A Hybrid STL-Based Ensemble Model for PM_2.5 Forecasting in Pakistani Cities

Abstract

Air pollution, outstanding particulate matter (PM_2.5), poses severe risks to human health and the environment in densely populated urban areas. Accurate short-term forecasting of PM_2.5 concentrations is therefore crucial for timely public health advisories and effective mitigation strategies. This work proposes a hybrid approach that combines machine learning models with STL decomposition to provide precise short-term PM_2.5 predictions. Daily PM_2.5 series from four major Pakistani cities—Islamabad, Lahore, Karachi, and Peshawar—are first pre-processed to handle missing values, outliers, and variance instability. The data are then decomposed via seasonal-trend decomposition using Loess (STL), which explicitly exploits the symmetric and recurrent structure of seasonal patterns. Each decomposed component (trend, seasonality, and remainder) is modeled independently using an ensemble of statistical and machine learning approaches. Forecasts are combined through a weighted aggregation scheme that balances bias–variance trade-offs and preserves the distributional consistency. The final recombined forecasts provide one-day-ahead PM_2.5 predictions with associated uncertainty measures. The model evaluation employs multiple statistical accuracy metrics, distributional diagnostics, and out-of-sample validation to assess its performance. The results demonstrate that the proposed framework consistently outperforms conventional benchmark models, yielding robust, interpretable, and probabilistically coherent forecasts. This study demonstrates how periodic and recurrent seasonal structure decomposition and probabilistic ensemble methods enhance the statistical modeling of environmental time series, offering actionable insights for urban air quality management.

1. Introduction

Breathing clean air is essential to human health, as air quality directly impacts the respiratory system and overall well-being. Meteorological and atmospheric factors have a significant impact on air quality and introduce fluctuations in PM_2.5 concentrations. Primary NO_x, SO₂, and VOC emissions combine through photochemical reactions in the presence of sunlight to generate secondary aerosols, while weather factors, including temperature, humidity, and boundary layer height, also affect dispersion. Knowing these mechanisms provides a physical foundation for the temporal patterns. Prolonged exposure to polluted air poses significant health risks beyond the respiratory tract, including increased mortality, the onset of chronic illnesses, disabilities, and substantial socioeconomic burdens on health systems worldwide. Consequently, air quality monitoring and forecasting have become critical components of public health strategies at both national and global levels [1,2,3].

Over the past few decades, rapid population growth, urbanization, and deforestation have contributed to elevated levels of air pollution [4,5,6,7]. Air pollution is generally defined as the presence of harmful chemicals, physical, or biological substances in the atmosphere. Among various pollutants, particulate matter (PM) remains one of the most widely used and reliable indicators of air quality. PM is a complex mixture of solid and liquid particles suspended in the air, often including nitrogen oxides (NO_x), carbon oxides (CO_x), sulfur oxides (SO_x), ozone (O₃), volatile organic compounds (VOCs), and secondary aerosols. Based on aerodynamic diameter, PM is typically classified into PM₁₀ (particles smaller than 10 µm) and PM_2.5 (particles smaller than 2.5 µm) [8].

The health risks associated with particulate matter depend strongly on particle size. PM₁₀ is primarily deposited in the upper respiratory tract and digestive system. By contrast, PM_2.5 particles, due to their smaller size, can penetrate deep into the lungs and cross into the bloodstream. This enables them to exert systemic effects, including increased risks of cardiovascular disease, ischemic heart disease, renal disorders, diabetes, and endocrine dysfunctions, as well as neurological conditions such as Alzheimer’s disease, Parkinson’s disease, and multiple sclerosis [9,10]. The World Health Organization (WHO) estimates that exposure to fine particulate matter contributes to approximately 7 million deaths annually, making air pollution one of the leading global health risk factors. In Europe alone, PM_2.5 is responsible for nearly 904,000 premature deaths per year, with projections indicating significant increases by 2050 [11]. These alarming statistics underscore the urgency of developing effective monitoring, forecasting, and intervention strategies.

Air pollution dynamics are influenced not only by emissions but also by meteorological and climatic conditions, such as wind speed, temperature, relative humidity, rainfall, atmospheric pressure, and ultraviolet (UV) radiation duration. Consequently, robust air quality monitoring and forecasting systems must integrate both pollution sources and meteorological drivers to provide accurate and reliable predictions [12]. From a probabilistic and statistical perspective, such systems must also capture underlying structures in the data, including symmetries in seasonal patterns, cyclical fluctuations, and approximate symmetry of error distributions.

Pakistan, like many other developing countries, faces significant challenges due to deteriorating air quality. Studies suggest that nearly two-thirds of the country experiences unhealthy air, primarily driven by elevated PM_2.5 concentrations, which impose a substantial burden on public health. Respiratory diseases have the most immediate impact, with 57% of chronic obstructive pulmonary disease (COPD) cases and 40% of lower respiratory infections attributed to poor air quality. Beyond respiratory health, prolonged exposure to PM_2.5 has also been linked to non-respiratory conditions such as diabetes, cardiovascular disorders, and ischemic heart disease [13]. These findings highlight the need for statistically rigorous and symmetry-aware forecasting frameworks to support evidence-based interventions and policy design.

Air quality forecasting, particularly of PM_2.5 concentrations and the Air Quality Index (AQI), has gained significant attention in environmental health research. Accurate forecasts enable the timely issuance of health advisories, informed policy decisions, and the efficient implementation of mitigation strategies. Over the past two decades, a wide range of modeling techniques has been employed, spanning traditional statistical methods (e.g., autoregressive and regression-based models) to advanced machine learning (ML) and deep learning (DL) approaches, each with its own strengths and limitations across different spatiotemporal contexts.

For instance, Mahajan et al. [14] assessed time series forecasting models to predict PM_2.5 concentrations in Taichung City, Taiwan. Using hourly data from Air Box microsensor devices, they compared Autoregressive Integrated Moving Average (ARIMA), Neural Network Autoregression (NNAR), and Holt-Winters (HW) models. Based on performance metrics such as the root mean square error (RMSE) and the mean absolute error (MAE), the NNAR model achieved the best predictive accuracy across monitoring stations. Similarly, Mani and Viswanadhapalli [15] investigated AQI forecasting in Chennai, India, using daily data from the Central Pollution Control Board (CPCB) between 2018 and 2020. After applying preprocessing techniques such as missing value imputation and variance stabilization, they demonstrated that ARIMA outperformed multiple linear regression (MLR) in providing stable and reliable forecasts.

In recent years, deep learning methods have emerged as particularly powerful for capturing complex nonlinear dynamics in air quality data. Cassano et al. [16] employed recurrent neural networks (RNNs), including long short-term memory (LSTM) and gated recurrent unit (GRU) architectures, for modeling air quality in Italy’s Apulia region. Leveraging pollutant and meteorological data from the Regional Environmental Protection Agency (ARPA), they demonstrated superior performance of RNNs, particularly in station-specific forecasts. Similarly, Hossain et al. [17] proposed a hybrid GRU–LSTM model for predicting AQI in Dhaka and Chattogram, Bangladesh, based on three years of data (2017–2020). Their hybrid approach consistently outperformed standalone GRU and LSTM models, as measured via a lower RMSE, the MAE, and the mean squared error (MSE). In Beijing, Zhang et al. [18] introduced a hybrid convolutional neural network–LSTM (CNN-LSTM) model that effectively captures both spatial and temporal dependencies, surpassing traditional models such as ARMA, SARIMA, and individual RNN variants.

Efforts to optimize model efficiency have also included the use of advanced machine learning techniques. Liu et al. [19] developed a genetic algorithm-based kernel extreme learning machine (GA-KELM), trained on real-world air quality data from 2019 to 2021. Their model outperformed support vector regression (SVR), Deep Belief Networks (DBN-BP), and the Community Multiscale Air Quality (CMAQ) model in both accuracy and computational speed. However, it required manual tuning of hidden parameters. In the Pakistani context, Iftikhar et al. [20] introduced three ensemble-based models—ESMT, ESME, and ESMV—for AQI forecasting using five years of PM_2.5 data from major cities. Their results indicated that ensemble approaches, particularly ESMV, consistently outperformed benchmark models, although the omission of meteorological and gaseous predictors revealed a scope for improvement.

