股價波動交互關係的時間特徵 - 政大學術集成
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(2) y. ‧. sit. 學. v n. engchi. i Un. Ch. al io. 股價波動交互關係的時間特徵. Nat. er. 碩士論文. ‧ 國 國立政治大學 應用物理研究所. 陳群衛. 104 7. 立. 政 治 大.
(3) 立. 政 治 大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v.
(4) 中文 本論文在分析 SP500 指數其中交易最為頻繁的 345 家公司在 1996 年各月份的股票數據, 市場模式的空間特性已經被證實出來 [3] [4] [5], 而我們利用卡忽南 -拉維展開來分解股價對數報酬的時間序列, 利用傅 立葉分析，並考慮股價對數報酬是否有時間序列重疊，與比較高頻移 動平均對時間序列的影響，藉由參考各股市系統的特徵參數來尋找相 似或不同之處。 關鍵字：高頻移動平均, 股票, 卡忽南 -拉維展開式, 指數頻譜. 立. 政 治 大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i Un. v.
(5) Abstract We present the results of our analysis of time series for a collection of 345 stocks listed in SP 500, to show that integrated information on collective fluctuations in financial data can be revealed quantitatively by combined. 政 治 大. analysis, focusing separately on either the deterministic or the stochastic con-. 立. tents of the system. In comparing the fluctuations of high frequency one-day. ‧ 國. 學. moving averages (HF1MA) of the original prices of individual stocks with those inherited in the trajectories of Brownian particles [1], also comparing. ‧. the log return with overlapping time interval with the log return without over-. sit. y. Nat. lapping time interval, we can quantify the time characteristic properties of the. io. al. er. whole system which would direct the motions of tracer particles. In this study,. n. we decompose the fluctuations in Karhunan-Loeve expansions and reveal the. Ch. i Un. v. system-specific collective properties by analyzing those collective modes in. engchi. their time-wise as well as the stock-wise bases, obtained for either the original prices or those of HF1MA, and for the log return with or without overlapping time interval. Key words:HF1MA,Stock,Karhunan-Loeve expansions,Power spectra. iii.
(6) Contents. i 中文. 立. Abstract. ‧ 國. y. sit. n. al. er. io. 1 Introduction. Nat. List of Tables. Ch. 2 Theorem background and Method 2.1. iii. ‧. List of Figures. 政 治 大. 學. Contents. ii. engchi. i Un. iv vi xiv 1. v. 3. Markov process and Stock . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1.1. Markov process . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1.2. Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.2. Random walk and Brownian motion . . . . . . . . . . . . . . . . . . . .. 5. 2.3. Random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.3.1. Wishart matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.3.2. Correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.4. Karhunen-Loeve expansion . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.5. Discrete Fourier Transformation . . . . . . . . . . . . . . . . . . . . . .. 9. 2.6. High frequency one day moving average . . . . . . . . . . . . . . . . . .. 9. iv.
(7) 3 Data analysis. 11. 3.1. Definition of log-return . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 3.2. The correlation matrix and of Stock price log-return. 12. 3.3. The eigenvalue and eigenvector distribution of correlation matrix. . . . .. 13. 3.4. The Karhunan-Loeve expansion for temporal part . . . . . . . . . . . . .. 19. 3.5. Fourier transform of the time-wise part from the Karhunan-Loeve expansion 23. 3.6. Power spectra from log-log scale discrete Fourier transformation . . . . .. 31. 3.7. Analysis in long time data . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 3.7.1. Calculation with overlapping . . . . . . . . . . . . . . . . . . . .. 38. 3.7.2. Calculation without overlapping . . . . . . . . . . . . . . . . . .. 立. 4 Discussion. 政 治 大. ‧ 國. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. v. i Un. 54 62. 學. Bibliography. . . . . . . . . . . .. v. 64.
(8) List of Figures 2.1. Log return with overlap. . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. The red line is the original data points and the blue line is the data with. 政 治 大. moving average. [Ma, Wang, Chen and Hu , 2013] . . . . . . . . . . . .. 立. 8. 9. Log return with overlap. . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 3.2. The distribution of identity and stock correlation matrix’s eigenvalue . . .. 14. 3.3. The eigenvalue of correlation matrix from raw data with different τ . . . .. 15. 3.4. The eigenvalue of correlation matrix from data with HF1MA with differ-. ‧. ‧ 國. 學. 3.1. y. sit. 15. Distribution of correlation’s elements, calculated with overlap. The whole. io. er. 3.5. Nat. ent τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. month’s length of data is 11700, due to our calculation method, the length. n. al. Ch. i Un. v. of data in different τ will be different. The length of the data for 612 Sec.. engchi. is 11683. For 3600 Sec. is 11600. For 18000 Sec. is 11200. For 36000 Sec. is 10700. For 54000 Sec. is 10200. For 72000 Sec. is 9700. (1996 January SP 500) 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. Distribution of correlation’s elements, calculated with overlapping. The length of data is 11700, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 11683. For 3600 Sec. is 11600. For 18000 Sec. is 11200. For 36000 Sec. is 10700. For 54000 Sec. is 10200. For 72000 Sec. is 9700. (Random case). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vi. 16.
(9) 3.7. The raw data correlation matrix eigenvector for the biggest eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan) . . . . . . . . . . . . . .. 3.8. 17. The data with HF1MA correlation matrix eigenvector for the biggest eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan) . . . . . . . . . .. 3.9. 17. The raw data’s correlation matrix eigenvector for the 144th eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan) . . . . . . . . . . . . . .. 18. 治 政 value. The length of data for 3600 second is 11600, 大 for 36000 second is 立 10700, and for 72000 second is 9700.(SP500 1996 Jan) . . . . . . . . . .. 19. 3.10 The data with HF1MA correlation matrix eigenvector for the 144th eigen-. ‧ 國. 學. 3.11 The Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA. The length of data for 3600 sec-. ‧. ond is 10950, for 36000 second is 10050, and for 72000 second is 9050.. Nat. 21. sit. y. (SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. er. io. 3.12 The Karhunan-Loeve expansion temporal part mode 1(with the smallest. al. iv n C for 36000 second is 10700, and h eforn72000 i U is 9700. (SP500 1996 Jan) g c hsecond n. eigenvalue) for the raw data. The length of data for 3600 second is 11600,. 22. 3.13 The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan) . . . . . . . .. 24. 3.14 The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan) . . . . . . . .. 24. 3.15 The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan) . . . . . . . .. vii. 25.
(10) 3.16 The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 3600Sec .The length of data for 3600 second is 10950. (SP500 1996 Jan) . . . . .. 26. 3.17 The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan) . . . . .. 26. 3.18 The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan) . . . . . .. 27. 治 政 345(with the biggest eigenvalue) for the raw data 大with τ = 3600Sec. The 立 length of data for 3600 second is 11600.(SP500 1996 Jan) . . . . . . . .. 28. 3.19 The Fourier transform of Karhunan-Loeve expansion temporal part mode. ‧ 國. 學. 3.20 The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 36000Sec.. ‧. The length of data for 36000 second is 10700.(SP500 1996 Jan) . . . . .. 28. Nat. sit. y. 3.21 The Fourier transform of Karhunan-Loeve expansion temporal part mode. al. er. io. 345(with the biggest eigenvalue) for the raw data with τ = 72000Sec.. v ni. n. The length of data for 72000 second is 9700.(SP500 1996 Jan) . . . . . .. Ch. engchi U. 29. 3.22 The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950.(SP500 1996 Jan) .. 29. 3.23 The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 36000Sec.The length of data for 36000 second is 10050.(SP500 1996 Jan). 30. 3.24 The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan). viii. 30.
(11) 3.25 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700. (SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 3.26 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. 3.27 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion. 治 政 τ = 36000Sec. The length of data for 36000大 second is 10700.(SP500 立 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . temporal part mode 345(with the biggest eigenvalue) for the raw data with. 32. ‧ 國. 學. 3.28 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with. ‧. HF1MA with τ = 36000Sec. The length of data for 36000 second is. Nat. 33. sit. y. 10050.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . .. er. io. 3.29 log-log scale plot of the Fourier transform of Karhunan-Loeve expan-. al. iv n C 72000Sec. h e The n glength c h i ofUdata for 72000 second is. n. sion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ =. 9050.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 3.30 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 3.31 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. 35.
