High-accuracy PM_2.5 prediction via mutual information filtering and Bayesian-Optimized Spatio-Temporal Convolutional Networks

Abstract

Air pollution, particularly fine particulate matter (PM_2.5), poses severe threats to human health and ecological sustainability, rendering accurate prediction of PM_2.5 concentrations imperative for proactive public health interventions and evidence-based policy-making. While deep learning models like LSTM, GRU, and CNN are widely adopted for their robust modeling capacities, the direct use of raw, unfiltered data introduces feature redundancy. This not only prolongs training duration and elevates overfitting risks but also degrades prediction accuracy by complicating model convergence. To address these challenges, this paper presents an advanced PM_2.5 prediction framework incorporating three key innovations.First, in contrast to conventional static threshold-based feature selection, a dynamic framework integrating Mutual Information (MI) and Adaptive Information Distance (AID) is proposed. By quantifying nonlinear feature correlations (via MI) and spatial redundancies (via AID), the framework adaptively prunes redundant inputs, thereby enhancing the information utility of the feature space for subsequent modeling.Second, a Bayesian optimizer guided by multimodal Gaussian distributions is designed to overcome the limitation of traditional unimodal optimization, which often stagnates at local optima. This optimizer explores multiple potential optima in the parameter space concurrently, enabling efficient global hyperparameter search and improving model robustness to noise.Third, an Information Screening-Enhanced Spatiotemporal Convolutional Network (MIBO-STCN) is introduced. Building on our prior STCN framework that integrates causal convolution and adaptive receptive fields, this architecture embeds an information screening layer to achieve synergistic optimization of spatiotemporal dependency modeling and redundancy reduction. Through this synergistic data preprocessing and optimization pipeline, the framework substantially boosts the prediction accuracy and accelerates convergence of the STCN model. Experimental results demonstrate that the proposed approach outperforms state-of-the-art models, exhibiting robust generalization capabilities in PM_2.5 concentration forecasting across diverse scenarios.

Introduction

As society and technology rapidly advance, the public’s awareness of their living environment has increased significantly. Air quality, as a crucial factor affecting human health, has garnered widespread attention. Studies have shown that long-term exposure to air pollution not only directly harms physical and mental health^1,2but may also impair cognitive abilities^3–5 and reduce work efficiency, particularly in high-value innovation tasks⁵. Furthermore, fine particulate matter (PM_2.5, etc.) in the air can enter indoor environments through various channels, exacerbating potential health risks⁶.

PM_2.5 is a significant component of the Air Quality Index (AQI) and one of the main harmful pollutants. Due to its small particle size, PM_2.5 can infiltrate indoor environments from the outdoors, penetrating the respiratory system and posing severe threats to human health. Therefore, accurately predicting PM_2.5 concentrations not only provides comprehensive air quality forecasts and scientific evidence for government agencies to develop pollution control policies but also helps the public better understand air quality conditions and offers reference for health protection.

Currently, methods for predicting PM_2.5 can be divided into three categories: deterministic methods, statistical methods, and machine learning methods. Deterministic methods simulate the formation and diffusion processes of atmospheric pollutants but rely heavily on theoretical assumptions, making it difficult to explain many nonlinear and heterogeneous influencing factors⁷, leading to high prediction costs and low accuracy. Statistical methods rely on historical data for predictions but are limited by smaller parameter scales and simple model structures, resulting in limited feature selection and nonlinear fitting capabilities, thus struggling to handle the high complexity of actual PM_2.5 data. Machine learning and deep learning methods have shown remarkable results in the era of big data, especially deep learning methods based on large-scale multivariate and multichannel time series, which exhibit powerful modeling capabilities across various applications. For instance, Long Short-Term Memory (LSTM)⁸, Gated Recurrent Unit (GRU), and Convolutional Neural Networks (CNN)⁹ have been successfully applied. Despite the broad prospects of deep learning in air quality prediction, the issues of noise and redundant information in the data remain urgent problems. Redundant features not only increase model training costs but also lead to overfitting. Therefore, it is essential to minimize unnecessary information to improve the model’s training efficiency and prediction accuracy. Standard feature selection methods include mutual information (MI)¹⁰, Spearman rank correlation coefficient¹¹, and Gaussian-based outlier detection and filtering¹². These methods can quantitatively analyze the relationships between factors in multivariate time series and remove inconsistent or conflicting factors. However, they cannot account for the redundancy between features. As is well-known, the inclusion of redundant feature information may cause deep neural networks to overfit during the modeling process, thereby lowering the overall prediction performance.

