多重分形理论及其在中国股票市场中的应用研究
|
摘要
股票市场的价格波动问题是当前金融研究中的一项基本且重要的课题,能否对股票市场的演化规律和股票价格的运行机理进行较为准确的刻画直接关系到资产定价、风险控制、市场监管和价格预测等一系列金融市场的重大问题。在对多重分形理论进行系统分析及对多重分形方法进行改进和完善的基础上,对中国沪深股票市场进行了实证研究,其主要内容如下：
1.中国股票市场的异象性特征研究
对中国股市收益率的基本统计量、长记忆性、聚类特征、多标度特征及股票价格的可预测性等异象性特征进行分析研究,结果发现：
(1)中国股票市场的收益率序列不服从正态分布且具有明显的尖峰厚尾现象,并且与深成指数收益序列相比,上证指数收益序列具有更尖的峰和更厚的尾。
(2)中国股票市场是一个非有效的市场,其收益率序列具有较为明显的“长记忆”效应,它们遵循有偏的随机游走,市场未达到弱式有效。
(3)中国股票市场的股票价格具有聚类特征和多标度特性,用单一的标度指数不能对其进行全面细致的刻画,因此应该选择多标度分析模型对其进行描述。
(4)中国股票市场股价的涨落不是完全随机的,有一定的规律可循,股票价格是可以预测的,至少是短期可以预测的。
2.中国股票市场的多重分形特性研究
运用MF-DFA方法、多仿射方法及q阶矩结构分割函数等方法,并在此基础上用二次函数拟合多重分形谱来对中国股票市场股指收益序列及股指序列进行了多重分形分析,发现了中国股票市场的多重分形特性及规律。
(1)运用MF-DFA方法,对中国沪深两市股市收益序列的研究发现,股票市场存在明显的多重分形特征。进一步地,发现股票市场的多重分形特征是由两个因素共同作用的,其中收益序列的波动相关性起主导作用,是形成多重分形特征的主要原因,收益序列的胖尾概率分布对多重分形特征的形成也起到了一定的作用；而多重分形过程的本质原因是股票市场信息倍增级联过程的结果。
(2)沪深股市收益序列存在较为明显的标度突变点s*,该突变点将整个时间标度分为两个部分,每个部分具有不同的多重分形特征及标度指数。在ss*的情况下股指收益序列的多重分形强度较弱,多重分形特征不明显。
(3)运用多仿射方法对沪深两市收益序列的研究发现,股票市场存在多仿射现象；另外,研究结果发现广义Hurst指数H(1)与H(2)的值与市场的发展状态之间具有一定的敏感性,发达国家股票市场的H(1)值均较小,而且H(1)及H(2)值均小于0.5,而新兴股票市场的H(1)值均较大,而且H(1)及H(2)值均大于0.5。上海股票市场及深圳股票市场均属于新兴金融市场一类。
(4)运用q阶矩结构分割函数法对股价波动的多重分形特性进行实证研究,结果发现多重分形谱参数与指数的变化趋势及对数收益率具有明显的关联性,并且在股价发生大幅波动的情况下,多重分形谱参数α0,αmax,αmin,W和C均具有较明显的变化特征,这为研究股票市场的运行规律提供了有力的依据。
3.中国股票市场的价格预测研究
基于多重分形理论,对上证指数股指价格进行方向预测及非线性预测,得到了较好的预测效果：
(1)基于日收盘指数价差(ΔI)的符号序列方法与基于日多重分形谱参数(Δf)的符号序列方法都能以一定的概率预测指数的涨落,而且将两种符号序列方法结合可以更好地预测指数的大涨落；在引入股价指数大涨落的阈值和条件平均增益后,发现大幅涨落时,条件与指数变化的关联性比小的涨落要强得多。
(2)提出基于多重分形谱的神经网络模型,并将其应用于股票价格的短期预测,发现该模型对于模拟股市的短期走势具有较好的预测效果,对防范和控制风险具有现实意义。
Stock price fluctuation in stock markets is a very important issue in financial researches. The right description to the evolution law of stock markets and to the running mechanism of stock prices is related to assets pricing、risk controlling、markets supervision and price prediction. After systemically analyzed the multifractal theory and deepened the methods of it, an empirical research on the stock markets in China is presented. The main research work of this paper is as follows:
1. The abnormal characteristics of stock markets in China
Through studying the basic statistics、long-term memory、clustering property、multi-scaling property of the stock price index returns and predictability of stock price index, some conclusions can be drawn:
(1) The returns series of Shanghai and Shenzhen stock markets are disobedient normally distribution but has obvious peak kurtosis and fat tails; And the returns series of Shanghai stock market has peaker kurtosis and fatter tail than those of Shenzhen stock market.
(2) The stock markets of China are not efficient markets, and the returns series had obvious long-term memory, indicating that the stock market did not reached the soft efficiency.