(12) 3.32 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 3.33 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 3.34 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion. 治 政 HF1MA with τ = 3600Sec. The length of data 大 for 3600 second is 10950. 立 (SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . temporal part mode 345(with the biggest eigenvalue) for the data with. 36. ‧ 國. 學. 3.35 log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with. ‧. τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996. Nat. 37. sit. y. Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. er. io. 3.36 log-log scale plot of the Fourier transform of Karhunan-Loeve expan-. al. iv n C HF1MA with τ = 3600Sec.hThe i Ufor 3600 second is 10950. e nlength g cofhdata n. sion temporal part mode 1(with the smallest eigenvalue) for the data with. (SP500 1996 Jan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 3.37 The eigenvalue of correlation matrix from raw data in different τ (SP500 1996 full year) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 3.38 The eigenvalue of correlation matrix from data with HF1MA in different τ . (SP500 1996 full year). . . . . . . . . . . . . . . . . . . . . . . . . .. x. 39.
(13) 3.39 Distribution of correlation’s elements, calculated with overlap. The whole year’s length of data is 162500, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 162583. For 3600 Sec. is 162400. For 18000 Sec. is 162000. For 36000 Sec. is 161500. For 54000 Sec. is 161000. For 72000 Sec. is 160500. (1996 SP 500 full year) . . . . . . . . . . . . . . . . . . . . . .. 40. 3.40 Distribution of correlation’s elements, calculated with overlap. The length of data is 162500, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 162583.. 治 政 161500. For 54000 Sec. is 161000. For 72000 大 Sec. is 160500. (Random 立 case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For 3600 Sec. is 162400. For 18000 Sec. is 162000. For 36000 Sec. is. 41. ‧ 國. 學. 3.41 The variance of correlation matrix element versus the τ , calculated with overlap. The length of the data for 612 Sec. is 12983. For 3600 Sec. is. ‧. 12900. For 18000 Sec. is 12500. For 36000 Sec. is 12000. For 54000. Nat. 42. sit. y. Sec. is 11500. For 72000 Sec. is 11000. (1996 SP 500 full year) . . . . .. er. io. 3.42 The variance of correlation matrix element, versus the largest eigenvalue. al. iv n C 612 Sec. is 12983. For 3600hSec. e nisg12900. c h iForU18000 Sec. is 12500. For n. of correlation matrix, calculated with overlap. The length of the data for. 36000 Sec. is 12000. For 54000 Sec. is 11500. For 72000 Sec. is 11000. (1996 SP 500 full year) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 3.43 The largest eigenvalue compute from real market data and random symmetric matrix.The real market data is calculated with overlapping. (1996 SP 500) The red line represent the slope =1 which means the largest eigenvalue compute from random symmetric matrix equal to the largest eigenvalue compute from the real market data. . . . . . . . . . . . . . . . . .. 45. 3.44 The eigenvector of correlation matrix from raw data in typical modes. The length of data for 3600 second is 11600. (SP500 1996 full year) . . . . .. xi. 46.
(14) 3.45 The eigenvector of correlation matrix from data with HF1MA in typical modes. The length of data for 3600 second is 10950. (SP500 1996 full year). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 3.46 The eigenvector of correlation matrix from raw data in typical modes. (SP500 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 3.47 The eigenvector of correlation matrix from data with HF1MA in typical modes. (SP500 1996 full year) . . . . . . . . . . . . . . . . . . . . . . .. 49. 3.48 The Karhunan-Loeve expansion temporal part from raw data in typical modes. (SP500 1996 full year) . . . . . . . . . . . . . . . . . . . . . . .. 50. 治 政 typical modes. (SP500 1996 full year) . . . . 大 . . . . . . . . . . . . . . . 立 3.50 The Karhunan-Loeve expansion temporal part from raw data in typical. 51. 3.49 The Karhunan-Loeve expansion temporal part from data with HF1MA in. ‧ 國. 學. modes. (SP500 1996 full year) . . . . . . . . . . . . . . . . . . . . . . .. 52. 3.51 The Karhunan-Loeve expansion temporal part from data with HF1MA in. ‧. typical modes. (SP500 1996 full year) . . . . . . . . . . . . . . . . . . .. 53. Nat. sit. y. 3.52 Distribution of correlation’s elements, calculated without overlap. The. er. io. whole year’s length of data is 162500, but in the calculation of no over-. al. iv n C 360 Sec is 14772. For 612 Sec Fori 1080 U Sec is 5241. For 1440 h eisn9027. gch n. lapping, the length of the data is quite different. The length of the data for. Sec is 3963. For 1800 Sec is 3186. (1996 SP 500 full year) . . . . . . . .. 54. 3.53 The variance of correlation matrix element, versus the largest eigenvalue of correlation matrix, calculated without overlap. The whole year’s length of data is 162500, but in the calculation of no overlapping, the length of the data is quite different. The length of the data for 360 Sec is 14772. For 612 Sec is 9027. For 1080 Sec is 5241. For 1440 Sec is 3963. For 1800 Sec is 3186. (1996 SP 500 full year) . . . . . . . . . . . . . . . . . . . .. xii. 55.
(15) 3.54 The largest eigenvalue compute from real market data and random symmetric matrix.The real market data is calculated without overlapping. (1996 SP 500 full year) The red line represent the slope =1 which means the largest eigenvalue compute from random symmetric matrix equal to the largest eigenvalue compute from the real market data. the length of the data is quite different. The length of the data for 360 Sec is 14772. For 612 Sec is 9027. For 1080 Sec is 5241. For 1440 Sec is 3963. For 1800 Sec is 3186. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56. 3.55 Distribution of correlation’s elements, calculate with or without overlap-. 治 政 the red line is 12983. The black and red line are大 calculating with overlap立 ping. The green line is calculating without overlapping and it’s length of ping in same τ = 612 Second. The length of the black line is 162483, and. ‧ 國. 學. data is 9027. (1996 SP 500 full year) . . . . . . . . . . . . . . . . . . . .. 57. 3.56 The Fourier transform of Karhunan-Loeve expansion temporal part mode. ‧. 1(with the smallest eigenvalue) for the raw data in τ = 612Sec .(SP 500. Nat. 57. sit. y. 1996 full year ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. er. io. 3.57 The Fourier transform of Karhunan-Loeve expansion temporal part mode. al. iv n C .(SP 500 1996 full year ) . h. e . . . . . . . i. .U ngch . . . . . n. 1(with the smallest eigenvalue) for the data with HF1MA in τ = 612Sec . . . . . . . . . .. 58. 3.58 The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the largest eigenvalue) for the raw data in τ = 612Sec .(SP 500 1996 full year ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 3.59 The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the largest eigenvalue) for the data with HF1MA in τ = 612Sec .(SP 500 1996 full year ) . . . . . . . . . . . . . . . . . . . . . . . . . .. 59. 3.60 The Karhunan-Loeve expansion temporal part from raw data without overlap in typical modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60. 3.61 The Karhunan-Loeve expansion temporal part from data without overlap in HF1MA in typical modes. . . . . . . . . . . . . . . . . . . . . . . . .. xiii. 61.
(16) List of Tables. 立. 政 治 大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. xiv. i Un. v.
(17) Chapter 1 Introduction 政 治 大. For many years, physicists have worked on complex systems and trying to figure out. 立. how these systems work. In the 1980s, while a large amounts of financial data challenged. ‧ 國. 學. economists, physicists who were interested in the social sciences. Physicists connect the real market data to the physical world start to use the method or theory designed for phys-. ‧. ical problems to explain the phenomena in financial market. Now days, econophysics has. sit. y. Nat. become an well recognized field in the study of complex systems . Where econophysics. io. er. approach to understand the phenomena in financial market. One is to build up a model to imitate the real world market. The other is to analyze the empirical data from the real. n. al. Ch. i Un. v. market. By describing the real market data by the concepts of physical world. Physicists. engchi. can dig out new information that economist didn’t tell us. The importance of fluctuations in financial market has been noticed by the trader for more than a century. If the fluctuations in financial market could be completely characterized by random walks, as the paper published by Bachelier in 1900, making profit without any risks would be possible [1]. However, fluctuations in financial market is not that simple. In 1995, Mantegna and Stanley showed that the scaling of the probability distribution of an economic index (SP 500) can be described by a non-Gaussian process with dynamics, correspond to a Levy stable process [2]. In 1999,Laloux et al. published ”Noise Dressing of Financial Correlation Matrices” in which they calculated the correlation matrix for 406 stocks in SP 500 in 1991 to 1996 with a time interval of one day [3] and Plerou et al. calculated the correlation matrix for 1000 stocks in USA in 1994 to 1995 with a time interval of 30 minutes [4] [5]. 1.
