## 6 Conclusions and Discussion  
This study aims to address the limitations of traditional temporal filters in extracting certain signals from discontinuous time series, which are commonly seen in meteorological and climate research. Making use of machine learning techniques, we applied a 1D-CNN temporal filter to daily temperature observations from the Hong Kong Observatory, examined its validity, and explored its ability to filter time series with missing data points. The key contributions of this paper are as follows:  
1. A 1D-CNN temporal filter, which can be transformed into a highpass, bandpass, or lowpass filter, is developed. 2. The 1D-CNN filter is shown to be good at handling discontinuous time series. 3. The 1D-CNN filter allows a maximum number of missing data points that is approximately 16.67% of the filter window length. In other words, say, for a 100-day lowpass filter, the 1D-CNN filter is able to give relatively accurate filtered results even if there are \(\sim\)17 missing values within a 100-day window. It is important to note here that the 1D-CNN bandpass filter presented in this study can be extended to other types of temporal filters, such as highpass and lowpass filters. Namely, a lowpass filter only requires one convolutional layer; a highpass one requires an additional subtraction layer after the convolutional layer; and a bandpass filter can be constructed by combining a highpass and a lowpass filter, as explained in Section 3. The corresponding code, written in Pytorch, is publicly available at ([[https://github.com/jeremycheung/1](https://github.com/jeremycheung/1) DCNN_Filter]([https://github.com/jeremycheung/1](https://github.com/jeremycheung/1) DCNN_Filter), accessed on 2 May 2024). The validity of the 1D-CNN temporal filter has been verified in Section 4 and also by [PERSON] and [PERSON] [34]. However, the ability of 1D-CNN highpass and lowpass filters to handle missing input data points was not explicitly discussed in this paper and needs further analyses. Another important remark is that the maximum number of missing data points allowed depends on various factors, such as variables, data types, regions, period, etc. Although our analyses revealed that the presented 1D-CNN bandpass filter allows a maximum number of missing data points of 16.67% of the filter window length, this conclusion may not be applicable to different application scenarios. It is recommended to re-examine the limits of the 1D-CNN filter beforehand. To conclude, this study demonstrates that the 1D-CNN filter can effectively address the limitations of Lanczos and other filters when handling time series with missing data points. Although the discussion of this paper is focused on bandpass filtering, the findings also hold for highpass and lowpass filters. This finding provides an alternative way to extract atmospheric variability signals from certain time scales, especially when traditional filters fail to work well for discontinuous time series. The introduced 1D-CNN filter will be particularly useful in signal extractions from datasets that contain missing values and also in real-time operational applications, such as S2S monitoring and forecasting and the filtering of numerical instability [46]. By exploring this innovative approach, this study also demonstrates another potential for applying machine learning algorithms in this field of research. Conceptualization, H.K. and J.C.-H.L. ; methodology, H.K. and J.C.-H.L. ; software, D.Y. ; validation, D.Y., H.K. and J.C.-H.L. ; formal analysis, D.Y., H.K. and J.C.-H.L. ; investigation, D.Y., H.K. and J.C.-H.L. ; resources, P.W.C. ; data curation, P.W.C. ; writing--original draft preparation, D.Y., H.K. and J.C.-H.L. ; writing--review and editing, D.Y., H.K., J.C.-H.L., P.W.C., C.F., Y.W. and B.Z. ; visualization, D.Y. ; supervision, H.K., J.C.-H.L. and P.W.C. ; project administration, H.K. and J.C.-H.L. ; funding acquisition, Y.W. and B.Z. All authors have read and agreed to the published version of the manuscript.