Additionally, collected air quality data often encounter another challenge: complex noise distributions. These distributions can obscure crucial time series characteristics, such as periodicity and seasonality, complicating the model and reducing prediction accuracy. As a result, it is essential to incorporate a denoising step to adjust the parameters before making predictions. The STCN model proposed by Zhang et al.³⁴ struggles with redundant spatial correlations, leading to a higher RMSE (19.47 g/m³) compared to our approach. In this paper, we focus on information selection and propose a method that integrates mutual information (MI), adaptive distance (AID), and multi-modal Gaussian distribution-based Bayesian hyperparameter optimization to filter redundant information and reduce noise interference, thereby improving model training efficiency and prediction accuracy. The contributions of this paper are as follows:

1. Unlike traditional static threshold-based methods, we propose a dynamic feature selection framework that integrates Mutual Information (MI) and Adaptive Information Distance (AID). This framework quantifies nonlinear feature correlations via MI and spatial redundancies between features via AID, enabling adaptive pruning of redundant inputs. By dynamically distinguishing high-informative signals from low-value noise in multi-dimensional data, it constructs a compact yet discriminative feature space for subsequent modeling.

2. To address the limitations of traditional unimodal optimization—such as getting trapped in local optima within complex parameter spaces—we design a Bayesian optimizer driven by a Gaussian Mixture Model (GMM). This approach models the hyperparameter space as a multimodal Gaussian distribution using Gaussian processes, allowing simultaneous exploration of multiple potential optimal regions. By synchronously optimizing threshold decisions and feature weights during the screening process, it adapts to the diverse feature correlation patterns in complex pollution scenarios, overcoming the myopia of single-modal optimization methods.

3. Building on our prior STCN architecture, we introduce MIBO-STCN by embedding an adaptive information filtering layer. While retaining the original STCN’s capabilities in modeling spatiotemporal dependencies via causal convolution and adaptive receptive fields, the filtering layer employs joint MI-AID computations to dynamically identify and suppress spatially redundant input data in real time. Critically, a backpropagation mechanism synchronizes loss gradients from the prediction layer to the filtering layer, enabling collaborative optimization of feature-selection parameters (e.g., MI thresholds, AID weights) and core network parameters (e.g., convolutional kernels, recurrent units). This end-to-end architecture forms a closed-loop feedback system, where feature screening and model prediction evolve synergistically to enhance the network’s representation of complex pollution dynamics.

The structure of this paper is as follows: Section 2 reviews related research in the field of air quality prediction; Section 3 introduces the proposed model framework; Section 4 discusses the research data, experiments, and analysis of experimental results; and Section 5 concludes the paper and provides future research directions.

Materials and methods

Dataset and data analysis

Dataset

This study utilizes hourly PM_2.5 and meteorological data for Beijing from May 1, 2014, to April 30, 2015. Each dataset contains 105,120 data points, with key features including air pollutants (PM10, NO2, CO, O3, SO2) and meteorological information (temperature, pressure, humidity, wind speed, wind direction). All data points were sampled at an hourly frequency,and the data description is shown in Table 1.

Table 1.

Dataset description in this study.

Variable type	Variable	Data type	Unit
Air quality data	PM2.5	Numeric	g/m3
	PM10	Numeric	g/m3
	NO2	Numeric	g/m3
	CO	Numeric	g/m3
	O3	Numeric	g/m3
	SO2	Numeric	g/m3
Meteorological data	Weather	Categorical
	Temperature	Numeric	°C
	Pressure	Numeric	hPa
	Relative humidity	Numeric	%
	Wind speed	Numeric	m/s
	Wind direction	Categorical

To eliminate differences in feature scales and reduce the results, data was processed with Z-score normalization to speed up model convergence and improve predictive accuracy. The Z-score normalization formula is as follows:

1

where represents the input data, is the mean of the observed data, and is the standard deviation of the observed data.

Analysis of data correlation and redundancy

The spatial correlation of air quality in different regions of Beijing is shown in Figure 1.

Fig. 1.

The relative locations and PM_2.5 concentration of four monitoring stations in Beijing, with Dongcheng Dongsi (1013) as the central station.

