Semiparametric hybrid air quality forecasting using prophet ensemble and feature selection

Abstract

Due to air quality management and public health planning, accurately predicting PM2.5 concentrations is absolutely imperative. Having many lagged variables, interactions between pollutants and other temporal features means that high-dimensional environmental datasets (that may include multiple features) may hinder forecasting performance. To mitigate this problem, we develop a hybrid forecasting framework combining the Prophet time series model with ensemble-based machine learning techniques for residual modelling. Residuals from prophet are modelled via bagging, boosting, stacking and voting regressors. To cope high-dimensionality, feature selection techniques such as Lasso, Shap and Recursive Feature Elimination, amongst others are applied such that the remaining influential features may be accurately modelled and interpreted. As benchmarks, the hybrids are also tested against stand along machine learning models like XGboost, SVR and LSTM’ s. Forecasting with the models is performed on both a high- dimensional simulated dataset and real daily air quality data from Seoul, South Korea. We found that the prophet with boosting ensemble-based model coupled with Lasso and Shap have lowest errors based on RMSE, compared to Prophet with other machine learning ensemble residual regressors and also standalone models in high-dimensional settings. We offer a statistically sound, interpretable and scalable solution for forecasting air pollution in high dimensions.

Introduction

There’s now (almost universally in any country) a need of forecasting air pollution especially fine PM.2.5 s for environmental policy making and public healthcare planning etc. The data sets in this field are becoming too common. It sometimes may include lagged pollutant concentrations with meteorological features and temporal patterns where input features can be large (~ dozens numbers) with huge feature spaces and dimensions which means we need effective prediction models along the lines of being Accurate, Interpretable and Robust with high dimensional features spaces. However classic simple linear regression models (Statistical type) lack the flexibility, but the complex machine learning model like ensemble trees (gradient boosting trees etc.) are too “black box”, where interpretability becomes a concern like in case environment policy making and healthcare planning where this may affect people’ s decision making if they are transparently given forecast results, this concern is addressed by the Prophet Model, which is one of the many time-series forecasting model which is Interpretable by breaking down trend, seasonality components of the time-series with robustness on missing value and outliers but its prediction is often limited when applied individually in high dimension and complicated feature spaces such as environmental data with residuals of Prophet often show some structural signals to use further which can be applied with another machine learning modelling technique know as hybrid modelling.

We propose a novel hybrid framework that uses a prophet model which then follows up to apply ensemble machine learning regressors for learning and predicting from the residuals. Specifically, we used four types of ensemble methods—bagging, boosting, stacking and voting to learn the errors of prophet to improve upon its predictions. To deal with the problem of handling high dimensionality, we employed some feature selection techniques. Feature selection techniques such as Lasso, SHAP values and recursive feature elimination (RFE) were used to find the right predictor, to improve generalization and to interpret the model better. We validate our Prophet ensemble method on both a high dimensional synthetic dataset as well as a real-world day PM2.5 dataset of Seoul. We show that the Prophet model that is enhanced using a Boosting based ensemble with good feature selection outperforms the models XGBoost, SVR and LSTM (measured using RMSE). Furthermore, our proposed ensemble is also interpretable and can therefore be used for environmental decision making, a critical area. We further contribute to the field of environmental and ecological statistics where we develop a robust and interpretable method using the best models in our arsenal to deal with the problem of having high dimensionality as well as high forecasting power in air quality prediction datasets.

