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| 研究生： |
陳國帥 Chen, Kuo Shuai |
|---|---|
| 論文名稱： |
臺灣股票市場非線性現象之研究：傅利葉轉換與小波轉換之應用 The Research of Nonlinear Phenomena of the Taiwan Stock Market: the Applications of Fourier Transform and Wavelet Transform |
| 指導教授： |
胡聯國
Hu, Len Kuo |
| 學位類別： |
碩士
Master |
| 系所名稱： |
商學院 - 國際經營與貿易學系 Department of International Business |
| 論文出版年： | 1995 |
| 畢業學年度： | 83 |
| 語文別： | 英文 |
| 論文頁數： | 89 |
| 中文關鍵詞： | 非線性 、碎形結構 、混沌 、傅利葉轉換 、小波轉換 、臺灣股票市場 |
| 外文關鍵詞： | Nonlinear, Fractal Structure, Chaos, Fourier Transform, Wavelet Transform, Taiwan Stock Market |
| 相關次數： | 點閱：406 下載：0 |
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本文採用傅利葉轉換與小波轉換以探討非線性現象：長期相依的碎形結構與混沌現象。藉由傅利葉轉換與小波轉換兩種研究方法，所得到臺灣股票市場加權股價指數的實證結論如下：1.藉由傅利葉轉換所得到的H值為0.4632；藉由小波轉換所得到的H值為0.4750。這兩種研究方法皆顯示臺灣股票市場具有負的長期相依的碎形結構。2.藉由傅利葉轉換的研究方法，臺灣股票市場加權股價指數的頻譜由初始向下與寬的連續的頻帶所組成；臺灣股票市場加權股價指數的自我相關函數則隨著時間差距的增加而遞減。此顯示臺灣股票市場具有混沌現象。3.小波轉換可以檢測出臺灣股票市場加權股價指數的奇異之處，並且指出存有一能說明臺灣股票市場碎形結構的複雜性的機制。藉由以上的實證結論，可以得知臺灣股票市場具有反持續性的碎形結構，股票價格的變動來自於臺灣股票市場尺度上的自我相似性。即使如此，由於混沌不可預測性的本質，使得股票價格的預測似乎是不可能的。
The Fourier transform and the wavelet transform are utilized in this research to explore the nonlinear phenomena: the fractal structure of long trem dependence and the phenomenon of chaos.
In terms of the two research methods of the Fourier transform and the wavelet transform, the empirical conclusions of the Taiwan stock exchange weighted stock index are derived as follows:
1. The $H$ value of the research method of the Fourier transform is 0.4632; the $H$ value of the research method of the wavelet transform is 0.4750. The two research methods show that the Taiwan stock market has a fractal structure of negative long term dependence.
2. In terms of the research method of the Fourier transform, the power spectrum of the Taiwan stock exchange weighted stock index consists of initially downward and wide continuous band of frequencies; the autocorrelation function of the Taiwan stock exchange weighted stock index decreases as the time lag increases. These observations show that there exists the phenomenon of chaos in the Taiwan stock market.
3. The wavelet transform can detect out the singularities of the Taiwan stock exchange weighted stock index and can point out the heirarchy that illustrates the complexity of the fractal sturcture in the Taiwan stock market.
By the above empirical conclusions, there exists the antipersistent fractal structure in the Taiwan stock market. The variations of stock prices result from the self-similarity of the scales of the Taiwan stock market. Even so, the prediction of stock prices seems very impossible as a result of the unpredictability of chaotic nature.