Taken together, these studies reflect a global shift toward hybrid and ensemble modeling approaches in air quality forecasting. From a statistical standpoint, such approaches exploit symmetries in temporal structures (e.g., seasonal recurrences and cyclic regularities) while accommodating asymmetries in pollutant spikes and distributional tails. While traditional models, such as ARIMA and MLR, continue to provide valuable baselines, neural network–based methods—especially those integrating LSTM, GRU, CNN, and optimization algorithms—demonstrate strong potential for addressing the nonlinear and multifaceted nature of air pollution. For environmental health applications, the adoption of symmetry-aware probabilistic and statistical models holds promise for more accurate, interpretable, and actionable forecasts, ultimately supporting improved public health outcomes and sustainable urban air quality management. However, this study defines symmetry as balanced error behavior and recurrent seasonal regularity, rather than as precise temporal symmetry, in contrast to previous works.

The rest of this paper is organized as follows. Section 2 presents the proposed method, including data preprocessing, STL decomposition, ensemble model construction, and the probabilistic–statistical framework for forecasting. Section 3 reports the experimental setup, evaluation metrics, and comparative results with benchmark models, highlighting improvements in accuracy, robustness, and symmetry-based statistical interpretation. Section 4 provides an in-depth discussion of the findings, their implications for environmental health management, and comparisons with prior studies. Finally, Section 6 concludes the paper by summarizing the main contributions, outlining policy implications, and suggesting avenues for future research.

2. Methods and Materials

This study proposes a probabilistic decomposition-based ensemble forecasting framework for predicting daily PM_2.5 concentrations in four major Pakistani megacities—Islamabad, Lahore, Karachi, and Peshawar—over a one-day horizon. Rapid variations in emissions, boundary-layer dynamics, and climatic factors influencing short-term dispersion and accumulation are the causes of PM_2.5 volatility. The approach is motivated by the theme of symmetry; structural decomposition reveals balanced patterns (trend, seasonality, irregularity), and ensemble integration exploits the complementary strengths of machine learning models through statistical aggregation.

The methodology is organized into three stages: data preprocessing, probabilistic decomposition, and component-wise forecasting with symmetric reconstruction.

• Data Preprocessing: The observed daily PM_2.5 concentrations (2019–2023) are preprocessed to ensure reliability. Missing observations are imputed using the Multiple Imputation by Chained Equations (MICE) procedure, which draws imputations from conditional distributions to preserve statistical symmetry and variability in the data (Section 2.1).
• Probabilistic Decomposition: Seasonal-trend decomposition based on Loess (STL) is applied to partition the original series into three additive components: trend ($T_t$), seasonal ($S_t$), and remainder ($R_t$). This decomposition enforces structural decomposition by isolating deterministic and stochastic parts of the signal [21].
• Component Forecasting and Temporal Reconstruction: Each component is independently modeled using four probabilistic machine learning algorithms—support vector regression (SVR), extreme gradient boosting (XGBoost), the Neural Network Autoregressive model (NNETAR), and extreme learning machine (ELM). Their forecasts are symmetrically aggregated to reconstruct the final PM_2.5 prediction (Section 2.3).

This design enhances robustness by aligning probabilistic inference with symmetry-based decomposition.

2.1. Multiple Imputation via Chained Equations (MICE)

Let the dataset be $\mathbf{Y}=\{Y_1,Y_2,\dots ,Y_p\}$, where each variable $Y_j$ may contain missing values. The MICE algorithm generates m plausible completed datasets by iteratively imputing missing entries from predictive distributions conditioned on the observed data:

$$Y_j^{\left(\mathrm{mis}\right)}\sim \mathrm{PMM}(Y_j^{\left(\mathrm{obs}\right)}\mid {\mathbf{Y}}_{-j}),$$

(1)

where predictive mean matching (PMM) preserves the distributional consistency between observed and imputed values. The multiple imputations

$$\{{\widehat{\mathbf{Y}}}^{\left(1\right)},{\widehat{\mathbf{Y}}}^{\left(2\right)},\dots ,{\widehat{\mathbf{Y}}}^{\left(m\right)}\}$$

(2)

represent posterior samples, thereby accounting for uncertainty in a symmetric Bayesian framework [22].

2.2. Mathematical Formulation of Forecasting Models

Each decomposed component ($T_t$, $S_t$, $R_t$) is modeled separately using the following algorithms, with each providing probabilistic estimates respecting data symmetry.

2.2.1. Support Vector Regression (SVR)

SVR constructs a symmetric $\epsilon$-insensitive loss function, balancing under- and over-estimation:

$$\begin{array}{cc}\hfill f\left(x\right)& ={\mathbf{w}}^T\varphi \left(x\right)+b,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \underset{\mathbf{w},b,{\xi }_i,{\xi }_i^{*}}{min}& \left(\frac{1}{2}{\parallel \mathbf{w}\parallel }^2+C\sum _{i=1}^n({\xi }_i+{\xi }_i^{*})\right),\hfill \end{array}$$

(4)

Subject to symmetric deviation constraints,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_i-f\left(x_i\right)\le \epsilon +{\xi }_i,\hfill \\ f\left(x_i\right)-y_i\le \epsilon +{\xi }_i^{*},\hfill \\ {\xi }_i,{\xi }_i^{*}\ge 0.\hfill \end{array}\right.\end{array}$$

(5)

2.2.2. Extreme Gradient Boosting (XGBoost)

XGBoost ensembles regression trees via additive updates under a regularized objective:

$$\begin{array}{cc}\hfill {\widehat{y}}_i& =\sum _{k=1}^K{f}_k\left(x_i\right),f_k\in \mathcal{F},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\left(t\right)}& =\sum _{i=1}^{n}l\left(y_i,{\widehat{y}}_i^{(t-1)}+f_t\left(x_i\right)\right)+\mathrm{\Omega }\left(f_t\right),\hfill \end{array}$$

(7)

where the regularization $\mathrm{\Omega }\left(f_t\right)$ ensures balanced complexity–fit symmetry.

2.2.3. Neural Network AutoRegressive Model (NNETAR)

NNETAR introduces nonlinear mapping through symmetric activation functions, $h(·)$:

$$y_t={\alpha }_0+\sum _{i=1}^k{\alpha }_{i}h\left(\sum _{j=1}^p{\beta }_{ij}{y}_{t-j}+{\beta }_{i0}\right)+{\epsilon }_t.$$

(8)

2.2.4. Extreme Learning Machine (ELM)

ELM achieves computational symmetry by randomly assigning input weights and estimating output weights via a pseudoinverse:

$$\begin{array}{cc}\hfill \widehat{y}& =H\beta ,\beta =H^{†}Y,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill H_{ij}& =g({\mathbf{w}}_j^T{x}_i+b_j).\hfill \end{array}$$

(10)

In this work, the balanced structural behavior of PM_2.5 time series over equivalent temporal intervals is indicated by temporal symmetry. Auto-correlation pattern stability and the almost zero skewness of residuals derived from STL-decomposed components are used to measure symmetry. The directional neutrality between positive and negative forecast deviations is maintained by using point estimates from each model, rather than a weighted aggregate. The theoretical idea of temporal equilibrium in environmental dynamics is reflected in this method’s preservation of structural stability and cyclical regularity.

2.3. Proposed STL–Decomposition Forecasting Framework

By combining point estimates from several base learners, the proposed ensemble framework produces probabilistic projections. The combined outputs capture the uncertainty in model predictions by producing a central tendency (mean forecast).

2.3.1. STL Decomposition

The observed PM_2.5 series $y_t$ is decomposed as follows:

$$y_t=T_t+S_t+R_t,$$

(11)

with symmetric additive partitioning.

2.3.2. Component Forecasting

For each component $C\in \{T,S,R\}$ and model $i\in \{\mathrm{SVR},\mathrm{XGB},\mathrm{NNETAR},\mathrm{ELM}\}$:

$${\widehat{C}}_t^{\left(i\right)}=f_{{\mathrm{Model}}_i}^{\left(C\right)}\left(X_t\right).$$

(12)

2.3.3. Final Symmetric Reconstruction

The overall PM_2.5 forecast is reconstructed by summing the aggregated component forecasts:

$${\widehat{y}}_t={\widehat{T}}_t+{\widehat{S}}_t+{\widehat{R}}_t.$$

(13)

This ensures structural symmetry between decomposition and reconstruction.

2.4. Evaluation Metrics

To assess forecasting accuracy, five symmetric and probabilistic metrics are employed (

Table 1. Evaluation metrics and their mathematical formulations.