To further expand the research of literature study it was noticed that, using the integration of Random Forest, Landsat OLI 8 spectral data and land use indices, particulate matter (PM) pollution in Ras Garib, Egypt, can be predicted by the work made by El-Magd, Abu El-Magd and Awam¹. Based on the study’s findings, it helped in the reliable susceptibility maps for aiding air quality and urban planning can be developed thereby resulting this research to be much helpful. However, the study could have been greatly strengthened by the incorporation and comparison of other machine learning models. Agbehadji and Obagbuwa utilized a PRISMA-based methodology to systematically review machine learning (ML) and deep learning techniques for spatiotemporal air quality prediction². The analysis of over 374 articles was narrowed focusing on 80 relevant articles and on the reviews it was indicated that due to the advancement on techniques such as federated and transfer learning the efficiency of hybrid ML-DL approaches, especially coupling RandomForest and deep neural networks when dealing with nonlinear pollutant patterns has resulted as an advantage whereas and a limitation highlighted by the study include; the limited databases employed and keyword-based filtering leading to the potential exclusion of certain other relevant research. Improved extreme Learning Machine (ELM), outlier detection, data decomposition and a multi-objective optimization algorithm was introduced and combined by Bai, Liu and Wang to form a novel hybrid system for both deterministic and interval air pollution prediction³. Thus with the above-mentioned combination made and with data from the PM2.5 concentration spanning across three cities the model obtained a much superior forecasting accuracy, also effectively quantify prediction uncertainties; hence an integrated tool can also thereby provide advancements for improving forecasting accuracy and reliability air quality predictions with the newly proposed approach. From the previous research of other journal studies, using pollutant, meteorological and nearby station data the study proposed framework integrating spatial–temporal features used for hourly PM 2.5 and PM10 prediction for Beijing⁴. Thus it was noticed and established with their research on Deep Learning being; the higher accuracy model they used such as hybrid CNN-LSTM (Convolutional Neural Network-Long Short Term Memory) model in comparison, LSTM, GRU and CNN models hence by linking convolutional and recurrent layers a much higher success was achieved in improving the accuracy of air quality forecasting thereby this study too will be proven to be effective and successful for other cities. In Beijing et al. Douzi created a deep learning framework to predict hourly PM2.5 levels using spatial–temporal features from pollution, meteorology and neighbouring stations⁵. A combination of CNN and LSTM (Hybrid CNN-LSTM) showed better prediction results compared to deep learning models applied separately (LSTM, GRU and CNN). Their work underlines the importance of combining convolutional layers with recurrent layers in forecasting air quality in city environment. A total of 47 studies on air pollution have been reviewed identifying recent developments in using data mining and machine learning methods according to Bellinger et al.⁶. They categorised these into source apportionment, estimation and prediction and hypothesis generation studies. Artificial neural network (ANN) was once widely prevalent in this field. It has now transformed into a palette of methods such as decision trees, support vector machines (SVMs), clustering and association rule mining, replacing the earlier used methods. Also, opportunities for deep learning and geospatial pattern mining was discovered in their article. As environmental health research and computational health come closer together, Jabbar, Zaïane and Osornio-Vargas’s work provides a compelling roadmap to tackle health concerns. IMDA-VAE was invented by Dairi et al. to enhance air pollution forecasting⁷. The innovative model created uses variational autoencoders with a multi-attention mechanism to deal with spatial–temporal dependencies and provide enhanced accuracy than models like LSTM, GRU and ConvLSTM. Delavar et al., created a unique framework to predict PM10 and PM2.5 levels in Tehran using decision trees (regression SVM and geographically weighted regression), ANN and autoregressive non-linear neural networks with external input⁸. Furthermore, Delavar et al. were able to decrease error rates for up to 94% using this framework. Moreover, the study was able to identify the factors affecting particulate matter levels that had the most influence with its framework by utilizing genetic algorithms (GA) that also identified the best predicting model among the four (autoregressive non-linear neural networks, average error one-day was 1.79 µg/m³).