Figure 1 illustrates the relative locations of four monitoring stations in Beijing, with Dongcheng Dongsi (1013) serving as the central station. It compares PM_2.5 concentrations between Dongcheng Dongsi and the other three stations: South Third Ring West Road, Chaoyang Agricultural Exhibition Hall, and East Fourth Ring North Road. The comparisons between each pair of stations highlight the spatial correlation of air quality:

1. shows the geographical locations of the four monitoring stations in Beijing, listed in order: South Third Ring West Road (1011),Chaoyang Agricultural Exhibition Hall (1012),Dongcheng Dongsi (1013),and East Fourth Ring North Road (1014).Dongcheng Dongsi Station acts as the central station, and the spatial correlation of air quality at the other stations is analyzed with respect to this central point;

2. Presents data from 8,760 samples collected at the Chaoyang Agricultural Exhibition Hall station and the Dongcheng Dongsi station.(e) Shows the concentration distribution and similarity analysis between the Chaoyang Agricultural Exhibition Hall station and the Dongcheng Dongsi station. These data reveal that the PM_2.5 concentrations at these two stations are highly similar, and there is a strong spatial correlation. This high degree of similarity indicates significant redundancy between the two stations. Directly feeding redundant features into the model may lead to overfitting, slow down the training speed, and complicate the convergence process, thus having a negative impact on the performance of the model;

3. illustrates the correlation in air quality data between the South Third Ring West Road station and the Dongcheng Dongsi station.(f) presents the concentration distribution and similarity analysis between these two stations. While their similarity is slightly lower than that between the East Fourth Ring North Road station and Dongcheng Dongsi (with similarity scores of 0.93 and 0.92, respectively), a measurable correlation still exists. Given their close proximity, air pollutants can be transported between the stations under specific weather conditions, such as windy episodes. Despite this correlation, the level of redundancy remains relatively low;

4. illustrates the spatial correlation of air quality data between the East Fourth Ring North Road station and the Dongcheng Dongsi station.(g) presents the concentration distribution and similarity analysis between these two stations. When compared to Figures 1b and 1c, their similarity is notably lower (with a similarity score of 0.90 for East Fourth Ring North Road and Dongsi). Due to the greater distance between them, air pollutants have less mutual influence, resulting in a low correlation. Therefore, when predicting air quality at the Dongcheng Dongsi station, reducing the input of data from the South Third Ring West Road station may be beneficial.

The role of meteorological factors (e.g., wind) in influencing prediction residuals will be analyzed in in Experimental settings, which focuses on The Combined Effects of Wind Direction and Wind Speed on PM_2.5 Prediction Residuals.

Based on this analysis, it is possible to optimize model performance by reasonably selecting input features, thus effectively reducing the impact of redundant features.

Architecture of the proposed network

Based on the previous analysis, this paper proposes a Spatio-Temporal Convolutional Network (STCN) model with feature information filtering. The process is illustrated in Figure 2:

Fig. 2.

Information filtering and optimization framework (Module A: Feature Screening, Module B: STCN, Module C: Bayesian Optimization). Module B: STCN and Module C: Bayesian Optimization architecture are detailed in Figure 3. The information filtering layer (a) adopts a two-stage feature selection process: First, Mutual Information (MI) is used to calculate the non-linear correlation between each feature and the target variable (PM_2.5 concentration). Features with high MI values are retained to effectively eliminate irrelevant variables. Second, Adaptive Inter-Distance (AID) evaluates redundant data among features to further screen and refine the feature set. The filtered features are then fed into a Spatio-Temporal Convolutional Network (STCN, b), which employs causal convolutions to enforce temporal causality and adaptive receptive fields, enabling flexible modeling of temporal and spatial dependencies. Finally, Bayesian hyperparameter optimization (c) utilizes multimodal Gaussian distributions to iteratively fine-tune the AID redundancy threshold (AID, a) and STCN parameters (STCN, b), maximizing prediction performance through holistic model configuration.

Information filtering layer

Research by Chen et al. (2020) demonstrated complex nonlinear relationships between PM_2.5 concentration and meteorological factors in China. For example, temperature influences chemical reaction rates—some atmospheric reactions accelerate at higher temperatures, affecting PM_2.5 formation or reduction. Wind speed and direction also affect PM_2.5 transmission and dispersion; low wind speed can lead to local pollutant accumulation, whereas strong winds disperse pollutants. These factors, along with spatial interactions in PM_2.5 concentration, make modeling challenging due to the volume and complexity of information involved.