(18) Both groups found that there are some discrete and larger eigenvalues beyond the continuous component predicted by the random walk model. And the eigenvector correspond to the largest eigenvalue λM of the correlation matrix, all stocks , which decomposed by the Karhunen-Loeve expansion are activated in the same direction [5] [6]. The mode corresponds to the largest eigenvalue λM of the correlation matrix is called market mode [4] [5]. Correlation in such fluctuations have been demonstrated in recent studies, by analyzing the distribution of eigenvalue of the correlation matrix, which made by the log return of stock price in real market, and analyzing the typical mode expands by the Karhunen-Loeve expansion would give us more information inside the data. The market mode take a big. 治 政 tion obtained by analyzing the mean square log-return of大 USA SP500 stocks and Taiwan 立 TAIEX stocks. Comparing two different markets revealed the difference and similarity role in the market tendency. In 2013, Ma, Wang, Chen and Hu, compare the informa-. ‧ 國. 學. between two different markets [7] [6]. With or without the manipulation of taking moving averages(HF1MA) represent the stochastic scenario and the deterministic scenario. For. ‧. stock market data, HF1MA made the zig-zag trajectory into smooth trajectory.. Nat. sit. y. In this study, in calculating correlation coefficients. The time interval of log-returns in. er. io. the time series of stocks are allowed to have overlapping. We try to characterize the time. al. iv n C Karhunen-Loeve expansion. We use h discrete to analyze the temU e n gFourier c h i transformation n. characteristic in cross correlation of stock fluctuations. We compute different modes from. poral part from K-L expansion, and use the power spectra of temporal part to figure out the features of the bulk in major continuous band. We find well recognized regular packs, going along with the exponent −1 feature in the log-log scale power spectra whether it is in the market modes or in the other modes, and we analyze the relationship between such features and the overlapping in time sequence.. 2.
(19) Chapter 2 Theorem background and Method 政 治 大. In this chapter is going to introduce the theorem and method we have use in this paper:. 立. • Markov process and Stock. ‧ 國. 學. • Random walk and Brownian motion. ‧. • Random matrix theorem. y. Nat. er. io. sit. • Karhunen-Loeve expansion • Discrete Fourier Transformation. n. al. ni C h average • High frequency one day moving U engchi. v. 2.1 Markov process and Stock In this section will briefly introduce the Markov process and stock fluctuation : • SubSection 2.1.1: Markov process • SubSection 2.1.2 Stock. 2.1.1 Markov process In statistics, a Markov process, which named by Russian mathematician Andrey Markov, is a process that satisfies the Markov property. The Markov property defined a random 3.
(20) process whose past has no influence on the future if its present is specified .We call this phenomenon as ”memoryless” [8]. We can describe the Markov by following equation :If. tn−1 < tn. (2.1). p(x(tn ) ≤ xn | x(t), t ≤ tn−1 ) = p(x(tn ) ≤ xn | x(tn−1 )). (2.2). then. From Eq. 2.2 ,it follows that if t1 < t 2 < · · · < t n. (2.3). p(x(tn ) ≤ xn | x(tn−1 ), . . . , x(t1 )) = p(x(tn ) ≤ xn | x(tn−1 )). (2.4). 立. then. 政 治 大. ‧ 國. 學. From Eq. 2.4,it follows that. ‧. n. al. y. er. io. We thus obtain. (2.5). sit. Nat. f (xn | xn−1 , . . . , x1 ) = f (xn | xn−1 ). Ch. engchi. i Un. v. f (x1 , . . . , xn ) = f (xn | xn−1 )f (xn−1 | xn−2 ) . . . f (x2 | x1 )f (x1 ). (2.6). 2.1.2 Stock We can track market stock back to Roman Republic, when the state contracted out many of its services to private. In this century, stock has became a very important part of our human economy systems. It has been verified by real market data that stock , interest rate and exchange rate change in time are random fluctuation process which we can not predict their future. We can also view their time evolution as Markov process. By combining many assets, bank or securities institutes can create some financial products,which we called financial derivatives.. 4.
(21) 2.2 Random walk and Brownian motion In 1828 ,the Scottish botanist Robert Brown observed through his microscope that tiny grains of pollen suspended in water underwent ceaseless random motion.We now call this phenomenon as Brownian motion or Random walk. Brown also noted that this same ’dancing’ motion occurred when particles of finely powdered coal, rocks and glass,and various minerals were suspended in a fluid [9]. In 1900,the French mathematician Louis Bachelier published his PhD thesis ’The theory of speculation’,which is the first paper to use advanced mathematics to discussed the use of stochastic process, which is called Brownian motion, to evaluate stock options. In. 政 治 大 Stock Exchange are innumerable. 立 Events past, present or even anticipated, often showing. his paper Bachelier mentioned that:”The influences which determine the movements of the. ‧ 國. 學. no apparent connection with its fluctuations, yet have repercussions on its course.” Thus, Bachelier is considered a pioneer in the study of financial mathematics and stochastic. ‧. processes [1].. y. Nat. In 1905, Albert Einstein predicted that the Brownian motion should occur and pre-. io. sit. sented it as direct evidence that the fluid in which the particle is suspended is made up. n. al. er. of atoms. A particle suspended in a fluid is bombarded on all sides by the atoms of the. Ch. i Un. v. fluid ,which are in constant motion of thermal agitation. Let N be the average number of. engchi. particle-atoms collisions on any one side of the particle during a short time interval ∆t .On average ,the same number of collision will occur on the other side of the particle .However,because the collision occur randomly ,there will be fluctuation about this average on each side . Thus in many particular interval ∆t there will be slightly more collision on one side of the particle than on the other. These random unbalances occur in three dimensions so the bombarded particle,which typically is many orders of magnitude more massive than the atoms that bombard it,jitters about in the erratic manner that characterizes Brownian motion [9] [10].. 5.
(22) 2.3 Random matrix theory Random matrix theory (RMT), which were used in nuclear physics by Eugene Wigner to model the spectra of heavy atoms. In solid-state physics, random matrix model the behaviour of large disorders Hamiltonians. In multivariate statistics, random matrices were introduced by John Wishart for statistical analysis of large samples. For recent years, random matrix theory has been applied to filter the relevant information from statistical fluctuations. By using random matrix approach to analyse various financial time series, one compare varies quantities with the counterpart from random matrix ensembles. Significant deviations from RMT eigenvalue predictions would provide genuine information,. 政 治 大 the genuine information on the 立correlation structure of the system. [11]. which is hidden in the market data. We can reduce the noise with the help of RMT to find. ‧ 國. 學. The following features are introduced in the coming subsections: • Subsection 2.3.1 Wishart matrix. ‧. • Subsection 2.3.2 Correlation matrix. sit. y. Nat. n. al. er. io. 2.3.1 Wishart matrix. i Un. v. Suppose we have the T × N random matrix C ∈ Wishart matrix, where T is the length. Ch. engchi. of the time series and N is the number of our ensemble. According to the random matrix theory, the spectrum of the eigenvalues for totally random case is given by. √. ρ(λ) =. Q 2πσ 2. (λmax − λ)(λ − λmin ) λ. where. (2.7). √. λmax min. 1 1 = σ (1 + ± 2 ) Q Q. (2.8). T N. (2.9). 2. Q=. The time series are gaussian distribution,and σ is its standard deviation. 6.
(23) From eq. 2.7 and eq. 2.8, we can capture some features that 1. The distribution of eigenvalue is relate to the length of the time series (Q =. T ) N. 2. The boundary of the eigenvalue can be compute from the Q and variance 3. The distribution of eigenvalue is scaled by the variance of the original time series. We can set σ as 1 and ignore it.. 2.3.2 Correlation matrix In order to discuss the variations of different stocks, we have to construct the cross correlation from the log returns, which are the relative changes in price different between. 政 治 大 We consider the time series {s (t, τ ), t = 1, 2,· · · , T } of the normalized fluctuations 立 stocks.. si (t, τ ) =. 學. ‧ 國. i. (ri (t, τ )− < ri (t, τ ) >T ). (2.10). 1. < (ri (t, τ )− < ri (t, τ ) >T )2 > 2. ‧. of log-returns ri (t, τ ) = logPi (t + τ ) − logPi (t) for T time spots in intervals of τ , for. Nat. sit. y. ˙ t denotes an average over the T time spots. price Pi of stock i. The expression < >. n. al. er. io. We can have the correlation coefficient. i < δri δrj >T n C Cij = U h e n gσiσcj h i. v. (2.11). δri (t) = ri (t)− < ri >T , σi2 =< (δri )2 >T. Correlation matrix as . c11 . .. . cN 1. . . . · · · c1N s1 (1) · · · s1 (T ) s1 (1) · · · sN (1) . 1 .. .. .. .. . ··· . = . ··· . . ··· . T · · · cN N sN (1) · · · sN (T ) s1 (T ) · · · sN (T ). 7. (2.12).