Metric	Formula
Mean Absolute Error (MAE)	${\displaystyle \mathrm{MAE}=\frac{1}{n}\sum _{t=1}^n|y_t-{\widehat{y}}_t|}$
Root Mean Squared Error (RMSE)	${\displaystyle \mathrm{RMSE}=\sqrt{\frac{1}{n}\sum _{t=1}^n{(y_t-{\widehat{y}}_t)}^2}}$
Correlation Coefficient (r)	${\displaystyle r=\frac{{\sum }_{t=1}^n(y_t-\overline{y})({\widehat{y}}_t-\overline{\widehat{y}})}{\sqrt{{\sum }_{t=1}^n{(y_t-\overline{y})}^2}\sqrt{{\sum }_{t=1}^n{({\widehat{y}}_t-\overline{\widehat{y}})}^2}}}$
Symmetric Mean Absolute Percentage Error (sMAPE)	${\displaystyle \mathrm{sMAPE}=\frac{100\%}{n}\sum _{t=1}^n\frac{|y_t-{\widehat{y}}_t|}{\left(\right|y_t|+|{\widehat{y}}_t\left|\right)/2}}$
Mean Directional Accuracy (MDA)	${\displaystyle \mathrm{MDA}=\frac{1}{n-1}\sum _{t=2}^n\mathbb{I}[(y_t-y_{t-1})({\widehat{y}}_t-{\widehat{y}}_{t-1})>0]}$

). These metrics quantify both scale-dependent error (MAE, RMSE) and relative symmetric performance (sMAPE, MDA), while the correlation coefficient r captures probabilistic dependence [23,24,25,26].

3. Results

Table 2. Details about the datasets considered in this work.

Attribute	Islamabad	Lahore	Karachi	Peshawar
Data period	2019–2023	2019–2023	2019–2023	2019–2023
Total	1461	1461	1461	1461
Observed	1430	1418	1426	1433
Missing	31	43	35	28
Imputed (%)	2.12	2.94	2.40	1.92

presents detailed information on the daily PM_2.5 datasets collected from four major Pakistani cities—Islamabad, Lahore, Karachi, and Peshawar—covering the period from 2019 to 2023. Each dataset contains 1461 daily entries, ensuring temporal uniformity across locations. The number of observed (non-missing) data points varies slightly, with Islamabad recording 1430, Lahore 1418, Karachi 1426, and Peshawar 1433 valid observations. Consequently, the proportion of missing values remains relatively small, ranging between $1.92\%$ (Peshawar) and $2.94\%$ (Lahore). These missing observations were subsequently imputed using the MICE algorithm, which preserves the underlying statistical distribution by generating plausible values from conditional probability models. The low percentage of imputation ensures high data reliability and symmetry of information across all cities, thereby enhancing the robustness of subsequent inferential analysis and comparative modeling.

Table 3. Descriptive statistics of PM_2.5 (raw and logarithmic values).

City/Statistic	Minimum	Maximum	25%	50%	75%	Mean	Variance	Std. Dev.
Islamabad	10.00	270.00	84.00	106.00	139.00	112.34	1576.95	39.71
Lahore	5.00	503.00	128.00	162.00	218.00	183.15	7257.72	85.19
Karachi	9.00	277.00	77.00	97.00	143.00	109.97	1795.17	42.37
Peshawar	16.00	557.00	112.00	142.00	168.00	146.01	2662.09	51.60
City/Statistic	Skewness	Kurtosis	ADF Statistic	D_ADF Statistic	log(Min)	log(Max)	log(25%)	log(50%)
Islamabad	0.43	2.94	−4.05	−16.97	2.30	5.60	4.43	4.66
Lahore	1.12	4.18	−4.24	−16.12	1.61	6.22	4.85	5.09
Karachi	0.67	2.76	−3.66	−16.82	2.20	5.62	4.34	4.57
Peshawar	1.44	8.60	−4.68	−17.38	2.77	6.32	4.72	4.96
City/Statistic	log(75%)	log(Mean)	log(Variance)	log(Std. Dev.)	log(Skewness)	log(Kurtosis)	log(ADF)	log(D_ADF)
Islamabad	4.93	4.65	0.15	0.39	−1.03	6.23	−4.54	−17.42
Lahore	5.38	5.10	0.25	0.50	−1.43	11.44	−5.02	−15.85
Karachi	4.96	4.62	0.16	0.40	−0.66	6.43	−4.34	−16.57
Peshawar	5.12	4.92	0.13	0.36	−0.92	7.95	−5.37	−16.20

provides a comprehensive set of descriptive statistics for each city, including raw values and their logarithmic transformations. All variable units are listed in Table 3; the squared concentration units (µg/m³)² generate the high variance values, which indicate dispersion, rather than magnitude. The raw distributions reveal heterogeneity across cities: Lahore exhibits the widest range (minimum $5.00$, maximum $503.00$) with the most significant variance ($7257.72$) and standard deviation ($85.19$), suggesting pronounced dispersion and potential outliers. Peshawar records the overall maximum PM_2.5 value ($557.00$), reflecting extreme pollution episodes and elevated variability. In contrast, Islamabad and Karachi exhibit comparatively moderate levels (means of $112.34$ and $109.97$, respectively) and more minor variances, indicating more stable pollution dynamics. From a probabilistic perspective, skewness and kurtosis statistics highlight the symmetry properties of the distributions. Islamabad and Karachi exhibit relatively balanced distributions (skewness of $0.43$ and $0.67$, respectively, and kurtosis of $2.94$ and $2.76$), indicating near-symmetric patterns with moderate tails. Conversely, Lahore and Peshawar show stronger positive skewness ($1.12$ and $1.44$) and higher kurtosis ($4.18$ and $8.60$), consistent with asymmetric and leptokurtic behavior, where extreme events occur more frequently than in Gaussian processes.

Logarithmic transformation introduces statistical symmetry by stabilizing variance and reducing extreme values, as evidenced by smaller variances and lower dispersion in log-space. The negative skewness of the log-transformed variables further suggests that the transformation mitigates right-skewness, restoring distributional balance. This aligns with the symmetry theme by demonstrating how transformations recover equilibrium in statistical moments and improve interpretability for probabilistic modeling. However, the results of augmented Dickey–Fuller (ADF) and differenced ADF (D_ADF) tests confirm strong stationarity across all series, with test statistics remaining strongly negative. This indicates that the PM_2.5 series, once detrended or differenced, attain stable statistical symmetry in their stochastic structure, a crucial prerequisite for time series forecasting.

The concept of symmetry, central to the theme of this study, is examined both statistically and temporally within the PM_2.5 time series. From a statistical perspective, symmetry refers to the balanced distribution of concentration values around the central tendency. To assess this property, descriptive statistics such as Figure 1 present seasonal subseries plots to evaluate temporal symmetry in PM_2.5. Karachi and Peshawar show inconsistencies later captured in the STL remaining components, while Lahore and Islamabad show a moderately similar pattern due to the occurrence of high-pollution episodes. However, the application of logarithmic transformation substantially reduces skewness and excess kurtosis across all cities, thereby improving the symmetry of the distributions. This enhanced statistical symmetry stabilizes the variance and mitigates the influence of outliers, improving the performance of subsequent machine learning models.

From a temporal perspective, symmetry manifests through the repetitive, cyclic patterns captured via the STL decomposition (Figure 1). The seasonal component exhibits a nearly symmetric oscillation around the long-term trend, reflecting recurring emission cycles and meteorological regularities. The decomposition thereby isolates symmetric temporal fluctuations from irregular residual variations, which enables more interpretable and stable forecasting. In addition, the proposed STL-based hybrid framework exploits these symmetrical characteristics through its ensemble design. Models with distinct inductive biases—SVR and ELM for linear relationships, and NNAR for nonlinear dynamics—are integrated to maintain equilibrium between underfitting and overfitting tendencies. This balanced modeling structure effectively enforces a probabilistic form of symmetry in the error distribution, as reflected in the nearly unbiased residuals and improved directional accuracy (MDA) across all cities. Collectively, these findings confirm that both statistical and temporal symmetry are intrinsic to the PM_2.5 series and are strategically leveraged within the proposed hybrid framework to enhance predictive precision and interpretability.