謝辭
Abstract
Contents-----i
List of Figures-----iii
List of Tables-----vi
1 Introduction-----1
1.1 Motivation of Research-----1
1.2 Purposes of Research-----1
1.3 Scope of Research-----3
1.4 Structure of Research-----3
2 Review of Literature-----4
2.1 Fractional Brownian Motion and Fractal Structure-----4
2.2 Introduction to Fractals-----9
2.3 Introduction to Chaos-----11
3 Research Method (I) Fourier Transform-----18
3.1 Fourier Series-----18
3.2 Fourier Transform-----23
3.3 Fourier Transform and Fractional Brownian Motion-----27
3.4 Fourier Transform and Chaotic Signals-----30
4 Research Method (II) Wavelet Transform-----36
4.1 Wavelet Transform-----36
4.2 Comparison between Fourier Transform and Wavelet Transform-----45
4.3 Wavelet Transform and Fractional Brownian Motion-----47
4.4 Wavelet Transform and Fractallike Signals-----48
5 Empirical Results and Analyses-----53
5.1 Empirical Studies of Fourier Transform-----53
5.1.1 Empirical Analysis of Fractal Structure-----55
5.1.2 Empirical Analysis of Chaos-----57
5.2 Empirical Studies of Wavelet Transform-----59
5.2.1 Empirical Analysis of Fractal Structure-----60
5.2.2 Empirical Analysis of Fractallike Data-----64
5.3 Empirical Conclusions-----72
6 Conclusions and Suggestions-----73
6.1 Conclusions of Research-----73
6.2 Suggestions of Research-----74
List of Figures
2.1 Cantor ternary set. Data resource: Gulick (1992), pp.192-----9
2.2 von Koch curve. Data resource: Gulick (1992), pp.195-----10
3.1 The relationship between data, Fourier transform, power spectrum and autocorrelation function. Data resource: This research-----26
3.2 Four attractors: (a), point attractor, (b). limit cycle,(c). torus, (d). strange attractor. Date resource: Gulick (1987), pp-50-----33
3.3 Point attractor. Data resource: Argyris, Faust and Haase (1994), pp.149-----33
3.4 Limit cycle. Data resource: Argyris, Faust and Haase (1994), pp.149-----34
3.5 Torus. Data resource: Argyris, Faust and Haase (1994), pp.149-----34
3.6 Strange attractor. Data resource: Argyris, Faust and Haase (1994), pp.149-----35
3.7 White noise. Data resource: Argyris, Faust and Haase (1994), pp.149-----35
4.1 The comparison between Fourier transform and wavelet transform. (a) Fourier transform. Perfect wavenumber-space resolution, no physical-space resolution, (b) Wavelet transform. Balance between wavenumber- and physical-space resolution varies with length-scale. Smaller length-scales are more finely resolved: mathematical microscope. Data resource: Farge, Hunt and Vassilicos (1993), pp.19-----38
4.2 Flexible time-frequency windows, a<sub>1</sub> < a<sub>2</sub>. Data resource: Chui (1992), pp.9-----40
4.3 Haar wavelet. Data resource: Wei (1994)-----42
4.4 Hat wavelet. Data resource: Wei (1994)-----43
4.5 Mexican hat wavelet. Data resource: Wei (1994)-----43
4.6 Gaussian wavelet. Data resource: Wei (1994)-----44
4.7 Morlet wavelet. Data resource: Wei (1994)-----44
4.8 The question of singular points. Data resource: This research-----45.
4.9 The wavelet transform of the Cantor ternary set. Data resource: Argoul et al. (1989)-----50
4.10 The construction rule of the Cantor ternary set. Data resource: Argoul et al. (1989)-----51
4.11 The wavelet analysis of the wind tunnel data. The top graphs show the signal being analyzed, (a), the wavelet transform of a 852 m-long sample from the scale 28 l<sub>o</sub> to the scale l<sub>o</sub>/10; (b). magnification x20 of the central position indicated by the arrow in the top graph of (a); (c). magnification x20 of the central position indicated by the arrow in the top graph of (b).Data resource: Argoul et al. (1989)-----52
5.1 The Taiwan stock exchange weighted stock index. Data resource: This research-----54
5.2 The original power spectrum of the Taiwan stock exchange weighted stock index. Data resource: This research-----55
5.3 The power spectrum of the Taiwan stock exchange weighted stock index. Data resource: This research-----56
5.4 The log-log plot of the power spectrum versus frequency. Data resource: This research-----57
5.5 The autocorrelation function of the Taiwan stock exchange weighted stock index. Data resource: This research-----58
5.6 The orthonormal Maxican hat wavelet. Data resource: Wei (1994)-----59
5.7 The wavelet transform of the Taiwan stock exchange weighted stock index of 4096 trading days. Data resource: This research-----65
5.8 The contour map of the wavelet transform of the Taiwan stock exchange weighted stock index of 4096 trading days. Data resource: This research-----66
5.9 The Taiwan stock exchange weighted stock index between the 2000th and the 3000th trading days. Data resource: This research-----69
5.10 The wavelet transform of the Taiwan stock exchange weighted stock index between the 2000th and the 3000th trading days. Data resource: This research-----70
5.11 The contour map of the wavelet transform of the Taiwan stock exchange weighted stock index between the 2000th and the 3000th trading days. Data resource: This research-----71
List of Tables
5.1 The wavelet coefficients of the Taiwan stock exchange weighted stock index of 4096 trading days. Data resource: This research-----62
5.2 The wavelet coefficients of the Taiwan stock exchange weighted stock index between the 2000th and the 3000th trading days. Data resource: This research-----67
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