本論文中，我們嘗試要去判斷一特定曲線的轉折處所在的位子。首先我們先回顧了一些曲線平滑方法，包括移動平均法以及Savitzky-Golay 濾波法。接著我們引進了一個指標，使尋找轉折處的問題變成可用一個閥值來判斷。然而當我們在曲線上進行一些擾動時，該指標會變得不太穩定，即使擾動僅在很小的區域內發生。最後我們使用一開始提到的平滑方法來減少擾動的影響，但我們可看見該指標仍舊不太穩定。

In this paper, we are trying to characteristic the turning point of a specific curve. To do this, we first review some smoothing techniques, including Moving averaging, and Savitzky-Golay filters. Then we introduce an index for characterizing the curve, show that it can solve the problem by a threshold. Next, we show that this index is unstable with respect to a class of perturbations, even when the perturbations only occur in a small segment of the curve. Finally, we use the smoothing techniques in the first section to reduce the noise, but we can see that the index is still unstable.

目錄
1 Introduction----------------------------- 1
2 Smoothing techniques
2.1 Moving average------------------------------- 1
2.2 Savitzky-Golay filtering--------------------- 3
3 Characteristics of a curve
3.1 Characterization for jump continuity--------- 6
3.2 Index for a curve: Linear tail case---------- 8
3.3 Index for a curve: Quadratic tail case------ 19
4 Noise effect
4.1 Two notes on convex function---------------- 33
4.2 The effect of large gap--------------------- 35
4.3 The effect of localized perturbation-------- 40
4.4 The effect of smooth technique-------------- 42
5 Elementary case------------------------- 46

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