Since adjacent areas often contain redundant information, we need to consider data correlations and apply filtering methods. However, traditional data analysis methods often overlook redundancy and noise, which can lead to reduced model generalization and predictive accuracy. To address these challenges, we designed an information filtering layer composed of mutual information, adaptive distance, and Bayesian hyperparameter optimization. The filtering process is as follows:

• Step 1:
  Mutual information is used to calculate the association between dataset features, which include temperature, wind speed, and other atmospheric chemicals measured across different stations. The formula for calculating mutual information (MI) is given by Equation (2):

  2

  Among them, denotes the feature variable, representing spatiotemporal sequence data from multiple monitoring stations in this scenario, including pollutants (PM, PM, NO, etc.) and meteorological variables (temperature, humidity, wind speed, etc.), with the dimension . Here, is the number of monitoring stations, is the time step, and is the feature dimension. denotes the target variable, i.e., the PM concentration sequence of the central monitoring station, with . is the joint probability distribution, while and are the marginal probability distributions of and , respectively. Higher mutual information indicates stronger correlation. However, data with high correlation often contain redundant information, which may impose a burden on the network and lead to overfitting. We calculate the mutual information value between each feature and the target . Features with are retained (initial threshold ), yielding a high-correlation feature set (). Next, we will further discuss the “effective” mutual information.

  Higher mutual information indicates stronger correlation. However, data with high correlation often contain redundant information, which may impose a burden on the network and lead to overfitting. Through the calculation of mutual information, we obtain a highly correlated feature set ().

  Below, we will further discuss the “effective” mutual information.

• Step 2:
  Adaptive distance is incorporated to filter input variables, selecting those with high relevance and low redundancy.

  3

  Among them, is the set of variables selected based on mutual information (Equation (2)), where is the covariance matrix dynamically updated via Bayesian optimization, used to balance relevance and redundancy among the selected features, with as the target variable. We calculate the AID value for feature pairs using AID (Equation (3)), and remove redundant features with (initial threshold ) to preliminarily filter a non-redundant feature set ()).

• Step 3:
  Bayesian hyperparameter optimization is used to refine parameters in adaptive distance that directly affect predictive accuracy, ensuring reliability and efficiency in feature selection. Root mean square error (RMSE) is used as the objective function for optimization:

  4
  Where is the parameter to be optimized, is the number of input samples, is the target variable, is the predicted value. Minimizing this loss yields an optimal parameter set:

  5

  where is the optimal parameter from Bayesian optimization, represents input parameters (such as the initial threshold for MI, the initial threshold for AID, and parameters in the STCN model like the dilation factor , the convolution kernel weight ), and spans the multi - dimensional parameter space.

The information filtering layer integrates mutual information, adaptive distance filtering, and Bayesian optimization. Mutual information identifies features strongly related to the target variable, while adaptive distance filters low-redundancy variables. Bayesian optimization learns parameters based on input data, yielding refined input data that enhances prediction accuracy.

Spatio-Temporal Convolutional Network prediction layer

The proposed prediction model uses a Bayesian-optimized Spatio-Temporal Convolutional Network (BO-STCN). BO-STCN employs STCN as its main structure, training on the spatiotemporal characteristics of PM_2.5 data. The network structure of BO-STCN is shown in Figure 3.

Fig. 3.

The interaction architecture between STCN (Module B) and the Bayesian optimizer (Module C) corresponds to Module B and Module C in Figure 2. Spatio-temporal Convolutional Network (STCN) and Bayesian Optimizer Architecture (Module B and Module C). The deep prediction network process. Module (b) integrates spatiotemporal features from input data, while module (a) performs multi-modal Gaussian Bayesian hyperparameter optimization on module (b).

Compared to traditional networks, STCN extends the receptive field by utilizing dilated convolutions to capture long-term temporal dependencies. STCN’s core lies in causal and dilated convolutions, ensuring the model only relies on past data points, preventing data leakage. Given a filter function and time-series input , the dilated convolution is defined as:

6

where is the input time series, is the dilation factor controlling the kernel spacing, and denotes the time step. This operation, known as the dilated convolution operator, expands the CNN’s receptive field to capture dependencies over longer time scales. In this study, the outer loop of Bayesian optimization adjusts key hyperparameters, including the dilation factor , learning rate , mutual information threshold , and adaptive distance covariance matrix . Simultaneously, it coordinates with the inner loop to update the model parameters and via Adam optimization.

