- 1Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan
- 2Escuela de Posgrado, Universidad Peruana Unión, Lima, Peru
- 3Government Degree College Tandojam, Hyderabad, Pakistan
- 4Faculty of Management, Czestochowa University of Technology, Czestochowa, Poland
- 5Department of Statistics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia
Particulate matter with a diameter of 2.5 microns or less (
1 Introduction
Maintaining a healthy atmosphere is vital for both humans and other living creatures. Air quality refers to the absence of harmful pollutants in the air. However, there are several deadly and fatal contaminants in the air nowadays, including
In the past, many researchers have developed solutions to this problematic air pollution issue. In this context, many researchers have applied classical regression, time series, machine learning, and hybrid models to find the optimum solution according to the nature of the data (Zhan et al., 2017; Xu et al., 2018; Abdullah et al., 2019; Dutta and Jinsart, 2021). For instance, the work (Geetha and Prasika, 2019) provides an LSTM in addition to two conventional forecasting models for estimating the levels of air pollutants (NO2, NOx, CO, SO2, O3, PM2.5, and PM10) in metropolises. The outcomes demonstrated that LSTM outperformed LR and ARIMA. Zaman et al. (2024) attempt intends to create machine learning models that are computationally efficient and simpler than those found in earlier studies. Findings indicated that the RF methodology performed marginally better than the XG-Boost and SVR techniques. In another research work, Zaman et al. (2021) predict
This work (Freeman et al., 2018), a forecast model for ozone levels averaged over 8 hours, was trained using novel deep learning techniques. A new technique for imputation was used to replace missing data and outliers in the collected data set. This method produced computed values based on the time and season closer to the predicted value. (Carbo-Bustinza et al., 2023). Using a comparison of numerous hybrid varieties of time series models, this work extensively studies projecting ozone concentrations. According to the study, the suggested models perform noticeably superior to the standard models that were considered. However, this research (Xie et al., 2021) aims to apply the Hausdorff distance method to huge data to improve future cyclone effect predictions (Rakholia et al., 2023). Analyzed to develop a multi-step-ahead with multi-output type multivariate statistical model (NBEATS) to estimate the air quality with the auxiliary information. The data set was collected from six healthcare air quality centers in Vietnam. To compare the results and efficiency with the existing model, accuracy indices such as MAPE and RMSE were used. The result indicated that the developed (d-BEATS) multi-dimensional co-variate model outperforms the existing models. In the work Bhatti et al. (2021), the researcher conducted a comparative study to estimate the air quality index for Pakistan. They used the SARIMA and factor analysis approaches to achieve this end. The study found that the SARIMA model outperforms others in attaining better estimation accuracy for Pakistan’s air quality index. In another work, Ashraf et al. (2022) conducted an analysis based on the comparative study of machine learning and classical forecasting models to forecast air pollution data. This article uses the metrics, i.e., RMSE, MAE, and MAPE, to compare the classical and traditional models. The study suggested that the machine learning model performed more efficiently than the existing time series methods.
In the same way, Lin et al. (2019) performed an analysis based on machine learning and integrated assimilation data techniques to expand the air quality prediction for the Netherlands. The findings disclosed that the developed approach, which incorporates data-driven machine learning and a physics-based model, significantly improved the air quality forecast statistically. However, Kleine Deters et al. (2017) modeled
On the other hand, Wang et al. (2019) proposed a hybrid model for air quality variables forecasting. This proposed method is a hybridization of the Long Short-Term Memory Neural Network and Gated Recurrent Unit (LSTM and GRU), an enhancement of ordinary LSTM. This work uses the data set of 74 cities in China for a comparative study. The outcomes disclosed that the proposed hybrid model outperformed the existing approaches. In the same way, Ejohwomu et al. (2022) conducted a comprehensive study to model the
In contrast to the research mentioned earlier, this study introduces a comparatively simple and easily implemented new time series ensemble technique based on various linear (autoregressive, simple exponential smoothing, autoregressive moving average, and theta) and nonlinear (nonparametric autoregressive and neural network autoregressive) models to accurately and efficiently forecast short-term
2 The proposed time series ensemble forecasting technique
This section elucidates the proposed time series ensemble technique for short-term (one-day-ahead)
2.1 Preparation of raw data
This work uses daily
That is, the log
Thus, once the air pollutant (
2.2 Forecasting models
This section briefly overviews the forecasting models and their proposed ensemble models: the autoregressive, the simple exponential smoothing, the autoregressive moving average, the Theta, the nonparametric autoregressive, and the neural network autoregressive models.
