## 5 Conclusions  
In this paper, we proposed a multi-model framework for PM\({}_{2.5}\) forecasting. The framework combines four diverse learning models, namely, cluster linear regression (CLR), long-term multi-layer perceptron (LMLP), Fourier series descriptor (FSD), and the short-term multi-layer perceptron (SMLP). The feature set fed into the multi-model framework is selected by a spiral validation process which evaluates the risk for eliminating every feature. We explore the collaborations between parametric and non-parametric learning, short-term and long-term learning. Our experiments with Puli dataset spanning 1 January 2008 to 31 December 2019, show that the proposed multi-model framework can synergize the advantages of the embedded models and can obtain an improved forecasting result. We also show that the performance obtained by a mixture of short-term and long-term learning can surpass that obtained by applying just a single type of learning. A promising direction  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sources** & **Models** & **RMSE** & **MAE** & **MAPE** \\ \hline Single models & CLR & 28.23 & 19.73 & 0.41 \\  & LMLP & 29.43 & 21.45 & 0.41 \\  & FSD & 40.51 & 25.38 & 0.48 \\  & **SMLP** & **29.32** & **19.12** & **0.40** \\ \hline Ensembles & Long-term ensemble & 27.16 & 19.17 & 0.38 \\  & Short-term ensemble & 27.13 & 18.82 & 0.38 \\  & Averaged & 26.75 & 17.81 & 0.34 \\  & **Weighted** & **25.26** & **16.93** & **0.32** \\  & Max\_Avg\_Min & 26.95 & 18.17 & 0.35 \\  & Max\_Wgt\_Min & 25.52 & 17.34 & 0.33 \\  & Adpt\_Wgt & 25.36 & 17.38 & 0.34 \\  & Lower Bound & 16.58 & 9.30 & 0.16 \\ \hline [PERSON] et al. & Decision trees (DT) & 38.13 & 22.18 & \(-\) \\ (2020) & Random forest (RF) & 25.83 & 15.21 & \(-\) \\  & Extra trees (ET) & 25.37 & 15.04 & \(-\) \\  & DT + AdaBoost & 25.40 & 14.46 & \(-\) \\  & RF + AdaBoost & 25.30 & 14.99 & \(-\) \\  & **ET + AdaBoost** & **25.11** & **14.79** & \(-\) \\  & LSTM & 28.97 & 16.66 & \(-\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparative performances of various methods on Delhi dataset.  
for future research is the application of evolutionary instance selection [37] to appropriately train the forecasting system under the environmental scenarios of current testing.  
Conceptualization, P.-Y.Y., A.Y.Y. and R.-F.D.; methodology, P.-Y.Y. and A.Y.Y.; software, S.-E.C.; validation, S.-E.C.; writing--original draft preparation, P.-Y.Y. and B.B.; writing--review and editing, P.-Y.Y. and B.B.; visualization, S.-E.C.; funding acquisition, P.-Y.Y. and R.-F.D. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Science and Technology of ROC, under grant MOST 110-2410-H-163-001-MY2, grant MOST 107-2410-H-260-015-MY3, grant MOST 107-2420-H-260-002-H3, and the Environmental Protection Administration of ROC, under grant EPA-107-FA12-03-A150.  
The raw data used in this study can be access at [[https://airtw.epa.gov/twr/CHT/Query/His_Data.aspx](https://airtw.epa.gov/twr/CHT/Query/His_Data.aspx)]([https://airtw.epa.gov/twr/CHT/Query/His_Data.aspx](https://airtw.epa.gov/twr/CHT/Query/His_Data.aspx)) and [[http://e-service.cwb.gov.tw/HistoryDataQuery/](http://e-service.cwb.gov.tw/HistoryDataQuery/)]([http://e-service.cwb.gov.tw/HistoryDataQuery/](http://e-service.cwb.gov.tw/HistoryDataQuery/)), accessed on 16 December 2021.  
The authors declare no conflict of interest.