In air quality, temporal symmetry refers to recurrent seasonal or diurnal patterns that exhibit steady structural repetition throughout time. Figure 1 presents seasonal subseries plots to evaluate temporal symmetry in PM_2.5: Karachi and Peshawar show inconsistencies later captured in the STL remaining components, while Lahore and Islamabad show moderate similar pattern. A comparative assessment of the original and MICE-imputed time series of PM_2.5 concentrations for the four major cities, Islamabad, Karachi, Lahore, and Peshawar, together with their corresponding STL decompositions, is provided in Figure 1. The imputed series, as displayed in Figure 1a,c,e,g, exhibit a high degree of concordance with the original data, successfully preserving underlying temporal patterns and abrupt fluctuations. This alignment demonstrates that the MICE procedure accurately restored missing values while maintaining the statistical symmetry of the series, i.e., the balance between smooth cyclic components and irregular stochastic variations. Notably, the imputation preserved the integrity of extreme values in more volatile environments such as Lahore and Peshawar, where sudden pollution spikes are characteristic of episodic urban activity and meteorological shocks. However, the term “volatile environments” refers to urban contexts characterized by high temporal variability and abrupt fluctuations in PM_2.5 concentrations, typically driven by short-term anthropogenic and meteorological influences such as irregular industrial emissions, traffic congestion peaks, biomass burning, dust storms, and temperature inversions. Notably, the imputation preserved the integrity of extreme values in such volatile environments, particularly in Lahore and Peshawar—where sudden pollution spikes reflect episodic urban activity and meteorological shocks. In contrast, relatively stable environments like Islamabad exhibit smoother and more symmetric seasonal cycles with limited short-term disturbances. Recognizing these differences is essential, as hybrid ensemble models must adapt to both symmetric and asymmetric fluctuation patterns to maintain forecasting accuracy across diverse urban conditions.

The subsequent STL decompositions, shown in Figure 1b,d,f,h, further validate the reliability of the imputation step. In each case, the decomposition successfully disentangled the trend, seasonal, and remainder components, ensuring that the structural symmetry between long-term drift, cyclic oscillations, and irregular shocks was preserved. Islamabad and Lahore reveal smooth seasonal cycles and steady trend evolution, suggesting symmetric and stable temporal dynamics. By contrast, Karachi and Peshawar exhibit greater short-term variability, implying asymmetries introduced via external factors such as industrial activity or meteorological dispersion effects.

Taken together, these results confirm that the MICE-based imputation not only filled missing values without distorting the temporal structure but also maintained the probabilistic and structural symmetry required for reliable downstream modeling. The preservation of trend–seasonal balance and the accurate representation of extreme events ensures that the reconstructed series provides a statistically consistent foundation for subsequent hybrid ensemble forecasting of PM_2.5 concentrations. However, it is acknowledged that validating the imputation using the same dataset may not provide a fully independent measure of reliability. Therefore, in this study, the evaluation of MICE-imputed PM_2.5 values emphasizes internal consistency, rather than external validation. Specifically, we assessed the reliability of the imputation by comparing the statistical distributions, temporal dynamics, and decomposed STL components between the original and imputed series. The close alignment of these characteristics suggests that the MICE procedure effectively preserves the intrinsic temporal and probabilistic structure of the data without introducing artificial bias. Nevertheless, future studies could further strengthen imputation validation by employing external datasets, such as concurrent meteorological observations or adjacent monitoring station records, to independently assess imputation fidelity.

3.1. Islamabad

For Islamabad, several statistical indicators—MAE, RMSE, correlation, sMAPE, and MDA—were employed to evaluate the STL-decomposed forecasting frameworks. The comparative results, summarized in

Table 4. Islamabad: accuracy mean error.

STL Model Combinations	MAE	RMSE	Correlation	sMAPE	MDA
SVRt + SVRr + SVRs	10.324	12.832	0.784	11.661	0.495
SVRt + SVRr + XGBs	11.895	14.498	0.721	13.262	0.475
SVRt + SVRr + NNARs	6.584	8.518	0.918	7.562	0.737
SVRt + SVRr + ELMs	7.773	9.979	0.897	8.931	0.737
SVRt + XGBr + SVRs	11.955	14.759	0.716	13.498	0.485
SVRt + XGBr + XGBs	12.516	15.667	0.677	14.082	0.475
SVRt + XGBr + NNARs	10.163	12.828	0.794	11.332	0.646
SVRt + XGBr + ELMs	10.980	13.437	0.772	12.335	0.667
SVRt + NNARr + SVRs	12.845	15.729	0.690	14.062	0.485
SVRt + NNARr + XGBs	13.910	16.944	0.639	15.216	0.485
SVRt + NNARr + NNARs	9.158	11.467	0.832	10.347	0.707
SVRt + NNARr + ELMs	10.031	12.450	0.795	11.160	0.616
SVRt + ELMr + SVRs	11.099	13.445	0.780	12.443	0.566
SVRt + ELMr + XGBs	12.897	15.526	0.697	14.216	0.535
SVRt + ELMr + NNARs	6.277	7.912	0.969	7.559	0.838
SVRt + ELMr + ELMs	8.134	10.035	0.951	9.589	0.869
XGBt + SVRr + SVRs	10.321	12.854	0.784	11.656	0.495
XGBt + SVRr + XGBs	11.904	14.524	0.720	13.267	0.475
XGBt + SVRr + NNARs	6.612	8.548	0.918	7.597	0.737
XGBt + SVRr + ELMs	7.808	10.017	0.897	8.976	0.737
XGBt + XGBr + SVRs	11.973	14.790	0.716	13.512	0.485
XGBt + XGBr + XGBs	12.543	15.702	0.677	14.108	0.475
XGBt + XGBr + NNARs	10.200	12.861	0.794	11.372	0.646
XGBt + XGBr + ELMs	11.006	13.479	0.772	12.365	0.667
XGBt + NNARr + SVRs	12.843	15.733	0.690	14.056	0.485
XGBt + NNARr + XGBs	13.908	16.953	0.639	15.207	0.485
XGBt + NNARr + NNARs	9.162	11.469	0.832	10.349	0.707
XGBt + NNARr + ELMs	10.040	12.463	0.795	11.172	0.616
XGBt + ELMr + SVRs	11.131	13.492	0.780	12.477	0.566
XGBt + ELMr + XGBs	12.936	15.572	0.697	14.255	0.535
XGBt + ELMr + NNARs	6.333	7.986	0.969	7.631	0.838
XGBt + ELMr + ELMs	8.194	10.106	0.951	9.660	0.869
NNARt + SVRr + SVRs	10.321	12.847	0.784	11.657	0.495
NNARt + SVRr + XGBs	11.901	14.515	0.720	13.265	0.475
NNARt + SVRr + NNARs	6.603	8.539	0.918	7.586	0.737
NNARt + SVRr + ELMs	7.797	10.005	0.897	8.962	0.737
NNARt + XGBr + SVRs	11.967	14.780	0.716	13.507	0.485
NNARt + XGBr + XGBs	12.535	15.691	0.677	14.100	0.475
NNARt + XGBr + NNARs	10.189	12.851	0.794	11.359	0.646
NNARt + XGBr + ELMs	10.998	13.466	0.772	12.356	0.667
NNARt + NNARr + SVRs	12.843	15.731	0.690	14.057	0.485
NNARt + NNARr + XGBs	13.909	16.950	0.639	15.209	0.485
NNARt + NNARr + NNARs	9.161	11.468	0.832	10.348	0.707
NNARt + NNARr + ELMs	10.037	12.459	0.795	11.168	0.616
NNARt + ELMr + SVRs	11.121	13.477	0.780	12.466	0.566
NNARt + ELMr + XGBs	12.924	15.557	0.697	14.243	0.535
NNARt + ELMr + NNARs	6.316	7.963	0.969	7.609	0.838
NNARt + ELMr + ELMs	8.175	10.084	0.951	9.638	0.869
ELMt + SVRr + SVRs	10.321	12.847	0.784	11.657	0.495
ELMt + SVRr + XGBs	11.901	14.516	0.721	13.265	0.475
ELMt + SVRr + NNARs	6.603	8.539	0.918	7.586	0.737
ELMt + SVRr + ELMs	7.797	10.005	0.897	8.962	0.737
ELMt + XGBr + SVRs	11.967	14.780	0.716	13.507	0.485
ELMt + XGBr + XGBs	12.535	15.691	0.677	14.100	0.475
ELMt + XGBr + NNARs	10.189	12.851	0.794	11.359	0.646
ELMt + XGBr + ELMs	10.998	13.466	0.772	12.356	0.667
ELMt + NNARr + SVRs	12.843	15.731	0.690	14.057	0.485
ELMt + NNARr + XGBs	13.908	16.950	0.639	15.209	0.485
ELMt + NNARr + NNARs	9.161	11.468	0.832	10.348	0.707
ELMt + NNARr + ELMs	10.037	12.459	0.795	11.168	0.616
ELMt + ELMr + SVRs	11.121	13.477	0.780	12.466	0.566
ELMt + ELMr + XGBs	12.924	15.558	0.697	14.243	0.535
ELMt + ELMr + NNARs	6.316	7.963	0.969	7.609	0.838
ELMt + ELMr + ELMs	8.176	10.085	0.951	9.638	0.869

, reveal that the four leading hybrid configurations were as follows: {SVR_t + ELM_r + NNAR_s}, {NNAR_t + ELM_r + NNAR_s}, {ELM_t + ELM_r + NNAR_s}, and {XGB_t + ELM_r + NNAR_s}. Among these, the {SVR_t + ELM_r + NNAR_s} model emerged as the most effective, attaining the lowest sMAPE (7.56) and RMSE (7.91), the highest MDA (0.838), and a strong correlation (0.969). This balanced performance illustrates its ability to capture both the symmetry of forecast alignment and probabilistic directional consistency. The near-identical accuracy of the second- and third-ranked models, {NNAR_t + ELM_r + NNAR_s} and {ELM_t + ELM_r + NNAR_s}, highlights the robustness of ELM and NNAR integration. Finally, the {XGB_t + ELM_r + NNAR_s} specification also performed competitively (RMSE: 7.99; correlation: 0.969), underscoring the utility of boosting strategies for trend representation. Overall, these findings confirm that statistical symmetry between error minimization and directional forecasting can be effectively achieved using ensemble architectures.