2.2.1 The auto-regressive model
A linear autoregressive (AR) model is used to understand the short-term dynamics of
In Equation 3,
2.2.2 The exponential smoothing model
The Exponential Smoothing Model (ESM) is a group of forecasting models that apply exponentially decreasing weights to previous observations. It is a time-series forecasting model that uses a weighted average of past observations to predict the future value of a variable. The ES model assumes that a variable’s future value depends on its past values, with greater emphasis placed on recent values than on older ones. The ESM model can be expressed as follows:
In the given Equation 4,
2.2.3 The autoregressive moving average model
The autoregressive moving average (ARMA) models incorporate lagged values from a time series and factor in error terms passed into the model. This study utilized a model representing the residual series
In Equation 5, where
2.2.4 The neural network autoregressive model
The Neural Network Autoregressive (NNA) model is a machine learning approach that uses historical observations to predict future values in a time series. It does this by analyzing a mathematical function that considers the previous values, denoted by
2.2.5 The nonparametric autoregressive model
The nonparametric autoregressive model (NPAR) presents an alternative to the conventional parametric AR model, departing from the latter’s reliance on specific mathematical equations to elucidate the relationship between past and future values. In contrast, NPAR models employ flexible and adaptive techniques, such as kernel regression or spline functions, to capture dynamic patterns in the data without explicit parameter estimation. These models are distinguished by their flexibility, absence of predefined parameters, emphasis on local relationships, and reliance on data-driven structures to address intricate and nonlinear dependencies within time series data. This model’s association between
here in Equation 6,
2.2.6 The theta model
The Theta Model is a forecasting method that predicts future values based on the average change in the time series data. It involves calculating the average change between consecutive time points and extrapolating it into the future. The equation for the Theta Model is given by in Equation 7:
2.2.7 The proposed ensemble models
At its core, an ensemble technique integrates outcomes from various models, each meticulously calibrated before unity. This approach capitalizes on the inherent strengths of individual models while compensating for their inherent limitations. Within the scope of this study, ensemble techniques are initially employed to compute weights for the results derived from individual models (Iftikhar et al., 2024a; Gonzales et al., 2024). Consequently, the proposed ensemble encompasses three distinct weighting strategies: a) equitable distribution of weight among all single models, denoted as ESME; b) weight assignment based on training average accuracy errors (1), designated as ESMT; and c) weight assignment based on validation mean accuracy measures, denoted as ESMV. The model allocates greater weight to the ensemble model for training and validation datasets with lower mean accuracy errors, while models exhibiting higher mean accuracy errors contribute comparatively less weight to the ensemble. Notably, the model weights assume small positive values, and their accumulation equates to one, signifying the percentage of reliance or anticipated performance from each model.
Thus, after estimating the linear trend component and annual periodicity using the multiple regression model discussed above, the next step is forecasting the remaining part
2.3 Evaluation criteria
This study examines two evaluation criteria for the proposed time series ensemble forecasting technique: accuracy average errors and an equal forecast accuracy test.
2.3.1 Accuracy average errors
Primarily, Table 1 presents the accuracy average errors, outlining the formulas for computing each metric. The metrics encompass the mean absolute error (MAE), an indicator of errors within pair samples reflecting the same phenomena. The mean absolute percent error (MAPE) is a metric used to assess how accurate a forecasting system is in making predictions. The mean scaled absolute error (MASE), it is calculated by dividing the mean absolute error of the prediction values by the mean absolute error of the one-step naive forecast made in the sample. The root mean squared error (RMSE) calculates the average disparity between the values a statistical model predicts and the observed values. The root relative squared error (RRSE) root of the squared prediction error in comparison to a simple model that predicts the mean. After applying the log to both, the root mean log squared error (RMSLE) is computed by considering the differences between the actual and anticipated values, Iftikhar et al. (2024b).
The table presents the actual
2.3.2 Equal forecast accuracy test
Second, a statistically equal forecast test, the Diebold–Marino (DM) test (Diebold, 2015), is performed to evaluate the forecasting ensemble time series proposed approach. In the literature, It is used to evaluate time series forecasting models, determining whether the forecast errors from one model are statistically different from another model’s forecast errors (Iftikhar et al., 2023a; Shah et al., 2019; Iftikhar et al., 2023b). To perform the DM test, the forecast errors of each model are calculated using a loss function. Then, a statistical value is computed by comparing the errors of each model. The test statistic is based on the difference between the mean squared errors of the two models. Suppose the test statistic is above a certain threshold and the p-value is below a significance level
To complete this section, the main steps, including the introduced time series ensemble forecasting approach in bullet form, are listed below, and the flowchart is presented in Figure 1.
