## 5 Application of 1D-CNN Bandpass Filter to Time Series with Missing Data Points  
As mentioned in the Section 1, because the Lanczos filter relies on a certain number of neighboring data points to reduce the artificial oscillations during filtering calculations, this leads to a loss of data at the very beginning and the end of the filtered output, as well as nearby missing values in the input series. [PERSON] and [PERSON] have successfully shown that the 1D-CNN filter is one of the best methods to handle data edges and mitigate this limitation by shortening the length of window [34] without sacrificing the accuracies of filtering calculations, which is also verified in our results in Section 4. However, it remains unclear how well the 1D-CNN filter works with the existence of missing data points in the original input. In this section, a series of tests are carried out to examine the abilities and limitations of the 1D-CNN filter when dealing with time series with missing values. As with most in situ observations, the daily temperature data of most stations employed in this study do have missing data points due to reasons such as instrument failure, maintenance and replacement, human error, communication failures, outbreak of war, etc. At the positions of the missing points, the leading edge, and the trailing edge, the edge effects of the Lanczos filter can affect the filtering effects in their vicinity. Therefore, a common way is to simply ignore these missing points and return NaN values in the filtering outputs. This causes difficulties for us to examine the accuracies of 1D-CNN filter outputs by taking the Lanczos filter as the ground truth. To address this problem, we employ the HKO station observations, of which the data remain continuous throughout the period from 1947 to 2022. By applying the 10-60-day Lanczos filter to the HKO daily temperature data in this period, we first obtain the ground truth of the QBWO-to-ISO signals. After that, we manually replace data points in the original time series with missing values according to a predetermined interval (from 1 to 60 days). Subsequently, the pre-trained 1D-CNN bandpass filter, by the procedures introduced in Section 3, is applied to series with manually entered missing data points. Finally, the results between the ground truth (i.e., Lanczos filter output) and the 1D-CNN filter outputs with different numbers of missing values are compared based on the _MAE_, _RMSE_, and _R_\({}^{2}\). These procedures give clues about the impact of missing values on 1D-CNN filtering performance. In the following discussion, one missing data point refers to one missing value for every 60 days, two missing data points refer to two missing values for every 60 days, and so on. Comparison is carried out for the period of 2010-2020. Our results reveal that the 1D-CNN filter still maintains a high level of accuracy even when some data points are missing. However, as the number of missing values increases, the model's errors generally show a growing trend. Figure 5 clearly shows that as the number of missing values increases, the _MAE_ and _RMSE_ increase rapidly when the number of missing values is fewer than 10, and the increasing trend remains rather steady after that. \(R^{2}\) exhibits