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研究生：	周彥洵 Zhou, Yan-Xun
論文名稱：	小波轉換的實作及其在睡眠期別分類中的應用 Implementation of Wavelet Transforms and Their Applications on Sleep Stage Classification
指導教授：	劉聚仁 Liu, Gi-Ren
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2022
畢業學年度：	110
語文別：	英文
論文頁數：	71
中文關鍵詞：	小波轉換、時頻分析、睡眠期別
外文關鍵詞：	Wavelet transform, Time-frequency analysis, Sleep stages
相關次數：	點閱：101 下載：6
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我們從時頻分析的角度去回顧連續小波轉換(CWT)，並詳盡地說明如何設計小波和尺度函數。接著利用離散小波轉換(DWT)的演算法設計理念，我們提出一個速度更快的CWT算法，並將它稱之為遞迴連續小波轉換(RCWT)。我們把RCWT應用在分類睡眠期別上，並觀察參數變化對訓練結果的影響。我們最終得到一個睡眠期別分類模型，整體準確度大約為84.8%、Macro F1大約為78.6%，而Cohen's Kappa 大約為78.9%。

We review the continuous wavelet transform (CWT) from the perspective of time-frequency analysis and give a detailed guide on how to design wavelets and scaling functions. Then, following the idea of discrete wavelet transform (DWT), we propose a faster CWT algorithm and call it recursive continuous wavelet transform (RCWT). We use RCWT to train a model to classify sleep stages and observe the effect of parameter changes on the model. In the end, we obtained a model with an overall accuracy of around 84.8%, Macro F1 around 78.6%, and Cohen's Kappa around 78.9%.

中文摘要 i
Abstract ii
Contents iii
List of Tables v
List of Figures vi
1 Introduction 1
1.1 Wavelet Transform 1
1.2 Sleep Stages 2
2 Wavelet Transform Theorem 3
2.1 Continuous Wavelet Transform 3
2.2 Some Properties of Wavelets 4
2.3 Analytic Wavelets 7
2.4 Scaling Functions 9
2.4.1 Inverse Wavelet Transform and Admissibility Condition 10
2.4.2 Scaling Functions 13
2.5 Frequency Bands of Scaling Functions 16
2.6 Using Gamma Distribution to Design an Analytic Wavelet and its Scaling Function 19
3 Recursive Continuous Wavelet Transform 25
3.1 Recursive Continuous Wavelet Transform 26
3.2 RCWT with multiple wavelets 32
4 Designing Algorithm 35
4.1 Some Notations of Discrete Time Signals 35
4.2 Discrete Time Fourier Transform (DTFT) v.s. Discrete Fourier Transform (DFT) 36
4.3 Convolution Theorem 37
4.4 The Algorithm of CWT and RCWT 40
5 The Sleep Stage Classification 48
5.1 Scattering Transform 49
5.2 Replace Scattering Transform with Recursive Continuous Wavelet Transform 50
5.3 Results 51
6 Conclusions 59
6.1 Wavelet Transform 59
6.2 Sleep Stage Classification 60
References 61
Appendix A: Various versions of convolution therorem 62
A.1 Various Versions of Convolution and Fourier Transform 62
A.2 Various versions of convolution theorems (CT) in practice 64
A.2.1 Convolution theorem of periodic functions 64
A.2.2 Generalize convolution theorem with algebra 66
Appendix B: Parseval and Plancberel Formulas 71
                                    

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[4] Gi-Ren Liu, Yu-Lun Lo, John Malik, Yuan-Chung Sheu, and Hau-Tieng Wu. Diffuse to fuse eeg spectra-intrinsic geometry of sleep dynamics for classification. Biomedical Signal Processing and Control, 55:101576, 2020.
[5] Stephane Mallat. A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, 2008.
[6] The MathWorks, Inc. cwt (wavelet toolbox). 2019.

校內：2024-01-01公開
校外：2024-01-01公開

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