扩散熵和概率流的经验模式分解模型
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摘要
标度不变性对于复杂系统的研究越来越受到学者们的重视,而近年来测定复杂动态时间序列的标度化指数主要都是一些基于方差计算的方法,这些方法可以很好的测定那些方差有限的序列,而对于方差无限或者不存在的序列却无能为力。然而在很多复杂系统中的时间序列都可能有无限方差或是方差不存在的情况,本文将主要致力于基于时间序列产生的扩散过程熵的统计分析方法和基于累积概率的统计方法进行探讨和研究。这两种方法都可以很好的测定方差无限或者不存在的序列。在本文中,首先对于扩散熵分析法(DEA)的盒子过程我们讨论了其算法的改进,然后讨论了扩散熵分析法与概率流分析法(PFA)在高斯模型和勒维模型中的差异。之后尝试使用经验模式分解(EMD)与这两种方法结合,再将结合后的方法应用在高斯模型和勒维模型中。比较结合前后两种方法在两种模型中的差异,讨论其特性。将研究得出的结果应用于实际的股票数据中,研究股票指数的标度特性,从而进一步研究股票指数的统计特性。
The study of scale invariance has been payed more and more attention to for complex systems. The methods currently used to determine the scaling exponent of a complex dynamic time series are based on numerical evaluation of variance. That means all this methods can be safely applied to the case where the variances are finite but fail to determine the scaling exponent when the variances are infinite or don't exit. Most time series of the complex dynamic system have infinite variances or don't have variances. In this paper we study a statistic method based on the Shannon entropy of the diffusion process generated by a time series and a statistic method based on the cumulative probabilities. Both methods can well determine the scaling exponent of the series with infinite variances or without variances. In this paper, first we will improve the binning algorithm of the diffusion entropy analysis. And then we will discuss the distinction between diffusion entropy analysis and probability flux analysis applied to Guass model and Levy model. We then try to combine empirical mode decompose and the two methods separately. After the combination we apply the two methods to Guass model and Levy model. We study the differences between before and after the combination. We use the conclusions to study the scaling properties and statistic properties of the stocks index datas.
The study of scale invariance has been payed more and more attention to for complex systems. The methods currently used to determine the scaling exponent of a complex dynamic time series are based on numerical evaluation of variance. That means all this methods can be safely applied to the case where the variances are finite but fail to determine the scaling exponent when the variances are infinite or don't exit. Most time series of the complex dynamic system have infinite variances or don't have variances. In this paper we study a statistic method based on the Shannon entropy of the diffusion process generated by a time series and a statistic method based on the cumulative probabilities. Both methods can well determine the scaling exponent of the series with infinite variances or without variances. In this paper, first we will improve the binning algorithm of the diffusion entropy analysis. And then we will discuss the distinction between diffusion entropy analysis and probability flux analysis applied to Guass model and Levy model. We then try to combine empirical mode decompose and the two methods separately. After the combination we apply the two methods to Guass model and Levy model. We study the differences between before and after the combination. We use the conclusions to study the scaling properties and statistic properties of the stocks index datas.
引文
[1]H.E. Stanley, L.A.N. Amaral, P. Gopikrishnan, P.Ch. Ivanov, T.H. Keitt, and V. Plerou. Scale invariance and universality:organizing principles in complex systems[J]. Physica A 281,60 (2000).
[2]Amir Bashan, Ronny Bartsch, Jan W Kantelhardt, et al. Comparison of detrending methods for fluctuation analysis[J]. Physica A,2008,387:5080-5090.
[3]Jan W Kantelhardt, Stephan A Zschiegner, Eva Koscielny-Bunde, et al. Multifractal detrended fluctuation analysis of nonstationary time series[J]. Physica A,2002,316:87-114.
[4]Kun Hu, Plamen Ch Ivanov, Zhi Chen, et al. Effect of trends on detrended fluctuation analysis[J].
[5]N. Scafetta, P. Hamilton, and P. Grigolini, The thermodynamics of social processes:The teen birth phenomenon[J]. Fractals 9,193 (2001).
[6]P. Grigolini, L. Palatella, and G. Raffaelli, Asymmetric anomalous diffusion:an efficient way to detect memory in time series[J]. Fractals 9,439 (2001).
[7]B.B. Mandelbrot, Fractals; form, chance and dimension[M]. San Francisco:W.H. Freeman and Co.,1977.
[8]刘增东,刘建国,陆亦怀,等.基于EMD的激光雷达信号去噪方法[J].光电工程,2008,35(6)：79-83.