Du et al. devised an ensemble-based system for air quality early-warning in China. This system combined clustering with advanced data pre-processing and forecast using a length-changeable incremental extreme learning machine⁹; a qualitative evaluation via the fuzzy methods. The test datasets from three different cities were used; system indicated better forecast with accuracy/generalisation. Tests, such as DM and SPA, for statistical analysis were used to confirm findings concerning models. The proposed ensemble-based early warning system links quantitative forecast with fuzzy evaluation. Fang et al. developed the regional feature selection based machine learning system (RFSML) v1.0¹⁰. This improves air pollution forecast at across China. RFSML integrates a modified ensemble-Shapley additive importance that automatically generates/screens regional scale features based on a feature importance score; this makes the model interpretable while also reducing its training time and chance of overfit. They performed multi-city and national scale validations and the results demonstrated that the approach has the high prediction stability and generalisability. The study by Gao et al. analyses the effectiveness of air pollution control regulation and policy of the Chengdu-Chongqing region¹¹. They examine control policy efficacy using combined historical evaluation and deep learning-based air quality forecasting. The PCA and TOPSIS was utilised to combine pollutants into a monthly air quality for evaluation purposes spanning from 2015 to 2019. To create and validate an air quality forecasting model (from 2019 to 2020), LSTM-DL models were implemented; significant improvement was established in cities’ scores of air quality and forecasting demonstrated promising results for future. Gu et al. developed the Hybrid Interpretable predictive machine learning model based on deep neural network in combination with Nonlinear Auto-Regressive Moving Average with Exogenous Input (NARMAX) for PM₂ concentration forecast¹². The interpretable models can be directly modelled and allow automated feature generation; the best predictive model was chosen on performance and interpretability scores based upon accuracy of peak values; the new interpretable hybrid approach offers an accurate forecast with correlations exceeding 0.9. Gugnani and Singh reviewed the deep-learning Architectures for the purpose of prediction on air quality, describing their advantages and current shortcomings¹³. They identify architectures that are well suited for air pollutant forecasting; examine the role of richer datasets towards more accurate prediction, while discussing how better DL models can contribute to an improved forecast accuracy from present models over richer data. Habeebullah, Munir et al. investigated the impact of the lockdown of the COVID-19 pandemic on air quality focusing on O₃, NO₂ and PM₁₀ in Makkah¹⁴. It analyses the air quality during lockdown using direct comparison (weather-corrected machine learning). There was reduction in NO₂ and PM₁₀ concentrations after lockdown in the selected cities and increase in O_3. Using meteorological data, the prediction error for all pollutants becomes smaller. The changes are higher without meteorological adjustments and lower when adjusted for meteorology, highlighting the need for meteorological correction when assessing the impact of COVID-19, lockdowns, etc. on air quality. Beyond the limitation of the PDE-based numerical solvers, Haehnel, Marecek, Monteil and O’Donncha have proposed a new model framework of deep learning, extending the problem from the limitations of the numerical solvers, by introducing a domain decomposition technique which allows to train over several domain pieces and accelerates the computational time twice the orders of magnitude at run-time, simultaneously making the entire scheme scalable and computationally efficient This first-of-its-kind effort provides a path to solving real-time, global-scale environmental forecasting tasks¹⁵. Heydari et al. proposed a new short term forecasting framework combining long- and short-term memory (LSTM) neural networks with multi-verse optimization (MVO) algorithm to forecast the emissions from a combined cycle power plant¹⁶. LSTM was set to act as the forecasting engine while MVO optimizes the parameters towards minimal forecasting errors. By validating the model with real measurement data from a power plant in Kerman, Iran, it has outperformed other benchmark hybrids (ENN-PSO, ENN-MVO, LSTM-PSO) in terms of accuracy in two sets of input (with meteorology or with pollutants only). The idea and model show promising ability to improve the precision of gas and pollutants emission forecasting from the industrial plants, hence air quality with such pollutants can be forecast with enhanced precision.

Inam et al. pioneered PM2.5 forecasting with the PR-FCNN framework combining Polynomial Regression and Fully Connected Neural Network¹⁷. Drawn from the WAQI and SOGA databases, their study showed enhanced accuracy by merging statistical and deep learning approaches, outperforming standalone models. Capturing linear and non-linear pollutant relationships, their hybrid architecture offered improved predictions. In pursuit of superior prediction capability, Inam et al. introduced a transformer-based PM2.5 forecasting framework integrating country-level embeddings and xAI¹⁸. The transformer architecture with the capacity to cope with long temporal dependencies and regional heterogeneity was explored in the SHAP and LIME analyses to render explanations about model predictions comprehensible assisting decision makers in decoding air pollution evolution. The study confirmed that it could perform better than traditional machine learning models and was suitable for dealing with air pollution. A machine learning framework using hybridized learning and LightGBM, neural networks and ridge regression as the meta-learner was proposed and utilized in the study to predict particulate matter (PM) 10 pollution. Data amounting to 1.8 M was obtained and studied of approximately 380 cities. Upon comparison with traditional machine learning, tree, random forest and boosting algorithms, it was rendered that the framework was efficient among them and predicted efficiently as opposed to all other methods. Lagged PM10 variables were indicated to be the most influential predictors, highlighting the essence of temporal persistence. Acknowledging the importance of forecasts on larger scale, Inam et al. developed a transformer-based deep learning architecture for forecasting population-weighted PM2.5 exposure for 204 countries and territories with an aim for information leakage prevention and thus model reliability improvement via a temporally constrained evaluation strategy for model evaluation¹⁹. The developed architectural set can easily account for the trends in long-term exposures together with existing variations in the region under study, making transformer models suitable for applications relating to and in the favor of global environmental health without fail. Attention-based explanations helped provide insightful and valued interpretation about all patterns affecting the underlying forecasts. Using models like LightGBM, neural network and Ridge regression, the use of hierarchical country and city representations has effectively enhanced performance against various machine learning and deep learning baselines by enhancing the PM10 forecasts of a multi country scale employing, for the first time on such a scale, ensemble learning at first together together with chronological splits validation for future forecasting, as recently found out by Inam et al.^20,21. in their most recent research which is also aimed to address the problems mentioned in related previous researches. Conclusively, there has been a considerable transition from traditional machine learning methods to more sophisticated hybrid, ensemble and transformer-based architectures for forecasting particulate matter as witnessed. Given these leaps, certain challenges like uncertainty quantification, region-specific transferability and development of models offering physically insightful predictions remain. Further research, therefore, should be directed towards leveraging explainable, physics-informed and ensemble learning to advance the resilience and trustworthiness of air quality prediction models thereby.