Where
3 Case study results
In order to obtain short-term
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Table 2. Details about the considered original, missing, and imputed datasets are provided in this work.
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Figure 3. Time Series Plot: Original time series (top) and the first ordered difference time series for all four megacities (bottom).
3.1 Data description
On the other hand, Table 3 represents a comprehensive overview of the statistical properties (with and without log descriptive statistics) for
3.2 forecasting outcomes
The given steps must be followed to obtain the forecast for
Hence, for all nine models for the four monitoring megacities, including Lahore, Islamabad, Karachi, and Peshawar, one-day-ahead out-of-sample forecast outcomes (MAE, MASE, MAPE, RMSE, RRSE, and RMSLE) are listed in Table 4. Table 4 shows that the ESMV produced the best forecasting results compared to all nine forecasting models within the proposed time series ensemble forecasting approach in all four monitoring megacities. For instance, the average accuracy errors for these magacities are the following: Islamabad (MAPE = 0.1739, MAE = 15.2718, MASE = 0.9237, RMSE = 20.3203, RRSE = 0.4830, and RMLSE = 0.2354); Lahore (MAPE = 0.2167, MAE = 34.1786, MASE = 0.9207, RMSE = 48.3090, RRSE = 0.5837, and RMLSE = 0.2701); Karachi (MAPE = 0.1679, MAE = 16.0982, MASE = 0.9122, RMSE = 22.9913, RRSE = 0.5464, and RMLSE = 0.2215); and Peshawar (MAPE = 0.1973, MAE = 24.1188, MASE = 0.8858, RMSE = 35.0552, RRSE = 0.5998, and RMLSE = 0.2594). However, the ESMT model shows the second-best forecasting results among all nine forecasting models in all four monitoring megacities, while the third-best forecasting accuracy average error results are given in the following manner: Islamabad (the Theta model; MAPE = 0.1835, MAE = 16.3457, MASE = 0.9887, RMSE = 21.2335, RRSE = 0.5047, and RMLSE = 0.2370); Lahore (the Theta model; MAPE = 0.2179, MAE = 35.0539, MASE = 0.9442, RMSE = 49.9617, RRSE = 0.6037, and RMLSE = 0.2723); Karachi (the ARMA model; MAPE = 0.1702, MAE = 16.3909, MASE = 0.9287, RMSE = 23.3728, RRSE = 0.5555, RMSLE = 0.2226); and Peshawar (the NPAR model; MAPE = 0.2010, MAE = 25.0597, MASE = 0.9204, RMSE = 36.1912, RRSE = 0.6037, and RMLSE = 0.2723); Karachi (the ARMA model; MAPE = 0.1702, MAE = 16.3909, MASE = 0.9287, RMSE = 23.3728, RRSE = 0.5555, RMSLE = 0.2226); and Peshawar (the NPAR model; MAPE = 0.2010, MAE = 25.0597, MASE = 0.9204, RMSE = 36.1912, RRSE = 0.6192, RMSLE = 0.2610). Therefore, it is seen that within all nine forecasting models, the proposed ensemble models (the ESMV and the ESMT models) generally perform better than single models; however, within the single models, different cities have different single best models, as mentioned previously. Note that the best model is an ESMV or equivalent for all four mountaineering megacities. Also, using the proposed ensemble learning leads to marked error reduction (see Table 4). The proposed ensemble learning approach, thus, proves to be particularly effective in forecasting short-term
On the other hand, after the evaluation of the proposed time series ensemble modeling and forecasting technique by the average mean errors and the DM test, another check can be made to evaluate the accuracy of the selected best ensemble models by the graphical representation of the observed data and the predicted data. To do this, each city’s scatter plot (correlation) is drawn, and the correlation coefficient is calculated. Figures 4A–D shows the graph for each megacity. In Figure 5A for Islamabad city, it is noticed that the correlation coefficient value between the forecasted and actual data set is 0.97, indicating a strong and positive correlation. Moreover, from Figure 4B for Lahore city, the coefficient value between forecasted and actual data is 0.86, which shows a strong positive correlation between forecasted and actual data. Figure 4C for Karachi city and Figure 4D for Peshawar city, the coefficient values are 0.85 and 0.94, which shows a strong positive correlation between forecasted and actual data. In addition, the diagnostic checking (final residuals) plays an essential role in model selection, and this is tracked by the auto-correlation (ACF) plot and the partial auto-correlation plot (PACF), also known as the correlogram plot. As stated earlier, the proposed ensemble model is significant and efficient in forecasting the
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Figure 4. The correlation plots for the best models among all nine considered models in each city: Islamabad (a), Lahore (b), Karachi (c), and Peshawar (d).