3.2. Karachi

For Karachi, a consistent performance pattern emerges across STL-based hybrid configurations, as detailed in

Table 5. Karachi: accuracy mean error.

STL Model Combinations	MAE	RMSE	Correlation	sMAPE	MDA
SVRt + SVRr + SVRs	9.708	13.652	0.851	11.775	0.556
SVRt + SVRr + XGBs	10.303	14.726	0.823	12.521	0.545
SVRt + SVRr + NNARs	6.891	8.351	0.951	8.629	0.677
SVRt + SVRr + ELMs	7.990	10.388	0.930	10.000	0.697
SVRt + XGBr + SVRs	11.387	15.763	0.794	13.759	0.444
SVRt + XGBr + XGBs	11.464	16.549	0.775	13.922	0.414
SVRt + XGBr + NNARs	10.197	12.346	0.881	12.880	0.545
SVRt + XGBr + ELMs	11.039	13.740	0.850	13.681	0.566
SVRt + NNARr + SVRs	13.456	17.774	0.733	16.221	0.434
SVRt + NNARr + XGBs	14.150	18.863	0.708	16.982	0.404
SVRt + NNARr + NNARs	9.866	13.149	0.861	12.202	0.515
SVRt + NNARr + ELMs	10.824	14.342	0.833	13.212	0.505
SVRt + ELMr + SVRs	10.456	14.052	0.851	12.702	0.616
SVRt + ELMr + XGBs	11.420	15.276	0.813	13.898	0.596
SVRt + ELMr + NNARs	6.073	7.706	0.975	7.694	0.798
SVRt + ELMr + ELMs	8.224	10.161	0.962	10.443	0.869
XGBt + SVRr + SVRs	9.829	13.718	0.851	11.955	0.556
XGBt + SVRr + XGBs	10.405	14.787	0.823	12.652	0.545
XGBt + SVRr + NNARs	7.001	8.476	0.951	8.781	0.677
XGBt + SVRr + ELMs	8.120	10.532	0.930	10.199	0.697
XGBt + XGBr + SVRs	11.412	15.818	0.794	13.783	0.444
XGBt + XGBr + XGBs	11.534	16.601	0.775	14.011	0.414
XGBt + XGBr + NNARs	10.280	12.427	0.881	12.960	0.545
XGBt + XGBr + ELMs	11.118	13.846	0.850	13.765	0.566
XGBt + NNARr + SVRs	13.454	17.792	0.732	16.176	0.434
XGBt + NNARr + XGBs	14.157	18.879	0.707	16.953	0.404
XGBt + NNARr + NNARs	9.903	13.185	0.861	12.217	0.515
XGBt + NNARr + ELMs	10.862	14.406	0.833	13.258	0.505
XGBt + ELMr + SVRs	10.641	14.199	0.851	12.971	0.616
XGBt + ELMr + XGBs	11.543	15.411	0.813	14.061	0.596
XGBt + ELMr + NNARs	6.406	7.990	0.975	8.182	0.798
XGBt + ELMr + ELMs	8.540	10.422	0.962	10.885	0.869
NNARt + SVRr + SVRs	9.703	13.653	0.851	11.768	0.556
NNARt + SVRr + XGBs	10.298	14.728	0.823	12.515	0.545
NNARt + SVRr + NNARs	6.892	8.351	0.950	8.631	0.677
NNARt + SVRr + ELMs	7.992	10.388	0.930	10.001	0.697
NNARt + XGBr + SVRs	11.387	15.763	0.794	13.759	0.444
NNARt + XGBr + XGBs	11.465	16.551	0.775	13.923	0.414
NNARt + XGBr + NNARs	10.199	12.345	0.881	12.883	0.545
NNARt + XGBr + ELMs	11.043	13.739	0.850	13.685	0.566
NNARt + NNARr + SVRs	13.452	17.777	0.733	16.217	0.434
NNARt + NNARr + XGBs	14.148	18.867	0.708	16.981	0.404
NNARt + NNARr + NNARs	9.869	13.152	0.861	12.210	0.515
NNARt + NNARr + ELMs	10.828	14.344	0.833	13.218	0.505
NNARt + ELMr + SVRs	10.453	14.051	0.851	12.698	0.616
NNARt + ELMr + XGBs	11.419	15.276	0.813	13.898	0.596
NNARt + ELMr + NNARs	6.074	7.707	0.975	7.693	0.798
NNARt + ELMr + ELMs	8.225	10.161	0.962	10.445	0.869
ELMt + SVRr + SVRs	9.685	13.644	0.851	11.739	0.556
ELMt + SVRr + XGBs	10.283	14.721	0.823	12.495	0.545
ELMt + SVRr + NNARs	6.871	8.332	0.950	8.601	0.677
ELMt + SVRr + ELMs	7.971	10.364	0.930	9.968	0.697
ELMt + XGBr + SVRs	11.386	15.756	0.794	13.758	0.444
ELMt + XGBr + XGBs	11.452	16.544	0.775	13.907	0.414
ELMt + XGBr + NNARs	10.184	12.332	0.881	12.868	0.545
ELMt + XGBr + ELMs	11.028	13.721	0.850	13.669	0.566
ELMt + NNARr + SVRs	13.454	17.777	0.733	16.228	0.434
ELMt + NNARr + XGBs	14.153	18.867	0.708	16.997	0.404
ELMt + NNARr + NNARs	9.867	13.150	0.861	12.215	0.515
ELMt + NNARr + ELMs	10.825	14.336	0.833	13.215	0.505
ELMt + ELMr + SVRs	10.421	14.026	0.851	12.650	0.616
ELMt + ELMr + XGBs	11.395	15.253	0.813	13.866	0.596
ELMt + ELMr + NNARs	6.005	7.652	0.975	7.590	0.798
ELMt + ELMr + ELMs	8.164	10.110	0.962	10.357	0.869

. The top four models were identified as follows: {ELM_t + ELM_r + NNAR_s}, {NNAR_t + ELM_r + NNAR_s}, {SVR_t + ELM_r + NNAR_s}, and {XGB_t + ELM_r + NNAR_s}. For the most effective among these, {ELM_t + ELM_r + NNAR_s}, the lowest RMSE (7.652) and MAE (6.005) were reported, together with a favorable sMAPE (7.590) and high correlation (0.975). By employing ELM for both trend and residual components and NNAR for seasonality, this model effectively balanced nonlinear and linear dynamics, reflecting probabilistic symmetry in capturing both smooth cycles and stochastic fluctuations. The next two models, {NNAR_t + ELM_r + NNAR_s} and {SVR_t + ELM_r + NNAR_s}, attained slightly higher RMSE values but exhibited identical correlation (0.975) and MDA (0.798), demonstrating equivalent directional symmetry. The {XGB_t + ELM_r + NNAR_s} model also achieved stable correlation and MDA scores, though with modestly higher error magnitudes. These outcomes reinforce the effectiveness of combining statistical learning algorithms (ELM, SVR, NNAR, XGBoost) in decomposition-based hybrid structures for air quality forecasting.

3.3. Lahore

In Lahore, model evaluation emphasized RMSE due to its sensitivity to large deviations, which is particularly relevant, given the volatility of PM_2.5 levels in the region. As reported in

Table 6. Lahore: accuracy mean error.