In a similar manner, hybrid deep learning architectures coupling convolutional and recurrent neural networks have taken roots and gained popularity owing to recent success in air-quality forecasting with the capacity to capture spatio-temporal dynamics driving the pollution. Ahmad and Kumar proposed a hybrid BiGRU-1DCNN for forecasting of PM2.5 in multiple monitoring stations across Delhi thereby benefiting from integrating spatial information with temporal information²². Likewise taking cue from recent success Ahmad and Kumar proposed a WaveNet based hybrid architecture coupled with a residual correction from XGBoost and reported significant improvement in forecasting accuracy of PM2.5 in monitoring stations of urban settings²³. Since more often than not environmental datasets suffer from challenge of missing observations, given this Ahmad and Kumar proposed a novel random-imputation based RNN-BiGRU frameworks that showcases how a proper pre-processing and imputation method can improve the forecasts²⁴. The rest of this paper is organized as follows. “Methodology” section presents the proposed methodology including data pre-processing, feature engineering and selection and application of prophet to decompose the time-series forecasting. “Results and discussion” section describes the experimental setup, evaluation metrics along with validation methods employed.

Methodology

This study proposes a hybrid modelling framework that integrates the statistical decomposition capabilities of the Prophet model with the predictive power of machine learning techniques, specifically feature selection methods and ensembled techniques. The methodology is structured in two sequential stages. The first stage uses the Prophet model (Taylor and Letham, 2018), which decomposes the observed series into interpretable components: a trend component g(t), seasonal component s(t), Holiday component h(t) and an irregular error component . Mathematically, the prophet model is expressed as:

where trend component have two further types, one is piece wise linear trend and other is logistic growth. Piece wise liner trend is as follows.

where k is the growth rate, m is the initial offset, J is the number of changepoints,

adjusts the growth rate at each changepoint, and is an indicator function for the changepoints.

Similarly logistic growth is described as follows

where C is the carrying capacity, k is the growth rate, and t₀ is the midpoint. The abbreviations used throughout this article are listed in Table 1.

Table 1.

List of abbreviations.

Abbreviation	Description
PM2.5	Particulate matter with diameter ≤ 2.5 μm
PM10	Particulate matter with diameter ≤ 10 μm
ML	Machine learning
AI	Artificial Intelligence
SVR	Support vector regression
LSTM	Long short-term memory
RFE	Recursive feature elimination
SHAP	Shapley additive explanations
RMSE	Root mean square error
MAE	Mean absolute error
R2	Coefficient of determination
MI	Mutual information
ACF	Autocorrelation function
PACF	Partial Autocorrelation Function
CNN	Convolutional neural network
GRU	Gated recurrent unit
BiGRU	Bidirectional gated recurrent unit
XGBoost	Extreme gradient boosting
DSS	Decision support system
UQ	Uncertainty quantification

Seasonal effects are modelled using a Fourier series:

where P is the periodicity, and N is the number of fourier terms.

Holiday captures some special events it is modelled as a sum of indicator functions:

where represents the effect of holiday k, and is an indicator function for the holiday.

After extracting the structured components, the residuals are computed as:

These residuals are modelled using ensembled strategy.

Ensemble strategy

Bagging:

Boosting: Residuals are iteratively updated:

Stacking:

Meta-model learns:

Set of X(t) is selected using feature selection methods which are variance threshold, SelectKBest using F-statistics, SelectKBest using Mutual Information, Recursive Feature Elimination (RFE), Lasso (L1 regularization), Ridge (L2 regularization), ElasticNet (combined L1 and L2), Tree-based feature importance (Random Forest), and SHAP (Shapley Additive Explanations) values. For example, in LASSO L1 regularization we have

where:

λ: Regularization parameter controlling the sparsity of β.

p: Number of predictors in .

The L₁ penalty ensures that irrelevant predictors are excluded by shrinking their coefficients to zero.

To enhance the predictive performance of base learners, we adopted the gradient boosting technique, a powerful ensemble learning approach originally proposed by Friedman (2001). This method builds models in a stage-wise fashion by iteratively minimizing a specified loss function using the principles of functional gradient descent.