Multi-Modal Gaussian Bayesian hyperparameter optimization layer

STCN’s predictive performance heavily relies on hyperparameters (e.g., weight and bias in convolution layers). Traditional optimization methods like grid or random search are inefficient in high-dimensional spaces and may get trapped in local optima. By introducing a multi-modal Gaussian distribution, we can better handle complex functions with multiple local maxima, enhancing global search capability.

This model uses a Gaussian Mixture Model (GMM) to replace a single Gaussian process for optimization, modeling the objective function in a multi-modal fashion. The optimization process is as follows:

• Step 1:

  Gaussian Process Regression (GPR) establishes a prior distribution using training data, which is updated to obtain a posterior distribution after incorporating new data.

• Step 2:

  Multi-modal Gaussian process modeling is performed. GMM allows the prediction to capture multiple peaks, thus avoiding local optima.

• Step 3:

  An Expected Improvement (EI) acquisition function selects the next hyperparameter combination to sample in regions of uncertainty or potential high value.

• Step 4:

  At the selected combination, STCN is trained and RMSE evaluated. New data is added to update the mean and variance distribution of the GMM.

• Step 5:

  Repeating steps 3 and 4 iteratively explores local optimal regions in the parameter space, outputting the optimal combination.

This approach adaptively adjusts parameters in the filtering layer, enabling efficient information extraction even with complex distributions containing multiple local optima. GMM updates improve the model’s ability to fit the target function, reducing overfitting risks. The method identifies and retains high-value features, enhancing accuracy and efficiency in information filtering. This innovation strengthens the model’s feature selection capability and optimizes overall performance, ensuring robustness when handling complex air quality data.

Experimental settings

In the experimental validation, the dataset was divided into training, validation, and test sets, distributing them evenly across time to ensure reliable validation. Specifically, the training set consists of the first 60% of data for each month, the next 20% is used as the validation set, and the remaining 20% serves as the test set. We centered the analysis on monitoring station 1013, selecting data from surrounding stations as input features for modeling. During training, mean squared error (MSE) was used as the loss function, and the Adam optimizer²⁴ was applied for gradient descent optimization, facilitating efficient convergence of the model’s random objective function. We set the training epochs to 200 and the batch size to 32.

Model evaluation metrics

To comprehensively assess the model’s performance in PM_2.5 concentration prediction, we employed three metrics: root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (). The formulas are as follows:

7

8

9

Here, and represent the observed and predicted values, respectively, is the mean of the observed data, and is the number of samples. Ideally, lower and values indicate better performance, while an value closer to 1 indicates higher prediction accuracy.

Model comparison experiments

To validate the performance of the proposed model, we conducted comparison experiments using three widely adopted baseline models in air pollution forecasting: Long Short-Term Memory (LSTM), Recurrent Neural Network (RNN), and Temporal Convolutional Network (TCN). These models were evaluated alongside the proposed Spatio-Temporal Convolutional Network with Mutual Information-Based Feature Filtering (MIBO-STCN), which incorporates MI feature selection, aid feature selection, and multi-modal Bayesian hyperparameter optimization. Additionally, to assess the impact of selecting high-correlation, low-redundancy features, we tested the STCN model without the information filtering layer and Bayesian hyperparameter optimization. This allowed us to evaluate the significance of these components in improving the model’s predictive performance.

Experimental settings

PM_2.5 concentration prediction results of STCN with information filtering layer

The PM_2.5 concentration predictions of STCN with Information Filtering Layer (MIBO -STCN) on the test dataset, shown in Figure 4.

Fig. 4.

Comparison of Predicted and Actual PM_2.5 Concentrations at Dongcheng Dongsi Station (August 2014 – April 2015).

Figure 4 illustrates the relationship between the predicted and actual observed values. Apart from relatively large errors caused by missing data, the predictions closely follow the trend of the observations, verifying the model’s effectiveness in capturing temporal variations. Overall, prediction errors are within an acceptable range, demonstrating good performance. Additionally, the model effectively captures concentration fluctuations, despite some minor deviations in specific time periods. Specifically, it tends to overestimate lower PM_2.5 concentrations (below 100 g/), such as in November 2014, while underestimating higher concentrations, as observed in August 2014. In these periods, the gap between observed and predicted values is slightly larger than the overall prediction error.

The combined effect of wind direction and wind speed on the prediction residuals of PM_2.5

To investigate the impact of meteorological variables on PM_2.5 prediction residuals, the model incorporates wind speed, wind direction, and their interaction term (e.g., wind speed wind direction). Wind direction is encoded as a categorical variable (e.g., North, East, West) and processed using dummy variables to retain directional information. The interaction term is constructed through element-wise multiplication to capture the nonlinear combined effects of wind speed and direction.

