6.8074%	13.4553	0.9079
	Transformer–BiLSTM	5.1282	6.0585%	11.8790	0.9282
	CEEMDAN–SE–Transformer–BiLSTM	4.0205	5.1519%	9.2280	0.9567
	CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM	2.7829	3.9951%	4.8375	0.9881
	CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM–LSSVM	2.5857	3.7771%	4.5978	0.9892
Autumn	BiLSTM	5.5379	12.8260%	7.2512	0.9691
	Transformer	2.5000	6.0889%	4.2300	0.9895
	Transformer–BiLSTM	2.3396	5.8509%	3.9183	0.9910
	CEEMDAN–SE–Transformer–BiLSTM	1.4612	3.9717%	2.2165	0.9971
	CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM	1.0043	2.6157%	1.4095	0.9988
	CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM–LSSVM	0.9991	2.5125%	1.3599	0.9995
Winter	BiLSTM	4.3846	8.5386%	6.4984	0.9705
	Transformer	2.8472	6.4984%	4.3809	0.9866
	Transformer–BiLSTM	2.3753	5.0917%	3.8621	0.9896
	CEEMDAN–SE–Transformer–BiLSTM	1.4743	3.1120%	2.2201	0.9966
	CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM	1.1329	2.3427%	1.5960	0.9978
	CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM–LSSVM	1.0577	2.3090%	1.4663	0.9985

:

Based on validation results across different seasons in Beijing during 2023, the proposed improved model CEEMDAN–SE–CPO–VMD–Transformer–BiLSTM–LSSVM consistently maintained low values for MAE, RMSE, and MAPE, with R² values approaching 1. It demonstrated stable and outstanding predictive performance even amid spatiotemporal variations in meteorological conditions across seasons, fully reflecting its exceptional robustness in cross-seasonal contexts. By contrast, reference models such as BiLSTM and Transformer exhibited greater performance variability, further highlighting the robust adaptability of the proposed model across diverse seasonal conditions.

5. Conclusions

This paper addresses the challenges of strong nonlinearity and non-stationarity in AQI time series, alongside the limitations of single models in prediction accuracy. We constructed a hybrid forecasting model integrating CEEMDAN-SE-CPO-VMD secondary decomposition and Transformer–BiLSTM, and incorporated LSSVM for error correction. Experimental validation using hourly AQI data from Beijing in summer 2023 led to the following conclusions:

• Secondary decomposition effectively enhances feature quality: The CEEMDAN algorithm decomposed the raw AQI sequence into 11 IMF components. By quantifying their complexity using SE and applying K-means clustering, these components were grouped into high-, medium-, and low-frequency subsequences. We further used CPO to optimise VMD parameters (K = 6, $\alpha$ = 110.08) for secondary de-composition of high-frequency components, which substantially reduced data complexity and provided high-quality inputs for subsequent prediction.
• Hybrid deep learning model enhances temporal modelling capability: For the Transformer–BiLSTM architecture, the Transformer effectively captures global long-term dependencies, while the BiLSTM focuses on extracting local short-term dynamics. Their synergy between the two substantially improves feature learning capacity. This model outperforms standalone BiLSTM and Transformer models in MAE, MAPE and RMSE, while achieving a higher R², demonstrating superior predictive performance.
• Error correction further optimises prediction accuracy: After introducing LSSVM to correct prediction errors, all model metrics improved. MAE decreased from 0.7963 to 0.7102, RMSE decreased from 1.0544 to 0.9154, MAPE decreased from 2.0808% to 1.8543% and R² increased from 0.9958 to 0.9968. These results were significantly superior to those of the reference models without error correction, demonstrating enhanced stability and robustness.
• The model demonstrates robust cross-seasonal adaptability: In extended experiments across spring, autumn, and winter, it maintained consistently superior performance across key metrics including MAE, RMSE, MAPE, and R², significantly outperforming reference models. This confirms its strong universality and stability in addressing variations in meteorological conditions and pollution characteristics across different seasons.

This model provides an effective technical solution for high-precision AQI forecasting, providing support for air quality early warning systems and pollution control decision-making. Future research may focus on validating the model’s generalisation capability using multi-regional data, or incorporating more advanced dynamic feature extraction mechanisms to enhance predictive performance during sudden pollution events.