(24) In general, we have two way to compute log return. One has log returns with overlapping time intervals and the other does not has. Time intervals of log-returns with overlapping time interval are shown in figure 3.1.. 治 政 大 Figure 2.1: Log return with overlap. 立 ‧ 國. 學. The distribution of correlation of a matrix elements will be influenced by the length of the data and τ . We will discuss it in Chapter 3.. ‧. io. sit. y. Nat. 2.4 Karhunen-Loeve expansion. n. al. er. Karhunen-Loeve expansion(K-L expansion) is a powerful method to represent a stochas-. Ch. i Un. v. tic process as an infinite linear combination of orthogonal functions, we can compute dif-. engchi. ferent mode from K-L expansion, each mode has stock part and temporal part. Here is the temporal part from K-L expansion. si (t, τ ) =. N √ ∑. T λk ak (i)bk (t),. (2.13). k=1. si (t, τ ) is a time series and for this study. si (t, τ ) =. (ri (t, τ )− < ri (t, τ ) >T ) 1. < (ri (t, τ )− < ri (t, τ ) >T )2 > 2. (2.14). From 3.16, we have N modes, and each mode has stock part (ak (i)) and temporal part (bk (t)), where λk and ak (i) are the eigenvalue and its eigenvector for the mode k obtained by diagonalizing the correlation matrix 8.
(25) 2.5 Discrete Fourier Transformation In electrical engineering and complex system, frequency analysis of signals and systems is a central topic of signal processing, in using Fourier transformation intelligently,it can be very revealing, even though the signals to be analyzed are very complex. [8] The Discrete Fourier Transformation(DFT) and its inverse(IDFT) are defined by. DF T : X(k) =. N −1 ∑. x(n)e. −j2πkn N. K = 0, 1, 2, . . . , N − 1. (2.15). n=0. −1 j2πkn 1 N∑ IDF T : x(n) = X(k)e N n = 0, 1, 2, . . . , N − 1 N k=0. 立. 政 治 大. (2.16). The derivation of DFT and IDFT involve periodic extension of the signal sequence x(n), n =. ‧. ‧ 國. 學. 0, 1, 2, . . . , N − 1. 2.6 High frequency one day moving average. y. Nat. io. sit. In statistics, a moving average is a calculation that analyze data points by taking a. n. al. er. average of the full data set. If we view the price return as a particle displacement, by. i Un. v. taking a moving average, we can turn the zig-zag trajectory into smooth trajectory. Which. Ch. engchi. means we turn the stochastic scenario into the deterministic scenario.. Figure 2.2: The red line is the original data points and the blue line is the data with moving average. [Ma, Wang, Chen and Hu , 2013]. 9.
(26) In 2013, Ma, Wang, Chen and Hu, they found that the mean square log-returns calculated from US and Taiwan stocks. They show some crossover behavior can be well described by the mean square displacements of particles governed by the Langevin equation of motion [7]. The high frequency one-day moving averages (HF1MA) is one form of moving average in financial econometrics, but higher frequency. In financial econometrics moving averages are a ubiquitous tool, especially dominant in both technical analysis and high frequency trading. In this study, the length of the MA is one trading day. The reason we take the length as one trading day (6.5 hour) is that the length of the MA will influence the. 治 政 the smaller MA. And for our method, we collect the price大 for individual stocks, by taking 立 simple averages over a shifting window which is one-trading-day wide (23400 seconds, or slope of the mean square log return. The larger MA would show more diffusion-like than. ‧ 國. 學. 650 intervals).To retain the high-frequency feature of the data, the window shifts in steps of 36 seconds. By taking HF1MA, we can reveal some information and find out some. ‧. bulk structure inside the data. [7] [6]. n. er. io. sit. y. Nat. al. Ch. engchi. 10. i Un. v.
(27) Chapter 3 Data analysis 政 治 大. Our data source is from US stock SP500 in 1996-1999. We select 345 stocks, which. 立. has higher trading frequency in SP500. We build up correlation matrix and analyze the. ‧. ‧ 國. 學. eigenvalues, with the temporal part and stock part in K-L expansion.. 3.1 Definition of log-return. y. Nat. n. al. er. io. in time t is define as :. sit. Log-return is used to describe the price different in time series, the i th stock log-return. Ch. engchi. i Un. v. ri (t) = log(Pi (t + ∆t)) − log(Pi (t)). (3.1). = log(Pi (t) + ∆Pi (t)) − log(Pi (t)). (3.2). = log(. ∆Pi Pi (t) + ∆Pi (t) ) = log(1 + ) Pi (t) Pi. ≈. ∆Pi Pi. (3.3). (3.4). where ri (t) is the stock price log-return in time t, Pi (t) is the original stock price in time t In this study, log returns are calculated with overlapped time interval, as is sown in 11.
(28) figure 3.1 .. Figure 3.1: Log return with overlap.. 3.2 The correlation matrix and of Stock price log-return Variance: σi2 deviation:. t=1 (ri −. √∑. E(ri − < ri >T. )2. =. T t=1 (ri −. < r i >T ) 2 T. 學. ‧ 國. 立. < ri >T )2 T. 政 治 大. = E(ri − < ri >T ) =. √. σi =. ∑T. 2. (3.5). (3.6). ‧. Covariance:The covariance is a measure of how two random variables relate to each other in change, if the covariance is positive it means that, they tend to change together core-. y. Nat. sit. spondingly, if the covariance is negative it means that, they tend to change oppositely. We. er. io. can describe the covariance as:. al. n. iv n C Cov(ri , rj ) = E(r i − < ri >T )(rj − he n g c h i U< rj >T ) = E(ri rj ) − ri rj =. ∑T. t=1 (ri −. < ri >T )(rj − < rj >T ) T. (3.7). (3.8). If we normalize the covariance individually, we can compute the correlation function,which can describe as: Cij = ∑T. =. Cov(ri , rj ) E(ri rj ) − ri rj = σi σj σi σj. t=1 (ri −. < ri >T )(rj − < rj >T ) T σi σ j. 12. (3.9). (3.10).
(29) ∑T. = √∑. t=1 (ri −. T t=1 (ri −. < ri >T )(rj − < rj >T ). < ri >T )2. √∑ T. t=1 (rj −. (3.11). < rj >T )2. ri :the ith stock log-return price < ri >T :the ith stock average of log-return price in time T The properties of correlation coefficient: 1.−1 ≤ Cij ≤ 1 2.Cij = Cji , and Cii = 1 We can build up the N × N correlation matrix by the log-return correlation function as: . c 政· · · 治 .. 大 ··· . . c11 . .. . 1N. cN 1 · · · cN N. (3.12). . 學. ‧ 國. 立. . 3.3 The eigenvalue and eigenvector distribution of corre-. ‧. lation matrix. sit. y. Nat. er. io. We have provided the formula of distribution of eigenvalue of RMT in Eq. 2.7 .. al. n. iv n C Here we provided the formula from h the distribution ofU e n g c h i correlation coefficient, we consider the case of identical correlation coefficient matrix that all diagonal elements are 1 and the other elements equal to c. If the rank is 3 we can show by the following matrix . . 1 c c 1 . c c . (3.13). c c 1. If the rank of the matrix is N , we have the eigenvalue λM = 1 + (N − 1)C. 13. (3.14).
(30) and (N − 1)degenerate eigenvalue λ=1−C. (3.15). But if the correlation coefficient matrix is not identical, like as it is in stock correlation matrix. The distribution will be different with the identity correlation matrix. Figure 3.2 shows the distribution of eigenvalues for identical and for stock correlation matrix. The identical correlation matrix will have one largest eigenvalue λM and degenerate λ. As for the stock correlation matrix will have a continuous bulk structure and a discrete largest eigenvalue in the distribution of stock correlation matrix.. 立. 政 治 大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 3.2: The distribution of identity and stock correlation matrix’s eigenvalue We plot eigenvalues of correlation matrix from raw data(figure 3.3 ) or data with 14.