Figure 5 presents the partial dependence of wind speed on PM_2.5 prediction residuals. The x-axis represents wind speed (ranging from 2.5 to 17.5 m/s), while the y-axis shows the partial dependence value, indicating the average effect of wind speed on model prediction residuals. As wind speed increases from 2.5 m/s to 17.5 m/s, the residuals decrease from 20 g/ to 12 g/, suggesting that higher wind speeds significantly reduce prediction errors. This trend supports the hypothesis that higher wind speeds enhance prediction accuracy by promoting the dispersion of pollutants.

Fig. 5.

Partial Dependence Plot (PDP) of the joint effect of wind speed and wind direction on residuals.

Discussion

Comparison with other methods

With the advent of the big data era, deep learning has gradually demonstrated its superiority in processing large-scale data. However, increasing the volume of input data does not necessarily lead to better results for deep learning models. Excessive data often carries redundant information, which not only increases computational costs and slows model convergence but also potentially degrades predictive performance. This model introduces an adaptive information distance based on mutual information to screen for highly correlated and low-redundant data through nonlinear transformations. By fusing regional factors and meteorological attributes, the model filters out feature combinations highly correlated with PM2.5 while eliminating redundant information. Additionally, the overall model performance is further enhanced through Bayesian hyperparameter optimization with a multimodal Gaussian distribution.

As shown in Table 2, MIBO-STCN (STCN with an information filtering layer) significantly outperforms five baseline models (LSTM, RNN, Transformer, MultiChannelInformer, and TCN). The model achieves RMSE, MAE, and values of 16.0172, 10.3141, and 0.9462, respectively. Compared with LSTM, RNN, Transformer, MultiChannelInformer, and TCN, the RMSE is reduced by 76.30%, 62.71%, 76.83%, 76.75%, and 34.24%, respectively; the MAE is reduced by 81.67%, 68.70%, 81.80%, 81.57%, and 40.55%, respectively; and the is increased by 68.12%, 54.26%, (due to the baseline value being negative), 1781%, and 8.05%, respectively.

Table 2.

Comparison of Performance of STCN with Information Filtering and Other Models.

	LSTM	RNN	Transformer	MultiChannelInformer	TCN	STCN	MIBO - STCN
RMSE	67.5809	42.9497	69.1354	68.8780	24.3560	18.7333	16.0172
MAE	56.2791	32.9304	56.5849	55.9624	17.3164	11.7599	10.3141
	56.2791	0.6134	−0.0018	0.0503	0.8757	0.9264	0.9462

Compared to the STCN proposed by Zhang et al.³⁴, our model achieves a 17.74% reduction in RMSE (from 19.473 to 16.0172) and a 19.75% improvement in MAE (from 12.853 to 10.3141). This enhancement can be attributed to two key innovations: (1) the mutual information-based feature filtering layer that eliminates redundant spatial correlations, and (2) the multi-modal Bayesian optimization that adaptively tunes hyperparameters to resist noise interference.

Figure 6 shows the evaluation results of different models on test data:(a) presents the results of the LSTM model, which achieves an of 0.0427. While slightly better than GRU, the LSTM model still struggles to accurately predict PM_2.5 concentrations, particularly during periods of rapid fluctuation. This suggests that LSTM alone may not be sufficient for capturing the intricate dynamics of air quality data.(b) depicts the performance of the RNN model, with an of 0.6134. Although the RNN model shows some ability to track general trends, it fails to accurately predict peak concentrations and exhibits significant deviations during periods of high variability. This underscores the need for more advanced architectures to handle the complexity of PM_2.5 data.(c)illustrates the results of the TCN model, with an of 0.8757. The TCN model performs well in predicting general trends but exhibits slight underestimation during high-concentration periods. This indicates that while TCN is effective, it may benefit from additional mechanisms to handle extreme values.(d)demonstrates the performance of the STCN model, achieving an of 0.9264. The predicted values align well with actual observations, particularly in capturing long-term trends. However, minor deviations during peak concentrations suggest room for further improvement in handling extreme events.(e) illustrates the comparison between actual and predicted PM_2.5 concentrations using the proposed MIBO-STCN model. The predicted values closely follow the actual values, with an of 0.9462, indicating a strong correlation. The model demonstrates high accuracy in capturing both peak concentrations and long-term trends, validating its effectiveness in spatiotemporal PM_2.5 prediction.