Let the training dataset be:

The gradient boosting model approximates the true function as:

where:

M is the number of boosting iterations (tree). v is the learning rate between 0 and 1. is weak learner fitted at stage m. At each iteration the goal is to minimize the empirical risk:

where a differentiable function (e.g., squared error).

Initialize the model:

From m = 1 to M repeat Compute pseudo-residuals (negative gradients):

Fit the base learner to the residuals and determine the optimal step size then the final hybrid forecast at time t is given by:

where enabled the boosting mechanism and feature selection method like LASSO coefficients estimated from the Lasso model. This combination allows the model to leverage both the strong seasonality/trend decomposition of Prophet and the explanatory power of external variables.

Figure 1 presents the complete workflow of the proposed forecasting framework, highlighting the sequence of data preparation, model implementation, forecasting, and performance assessment. This flowchart summarizes the methodology adopted throughout the study.

Fig. 1.

Flowchart of the proposed prophet-based forecasting framework.

Results and discussion

In this section, I am going to analyse the working of the proposed forecasting framework by combining Prophet time series model, ensemble-based residual leaning and high dimensional feature selection approach with synthetic high dimension time series data along with the dataset of daily average PM2.5 concentration values from Seoul air quality dataset. Mean root mean squared error (RMSE) will be considered as evaluation metric as it is easily interpretable RMSE and more sensitive to larger values. The initial Seoul air quality dataset includes hourly readings from numerous monitoring stations. However, within this study, all hourly observations have been aggregated into daily mean values prior to modelling. As a result, the forecasting problem addresses daily PM2.5 forecasting, not hourly. The target variable is the daily mean PM2.5 concentration at monitoring station number 107. Predictor variables comprise daily pollutant measurements (SO2, NO2, O3 and PM10), spatial PM2.5 information from adjacent stations and synthetic indicators such as the PM10-to-PM2.5 ratio at station 107. This proposed hybrid framework conducts one-step-ahead daily forecasting employing only data available up to the forecasting date. The baseline Prophet model demonstrated excellence in tracking trend and seasonality, yet its predictive prowess on both datasets was lacking. This shortfall stemmed from its inadequacy in anticipating residual nonlinearities and high interaction effects generated by the plethora of input features. For the Seoul air quality dataset, the Prophet model performed poorly despite having accounted for the trend and seasonal patterns yielding an RMSE of 22.20, indicating unaccounted variability. Moreover it performed markedly poorer on the synthetic dataset (RMSE = 127.66 with a low R squared value of 0.44), stressing its ineffectiveness with high dimensional datasets. This inefficiency led to the approach of residual learning whereby the residual errors generated by Prophet were modeled using ensemble machine learning models to capture the nonlinear hidden patterns and dependencies that Prophet could not account for. Four state of the art ensemble methods—Bagging, Boosting, Voting and stacking were used to model the left out residuals of prophet against each of the ten feature selection techniques applied to the models, to identify which best captures the nonlinear patterns presented in the duo datasets.

To avoid leakage of information, all steps of pre-processing, feature engineering, scaling, feature selection and model training have been done within every training fold with rolling-origin evaluation. Feature selection including SHAP, selectkbest, Information gain, RFE, Lasso, Ridge, Elastic Net and Tree-based importance method have been trained using training data only and used on the test data. No futuristic observations were used during constructing features or selecting or ranking features or in using the models to estimate or forecast. Strictly out-of-sample evaluations were carried out based on a rolling-origin evaluation framework that respects the chronological order of the observations. With each fold, while the training set consists of all past observations before the test period, the test set contained only unseen future observations. The trailing window in all cases and folds were expanded over the folds, reflecting the real scenario as for model comparison there was no look-ahead. Prophet model has been used with additive seasonality by modelling weekly seasonality along with all other variables, with the rest as default model estimates of the trend and overall parameters. Several ensemble models of Residual learners, including bagging, boosting, voting and staking and the baseline models (of XGBoost, SVR and LSTM) trained purely over the residual of Prophet from the training folds for each of them. All hyperparameters were chosen using only the training data and fixed during testing for fair model comparison. Also, feature selection and modelling processes were all repeated independently with each Rolling origin fold. For the evaluation about the residual leaning, diagnostic information from residuals of the Prophet baseline model, Residual histogram and plots of autocorrelation function (ACF) and partial autocorrelation function (PACF). The distribution of the residual is centred over zero, though shows that there is some autocorrelation remaining at a few lags. The Ljung-Box test also rejected the null hypothesis that residuals are independently distributed (p < 0.001), showing some temporal structure in the residuals, not explain by Prophet decomposition. Based on these remaining systematic patterns/errors in model residual which in turn can be captured with good accuracy by a variety of ML models to improve the overall prediction of model and increase accuracy for future predictions is a good basis for residual learning framework.