(31) HF1MA(figure 3.4) with different τ .(SP500 1996 Jan). 立. 政 治 大. Figure 3.3: The eigenvalue of correlation matrix from raw data with different τ .. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 3.4: The eigenvalue of correlation matrix from data with HF1MA with different τ . From figure. 3.3 and figure. 3.4 we can find out that the biggest eigenvalue of the data with HF1MA is always larger than the raw data for different τ . It’s because that taking the average of the shifting windows (HF1MA) would strength the connection between each stock.. 15.
(32) Distribution of Correlation elements for data with overlap Data from S&P 500 in 1996 January 8 τ=612 Sec. τ=3600 Sec. τ=18000 Sec. τ=36000 Sec. τ=54000 Sec. τ=72000 Sec.. 6. 4. 2. 0 -0.5. 0 Value of element. 0.5. 1. 政 治 大 Figure 3.5: Distribution of correlation’s elements, calculated with overlap. The whole 立 month’s length of data is 11700, due to our calculation method, the length of data in. ‧ 國. 學. different τ will be different. The length of the data for 612 Sec. is 11683. For 3600 Sec. is 11600. For 18000 Sec. is 11200. For 36000 Sec. is 10700. For 54000 Sec. is 10200. For 72000 Sec. is 9700. (1996 January SP 500). ‧. Distribution of Correlation elements for random number with overlap Data from random number,length 11700. 20. Nat. y. sit. n. al. er. io. 15. τ=612 Sec. τ=3600 Sec. τ=18000 Sec. τ=36000 Sec. τ=54000 Sec. τ=72000 Sec.. 10. Ch. engchi. i Un. v. 5. 0. -0.5. 0 Value of element. 0.5. 1. Figure 3.6: Distribution of correlation’s elements, calculated with overlapping. The length of data is 11700, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 11683. For 3600 Sec. is 11600. For 18000 Sec. is 11200. For 36000 Sec. is 10700. For 54000 Sec. is 10200. For 72000 Sec. is 9700. (Random case) Comparing figure. 3.5 and figure. 3.6, we can easily find the difference between them. For random case, the mean value approach to 0. As for figure. 3.5 it shifts to right while 16.
(33) the τ grows. The length of random case is same as January.. Eigenvector mode 345 Raw data. τ =3600 Sec τ=36000 Sec τ=72000 Sec. 0.15. Eigenvector. 0.1. 0.05. 0. -0.05. 立. 0. 100. 政 治 大 200 Stock. 300. 400. ‧ 國. 學 ‧. Figure 3.7: The raw data correlation matrix eigenvector for the biggest eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan). 0.1 Eigenvector. sit. n. al. τ=3600 Sec τ=36000 Sec τ=72000 Sec. er. io. 0.15. y. Nat. Eigenvector mode 345 HF1MA. Ch. engchi. i Un. v. 0.05. 0. -0.05. 0. 100. 200 Stock. 300. 400. Figure 3.8: The data with HF1MA correlation matrix eigenvector for the biggest eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan). From Figure 3.7 and Figure 3.8 we can easily find out that the correlation matrix’s 17.
(34) eigenvector for the biggest eigenvalue towards the same direction, just like all eigenvectors are positive or all eigenvectors are negative,the effects cause in the eigenvectors of those eigenvalues isolated from the bulk,corresponding to the correlation among all stocks, which we call this phenomenon as market mode.. For comparing the market mode with the other mode, we randomly choose the mode 144 for comparison.. 政 治 大. Eigenvector mode 144 Raw data. 立. 0.4. τ=3600 Sec τ=36000 Sec τ=72000 Sec. ‧ 國. 學. 0. ‧. Eigenvector. 0.2. -0.2. sit. y. Nat. io. al. n. -0.6. er. -0.4. 0. Ch 100. 200 Stock. engchi. i Un. v. 300. 400. Figure 3.9: The raw data’s correlation matrix eigenvector for the 144th eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan). 18.
(35) Eigenvector mode144 HF1MA 0.2 τ=3600 Sec τ=36000 Sec τ=72000 Sec. Eigenvector. 0.1. 0. -0.1. -0.2. 0. 100. 200 Stock. 300. 400. 政 治 大. Figure 3.10: The data with HF1MA correlation matrix eigenvector for the 144th eigenvalue. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700.(SP500 1996 Jan). 立. ‧ 國. 學 ‧. From Figure 3.10 and Figure 3.9 we can find that the eigenvector does not toward the same way, it goes up and down with particular frequency, the frequency we will discus in. er. io. sit. y. Nat. the following section.. n. al v 3.4 The Karhunan-Loeve n i temporal part C expansion for hengchi U. The Karhunan-Loeve expansion of the time series {si (t, τ )} for N stocks. si (t, τ ) =. N √ ∑. T λk ak (i)bk (t),. (3.16). k=1. has N modes, where λk and {ak (i)} are the eigenvalue and its eigenvector for the mode k obtained by diagonalizing the correlation matrix . c11 . .. . cN 1. . . . · · · c1N s1 (1) · · · s1 (T ) s1 (1) · · · sN (1) . . 1 .. .. .. . . = ··· . . ··· . . ··· . T · · · cN N sN (1) · · · sN (T ) s1 (T ) · · · sN (T ). 19. (3.17).
(36) and time-wise part {bk (t)} is obtained by performing . . . · · · b1 (T ) 1 .. ··· . = √ T λk bN (1) · · · bN (T ). b1 (1) . .. . . a1 (1) . .. . . · · · a1 (N ) s1 (1) · · · s1 (T ) . .. .. . ··· . . ··· . . . aN (1) · · · aN (N ). . sN (1) · · · sN (T ). (3.18). To fully utilize the data sequences length T of bk (t) depends on τ . It can be describe by. 政 治 大. T = l(data) − τ /36. 立. (3.19). where T and l(data) represent the length of each data sequence.. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 20. i Un. v.
(37) Time part Mode1 with HF1MA from Karhunan-Loeve expansion. τ=3600 Sec τ=36000 Sec τ=72000 Sec. 0.04. b1 (t). 0.02. 0. -0.02. -0.04. 0. 5000. 10000 Time. 15000. 20000. Time part Mode144 with HF1MA from Karhunan-Loeve expansion 0.04. 立. 0.02. τ=3600 Sec τ=36000 Sec τ=72000 Sec. ‧ 國. 學. b144 (t). 政 治 大. 0. ‧. -0.02. 0. 5000. io. 10000 Time. 15000. sit. y. Nat. -0.04. 20000. n. al. er. Time part Mode345 with HF1MA from Karhunan-Loeve expansion 0.03. 0.02. Ch. engchi. i Un. v. τ=3600 Sec τ=36000 Sec τ=72000 Sec. b345 (t). 0.01. 0. -0.01. -0.02. -0.03. 0. 5000. 10000 Time. 15000. 20000. Figure 3.11: The Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA. The length of data for 3600 second is 10950, for 36000 second is 10050, and for 72000 second is 9050. (SP500 1996 Jan). 21.
(38) Time part Mode1 raw data from Karhunan-Loeve expansion. τ=3600 Sec τ=36000 Sec τ=72000 Sec. 0.04. b1 (t). 0.02. 0. -0.02. -0.04 0. 5000. 10000 Time. 15000. 20000. Time part Mode144 raw data from Karhunan-Loeve expansion. 政 治 大. 立. 0.04. ‧ 國. b144 (t). 學. 0.02. τ=3600 Sec τ=36000 Sec τ=72000 Sec. 0. ‧. -0.02. al. 10000 Time. sit. 5000. 15000. 20000. er. io. 0. y. Nat. -0.04. n. Time part Mode345 raw data from Karhunan-Loeve expansion. Ch. 0.02. engchi. i Un. v. τ=3600 Sec τ=36000 Sec τ=72000 Sec. b345 (t). 0. -0.02. -0.04. 0. 5000. 10000 Time. 15000. 20000. Figure 3.12: The Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data. The length of data for 3600 second is 11600, for 36000 second is 10700, and for 72000 second is 9700. (SP500 1996 Jan) 22.