[9]R Balocchi, D Menicucci, E Santarcangelo, et al. Deriving the respiratory sinus arrhythmia from the heartbeat time series using Empirical Mode Decomposition[J]. Preprint submitted to Elsevier Science,2008.
[10]Louis Yu Lu. Fast Intrinsic Mode Decomposition and Filtering of Time Series Data.
[11]曹志广,王安兴,杨军敏.股票收益率非正态性硇蒙特卡骂模拟检验[J].财经研究,2005(31).
[12]熊焰,赵铁山.金融资产收益率分布的混合高斯分布模型[J].理论新探,2004(12).
[13]N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group[M]. (Perseus, Reading, MA,1985).
[14]C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger. Mosaic organization of DNA nucleotides. Phys. Rev. E 49,1685 (1994).
[15]J. Feder, Fractals[M]. New York:Plenum Press,1988.
[16]M. Schroeder, Fractals, chaos, power laws[M]. San Francisco:W.H. Freeman,1991.
[17]A.I. Khinchin, Mathematical Foundations of Statistical Mechanics[M]. New York:Dover Publications, Inc.,1949.
[18]B.B. Mandelbrot, The Fractal Geometry of Nature[M]. New York:Freeman,1983.
[19]J. Klafter, M.F. Shlesinger, and G. Zumofen, Beyond brownian motion[J]. Phys. Today 49(2),33 (1996).
[20]M.F. Shlesinger, J. Klafter and B.J. West, Levy dynamics of enhanced diffusion:Application to turbulence[J]. Phys. Rev. Lett.58,1100-03 (1987).
[21]M.F. Shlesinger, Nature (London) 411,641 (2001).
[22]B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sum of Independent Random Variables (Addison-Wesley, Reading, MA,1954).
[23]T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev. Lett.54,616 (1985).
[24]M.F. Shlesinger and J. Klafter, Phys. Rev. Lett.54,2551 (1985).
[25]M.F. Shlesinger, B.J. West, and J. Klafter, Phys. Rev. Lett.58,1100 (1987).
[26]J.B. Bassingthwaighte, L.S. Liebovitch and B.J. West, Fractal Physiology[M]. Oxford: Oxford University Press,1994.
[27]C.E. Shannon, Bell Sys. Tech. J.27,379-423; ibid 623-656 (1948).
[28]N. Wiener, Time Series[M], Cambridge:MIT Press,1949.
[29]潘家柱,丁美春.GP分布模型与股票收益率分析[J].北京大学学报,2000(3).
[30]Mandelbrot B. New methods in Statistical Economics[J]. Journal of Political Economy,1963,71: 421-440.
[31]Hill B M. A Simple General Approach to Inference about the Tail of A distribution[J].1975(3).
[32]吴文峰,吴冲锋.股票价格波动模型探讨[J].系统工程理论与实践,2000(4).
[33]Flandrin P, Rilling G, Goncalves P. Empirical mode decomposition as a filter bank[J]. Signal Processing Letters,2004(2).
[34]Norden E Huang, Zheng Shen, Steven R Long. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[J]. The Royal Society,1998.
[35]蔡坤宝,王成良,陈曾汉.产生标准高斯白噪声序列的方法[J].中国机电工程学报,2004(24).
[36]蔡坤宝,罗汉文.产生高斯随机序列的新方法[J].上海交通大学学报,2004(38).
[37]Rosario N, Mantegna, H Eugene Stanley. Stochastic Process with Ultraslow Convergence to a Gaussian:The Truncated Levy Flight[J]. Phys. Rev. Lett.73,2946-2949 (1994).
[38]Hari M. Gupta, Jose R. Campanha. The gradually truncated Levy flight:stochastic process for complex systems[J]. Physica A,2007(275).
[39]庄新田,庄新路,田莹. Hurst指数及股市的分形结构[J].东北大学学报,2003(24).
[40]卢方元,中国股市收益率的多重分形分析[J].系统工程理论与实践,2004(6).
[2]Amir Bashan, Ronny Bartsch, Jan W Kantelhardt, et al. Comparison of detrending methods for fluctuation analysis[J]. Physica A,2008,387:5080-5090.
[3]Jan W Kantelhardt, Stephan A Zschiegner, Eva Koscielny-Bunde, et al. Multifractal detrended fluctuation analysis of nonstationary time series[J]. Physica A,2002,316:87-114.
[4]Kun Hu, Plamen Ch Ivanov, Zhi Chen, et al. Effect of trends on detrended fluctuation analysis[J].
