用於預測標籤的時間序列分類在現今被廣泛使用。舉例來說，長短期記
憶模型被應用於名叫星海爭霸二的複雜遊戲中。而散射變換以及支援向量
機也被運用於音樂摘錄上的分類，其準確度可高達87.5%。然而，只有少
數論文探討為何這些特徵提取之技巧，可以在理論上應用於非平穩過程。

本篇論文將會分成四個章節。在第一個章節中，我們將介紹一些時間序
列過程，分別為弱平穩過程、週期性平穩過程、及EPACS 過程。

在第二章節中，我們將介紹散射變換，其為一種截取訊號特徵的工具，
且可利用腦電圖分類睡眠階段[1]。為了更深入了解散射變換之定義，我們
必須先了解一些關於短時距傅立葉變換及小波變換之先備知識和限制。

在第三章節中，我們將介紹一些散射變換於弱相關隨機過程之統計性
質，包含散射矩（scattering moment）於卜瓦松過程上以及散射變換於白噪
音上。

最後，我們將先行介紹NAST ── 散射變換之推廣。接者，我們將會介
紹一些NAST 相似於散射變換之性質。最後，我們將說明NAST 於週期平
穩過程上的非擴散性質，並著重於解釋為何其為一個良好的提取訊號特徵
之工具。

Time series classification, aiming at predicting class labels, is widely used nowadays. For example, in 2019, the Long short­term memory (LSTM) model is used by DeepMind to excel at a complex video game called Starcraft II. The genre of a musical excerpt which is classified via wavelet scattering transform and
SVM with an accuracy of 87.5% is an another example. However, there are only a few articles state that why can these feature extraction techniques theoretically be applied well to non­stationary processes.

The whole thesis will be divided in four chapters. In the first chapter, we will introduce some types of time series, from specific to general, which are weak sta­tionary process, cyclostationary process, and evolving period and amplitude cyclo­stationary (EPACS) process.

In the second chapter, we will introduce the scattering transform, which is a tool for extracting the signals. It can be apply to EEG signals to classify sleep stages [1]. To motivate how and why the scattering transform was defined, prior knowledge and limitations about the most common transforms on the signals : the short-­time Fourier transform and the wavelet transform, are required.

In the third chapter, we will introduce statistical properties of scattering trans­form on weakly dependent random processes, including scattering moments on the Poisson process and scattering transform on the white noise.

In the fourth chapter, we will first introduce the generalization of the scat­tering transform called the neural activation scattering transform (NAST). Next, we will introduce some of the properties of the NAST that are similar to those of the scattering transform. Finally, we will state some of the results about the non­-expansiveness property of NAST on cyclostationary processes, partially explain
why it is a good tool as a feature extraction.

1. Types of Processes 1
1.1 Weakly stationary process, cyclostationary process, and EPACS process .... 1

2. From the Short-time Fourier Transform to the Scattering Transform 5
2.1 Short-time Fourier transform .... 5
2.2 Continuous wavelet transform .... 9
2.3 Mel-frequency spectrogram .... 9
2.4 (Wavelet) scattering transform .... 11

3. Scattering Transform of Weakly Dependent Random Processes 13
3.1 The Berk central limit theorem for $m$-dependent random variables ..... 13
3.2 Scattering moment .... 16
3.3. Scattering moment of the Poisson processes .... 17
3.4. Scattering transform on white noise .... 22

4. Neural Activated Scattering Transform of Cyclostationary Processes 25

4.1. Definition of NAST and its properties .... 25
4.2. Nonexpansiveness property of NAST on cyclostationary processes .... 26

Reference 31
Appendix 33

[1] G.­R. Liu, Y.­L. Lo, J. Malik, Y.­C. Sheu, and H.­T. Wu, “Diffuse to fuse eeg spectra – intrinsic geometry of sleep dynamics for classification,” Biomedical Signal Processing and Control, vol. 55, p. 101576, 2020.

[2] S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way. USA: Academic Press, Inc., 3rd ed., 2008.

[3] J. andén and S. Mallat, “Deep scattering spectrum,” IEEE Transactions on Signal Process­ing, vol. 62, 04 2013.

[4] S. Mallat, “Group invariant scattering,” Communications on Pure and Applied Mathemat­ics, vol. 65, 10 2012.

[5] K. N. Berk, “A Central Limit Theorem for m-­Dependent Random Variables with Un­bounded m,” The Annals of Probability, vol. 1, no. 2, pp. 352 – 354, 1973.

[6] J. Bruna, S. Mallat, E. Bacry, and J.­F. Muzy, “Intermittent process analysis with scattering moments,” The Annals of Statistics, vol. 43, no. 1, pp. 323 – 351, 2015.

[7] J. Bruna, S. Mallat, E. Bacry, and J.­F. Muzy, “Supplement to“Intermittent process anal­ysis with scattering moments.＂,” 2015.

[8] L. B. Koralov and J. G. Sinaj, Theory of probability and random processes. Springer, 2012.

[9] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, vol. 82. 01 2000.

[10] G.­R. Liu, Y.­C. Sheu, and H.­T. Wu, “Central and non­central limit theorems arising from the scattering transform and its neural activation generalization,” 2020.