(39) From the plots below we can easily separate into two part by significantly difference. For figure 3.12 and figure 3.11, we can tell that mode 345 (with the biggest eigenvalue) it has smaller frequency than other mode, no matter it is raw data or data with HF1MA. And the different between the data with HF1MA and the raw data for mode 345 (with the biggest eigenvalue) is that the HF1MA has more smooth curve, which is cause by the average of shifting windows, than the raw data. As the τ grows, the frequency of the bj (t) decrease. It suits for both raw data or data with HF1MA.. For figure 3.11, figure3.12, shows that they have much higher frequency than market. 治 政 to analyze them by the power spectra S(ω) under Fourier大 transform of the time-wise part 立 b (t) for the 345 modes in the next section.. mode. However, it’s hard to tell the different between them. As the result, we are going. k. ‧ 國. 學 ‧. 3.5 Fourier transform of the time-wise part from the Karhunan-. sit. y. Nat. Loeve expansion. n. al. er. io. In order to find out what’s in the figure 3.11 and figure 3.12, we are going to use the. i Un. v. Fourier transformation to expand the Karhunan-Loeve expansion temporal part bj (t) by the following equation. DF T : X(k) =. Ch. N −1 ∑. engchi. x(n)e. −j2πkn N. K = 0, 1, 2, . . . , N − 1. (3.20). n=0. IDF T : x(n) =. −1 j2πkn 1 N∑ X(k)e N n = 0, 1, 2, . . . , N − 1 N k=0. (3.21). And we sum the real part and imagine part as power spectra from Fourier transform by S(ω) =. √ Im2 + Re2. (3.22). First, we compare the Fourier transform of Karhunan-Loeve expansion temporal part. 23.
(40) for the raw data with the different τ in the mode 1(with the smallest eigenvalue) .. Discrete Fourier Transformation of Karhunen-Loeve expansion time part Raw data, Mode 1 τ= 3600 Sec 0.08. S(ω). 0.06. 0.04. 0.02. 0. 0. 1. 立. 政 治 大. 2. 3. ω. 4. 5. 6. ‧ 國. 學. Figure 3.13: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan). ‧. Discrete Fourier Transformation of Karhunen-Loeve expansion time part. 0.05. y. sit. n. al. er. io. 0.06. Raw data, Mode 1 τ= 36000 Sec. Nat. 0.07. S(ω). 0.04. Ch. engchi. i Un. v. 0.03. 0.02. 0.01. 0. 0. 1. 2. 3. ω. 4. 5. 6. Figure 3.14: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan). 24.
(41) Discrete Fourier Transformation of Karhunen-Loeve expansion time part Raw data, Mode 1 τ= 72000 Sec. 0.08. S(ω). 0.06. 0.04. 0.02. 0. 0. 1. 2. 3. ω. 4. 5. 6. Figure 3.15: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan). 立. 政 治 大. ‧ 國. 學. We can actually count the pack number while the τ is small, the pack number equals. ‧. to the τ /36. As the τ goes up, the pack number and the frequency also goes up. And we found out that there are some bulk structure inside the spectra, it will discuss in the next. y. Nat. n. er. io. al. sit. section using power spectra in log-log scale to analyze.. Ch. engchi. i Un. v. Second, we compare the Fourier transform of Karhunan-Loeve expansion temporal part for the data with HF1MA in different τ and in mode 1 (with the smallest eigenvalue).. 25.
(42) Discrete Fourier Transformation of Karhunen-Loeve expansion time part Data with HF1MA, Mode 1 τ= 3600 Sec 0.1. 0.08. S(ω). 0.06. 0.04. 0.02. 0. 0. 1. 2. 3. ω. 4. 5. 6. Figure 3.16: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 3600Sec .The length of data for 3600 second is 10950. (SP500 1996 Jan). 立. 政 治 大. Discrete Fourier Transformation of Karhunen-Loeve expansion time part. ‧ 國. 學. Data with HF1MA, Mode 1 τ= 36000 Sec. 0.2. ‧ y. sit. io. n. 0.05. 0. al. er. 0.1. Nat. S(ω). 0.15. 0. 1. Ch 2. engchi 3. ω. 4. i Un 5. v. 6. Figure 3.17: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan). 26.
(43) Discrete Fourier Transformation of Karhunen-Loeve expansion time part Data with HF1MA, Mode 1 τ= 72000 Sec 0.2. S(ω). 0.15. 0.1. 0.05. 0. 0. 1. 2. 3. ω. 4. 5. 6. Figure 3.18: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan). 立. 政 治 大. ‧ 國. 學. For the data with HF1MA, the difference between different τ is just same as the result. ‧. of the Fourier transform of Karhunan-Loeve expansion temporal part with raw data, as the τ goes up, the pack number and the frequency also goes up. But they show different fea-. y. Nat. sit. tures between data with HF1MA and raw data. While the data with HF1MA, the spectra. n. al. er. io. will concentrate to the both end sides, it’s because that taking the average of the shifting. i Un. v. windows (HF1MA) would eliminate the noise from the time series .. Ch. engchi. We are wondering that, what will be in the market mode, so we are going to do the same comparison as below, with the market mode (with the biggest eigenvalue) .. 27.
(44) Discrete Fourier Transformation of Karhunen-Loeve expansion time part Raw data, Mode 345 τ= 3600 Sec 0.2. 0.15 0.15. 0.1. S(ω). 0.05 0.1 0. 0. 0.2. 0.4. 0.6. 0.05. 0. 0. 1. 2. 3. w. 4. 5. 6. 政 治 大. Figure 3.19: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan). 立. ‧ 國. 學. Discrete Fourier Transformation of Karhunen-Loeve expansion time part Raw data, Mode 345 τ= 36000 Sec. 0.5. ‧. 0.5. 0.4. Nat 0.1. sit. al. n. 0.3. er. 0.2. io. S(ω). y. 0.3. 0.4. 0. -0.02. 0. 0.2. Ch. 0.02. 0.04. 0.06. engchi. i Un 0.08. v. 0.1. 0. 0. 1. 2. 3. ω. 4. 5. 6. Figure 3.20: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan). 28.
(45) Discrete Fourier Transformation of Karhunen-Loeve expansion time part Raw data, Mode 345 τ= 72000 Sec. 0.5 0.6 0.4 0.5. 0.3 0.2. S(ω). 0.4 0.1 0.3. 0. 0. 0.01. 0.02. 0.03. 0.2. 0.1. 0. 0. 1. 2. 3. ω. 4. 5. 6. 政 治 大. Figure 3.21: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan). 立. ‧ 國. 學. Discrete Fourier Transformation of Karhunen-Loeve expansion time part 0.5. Data with HF1MA, Mode 345 τ= 3600 Sec. ‧. 0.6. 0.5. Nat. 0.4. sit. y. 0.4. 0.1 0 0.2. al. n. S(ω). 0.2. io. 0.3. er. 0.3. 0. Ch. 0.02. 0.04. 0.06. 0.08. engchi. i Un 0.1. v. 0.1. 0. 0. 1. 2. 3. ω. 4. 5. 6. Figure 3.22: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950.(SP500 1996 Jan). 29.
(46) Discrete Fourier Transformation of Karhunen-Loeve expansion time part Data with HF1MA, Mode 345 τ= 36000 Sec 0.6 0.5. 0.6. 0.4 0.5 0.3 0.2. S(ω). 0.4. 0.1 0.3 0. 0. 0.05. 0.1. 0.15. 0.2. 0.1. 0. 0. 1. 2. 3. 4. ω. 5. 6. 政 治 大. Figure 3.23: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 36000Sec.The length of data for 36000 second is 10050.(SP500 1996 Jan). 學. ‧ 國. 立. Discrete Fourier Transformation of Karhunen-Loeve expansion time part Data with HF1MA, Mode 345 τ= 72000 Sec. 0.6. ‧. 0.6. 0.5. 0.4. y. Nat. 0.5. 0.1. al. n. 0.3. er. 0.2. io. S(ω). 0.4. sit. 0.3. 0. 0.2. 0.02. Ch. 0.1. 0. 0. 1. 2. 0.04. ω. v. 0.06. engchi 3. i Un. 4. 5. 6. Figure 3.24: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan). For market mode (with the biggest eigenvalue), we observed that not only the spectra will concentrate to the both end sides, it almost become a spectra with ultimate peak in both end sides and almost zero for other place.. In comparison with market mode and other mode, the market mode has the same phe30.