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Figure 5. The autocorrelation function and partial autocorrelation plots for the best models among all nine considered models in each megacity case: (A, B) Islamabad, (C, D) Lahore, (E, F) Karachi, (G, H) Peshawar.
4 Discussion
This section elaborates an overview of comparing the proposed best model of this work versus the literature that found the best forecasting models. On the other hand, it also explains the future
4.1 Comparatively study resutls
In this subsection, we compared the results of our best ensemble model with those reported in the literature models. Our model showed high comparability with the other methods. Our best (ESMV) model produced the most negligible mean errors and the highest correlation coefficient (MAPE = 0.1679, MAE = 16.0982, MASE = 0.9122, RMSE = 22.9913, RRSE = 0.5464, and RMLSE = 0.2215) compared to the best models reported in the literature. For example, the best autoregressive distributed lag model (the ARDL model) proposed in Qayyum et al. (2021) was applied to the dataset used in our study and showed accuracy measures (MAPE = 0.1835, MAE = 21.3457, MASE = 0.9887, RMSE = 21.2335, RRSE = 0.5047, and RMLSE = 0.2370) that were significantly greater than those of our best (ESMV) model. The best model proposed in another study (see Bhatti et al., 2021) - the seasonal autoregressive moving average factor analysis approach - was also applied to our dataset and obtained average mean errors (MAPE = 0.1823, MAE = 20.1038, MASE = 0.9483, RMSE = 36.1853, RRSE = 0.7214, and RMLSE = 0.2901) that were higher than those of our best (ESMV) model. Similarly, in reference Waseem et al. (2022), the best proposed LSTM encoder-decoder applied to our dataset obtained performance metrics (MAPE = 0.2010, MAE = 25.0597, MASE = 0.9204, RMSE = 36.1912, RRSE = 0.6192, RMSLE = 0.2610) worse than those obtained with our best combination model (ESMV). In conclusion, our study’s best final model (ESMV) showed high efficacy and accuracy compared to the best models reported in the literature.
4.2 Future short-term forecasting using the superior model
On the other hand, once the best models were assessed through average accuracy errors (MAPE, MAE, MASE, RMSLE, RRSE, and RMSE), an equal forecast statistical test (the DM test), graphical evaluation (the ACF, PCAF, and correlogram plots), and comparing with the literature best models this work proceeded to future short-term forecasting with the superior model (the ESMV). In this regard, the current work used the ESMV for the
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Table 6. Forecasted values exercise for all megacities for the next 15 days using the best model in each case.
As per the Air Quality Index by the Environment Protection Agency, US, the following ranges and their health level concerns: 0–12
5 Conclusion
This work proposes a novel time series ensemble approach using the daily
In addition, using the best ensemble model in this work, a forecast was made for the next 15 days (1 June to 15 June 2023); the forecast exercise shows that the elevated levels of
However, the study’s main limitation is that it only incorporates
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
HI: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing–original draft, Writing–review and editing. MQ: Data curation, Formal Analysis, Investigation, Writing–review and editing. JZ: Funding acquisition, Project administration, Supervision, Writing–review and editing. JL-G: Investigation, Project administration, Resources, Supervision, Writing–review and editing. OA: Investigation, Resources, Supervision, Writing–review and editing.
Funding
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Keywords: air pollution, concentration, short-term PM 2.5 forecasting, single time series models, ensemble time series models, sustainable development, early warning system, decision making
Citation: Iftikhar H, Qureshi M, Zywiołek J, López-Gonzales JL and Albalawi O (2024) Short-term
Received: 02 June 2024; Accepted: 23 August 2024;
Published: 10 September 2024.
Edited by:
Dimitris G. Kaskaoutis, National Observatory of Athens, GreeceReviewed by:
Nurul Amalin Fatihah Kamarul Zaman, University of Science Malaysia (USM), MalaysiaHamid Gholami, University of Hormozgan, Iran
Copyright © 2024 Iftikhar, Qureshi, Zywiołek, López-Gonzales and Albalawi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Hasnain Iftikhar, hasnain@stat.qau.edu.pk