Fig. 6.

Correlation analysis between the observed and predicted values of PM_2.5 concentrations by different models on the test data. The blue scatter points represent the actual values, and the red solid line represents the regression line.

Generalization analysis of the model

To assess the model’s generalizability, we selected additional monitoring stations in Beijing’s northwest, northeast, southwest, and southeast regions and compared the results to STCN without information filtering. This analysis evaluates whether the model can effectively adapt to the data distributions of new stations. Compared to station 1013, the proposed model also performed well on these four stations, achieving improved accuracy relative to STCN without redundancy filtering. The detailed comparisons for prediction accuracy at these stations are shown in Table 3.

Table 3.

Generalization Analysis Results.

Station ID	Model	RMSE	MAE
1001	STCN	18.036	10.865	0.944
	MIBO - STCN	16.1982	10.3832	0.9450
1023	STCN	23.741	14.037	0.931
	MIBO - STCN	16.4070	10.6765	0.9436
1005	STCN	18.055	10.977	0.938
	MIBO - STCN	16.6801	10.8816	0.9417
1018	STCN	22.742	13.979	0.939
	MIBO - STCN	17.8463	11.3042	0.9332

Despite the model’s superior predictive performance compared to other methods, certain limitations remain. First, the model’s focus is on predicting PM_2.5 concentration at a target monitoring station based on selected features derived from multiple stations and atmospheric chemical characteristics. Thus, the model’s performance is constrained by data quality; if monitoring data quality from stations is low, predictive accuracy may decrease. Additionally, while the model performs well on data from Beijing, its generalizability to other regions has not yet been fully validated. The applicability of the model across diverse geographic areas could be a direction for future research to further assess its generalization capabilities. Finally, we aim to develop more dynamic preprocessing strategies, such as time-sensitive feature weight adjustment mechanisms, to enable the model to respond in real time to sudden changes in air quality, improving its sensitivity to extreme weather or pollution events.

Currently, the model has only been applied to a single city in China due to limited access to hourly data. With the future release of more datasets, the proposed model could be tested across different regions or time periods, enabling broader validation and assessment. We also plan to explore the application of this model to predict other environmental variables (such as temperature, humidity, or pollutant distribution) to help build a more comprehensive urban environmental monitoring system.

Hyperparameter optimization analysis

To validate the effectiveness of multi-modal Bayesian hyperparameter optimization, we plotted the trend of the optimization results:

Figure 7 illustrates the decline in validation RMSE during the multi-modal Gaussian Bayesian optimization process. By constructing a multi-modal Gaussian Mixture Model (GMM), the optimizer can simultaneously explore multiple potential optimal regions in the hyperparameter space (e.g., Trial 2 and Trial 46), preventing the algorithm from getting trapped in local optima. Compared to the unimodal Gaussian process (dashed reference line), the multi-modal strategy identifies a high-quality solution (RMSE < 25) early on (Trial 2) and continues to approach the global optimum in later trials (Trial 46, RMSE = 20.67). As shown in Figure 7, during the 50 trials across the exploration, development, and stabilization phases, the multi-modal Bayesian optimization reduces the validation RMSE from 86.26 (Trial 0) to 20.67 (Trial 46), achieving a 62% faster convergence rate compared to traditional unimodal optimization. The advantage of multi-modal Bayesian optimization lies in its ability to retain potential optimization directions, even when initial trials perform poorly (e.g., Trial 4, RMSE = 89.89). In just 46 trials, it finds the global optimum. This feature makes it particularly well-suited for high-dimensional, multi-modal tasks such as air quality prediction.

Fig. 7.

Trend of validation RMSE during multi-modal Bayesian optimization (the dashed line represents the reference baseline of the unimodal Gaussian process, and the red star marks the global optimum).

Ablation study

To validate the independent contributions of the Mutual Information (MI)-based Adaptive Information Distance (AID) filtering layer and the multi-modal Bayesian optimizer (MBO), we designed four comparative experiments: the full model (MIBO-STCN) (with both filtering layer and Bayesian optimizer), the model without the filtering layer (STCN-BO) (using only the Bayesian optimizer), the model without the Bayesian optimizer (MIBO-STCN w/o BO) (retaining the filtering layer but using random search for hyperparameters), and the baseline model (STCN) (removing both components and using random search).

