## 5 Application of 1D-CNN Bandpass Filter to Time Series with Missing Data Points
\begin{table}
\begin{tabular}{c c c c c} \hline
**Automatic** & _MAE_ & _MSE_ & _RMSE_ & \(\mathbf{R^{2}}\) \\
**Weather Station** & _MAE_ & _MSE_ & _RMSE_ & \(\mathbf{R^{2}}\) \\ \hline Ta Kwu Ling (TKL) & 0.066 & 0.012 & 0.111 & 0.996 \\ Lau Fau Shan (LFS) & 0.063 & 0.012 & 0.108 & 0.996 \\ Wetland Park (WLP) & 0.061 & 0.010 & 0.101 & 0.996 \\ Shek Kong (SEK) & 0.064 & 0.011 & 0.104 & 0.997 \\ Tai Mo Shan (TMS) & 0.059 & 0.012 & 0.107 & 0.995 \\ Sha Tin (SHA) & 0.058 & 0.008 & 0.092 & 0.997 \\ Tate’s Cairn (TC) & 0.062 & 0.011 & 0.104 & 0.996 \\ King’s Park (KP) & 0.055 & 0.009 & 0.093 & 0.996 \\ Hong Kong International Airport (HKA) & 0.057 & 0.009 & 0.094 & 0.997 \\ Hong Kong Observatory (HKO) & 0.053 & 0.007 & 0.085 & 0.997 \\ [PERSON] (SLW) & 0.060 & 0.007 & 0.082 & 0.998 \\ [PERSON] (PEN) & 0.055 & 0.008 & 0.091 & 0.996 \\ [PERSON] (CCH) & 0.055 & 0.008 & 0.091 & 0.996 \\ Waglan Island (WGL) & 0.053 & 0.008 & 0.092 & 0.995 \\ \hline \end{tabular}
\end{table} 
Table 2: List of _MAE_, _MSE_, _RMSE_, and \(R^{2}\) values for the test set of all 14 stations from 2010 to 2020.  
similar patterns, but with opposite signs in the trend because of its inversely proportional relationship with error. This shows that as the number of missing values increases, the accuracy of the 1D-CNN filter gradually becomes lower. The above results indicate that the 1D-CNN filter is able to withstand a limited number of missing data points. What is the largest number of missing data points that the 1D-CNN filter can withstand? In this study, we set \(R^{2}\geq 0.8\) as the minimum accuracy criterion for an acceptable 1D-CNN filter, which indicates that the filtering output explains more than 80% of the ground truth-filtered signals. According to Figure 5c, R\({}^{2}\) drops from 0.991 for 1 missing data point to 0.811 for 10 missing data points. Therefore, based on this criterion, we conclude that the 1D-CNN filter can allow up to 10 days of missing values, which accounts for 16.67% of the length of filter window (i.e., 60 days, or the kernel size of the first convolutional layer). As an example, Figure 6 compares the time series of the filtered daily average temperature with the ground truth at the HKO site in 2018. Panels (a) to (f) present results with difference numbers of missing data points. The results show that the 1D-CNN filtering results are entirely consistent with the Lanczos filtering output when there are no missing data points (Figure 6a). When the number of data points is set to 10, while the 1D-CNN filter can capture most of the QBWO and ISO variability, it starts to produce some errors, such as slightly overestimating or underestimating some of the peaks and troughs (Figure 6b). In this case, the MAE, RMSE, and R\({}^{2}\) are 0.325, 0.554, and 0.885, respectively (Figure 7). As the number of missing data points increases to 20, the inconsistencies between the two series become larger (Figure 6c) and R\({}^{2}\) drops to 0.672. Based on the R\({}^{2}\geq 0.8\) criterion, the 1D-CNN-filtered series shown in Figure 6c is considered to have failed to accurately  
Figure 5: Changes in (**a**) MAE, (**b**) RMSE, and (**c**) \(R^{2}\) of the 1D-CNN filtering results against the number of missing values at the HKO station from 2010 to 2020.  
capture the overall 10-60-day signals. The same is also observed for 30, 40, and 50 missing data points (Figure 6d-f). Figure 6: Time series of Lanczos-filtered series (purple line) and 1D-CNN-filtered series (black line) of daily average temperatures at the HKO site in 2018. The 1D-CNN-filtered results are obtained based on (**a**) no missing values and (**b**) 10, (**c**) 20, (**d**) 30, (**e**) 40, and (**f**) 50 missing data points. Figure 7: Changes in (**a**) _MAE_, (**b**) _RMSE_, and (**c**) \(R^{2}\) of the 1D-CNN filtering results against the number of missing values at the HKO station in 2018.