STL Model Combinations	MAE	RMSE	Correlation	sMAPE	MDA
SVRt + SVRr + SVRs	19.053	24.054	0.748	14.824	0.616
SVRt + SVRr + XGBs	21.790	26.955	0.672	16.767	0.586
SVRt + SVRr + NNARs	12.631	15.424	0.933	9.701	0.747
SVRt + SVRr + ELMs	15.634	19.518	0.892	11.846	0.778
SVRt + XGBr + SVRs	22.790	28.856	0.626	17.178	0.515
SVRt + XGBr + XGBs	24.153	30.100	0.574	18.292	0.495
SVRt + XGBr + NNARs	18.411	23.224	0.768	13.741	0.636
SVRt + XGBr + ELMs	19.984	25.868	0.703	14.877	0.586
SVRt + NNARr + SVRs	23.766	31.430	0.556	17.875	0.424
SVRt + NNARr + XGBs	26.387	33.244	0.482	19.905	0.394
SVRt + NNARr + NNARs	18.575	24.197	0.747	14.001	0.596
SVRt + NNARr + ELMs	20.327	27.171	0.669	15.286	0.596
SVRt + ELMr + SVRs	18.447	23.318	0.772	14.463	0.576
SVRt + ELMr + XGBs	21.436	26.400	0.685	16.568	0.586
SVRt + ELMr + NNARs	10.151	12.521	0.979	8.386	0.818
SVRt + ELMr + ELMs	13.442	16.592	0.954	10.614	0.879
XGBt + SVRr + SVRs	18.844	23.947	0.749	14.680	0.626
XGBt + SVRr + XGBs	21.556	26.764	0.674	16.599	0.586
XGBt + SVRr + NNARs	12.456	15.187	0.934	9.638	0.747
XGBt + SVRr + ELMs	15.331	19.194	0.895	11.686	0.778
XGBt + XGBr + SVRs	22.638	28.818	0.626	17.053	0.515
XGBt + XGBr + XGBs	23.991	29.978	0.575	18.148	0.505
XGBt + XGBr + NNARs	18.306	23.132	0.769	13.674	0.636
XGBt + XGBr + ELMs	19.871	25.682	0.705	14.803	0.586
XGBt + NNARr + SVRs	23.664	31.344	0.557	17.755	0.424
XGBt + NNARr + XGBs	26.198	33.086	0.484	19.726	0.394
XGBt + NNARr + NNARs	18.557	24.042	0.748	13.962	0.596
XGBt + NNARr + ELMs	20.180	26.935	0.671	15.153	0.596
XGBt + ELMr + SVRs	18.549	23.524	0.772	14.551	0.576
XGBt + ELMr + XGBs	21.482	26.486	0.687	16.612	0.586
XGBt + ELMr + NNARs	10.349	12.819	0.979	8.641	0.808
XGBt + ELMr + ELMs	13.565	16.660	0.956	10.807	0.879
NNARt + SVRr + SVRs	19.050	24.055	0.748	14.823	0.616
NNARt + SVRr + XGBs	21.787	26.949	0.672	16.766	0.586
NNARt + SVRr + NNARs	12.625	15.414	0.933	9.701	0.747
NNARt + SVRr + ELMs	15.619	19.502	0.892	11.840	0.778
NNARt + XGBr + SVRs	22.790	28.865	0.625	17.176	0.515
NNARt + XGBr + XGBs	24.154	30.102	0.573	18.289	0.505
NNARt + XGBr + NNARs	18.410	23.228	0.768	13.739	0.636
NNARt + XGBr + ELMs	19.984	25.866	0.702	14.878	0.586
NNARt + NNARr + SVRs	23.755	31.423	0.556	17.865	0.424
NNARt + NNARr + XGBs	26.373	33.232	0.482	19.893	0.394
NNARt + NNARr + NNARs	18.571	24.181	0.747	13.997	0.596
NNARt + NNARr + ELMs	20.313	27.151	0.669	15.274	0.596
NNARt + ELMr + SVRs	18.465	23.344	0.771	14.479	0.576
NNARt + ELMr + XGBs	21.444	26.416	0.685	16.577	0.586
NNARt + ELMr + NNARs	10.175	12.556	0.979	8.414	0.808
NNARt + ELMr + ELMs	13.464	16.609	0.954	10.638	0.879
ELMt + SVRr + SVRs	19.108	24.090	0.748	14.863	0.616
ELMt + SVRr + XGBs	21.823	26.991	0.672	16.786	0.586
ELMt + SVRr + NNARs	12.657	15.460	0.933	9.736	0.747
ELMt + SVRr + ELMs	15.688	19.563	0.892	11.871	0.778
ELMt + XGBr + SVRs	22.829	28.950	0.625	17.188	0.515
ELMt + XGBr + XGBs	24.198	30.111	0.573	18.295	0.505
ELMt + XGBr + NNARs	18.443	23.259	0.768	13.724	0.636
ELMt + XGBr + ELMs	19.981	25.840	0.703	14.875	0.586
ELMt + NNARr + SVRs	23.774	31.435	0.555	17.883	0.424
ELMt + NNARr + XGBs	26.429	33.288	0.481	19.947	0.394
ELMt + NNARr + NNARs	18.601	24.215	0.747	14.018	0.596
ELMt + NNARr + ELMs	20.354	27.201	0.668	15.306	0.596
ELMt + ELMr + SVRs	18.493	23.369	0.771	14.511	0.576
ELMt + ELMr + XGBs	21.508	26.491	0.686	16.593	0.586
ELMt + ELMr + NNARs	10.287	12.720	0.978	8.566	0.808
ELMt + ELMr + ELMs	13.507	16.634	0.956	10.854	0.879

, the {ELM_t + ELM_r + NNAR_s} configuration yielded the lowest RMSE (12.475), alongside the lowest MAE (10.120) and sMAPE (8.340). Despite exhibiting a slightly lower MDA (0.808) than its nearest rival, this model demonstrated probabilistic superiority by consistently reducing both absolute and relative forecast errors. The second-best specification, {SVR_t + ELM_r + NNAR_s}, achieved a competitive RMSE (12.521), while recording the highest MDA (0.818) and a correlation of 0.979. This indicates enhanced directional symmetry despite marginally higher error variance. The other contenders, {XGB_t + ELM_r + NNAR_s} and {NNAR_t + ELM_r + NNAR_s}, posted RMSE values of 12.556 and 12.819, respectively, but they retained comparable correlation and MDA values. Collectively, these results highlight the probabilistic trade-off between minimizing error magnitudes and maximizing directional alignment, reinforcing the reliability of ELM-based ensemble schemes in volatile urban environments.

3.4. Peshawar

For Peshawar, four STL-hybrid frameworks, each employing SVR for the residual component, were evaluated (

Table 7. Peshawar: accuracy mean error.

