## 3 Proposed Methods
### Feature Selection
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Initial** & **Variable** & **Remova** & **Finally** \\
**Variables** & **Descriptions** & **Order** & **Retained** \\ \hline \hline \multicolumn{4}{c}{**(1) Periodic variables**} \\ \hline _sin\_d_ & \multicolumn{1}{c}{Sine of ordinal hour in a day} & 15 & \(\surd\) \\ _cos\_d_ & \multicolumn{1}{c}{Cosine of ordinal hour in a day} & 16 & \(\surd\) \\ _sin\_y_ & \multicolumn{1}{c}{Sine of ordinal hour in a year} & 3 & \\ _cos\_y_ & \multicolumn{1}{c}{Cosine of ordinal hour in a year} & 6 & \\ \hline \hline \multicolumn{4}{c}{**(2) Meteorological variables**} \\ \hline Temp & Temperature & 14 & \(\surd\) \\ RH & Relative humidity & 8 & \\ Prep & Precipitation & 7 & \\ WS & Wind speed & 10 & \(\surd\) \\ _sin\_w_ & \multicolumn{1}{c}{Sine of wind direction} & 2 & \\ _cos\_w_ & \multicolumn{1}{c}{Cosine of wind direction} & 5 & \\ \hline \hline \multicolumn{4}{c}{**(3) Short-term history meteorological variables**} \\ \hline ST\_Temp & Mean temperature in prior six hours & 17 & \(\surd\) \\ ST\_RH & Mean relative humidity in prior six hours & 19 & \(\surd\) \\ ST\_Prep & Mean precipitation in prior six hours & 9 & \(\surd\) \\ ST\_WS & Mean wind speed in prior six hours & 13 & \(\surd\) \\ _sinw\_WS & Sum of product of _sin\_w_ and WS in prior six hours & 1 & \\ _cosw\_WS & Sum of product of _cos\_w_ and WS in prior six hours & 4 & \\ ST\_WB & Rooted sum of square _sinw\_WS and square _cosw\_WS_ & 18 & \(\surd\) \\ \hline \hline \multicolumn{4}{c}{**(4) Short-term history autoregression variables**} \\ \hline L\_PM\({}_{2.5}\) & Last hour PM\({}_{2.5}\) in the preceding day & 20 & \(\surd\) \\ D1\_PM\({}_{2.5}\) & Mean hourly PM\({}_{2.5}\) in the preceding day & 11 & \(\surd\) \\ D2\_PM\({}_{2.5}\) & Mean hourly PM\({}_{2.5}\) in the day 24 h ahead & 12 & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial feature variables and their removal order in the spiral validation process. After we have prepared the initial feature set of 20 candidate features, the spiral validation process is applied to assist us to asymptotically converge to an ideal set of effective features. As previously noted, in each spiral iteration, we respectively remove each feature from the current set to conduct the production of respective new system prototypes with the same training set. The resulting risk of each system prototype is evaluated from system simulation over the validation set. The training set covers eight-year data for Puli Township from 2008 to 2015, while the validation set contains the entire year 2016 data. Based on the evaluation, the feature whose test removal has resulted in the remaining feature set having the minimum risk in terms of RMSE with the validation set is actually removed from the current feature set. Then the spiral feature validation process enters into the next spiral iteration until the current feature set contains only one feature. Finally, the best set of features can be determined by comparing the risk variations of generated prototypes along the spiral iterations. To illustrate, Figure 5 shows the variations of the risk in terms of RMSE obtained by the retained feature set in each spiral iteration by applying the spiral validation process. The order of removed features in the sequential spiral iterations is shown in Table 2. We observe that the RMSE stays at a relatively low level as the number of retained features in the feature set decreases from 12 to 7. We  
Figure 3: (**a**) The daily and (**b**) yearly periodic trends of PM\({}_{2.5}\) time series. finally determine to retain 12 features (i.e., removing 8 features) as indicated in Table 2 because then the final feature set contains some members from each feature category and it potentially has better generalization capability than other feature selections. Note that, our spiral validation process is a feature selection heuristic to asymptotically obtain the near-optimal feature set from 20 initial potential features.