(47) nomenon with the other mode while the τ is small, but for larger τ the temporal part of the Karhunan-Loeve expansion from market mode become a different spectra that has a ultimate peak in both end and almost zero for other place. To understand the phenomenon observed from power spectra, we make a log-log scale in power spectra in the next section.. 3.6 Power spectra from log-log scale discrete Fourier transformation. 政 治 大 end in market mode, so we try to dig out the information in log-log scale. 立. In last section, we observed the bulk structure in small mode, and ultimate peak in both. ‧ 國. 學. Discrete Fourier Transformation of Karhunen-Loeve expansion time part. ‧. Raw data, Mode 345 τ= 72000 Sec. S(ω). n. er. io. sit. y. Nat. al. Ch. 0.0001. 0.001. engchi. 0.01. ω. 0.1. i Un. v. 1. Figure 3.25: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700. (SP500 1996 Jan). 31.
(48) Discrete Fourier Transformation of Karhunen-Loeve expansion time part. S(ω). Data with HF1MA, Mode 345 τ= 72000 Sec. 0.0001. 0.001. 0.01. ω. 0.1. 1. Figure 3.26: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan). 學. ‧ 國. 立. 政 治 大. Discrete Fourier Transformation of Karhunen-Loeve expansion time part Raw data, Mode 345 τ= 36000 Sec. ‧ sit. n. al. er. io. S(ω). y. Nat 0.0001. 0.001. Ch. engchi. 0.01. ω. 0.1. i Un. v. 1. Figure 3.27: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan). 32.
(49) Discrete Fourier Transformation of Karhunen-Loeve expansion time part. S(ω). Data with HF1MA, Mode 345 τ= 36000 Sec. 0.0001. 0.001. 0.01. ω. 0.1. 1. Figure 3.28: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan). 立. 政 治 大. ‧ 國. 學. The power spectra S(ω) of the market modes(with the biggest eigenvalue) in log-log. ‧. scale for the largest τ considered in this study (τ = 72000 s). With or without the manipulation of taking moving averages(HF1MA) are shown in figure. 3.26 and figure. 3.25.. y. Nat. sit. The spectra share the same feature that they are straight line with a slope equal to −1 in the. n. al. er. io. log-log plot. And the different between with or without the manipulation of taking mov-. i Un. v. ing averages(HF1MA) is that data with HF1MA has less noise than raw data. The noise. Ch. engchi. in data with HF1MA has been eliminated by taking the average of the shifting windows. And for the raw data, the noise is still standing there. With or without the manipulation of taking moving averages(HF1MA) share the same feature, but the raw data has larger variation at high frequency regime. Same result is also seen in figure. 3.28 and figure. 3.27 .. 33.
(50) Discrete Fourier Transformation of Karhunen-Loeve expansion time part. S(ω). Data with HF1MA, Mode 1 τ= 72000 Sec. 0.0001. 0.001. 0.01. ω. 0.1. 1. Figure 3.29: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan). 立. 政 治 大. Discrete Fourier Transformation of Karhunen-Loeve expansion time part. ‧. ‧ 國. 學. Raw data, Mode 1 τ= 72000 Sec. y. sit. io. n. er. S(ω). Nat. al. 0.0001. 0.001. Ch 0.01. engchi ω. 0.1. i Un. v. 1. Figure 3.30: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan). 34.
(51) Discrete Fourier Transformation of Karhunen-Loeve expansion time part. S(ω). Data with HF1MA, Mode 1 τ= 36000 Sec. 0.0001. 0.001. 0.01. ω. 0.1. 1. Figure 3.31: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan). 立. 政 治 大. Discrete Fourier Transformation of Karhunen-Loeve expansion time part. ‧. ‧ 國. 學. Raw data, Mode 1 τ= 36000 Sec. y. sit. io. n. er. S(ω). Nat. al. 0.0001. 0.001. Ch 0.01. engchi ω. 0.1. i Un. v. 1. Figure 3.32: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan). As for the power spectra S(ω) of the first modes (with the smallest eigenvalue) in loglog plot, still retained some features in market modes,like they have straight line with a slope equal to −1 within the bulk of major continuous band, except the presence of a peak at low frequency regime.. 35.
(52) Discrete Fourier Transformation of Karhunen-Loeve expansion time part. S(ω). Raw data, Mode 345 τ= 3600 Sec. 0.0001. 政 治 大 Figure 3.33: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with 立 the biggest eigenvalue) for the raw data with τ = 3600Sec. 0.001. 0.01. ω. 0.1. 1. ‧ 國. 學. The length of data for 3600 second is 11600.(SP500 1996 Jan). ‧. Discrete Fourier Transformation of Karhunen-Loeve expansion time part Data with HF1MA, Mode 345 τ= 3600 Sec. S(ω). n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 0.0001. 0.001. 0.01. ω. 0.1. 1. Figure 3.34: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950. (SP500 1996 Jan). 36.
(53) Discrete Fourier Transformation of Karhunen-Loeve expansion time part. S(ω). Raw data, Mode 1 τ= 3600 Sec. 0.0001. 0.001. 0.01. 0.1. ω. 1. Figure 3.35: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan). 政 治 大. 立. Discrete Fourier Transformation of Karhunen-Loeve expansion time part. ‧. ‧ 國. 學. Data with HF1MA, Mode 1 τ= 3600 Sec. y. sit. io. n. al. er. b j(ω). Nat 0.0001. Ch. 0.01. engchi ω. 0.1. i Un. v. 1. Figure 3.36: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950.(SP500 1996 Jan). We found that smaller scale variations in S(ω) are in fact well-organized when we examine the spectra for smaller τ . Figures 3.33 and figures. 3.34 include the spectra for the market mode. And typical continuous band modes in figures.3.35 and 3.36.. We can see well recognized regular packs, going along with the exponent −1 feature 37.
(54) in the log-log scale power spectra whether it is in the market modes or in the other modes. Such characteristics provide evidences as the presence of intermittent fluctuations.. 3.7 Analysis in long time data 3.7.1 Calculation with overlapping To confirm the phenomenon we have observed, we use the whole year data to check it again. First, we check the eigenvalue of the cross correlation matrix. Here is the eigenvalue of the cross correlation matrix from 1996 whole year data.. 立. 政 治 大 Data from 1996 with overlap Raw data. 2000. ‧ 國. 學. 1900 1800 1700 1600 1500. ‧. 1400 1300 1200. Nat. y. 1100 1000. sit. τ. 900. io. 700 600. n. al. er. 800. 500 400 300 200 100. Ch. engchi. i Un. v. 0. 0. 10. 20. 30 eigenvalue. 40. 50. 60. Figure 3.37: The eigenvalue of correlation matrix from raw data in different τ (SP500 1996 full year). 38.
(55) Data from 1996 with overlap with HF1MA 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100. τ 1000 900 800 700 600 500 400 300 200 100 0. 0. 10. 20. 30 40 Eigenvalue. 50. 60. 70. 政 治 大. Figure 3.38: The eigenvalue of correlation matrix from data with HF1MA in different τ . (SP500 1996 full year). 立. ‧ 國. 學. From figure 3.37 and figure 3.38 we can find out that the biggest eigenvalue of the. ‧. data with HF1MA is always larger than the raw data for different τ . The result is same as. sit. y. Nat. the calculation we have done before. The reason is that taking the average of the shifting. io. er. windows (HF1MA) would strength the connection between each stock. The reason that the correlation matrix does not have the discrete largest eigenvalue is probably caused by. n. al. Ch. i Un. v. the length of the data and causing the mean of the correlation matrix elements close to. engchi. zero. It will be more clearly verified by the following figure. 39.
(56) Distribution of Correlation matrix elements for data with overlap Data from S&P 500 in 1996 12 τ=612 Sec. τ=3600 Sec. τ=18000 Sec. τ=36000 Sec. τ=54000 Sec. τ=72000 Sec.. 10. 8. 6. 4. 2. 立 -0.5. 0 Value of element. 0.5. 學. 1. ‧. ‧ 國. 0. 政 治 大. y. Nat. sit. n. al. er. io. Figure 3.39: Distribution of correlation’s elements, calculated with overlap. The whole year’s length of data is 162500, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 162583. For 3600 Sec. is 162400. For 18000 Sec. is 162000. For 36000 Sec. is 161500. For 54000 Sec. is 161000. For 72000 Sec. is 160500. (1996 SP 500 full year). Ch. engchi. 40. i Un. v.