The diagnostics of residuals show that the model prophet captured a great chunk of chronological structure in the data. Residual diagnostics for the baseline Prophet model are summarized in Table 2. The Ljung-Box test shows that the and p-values are insignificant at both lag = 10 and lag = 20 which mean no stronger residual autocorrelation presents in Prophet model residuals. Despite this, proposed framework of residual learning further enhanced forecasting accuracy implying that the machine learning models learned more non-linear relationship of data that existed beyond our Prophet forecasting model.

Table 2.

Residual diagnostics of the baseline prophet model.

Statistic	Value
Ljung-box statistic (Lag 10)	11.079
Ljung-box p-value (Lag 10)	0.351
Ljung-box Statistic (Lag 20)	18.614
Ljung-box p-value (Lag 20)	0.547
95% prediction interval coverage	1.000
Average prediction interval width	471.872

In Fig. 2, the synthetic target series can be seen which was used for validation. This series reveals high variability, sharp changes and nonlinearity over the training period. Such features were added to form a challenging environment to predict and to test the ability of the proposed hybrid framework in handling such complicated time behaviour.

Fig. 2.

Synthetic target time series.

The distribution of the synthetic target variable is shown in Fig. 3. The distributions of target variables can be found I graph on left side above. I think it is clearly visible there that this distribution is roughly symmetric, in centre for average value, but there are a few extreme observations on both sides. This spread proves the existence of a high variability for the synthesized process, representing a good domain for tests concerning robustness of forecasts.

Fig. 3.

Distribution of the target variable.

Figure 4 above illustrates the distribution of residuals from the Prophet baseline. The residuals are centered around zero, indicating that there is no bias. The spread out values suggest that mistakes are still committed and this provides motivation for conducting residual based correction using machine learning.

Fig. 4.

Prophet residual histogram.

The Prophet’s error autocorrelation plot is shown in Fig. 5 above. There is not a lot of serial correlation left as most of autocorrelation coefficients lie within the 95% confidence bounds. This weak autocorrelation can imply that prophet did identify the principal temporal pattern in our artificial series.

Fig. 5.

Prophet residual autocorrelation function (ACF).

Prophet residual partial autocorrelation functions are displayed in Fig. 6. Almost all of the lags are inside the intervals, which suggests that the strong linear dependencies were removed by the Prophet decomposition. The residual forecasting errors justify the residual modelling with machine learning.

Fig. 6.

Prophet residual partial autocorrelation function (PACF).

Figure 7 below shows the comparison of observed and forecasted values output from best hybrid model (i.e. model with Prophet + Boosting; RFE feature selection criteria). The forecasts show resemblance to trend of actual observations and the curve lies close on actual observation’s curve so depicting the capturingability of the trend that is built through Prophet and remaining non-linear residuals that is tried by boosting model.

Fig. 7.

Actual versus predicted values for the best hybrid model.

To confirm the applicability for all kinds of data, the model has also been tested over a synthetically simulated high-dimensional dataset generated through following the same scenario which is considered in dealing with the data corresponding to real-time environmental system—facing problems due tomulticollinearity and occurrence of sparse signals as well as nonlinear interactions etc.

The forecasting performance using best combinations of models on the synthesised dataset are shown in the result Table 3. In the leading five or three best-performing configurations, The proposed Prophet + Boosting models occupy most leading positions using multiple selection strategy. Meanwhile, Using prophet + boosting with RFE method could yield the best performance with mean RMSE of 65.592 and mean MAE of 52.445 among all the combinations of two models. While The RMSE value of baseline was 126.738 and MAE value of baseline was 98.699. This demonstrates that learning from residuals could improves performance. Moreover, The standard deviation of Prophet + Boosting methods showed lower values, which represents better performance stability.

Table 3.

Comparison of top forecasting models on synthetic dataset.

Rank	Feature selection	Model	Mean RMSE	Mean MAE
1	RFE	Prophet + Boosting	65.592	52.445
2	Lasso	Prophet + Boosting	65.631	51.933
3	ElasticNet	Prophet + Boosting	66.154	52.326
4	Ridge	Prophet + Boosting	66.434	52.969
5	ElasticNet	Prophet + Voting	66.880	54.254
6	None	Prophet only	126.738	98.699

The percentage improvements in RMSE and MAE achieved by the best hybrid model over the Prophet baseline are presented in Table 4. The Prophet + Boosting model with RFE feature selection reduces RMSE by 48.24% and MAE by 46.86% in comparison to the single Prophet model. Such result implies that a residual learning could recognize the nonlinear patterns that Prophet does not consider or recognize previously. And therefore, the forecast is much more accurate.

