分類問題在資料處理的應用上是一個相當重要的議題，特別是 TSC 問題。
在TSC 問題中，計算兩個時間序列之間的距離是一個核心的議題。其中一種知
名的計算距離的方法就是 DTW ，它建立在動態規劃的基礎上。然而，DTW 的
時間複雜度是 O(n2)。但是當資料長度越大時，它就必須花更多時間來計算。為
了要克服過度花費時間的問題，DDTW 將DTW 結合了扭曲視窗。這一個方法透
過限制可能的答案進而減少了計算時間，因此DTWW的答案可能不會是最佳解。
在這篇論文中，我們提出了一種方法，透過由小到大的順序來展開可能的答案。
我們的方法不僅可以減少所需的計算時間，還可以得到最佳解。

The classification problem is a critical issue in data processing field, especially emph{time series classification} (TSC) problem. In the TSC problem, the calculation of the distance of two time series is the kernel issue.

One of the famous methods for the distance calculation is the emph{dynamic time warping} (DTW), based on the dynamic programming. However, the time complexity of DTW is $O(n^2)$. When the data size is large, it takes too much time to calculate. In order to overcome time consuming problem, emph{dynamic time warping with window} (DTWW) combines the warping window into DTW calculation. This method reduces the computation time by restricting the number of possible solutions, so the answer of DTWW may not be the optimal solution. In this thesis, we present a method that expands the possible solutions in the minimum first order. Our method not only reduces the required computation time, but also gets the optimal answer.

THESIS VERIFICATION FORM . . . . . . . . . . . . . . . . . . . . . . i
THESIS AUTHORIZATION FORM . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 The Time Series Classi cation Problem . . . . . . . . . . . . . . . . . 4
2.2 The Distance Measurements . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Euclidean Distance . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Dynamic Time Warping Distance . . . . . . . . . . . . . . . . 5
2.2.3 Dynamic Time Warping with Window . . . . . . . . . . . . . 7
2.2.4 Derivative Dynamic Time Warping . . . . . . . . . . . . . . . 8
2.3 Similarity with the Longest Common Subsequence . . . . . . . . . . 9
Chapter 3. Our Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 An Example for Illustrating Our Method . . . . . . . . . . . . . . . . 11
3.2 The Irreplaceable Property . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The Minimum First Order . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 4. Experimental Results . . . . . . . . . . . . . . . . . . . . . . 20
4.1 The Experimental Datasets . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 The Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Analysis of Experimental Results . . . . . . . . . . . . . . . . . . . . 38
Chapter 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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