STL Model Combinations	MAE	RMSE	Correlation	sMAPE	MDA
SVRt + SVRr + SVRs	14.137	18.369	0.661	12.713	0.515
SVRt + SVRr + XGBs	15.908	20.258	0.592	14.253	0.485
SVRt + SVRr + NNARs	10.871	13.585	0.837	9.902	0.768
SVRt + SVRr + ELMs	12.182	15.462	0.797	11.046	0.798
SVRt + XGBr + SVRs	15.897	20.602	0.568	14.159	0.465
SVRt + XGBr + XGBs	17.526	22.244	0.510	15.572	0.444
SVRt + XGBr + NNARs	13.278	16.853	0.744	11.940	0.687
SVRt + XGBr + ELMs	14.132	18.316	0.692	12.642	0.657
SVRt + NNARr + SVRs	16.391	21.641	0.511	14.658	0.515
SVRt + NNARr + XGBs	17.749	23.133	0.455	15.867	0.414
SVRt + NNARr + NNARs	13.147	15.772	0.760	11.926	0.657
SVRt + NNARr + ELMs	14.343	18.145	0.667	12.877	0.606
SVRt + ELMr + SVRs	14.838	18.868	0.686	13.389	0.586
SVRt + ELMr + XGBs	16.750	21.260	0.596	14.952	0.515
SVRt + ELMr + NNARs	10.913	13.851	0.886	10.153	0.808
SVRt + ELMr + ELMs	13.431	16.237	0.862	12.321	0.909
XGBt + SVRr + SVRs	14.167	18.415	0.660	12.743	0.515
XGBt + SVRr + XGBs	15.952	20.314	0.592	14.288	0.485
XGBt + SVRr + NNARs	10.916	13.651	0.836	9.942	0.768
XGBt + SVRr + ELMs	12.258	15.547	0.796	11.118	0.808
XGBt + XGBr + SVRs	15.929	20.660	0.567	14.184	0.465
XGBt + XGBr + XGBs	17.565	22.311	0.510	15.594	0.444
XGBt + XGBr + NNARs	13.324	16.926	0.743	11.978	0.687
XGBt + XGBr + ELMs	14.213	18.407	0.690	12.711	0.657
XGBt + NNARr + SVRs	16.378	21.658	0.511	14.641	0.515
XGBt + NNARr + XGBs	17.776	23.162	0.454	15.881	0.424
XGBt + NNARr + NNARs	13.180	15.799	0.759	11.952	0.667
XGBt + NNARr + ELMs	14.393	18.192	0.666	12.925	0.606
XGBt + ELMr + SVRs	14.913	18.945	0.686	13.455	0.586
XGBt + ELMr + XGBs	16.826	21.343	0.595	15.013	0.515
XGBt + ELMr + NNARs	10.973	13.959	0.886	10.211	0.808
XGBt + ELMr + ELMs	13.509	16.357	0.860	12.391	0.909
NNARt + SVRr + SVRs	14.140	18.392	0.660	12.717	0.515
NNARt + SVRr + XGBs	15.918	20.279	0.592	14.261	0.485
NNARt + SVRr + NNARs	10.898	13.621	0.836	9.925	0.768
NNARt + SVRr + ELMs	12.206	15.489	0.797	11.071	0.808
NNARt + XGBr + SVRs	15.898	20.630	0.567	14.160	0.465
NNARt + XGBr + XGBs	17.531	22.270	0.510	15.572	0.455
NNARt + XGBr + NNARs	13.313	16.891	0.743	11.971	0.687
NNARt + XGBr + ELMs	14.158	18.348	0.691	12.667	0.657
NNARt + NNARr + SVRs	16.368	21.624	0.511	14.614	0.515
NNARt + NNARr + XGBs	17.743	23.110	0.455	15.856	0.424
NNARt + NNARr + NNARs	13.132	15.763	0.759	11.904	0.667
NNARt + NNARr + ELMs	14.317	18.129	0.666	12.846	0.606
NNARt + ELMr + SVRs	14.778	18.808	0.686	13.330	0.586
NNARt + ELMr + XGBs	16.678	21.196	0.595	14.932	0.515
NNARt + ELMr + NNARs	10.869	13.848	0.886	10.146	0.808
NNARt + ELMr + ELMs	13.413	16.212	0.860	12.308	0.909
ELMt + SVRr + SVRs	14.159	18.377	0.660	12.726	0.515
ELMt + SVRr + XGBs	15.926	20.285	0.592	14.267	0.485
ELMt + SVRr + NNARs	10.916	13.631	0.836	9.933	0.768
ELMt + SVRr + ELMs	12.206	15.478	0.797	11.063	0.808
ELMt + XGBr + SVRs	15.898	20.619	0.567	14.151	0.465
ELMt + XGBr + XGBs	17.531	22.251	0.510	15.562	0.455
ELMt + XGBr + NNARs	13.313	16.872	0.743	11.962	0.687
ELMt + XGBr + ELMs	14.158	18.332	0.691	12.657	0.657

). The leading model, {SVR_t + SVR_r + NNAR_s}, reported the lowest RMSE (13.585), MAE (10.871), and a competitive sMAPE (9.902), confirming its accuracy in capturing both absolute and relative forecast structures. The {ELM_t + SVR_r + NNAR_s} variant achieved a nearly identical RMSE (13.620) and MAE (10.898) while outperforming in terms of MDA (0.779), suggesting enhanced directional symmetry despite slightly higher error levels. The remaining specifications, {NNAR_t + SVR_r + NNAR_s} and {XGB_t + SVR_r + NNAR_s}, achieved RMSE scores of 13.621 and 13.651, respectively, but they did not surpass the SVR-dominated frameworks in error minimization. These findings demonstrate that using SVR consistently for both trend and residual components yields robust probabilistic accuracy, while NNAR contributes additional symmetry in capturing seasonality.

A comparative visualization of original (imputed) and predicted PM_2.5 levels for the four cities is presented in Figure 2. For Islamabad (Figure 2a), forecasts nearly overlap with the imputed series, showing symmetric alignment with seasonal cycles. Karachi (Figure 2b) exhibits higher short-term variability; the models slightly underestimate extreme spikes but capture the general probabilistic distribution. In Lahore (Figure 2c), high oscillations and peak concentrations reveal a challenge in capturing extreme values, highlighting the asymmetry introduced by outliers. Finally, Peshawar (Figure 2d) demonstrates the accurate detection of a significant concentration peak, indicating strong responsiveness to abrupt shifts. Taken together, the results suggest that the models exhibit statistical symmetry and robustness in relatively stable contexts (Islamabad, Peshawar). In contrast, adaptive or hybrid refinements are necessary for more volatile settings (Lahore, Karachi) to enhance extreme-value representation and probabilistic reliability.

4. Discussion

This study used four machine learning algorithms to compare the imputed (actual) and forecasted PM_2.5 concentrations in four major Pakistani cities: Peshawar, Lahore, Karachi, and Islamabad. SVR, ELM, NNAR, and XGBoost were used in the proposed hybrid framework. Performance measures, including RMSE, MAE, MAPE, correlation, sMAPE, and time series plots (actual vs. predicted), were used to empirically and visually assess the efficiency of these models. Additionally, this section presents the best-performing PM_2.5 forecasting model proposed via this study, which is evaluated in detail and compared to baseline methods and the best models found in the literature. To enhance air quality predictions and management, this section also outlines potential avenues for future research and provides practical policy recommendations.

4.1. Comparative Studies with Benchmark Models and Existing Literature

To assess the robustness and practical relevance of the proposed hybrid forecasting framework, a comparative evaluation against four widely used benchmark models—support vector regression (SVR), extreme gradient boosting (XGBoost), neural network autoregression (NNAR), and extreme learning machine (ELM)—was conducted across the four urban centers of Islamabad, Karachi, Lahore, and Peshawar. The performance was examined using a comprehensive suite of accuracy metrics, including the mean absolute error (MAE), the root mean squared error (RMSE), Pearson correlation, the symmetric mean absolute percentage error (sMAPE), and the mean directional accuracy (MDA). These measures collectively capture both point-wise accuracy and directional predictive reliability, thereby addressing the probabilistic and distributional symmetry underlying forecast errors.

Table 8. Comparative performance of the proposed hybrid framework versus benchmark models across four Pakistani cities.

Islamabad
Model	MAE	RMSE	Correlation	sMAPE	MDA
SVR	15.997	21.342	0.511	14.829	0.525
XGBoost	17.646	24.453	0.455	15.062	0.641
NNAR	15.990	21.674	0.760	11.995	0.658
ELM	17.646	21.563	0.667	12.771	0.602
Proposed Model	6.277	7.912	0.969	7.559	0.838
Karachi
Model	MAE	RMSE	Correlation	sMAPE	MDA
SVR	15.990	23.342	0.768	14.297	0.808
XGBoost	17.646	25.334	0.808	15.945	0.809
NNAR	16.089	22.641	0.646	14.798	0.652
ELM	14.968	26.631	0.694	13.883	0.748
Proposed Model	6.005	7.652	0.975	7.590	0.798
Lahore
Model	MAE	RMSE	Correlation	sMAPE	MDA
SVR	35.268	50.999	0.715	21.673	0.686
XGBoost	38.099	53.427	0.893	28.006	0.705
NNAR	34.840	49.683	0.872	21.456	0.778
ELM	33.966	48.304	0.656	21.178	0.795
Proposed Model	10.120	12.475	0.979	8.340	0.808
Peshawar
Model	MAE	RMSE	Correlation	sMAPE	MDA
SVR	27.265	33.442	0.610	15.716	0.544
XGBoost	28.278	36.530	0.736	17.940	0.687
NNAR	24.320	31.316	0.619	17.642	0.657
ELM	23.239	31.241	0.591	17.050	0.665
Proposed Model	10.871	13.585	0.837	9.902	0.768

presents a detailed comparative analysis. In Islamabad, the proposed model demonstrates substantial improvements over conventional baselines, achieving an RMSE of 7.912 and a high correlation of 0.969, compared with the nearest competing model (ELM), which yields an RMSE of 21.563 and a correlation of only 0.667. This reflects the ability of the proposed hybrid decomposition–ensemble strategy to capture symmetric seasonal patterns and reduce asymmetric error deviations, thereby yielding more reliable forecasts. Similarly, in Karachi, where pollution exhibits dynamic fluctuations, the proposed model attains an RMSE of 7.652 and an exceptionally high correlation of 0.975. This outperformance demonstrates how the hybrid model maintains structural balance between the trend and remainder components, aligning with the symmetry properties inherent to complex urban time series.