(57) Distribution of Correlation elements for random number with overlap Data from random number,length 162500 50 τ=612 Sec. τ=3600 Sec. τ=18000 Sec. τ=36000 Sec. τ=54000 Sec. τ=72000 Sec.. 40. 30. 20. 10. 立-0.2. -0.4. 0 0.2 Value of element. 0.4. 0.6. 學 ‧. ‧ 國. 0 -0.6. 政 治 大. n. al. er. io. sit. y. Nat. Figure 3.40: Distribution of correlation’s elements, calculated with overlap. The length of data is 162500, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 162583. For 3600 Sec. is 162400. For 18000 Sec. is 162000. For 36000 Sec. is 161500. For 54000 Sec. is 161000. For 72000 Sec. is 160500. (Random case). Ch. engchi. 41. i Un. v.
(58) Variance of correlation matrix element vs Tau 1996 S&P 500 , with overlapping 2000. 1500. τ 1000. 500. 立 0. 0.02. 0.04. 0.06. 0.08. 學. ‧ 國. 0. 政 治 大 Variance. ‧. n. al. er. io. sit. y. Nat. Figure 3.41: The variance of correlation matrix element versus the τ , calculated with overlap. The length of the data for 612 Sec. is 12983. For 3600 Sec. is 12900. For 18000 Sec. is 12500. For 36000 Sec. is 12000. For 54000 Sec. is 11500. For 72000 Sec. is 11000. (1996 SP 500 full year). Ch. engchi. 42. i Un. v. 0.1.
(59) Correlation matrix’s largest eigenvalue vs variance with overlap , 1996 S&P 500 60. 50. 40. λM 30. 20. 10. 立 0.02. 0.04. 0.06. 學. 0. Variance. 0.08. 0.1. ‧. ‧ 國. 0. 政 治 大. Nat. y. sit. n. al. er. io. Figure 3.42: The variance of correlation matrix element, versus the largest eigenvalue of correlation matrix, calculated with overlap. The length of the data for 612 Sec. is 12983. For 3600 Sec. is 12900. For 18000 Sec. is 12500. For 36000 Sec. is 12000. For 54000 Sec. is 11500. For 72000 Sec. is 11000. (1996 SP 500 full year). Ch. engchi. i Un. v. We can find some difference between the whole year data and monthly data in the distribution of correlation’s elements in figure 3.39 and figure 3.5. In figure 3.39 and figure 3.41, we can observed that as the τ grows, the variance of the distribution will grow too. The feature of figure. 3.39 is similar to figture. 3.40, but for figure 3.39 it’s mean value is not 0, it shift right from the zero line. The reason that variation grows with τ and λM is due to the random feature. In random case, while the τ grows the variation will grows, too. The correlation matrix element can be describe by Cij = C¯ + ηij σ 2 + b(τ ). (3.23). Where µ represent the means, η is a small constant, σ 2 represent the variance. b(τ ) is some. 43.
(60) factor related to τ . Therefore, we compare the largest eigenvalue from the real market data and the largest eigenvalue compute from random symmetric matrix. We can compute the largest eigenvalue from random symmetric matrix as λM = (N − 1)C¯ + 1 + σ 2 /C¯. (3.24). 治 政 ¯ where n represent the rank of the matrix, C represent 大 the means and σ 立 variance. [12] ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 44. i Un. v. 2. represent the.
(61) The largest eigenvalue from real data and symmetric matrix Real market data with overlap, 1996. 50. 40. λM. 30. 20. 立. 10. 學. 5. ‧ 國. 10. 政 治 大 15. λsymmetric. ‧ y. Nat. sit. n. al. er. io. Figure 3.43: The largest eigenvalue compute from real market data and random symmetric matrix.The real market data is calculated with overlapping. (1996 SP 500) The red line represent the slope =1 which means the largest eigenvalue compute from random symmetric matrix equal to the largest eigenvalue compute from the real market data.. Ch. engchi. i Un. v. From figure 3.42 we can know that the variance relates to the largest eigenvalue, but the largest eigenvalue compute from random symmetric matrix does not equal to the the largest eigenvalue compute from the real market data, moreover Eq.3.24 do not evaluated when C¯ ≈ 0. It might be that the random symmetric matrix does not capture the Wishart matrix’s structure. There may be some factor relate to τ behind it. Second, we check the stock part (eigenvector) from 1996 whole year data. We show the modes with the top four eigenvalue and the mode with the smallest eigenvalue in different τ .. 45.
(62) Eigenvector for typical mode,Raw, τ=3600 sec λM =12.01. 0.05 0 -0.05 -0.1 -0.15 -0.2 0. 200. 100. 300. 400. 300. 400. λ2=11.39 0.2 0.1 0 -0.1 -0.2. Eigenvector. 0. 200. 100. λ3=9.74 0.15 0.1 0.05 0 -0.05 0. 0.2 0.1 0. 立. -0.1 -0.2 0. 0 -0.1 -0.2. 400. 200. 300. 400. λmin=0.13. ‧. 0. 300. 政 治 大. 100. ‧ 國. 0.1. λ4=8.79. 學. 0.2. 200. 100. 200. 100. 300. Stock. 400. Nat. n. al. er. io. sit. y. Figure 3.44: The eigenvector of correlation matrix from raw data in typical modes. The length of data for 3600 second is 11600. (SP500 1996 full year). Ch. engchi. 46. i Un. v.
(63) Eigenvector for typical mode,HF1MA, τ=3600 sec λM =25.32. 0 -0.1 -0.2 0. 200. 100. 300. 400. 300. 400. λ2=22.64 0.2 0.1 0 -0.1 -0.2. Eigenvector. 0. 200. 100. λ3=21.11 0.2 0.1 0 -0.1 -0.2 0. 0.2 0.1 0. 立. -0.1 -0.2 0. 0 -0.1 0. 400. 200. 300. 400. λmin=1.45 E-004. 200. 100. Stock. ‧. -0.2. 300. 政 治 大. 100. ‧ 國. 0.1. λ4=19.61. 學. 0.2. 200. 100. Nat. y. 300. er. io. sit. Figure 3.45: The eigenvector of correlation matrix from data with HF1MA in typical modes. The length of data for 3600 second is 10950. (SP500 1996 full year). al. n. iv n C From figure 3.44 and figure 3.45, the U the same features in market h estock n gpart c hstilli remain mode, but some points in market mode cross the zero line. It is a little different from calculation in 1996 Jan. .. 47. 400.
(64) Eigenvector for typical mode,Raw, τ=72000 sec λM =54.87. 0.15 0.1 0.05 0 -0.05 -0.1 0. 200. 100. 300. 400. 300. 400. λ2=45.10 0.2 0.1 0 -0.1 -0.2. Eigenvector. 0. 200. 100. λ3=41.64 0.2 0.1 0 -0.1 -0.2 0. 0.2 0.1 0 -0.1. 立. -0.2 0. 0 -0.1 0. 400. 200. 300. 400. λmin=7.64 E-003. ‧. -0.2. 300. 政 治 大. 100. ‧ 國. 0.1. λ4=39.17. 學. 0.3 0.2. 200. 100. 200. 100. 300. T. 400. Nat. n. al. er. io. sit. y. Figure 3.46: The eigenvector of correlation matrix from raw data in typical modes. (SP500 1996). Ch. engchi. 48. i Un. v.
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In section 4, based on the cases of circular cone eigenvalue optimization problems, we study the corresponding properties of the solutions for p-order cone eigenvalue
volume suppressed mass: (TeV) 2 /M P ∼ 10 −4 eV → mm range can be experimentally tested for any number of extra dimensions - Light U(1) gauge bosons: no derivative couplings. =>
According to the Heisenberg uncertainty principle, if the observed region has size L, an estimate of an individual Fourier mode with wavevector q will be a weighted average of
incapable to extract any quantities from QCD, nor to tackle the most interesting physics, namely, the spontaneously chiral symmetry breaking and the color confinement..
• Formation of massive primordial stars as origin of objects in the early universe. • Supernova explosions might be visible to the most
Theorem 5.6.1 The qd-algorithm converges for irreducible, symmetric positive definite tridiagonal matrices.. It is necessary to show that q i are in
![TraditionalMLCalgorithmsmainlytacklethebatchMLCproblem,wheretheinputdataarepresentedinabatch[24,28].Nevertheless,inmanyMLCapplicationssuchase-mailcategorization[22],multi-labelexamplesarriveasastream.Onlineanalysisistherefore dimensionreducermotivatedbyma](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)