Table 4.

Improvement of best hybrid model over prophet.

Metric	Prophet only	Prophet + Boosting (RFE)	Improvement (%)
RMSE	126.738	65.592	48.24
MAE	98.699	52.445	46.86

A complete performance summary of the best hybrid model is provided in Table 5. Prophet + Boosting provided the best forecasting using feature selection from RFE compared to other configured models. Error was smaller and statistically significant (p < 0.001) in comparison with the prophet base model. Prediction Intervals were also reliable due to uncertainty analysis demonstrating complete coverage of observed values. Model is suitable over other models because of its empirical and simulated evaluation which shows the balance of accuracy, interpretability and scalability of an algorithm.

Table 5.

Best model performance summary.

Metric	Value
Best Model	Prophet + Boosting
Feature selection	RFE
Mean RMSE	65.592
RMSE Std. Dev	10.273
Mean MAE	52.445
MAE Std. Dev	9.635
95% coverage	1.000
Average interval width	471.872
Paired t-statistic	7.380
p-value	< 0.001

Figure 8 show us the PM2.5 daily concentrations from site 107. We aggregate the original hourly observations to daily average concentrations. The series obviously shows much variation from time to time. There are a few times it experiences high pollution episodes which are concentrations exceeding $80 \mu g/m^3$ for a period of time and there are times it keeps low concentrations. Much variation will result in the fact that it is complex and even non-stationary to predict PM_2.5. Moreover, it encourages us to use the hybrid-like forecasting model.

Fig. 8.

Daily PM2.5 time series.

The distribution of daily concentration of PM2.5 at the station 107 is shown in Fig. 9. We can observe heavily right-skewed distribution as most observations recorded can be found lying between 5 and 30 however there are still some outlier pollution events which shows heavy pollution episodes can happen and therefore make prediction task especially more difficult.

Fig. 9.

Distribution of daily PM2.5 concentrations.

Looking at Fig. 10, we observe the distribution of residuals from the baseline Prophet model. The residuals distribution is centred near zero with slight right skew, leading to a long positive tail. It means while the dominant trend and seasonal structural are extracted by Prophet, there can be forecasting errors still which can be captured by models for capturing residuals.

Fig. 10.

Prophet residual histogram.

Autocorrelation in the residual is observed through Fig. 11. Significant levels of autocorrelation are observed at the lag = 1 and some later lags as well implying that some time series dependent element still remains in the residue series after decomposition of Prophet. These findings are evidence against patterns among children being destroyed due tot decomposition and make it perfectly valid to use machine learning based residual modelling.

Fig. 11.

Prophet residual autocorrelation function (ACF).

The Figure 12 illustrates prophet residuals partial autocorrelation. Strong first lag dependency is observed as most of the lags are still with the confidence interval. Presence of short-term dependency suggests predictive information remaining in the residual sequence that also can be extracted via residual learning algo.

Fig. 12.

Prophet residual partial autocorrelation function (PACF).

Figure 13 compares actual PM2.5 concentrations and the predicted values by the Prophet + Boosting with SelectKBest-F. Its pattern follows closely to the actual observations and successfully catches major peaks and troughs. As the gap between the predicted values and the actual values is narrow, we find this hybrid framework performs well on capturing the linear temporal structure and nonlinear residuals.

Fig. 13.

Actual versus predicted PM2.5 for the best hybrid model.

Residual diagnostics for the baseline Prophet model on the Seoul air-quality dataset are summarized in Table 6. The residual diagnostics show that the residuals from Prophet exhibit a large amount of autocorrelation. The Ljung-Box test showed that the null hypothesis that residuals are independently distributed can be strongly rejected at both lag 10 and log 20 respectively (p < 0.001) i.e. structure is still remaining after the Prophet decomposition, strongly supporting the residual learning stage of the proposed hybrid forecasting framework.

Table 6.

Residual diagnostics of the baseline prophet model.