The experimental results are shown in the following table:

As shown in Table 4, it can be concluded that: The full model reduces the feature dimension from 5184 to 2048 (a 60% reduction) via the filtering layer, achieving a 27.2% reduction in validation RMSE compared to the baseline model (STCN). This demonstrates that the filtering layer effectively removes redundant spatial features, reducing computational complexity and mitigating overfitting. The model without the filtering layer (STCN-BO) required 50% more epochs to converge, confirming that redundant features slow down model convergence. The Bayesian optimizer reduced the number of hyperparameter trials from 100 (random search) to 50, while achieving a 17.6% reduction in validation RMSE (compared to the group without Bayesian optimization). As shown in Figure 7, the multi-modal strategy identified a suboptimal solution (RMSE < 25) by the 2nd trial, whereas random search required over 50 trials to approach similar performance. The group without the Bayesian optimizer (random search) was prone to local optima (e.g., Trial 46’s RMSE = 20.67 was an outlier), while Bayesian optimization systematically explored the multi-modal space via Gaussian process modeling, ensuring stable convergence to the global optimum.

Table 4.

Performance Comparison of Ablation Experiments.

Configuration	Feature dimension	Validation RMSE	Convergence Epoch
Full Model (MIBO-STCN)	2048	16.0172	30
w/o Filtering Layer (STCN-BO)	5184	18.7333	45
w/o Bayesian Optimizer	2048	19.4658	35
Baseline Model (STCN)	5184	22.0121	50

In summary, the full model combines the filtering layer and Bayesian optimizer to achieve dual optimization: the MI/AID filtering layer reduces feature redundancy, allowing the model to focus on high-correlation features, while the Bayesian optimizer adaptively tunes parameters for the compressed feature space. Their synergy results in a 27.2% RMSE reduction compared to the baseline model, validating the complementary roles of these components.

Conclusions

Given that air quality data is influenced by both temporal and spatial dimensions and contains information redundancy among its multiple variables, accurately predicting air quality poses a challenge. This study proposes an STCN model based on information filtering and multi-modal Gaussian distribution Bayesian hyperparameter optimization to predict future PM_2.5 concentrations. By leveraging the data self-selection layer to filter high-correlation, low-redundancy variables and combining it with the noise-resistant multi-modal Gaussian distribution Bayesian hyperparameter optimization method, this approach enhances the accuracy and robustness of air quality predictions. Experimental validation using air quality and meteorological data from Beijing, along with evaluation metrics such as RMSE, MAE, and , demonstrate the proposed method’s predictive advantage.

Compared to other deep learning models with RNN structures, this model incorporates an information filtering layer, which, through mutual information and a custom adaptive distance metric, filters out high-correlation, low-redundancy feature variables, reducing redundancy in the input data. This not only lowers the model’s training time (by reducing unnecessary computation) but also improves its robustness. Bayesian optimization further tunes the hyperparameters, particularly using a multi-modal Gaussian distribution optimization strategy, which enhances the model’s noise resistance and helps avoid overfitting. Consequently, the combined effects of the information filtering layer and Bayesian optimization allow the model to adapt more effectively to fluctuations in air quality data.

The model integrates spatial stations and meteorological features highly correlated with the target monitoring station, feeding them into the causal convolutional neural network, and applies multi-modal Gaussian distribution Bayesian hyperparameter optimization in backpropagation training to update key parameters, further enhancing model performance.

Case study results demonstrate that this approach excels in predicting hourly PM_2.5 concentrations in Beijing. Compared to other models, this model outperforms LSTM,RNN,Transformer,MultiChannelInformer and TCN with RMSE reductions of 76.30%, 62.71%, 76.83%, 76.75% and 34.24%, respectively.Besides,The proposed MIBO-STCN outperforms the state-of-the-art STCN³⁴ by reducing RMSE by 11.12%, demonstrating its superiority in handling spatiotemporal redundancy and complex noise distributions.

In conclusion, the results indicate that the proposed spatiotemporal causal convolutional neural network with an information filtering layer and Bayesian hyperparameter optimization based on a multi-modal Gaussian distribution (MIBO-STCN) achieves high accuracy in air pollution prediction.

Author contributions

The author, Wanyu Wang, contributed solely to this work.

Data availability

The dataset is http://research.microsoft.com/en-us/projects/urbanair (Urban Computing Team, Microsoft Research).

Declarations

Competing interests

The authors declare no competing interests.