[5]N. Scafetta, P. Hamilton, and P. Grigolini, The thermodynamics of social processes:The teen birth phenomenon[J]. Fractals 9,193 (2001).
[6]P. Grigolini, L. Palatella, and G. Raffaelli, Asymmetric anomalous diffusion:an efficient way to detect memory in time series[J]. Fractals 9,439 (2001).
[7]B.B. Mandelbrot, Fractals; form, chance and dimension[M]. San Francisco:W.H. Freeman and Co.,1977.
[8]刘增东,刘建国,陆亦怀,等.基于EMD的激光雷达信号去噪方法[J].光电工程,2008,35(6)：79-83.
[9]R Balocchi, D Menicucci, E Santarcangelo, et al. Deriving the respiratory sinus arrhythmia from the heartbeat time series using Empirical Mode Decomposition[J]. Preprint submitted to Elsevier Science,2008.
[10]Louis Yu Lu. Fast Intrinsic Mode Decomposition and Filtering of Time Series Data.
[11]曹志广,王安兴,杨军敏.股票收益率非正态性硇蒙特卡骂模拟检验[J].财经研究,2005(31).
[12]熊焰,赵铁山.金融资产收益率分布的混合高斯分布模型[J].理论新探,2004(12).
[13]N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group[M]. (Perseus, Reading, MA,1985).
[14]C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger. Mosaic organization of DNA nucleotides. Phys. Rev. E 49,1685 (1994).
[15]J. Feder, Fractals[M]. New York:Plenum Press,1988.
[16]M. Schroeder, Fractals, chaos, power laws[M]. San Francisco:W.H. Freeman,1991.
[17]A.I. Khinchin, Mathematical Foundations of Statistical Mechanics[M]. New York:Dover Publications, Inc.,1949.
[18]B.B. Mandelbrot, The Fractal Geometry of Nature[M]. New York:Freeman,1983.
[19]J. Klafter, M.F. Shlesinger, and G. Zumofen, Beyond brownian motion[J]. Phys. Today 49(2),33 (1996).
[20]M.F. Shlesinger, J. Klafter and B.J. West, Levy dynamics of enhanced diffusion:Application to turbulence[J]. Phys. Rev. Lett.58,1100-03 (1987).
[21]M.F. Shlesinger, Nature (London) 411,641 (2001).
[22]B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sum of Independent Random Variables (Addison-Wesley, Reading, MA,1954).
[23]T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev. Lett.54,616 (1985).
[24]M.F. Shlesinger and J. Klafter, Phys. Rev. Lett.54,2551 (1985).
[25]M.F. Shlesinger, B.J. West, and J. Klafter, Phys. Rev. Lett.58,1100 (1987).
[26]J.B. Bassingthwaighte, L.S. Liebovitch and B.J. West, Fractal Physiology[M]. Oxford: Oxford University Press,1994.
[27]C.E. Shannon, Bell Sys. Tech. J.27,379-423; ibid 623-656 (1948).
[28]N. Wiener, Time Series[M], Cambridge:MIT Press,1949.
[29]潘家柱,丁美春.GP分布模型与股票收益率分析[J].北京大学学报,2000(3).
[30]Mandelbrot B. New methods in Statistical Economics[J]. Journal of Political Economy,1963,71: 421-440.
[31]Hill B M. A Simple General Approach to Inference about the Tail of A distribution[J].1975(3).
[32]吴文峰,吴冲锋.股票价格波动模型探讨[J].系统工程理论与实践,2000(4).
[33]Flandrin P, Rilling G, Goncalves P. Empirical mode decomposition as a filter bank[J]. Signal Processing Letters,2004(2).
[34]Norden E Huang, Zheng Shen, Steven R Long. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[J]. The Royal Society,1998.
[35]蔡坤宝,王成良,陈曾汉.产生标准高斯白噪声序列的方法[J].中国机电工程学报,2004(24).
[36]蔡坤宝,罗汉文.产生高斯随机序列的新方法[J].上海交通大学学报,2004(38).
[37]Rosario N, Mantegna, H Eugene Stanley. Stochastic Process with Ultraslow Convergence to a Gaussian:The Truncated Levy Flight[J]. Phys. Rev. Lett.73,2946-2949 (1994).
[38]Hari M. Gupta, Jose R. Campanha. The gradually truncated Levy flight:stochastic process for complex systems[J]. Physica A,2007(275).
[39]庄新田,庄新路,田莹. Hurst指数及股市的分形结构[J].东北大学学报,2003(24).
[40]卢方元,中国股市收益率的多重分形分析[J].系统工程理论与实践,2004(6).