Statistic	Value
Ljung–Box statistic (Lag 10)	95.868
Ljung–Box p-value (Lag 10)	3.65 × 10⁻1⁶
Ljung–Box statistic (Lag 20)	105.255
Ljung–Box p-value (Lag 20)	1.43 × 10⁻13

For theSeoul air quality dataset, Table 7 presents the performance of the best hybrid forecasting configurations using the rolling-origin validation method. In all hybrid approaches, Prophet + Boosting with SelectKBest-Ffeature selection obtained the best performance with the lowest RMSE 4.016, lowest MAE 3.431 and highest R² value 0.705. The results indicate thatBoosting-based residual learning outperformsBagging and Voting for several feature selection techniques Overall performance differences were relatively small for the high ranking hybrid models suggesting robustness of the proposed framework to choice of features selection. Also due to the strong predictve abilities ofSelectKBest-F, SelectKBest-MI, Information Gain, this suggests that filterning out the irrelevant variables before residual modelling has good forecast accuracy and shows effective dailyPM2.5 forecasting capability using the Prophet decomposition combined with residual correcting using a machine learning algorithm.

Table 7.

Performance comparison of the top hybrid forecasting models on the Seoul air quality dataset.

Rank	Feature Selection	Model	Mean RMSE	Std RMSE	Mean MAE	Std MAE	Mean R2
1	SelectKBest-F	Prophet + Boosting	4.016	1.192	3.431	1.230	0.705
2	SelectKBest-MI	Prophet + Boosting	4.036	1.051	3.497	1.041	0.703
3	Information Gain	Prophet + Boosting	4.036	1.051	3.497	1.041	0.703
4	SelectKBest-F	Prophet + Voting	4.042	1.146	3.501	1.185	0.702
5	SelectKBest-F	Prophet + Bagging	4.133	1.149	3.592	1.153	0.688
6	Information Gain	Prophet + Voting	4.257	1.060	3.797	1.119	0.670
7	SelectKBest-MI	Prophet + Voting	4.257	1.060	3.797	1.119	0.670
8	Tree Importance	Prophet + Boosting	4.276	0.992	3.781	1.044	0.667
9	Tree Importance	Prophet + Voting	4.331	0.774	3.863	0.810	0.657
10	SHAP	Prophet + Boosting	4.407	1.135	3.791	1.071	0.646

The BPL-hybrid model (boosted prophet with lasso) leverages several advantages. Primarily, its interpretability is conserved by maintaining Prophet’s additive decomposability and additionally employing Lasso’s sparsity for feature selection. The model exhibits a high level of accuracy, outperforming traditional, ensemble and deep learning models across both real-world and simulated datasets. Additionally, its design is scalable for solving the low and high dimensional problems soon. It consistently achieves superior RMSE values over baseline and standalone ML approaches, under Lasso based feature selections. The ability of the model to generalize under the high noise and maintaining the explainability makes them preferred for forecasting of the environmental data. This work not only contributes to hybrid modeling in air quality, but it also offers a scalable benchmark for the interpretable forecasting in high-dimensional ecological time series analysis.

Author contributions

All authors contributed equally to this work. Conceptualization: Numan Yaqoob, Ibrahim Elbatal, Riffat Jabeen, Azam Zaka, Hatem E. Semary, Ahmed Z. Afify. Data curation: Numan Yaqoob, Ibrahim Elbatal, Riffat Jabeen, Azam Zaka. Formal analysis: Ibrahim Elbatal, Riffat Jabeen, Azam Zaka. Investigation: Numan Yaqoob, Azam Zaka, Hatem E. Semary. Methodology: Ibrahim Elbatal, Riffat Jabeen, Azam Zaka, Ahmed Z. Afify. Resources: Riffat Jabeen, Azam Zaka, Hatem E. Semary. Software: Numan Yaqoob, Riffat Jabeen, Azam Zaka. Validation: Azam Zaka, Hatem E. Semary, Ahmed Z. Afify. Visualization: Ibrahim Elbatal, Riffat Jabeen, Azam Zaka. Writing—original draft: Numan Yaqoob, Riffat Jabeen, Azam Zaka. Writing—review and editing: Ibrahim Elbatal, Riffat Jabeen, Azam Zaka, Hatem E. Semary, Ahmed Z. Afify.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2602).

Data availability

The dataset for Seoul air quality is openly available on various reliable platforms. Kaggle: https://www.kaggle.com/datasets/bappekim/air-pollution-in-seoul. This dataset contains hourly records of key air pollutants SO2,NO2, CO, O3,PM10 and PM2.5, originating from multiple monitoring stations throughout Seoul, South Korea. Air Korea (국가대기오염정보): https://www.airkorea.or.kr/eng. Air Korea is the official air quality data maintained by the Korean government. This portal provides access to real-time and historical air quality data from all monitoring stations nationwide. All data pre-processing and modelling scripts developed for this study are available from the corresponding author upon reasonable request.

Declarations

Completing interests

The authors declare no competing interests.