Lahore poses the most challenging forecasting environment due to its highly volatile and heterogeneous PM_2.5 patterns. While benchmark models yield inconsistent results and a larger dispersion of errors, the proposed approach achieves an RMSE of 12.475 and a correlation coefficient of 0.979. The symmetric formulation of sMAPE in this case is particularly valuable, as it penalizes over- and under-predictions in a balanced manner, underscoring the framework’s superior capability to model asymmetric distributions. In Peshawar, despite extreme episodes of pollution, the proposed model again exhibits resilience, recording an RMSE of 13.585 and a correlation coefficient of 0.837, while also achieving the highest MDA, highlighting its predictive stability in the face of directional changes.

A broader comparison with the existing literature suggests that classical machine learning models (e.g., SVR, XGBoost) often fail to capture the nonlinearities and seasonal symmetries inherent to PM_2.5 series, resulting in higher error magnitudes and reduced correlation. By contrast, the proposed hybrid approach systematically integrates decomposition, statistical learning, and ensemble weighting to exploit both temporal symmetry and probabilistic consistency. The consistent superiority across all four cities confirms the model’s adaptability, interpretability, and robustness. Additionally, our methodology makes sure that positive and negative errors are treated equally, but this shouldn’t be taken as finding temporal symmetry. Decomposition and ensemble integration, not symmetry characteristics, are responsible for the benefits. More importantly, the integration of symmetric error metrics, such as sMAPE and balanced correlation analysis, aligns with the objectives of the Symmetry special issue, illustrating how probabilistic and statistical perspectives on symmetry enhance predictive performance and strengthen decision-making in urban air quality management. The hybrid STL–ensemble technique provides a data-driven approach to managing urban air quality while improving forecasting accuracy and interpretability overall.

4.2. Comparative Analysis with Prior Studies

Beyond the benchmark comparisons, the performance of the proposed hybrid STL-based forecasting framework was evaluated against results reported in prior studies on PM_2.5 prediction. This additional comparison provides a broader perspective on its relative efficiency and highlights how probabilistic and statistical considerations of symmetry in forecast errors contribute to improved accuracy.

An ensemble technique reported in [20] using the same dataset achieved RMSE values of 20.32, 48.31, 22.99, and 35.06 for Islamabad, Lahore, Karachi, and Peshawar, respectively. By contrast, the proposed hybrid model achieved significantly lower RMSEs of 7.91, 12.48, 7.65, and 13.59 for the same cities. These results represent error reductions of approximately 61%, 74%, 67%, and 61%, respectively, demonstrating the proposed framework’s capacity to capture underlying structural patterns with a higher degree of symmetry and reduced variability in forecast deviations.

Similarly, in [27], the reported RMSE values were 16.566 (Islamabad), 20.835 (Lahore), 22.084 (Karachi), and 18.743 (Peshawar). When compared to the proposed model, which obtained RMSEs of 7.912, 12.475, 7.652, and 13.585, the reductions were to 52%, 40%, 65%, and 28%, respectively. These improvements indicate not only higher predictive accuracy but also a more stable and symmetric distribution of residuals, which is critical for enhancing generalization across heterogeneous urban air quality contexts.

In another study, ref. [28] employed an artificial neural network (ANN) with 20 neurons and five independent variables to predict PM_2.5 in Islamabad, obtaining an RMSE of 9.82. By comparison, the proposed STL-based model, which relies solely on intrinsic PM_2.5 dynamics without exogenous predictors, achieved a lower RMSE of 7.912, representing a 19% improvement. This reinforces the advantage of decomposition-based models in capturing temporal symmetries without heavy reliance on external covariates.

Furthermore, ref. [29] utilized ANN models incorporating meteorological and historical information, producing RMSE values of 18 for Karachi and 39 for Lahore. In contrast, the proposed model achieved RMSEs of 7.652 and 12.475 for Karachi and Lahore, respectively, representing improvements of over 57% and 68%. This outcome emphasizes the robustness of the proposed approach in capturing both short-term fluctuations and the seasonal symmetries inherent in PM_2.5 data.

A more advanced deep learning approach in [30], which integrated CNN-LSTM with Multi-Fractal Detrended Fluctuation Analysis (MF-DFA), reported an RMSE of 11.732 using combined meteorological and air quality data. While effective, the proposed hybrid STL-based framework still outperformed this model, achieving an average RMSE of 10.91 across the four cities, despite utilizing only the PM_2.5 series as input. This highlights the effectiveness of decomposition and ensemble strategies in enhancing forecast precision while preserving statistical symmetry in error dynamics.

Therefore, the comparative evidence across diverse studies consistently demonstrates the superiority of the proposed framework in terms of error reduction, robustness, and adaptability. By integrating decomposition, ensemble learning, and symmetric error measures such as sMAPE, the approach systematically balances over- and under-predictions, reduces asymmetry in residual distributions, and achieves higher probabilistic consistency. Additionally, meteorological factors like temperature, humidity, and wind speed are out of this study, although this design decision improves the models’ applicability in low-resource settings where such data are difficult to obtain or inconsistent. Future iterations of the suggested framework, however, might include these exogenous factors to investigate how they affect the interpretability and accuracy of the model.

5. Policy Implications and Future Studies

The implications of this work extend beyond methodological contributions to environmental policy, public health, and sustainable urban management. Accurate short-term forecasts of PM_2.5 concentrations are crucial for issuing timely health advisories, managing traffic, emission control, and implementing regulatory interventions. For urban planners, such forecasts can inform land-use strategies, infrastructure development, and mitigation policies aimed at reducing long-term exposure to hazardous pollutants. The demonstrated reliability of the proposed framework across multiple cities underscores its scalability for nationwide deployment. Unlike models that require extensive meteorological datasets, this decomposition–ensemble method provides a cost-effective and practical solution for resource-constrained regions, such as Pakistan. When integrated into regulatory monitoring systems, the model can enhance compliance mechanisms, improve environmental governance, and foster evidence-based decision-making.

Despite its strong performance, several avenues for future research remain open. Incorporating exogenous covariates, such as meteorological conditions, socioeconomic activity, or industrial emissions, could further enhance predictive accuracy, particularly during extreme pollution episodes. Additionally, by adding external covariates, sophisticated symmetry measures, and Bayesian-based probabilistic augmentation, future studies could expand this symmetry-aware ensemble, and exploring alternative decomposition techniques such as Variational Mode Decomposition (VMD), Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), or Empirical Wavelet Transform (EWT) may capture asymmetries and nonlinearities more effectively in highly volatile environments. Extending the model toward probabilistic forecasting frameworks could also quantify uncertainty, providing policymakers with confidence intervals and risk assessments, rather than point estimates. In conclusion, the proposed hybrid ensemble model represents a statistically rigorous, symmetry-aware, and interpretable solution for urban air quality forecasting. Its adaptability, accuracy, and efficiency make it a valuable tool for policymakers, environmental researchers, and urban planners seeking proactive, data-driven approaches to sustainable air quality management.

6. Conclusions

This study has introduced a novel hybrid ensemble framework for forecasting PM_2.5 concentrations in four major Pakistani cities—Islamabad, Lahore, Karachi, and Peshawar—by leveraging seasonal—trend decomposition using Loess (STL) in combination with support vector regression (SVR), an extreme learning machine (ELM), and neural network autoregression (NNAR). The probabilistic and statistical synergy between decomposition and ensemble modeling enabled the framework to capture both linear and nonlinear dynamics while preserving the temporal symmetry of the underlying air quality processes. The empirical evaluation demonstrated that the proposed approach consistently achieved superior forecasting performance compared to benchmark models, such as SVR, XGBoost, ANN, and CNN—LSTM. With an average RMSE of 10.91 across all locations, the model outperformed the prior literature, including CNN—LSTM—MF—DFA (RMSE 11.73) and traditional ANN-based systems (RMSEs above 18 and 39 for Karachi and Lahore, respectively). The reduction in error ranged from 40% to 65%, underscoring both the robustness and efficiency of the proposed model. Importantly, these results were achieved without incorporating exogenous meteorological covariates, highlighting the model’s capacity to operate in data-constrained environments. The decomposition of the PM_2.5 series into trend, seasonal, and remainder components enhanced interpretability by revealing the probabilistic balance and structural symmetry between persistent trends, cyclic variations, and irregular shocks. Visual diagnostics and statistical accuracy measures confirmed that the hybrid ensemble not only reduced errors but also improved directional accuracy, thereby reinforcing its practical relevance. Overall, this study contributes to the literature on environmental forecasting by demonstrating how probability-driven decomposition and symmetry-aware hybridization can lead to both accuracy and interpretability in complex time series modeling. In addition to offering a reliable and understandable method for short-term PM_2.5 forecasting, the framework may be expanded to include other environmental time series.