(3) The clustering structure and multi-scaling behavior of the time series were revealed. It indicates that a single scale exponent is insufficient to describe the scaling properties of the returns series and a multifractal model may be more suitable for describing the time series.
(4) The variation of the stock price in China is not totally random, but predictable, or at least predictable in a short period of time.
2. The multifractal characteristics of stock markets in China
Using MF-DFA、multi-affine、qth-order moment structure partition function and a quadratic function fitting to study the returns series and the stock price index series in China, some conclusions can be drawn:
(1) The returns series of stock markets in China showed pronounced multifractal characteristics. Furthermore, the sources of multifractality are analyzed. It is found that there are two different types of sources for multifractality in time series, namely, fat-tailed probability distributions and nonlinear temporal correlations. Most multifractality of the series is due to different long-term correlations, but the fat-tailed probability distributions also contribute to the multifractal behavior of the time series.
(2) The returns series showed a crossover time scale. This crossover divides the whole time scale into two different scale domains. There are different multifractal characteristics and scale exponents for these two different domains. The strength of multifractality in time scales ss*.
(3)The stock markets showed multi-affine phenomenon. In addition, it is found that the generalized Hurst exponents H(1) and H(2) show remarkable differences between developed and emerging markets:Emerging markets are associated with high value of H(1) and developed markets are associated with low values of H(1). Besides, it is found that all the emerging markets have H(2)>0.5 whereas all the developed have H(2)≤0.5. The stock markets in China belong to the category of emerging financial markets.
(4) There are pronounced statistical correlations between the parameters of the multifractal spectra and the variation of the closing index, the logarithmic return and the gain probability. And when the stock price index fluctuates sharply, a strong variability is clearly characterized by the multifractal parameters (α0,αmax,αmin, W and C).
4. Forecasting research of stock price in China
Through directional prediction and nonlinear prediction to the stock price in China, some conclusions can be drawn:
(1) Both of two sign sequence methods have the forecasting capacity. One is the symbol sequence of difference of daily closing quotation indexes; the other is the symbol sequence of multifractal spectrum parameter of the high frequency data. The relation between the condition of different symbol sequences and changes of index in larger fluctuating of stock price index is stronger than the relation in smaller fluctuating of stock price index when threshold values are introduced. And the combining of these two methods has remarkable forecasting capacity.
(2) A neural network model based on the multifractal spectrum is advanced and is applied to the stock price forecasting. The test results indicated that the model can simulate stock market trends in a short time. It is useful to prevent and control risks.
1.中国股票市场的异象性特征研究
对中国股市收益率的基本统计量、长记忆性、聚类特征、多标度特征及股票价格的可预测性等异象性特征进行分析研究,结果发现：
(1)中国股票市场的收益率序列不服从正态分布且具有明显的尖峰厚尾现象,并且与深成指数收益序列相比,上证指数收益序列具有更尖的峰和更厚的尾。
(2)中国股票市场是一个非有效的市场,其收益率序列具有较为明显的“长记忆”效应,它们遵循有偏的随机游走,市场未达到弱式有效。
(3)中国股票市场的股票价格具有聚类特征和多标度特性,用单一的标度指数不能对其进行全面细致的刻画,因此应该选择多标度分析模型对其进行描述。
(4)中国股票市场股价的涨落不是完全随机的,有一定的规律可循,股票价格是可以预测的,至少是短期可以预测的。
2.中国股票市场的多重分形特性研究
运用MF-DFA方法、多仿射方法及q阶矩结构分割函数等方法,并在此基础上用二次函数拟合多重分形谱来对中国股票市场股指收益序列及股指序列进行了多重分形分析,发现了中国股票市场的多重分形特性及规律。
(1)运用MF-DFA方法,对中国沪深两市股市收益序列的研究发现,股票市场存在明显的多重分形特征。进一步地,发现股票市场的多重分形特征是由两个因素共同作用的,其中收益序列的波动相关性起主导作用,是形成多重分形特征的主要原因,收益序列的胖尾概率分布对多重分形特征的形成也起到了一定的作用；而多重分形过程的本质原因是股票市场信息倍增级联过程的结果。
(2)沪深股市收益序列存在较为明显的标度突变点s*,该突变点将整个时间标度分为两个部分,每个部分具有不同的多重分形特征及标度指数。在s
(3)运用多仿射方法对沪深两市收益序列的研究发现,股票市场存在多仿射现象；另外,研究结果发现广义Hurst指数H(1)与H(2)的值与市场的发展状态之间具有一定的敏感性,发达国家股票市场的H(1)值均较小,而且H(1)及H(2)值均小于0.5,而新兴股票市场的H(1)值均较大,而且H(1)及H(2)值均大于0.5。上海股票市场及深圳股票市场均属于新兴金融市场一类。
(4)运用q阶矩结构分割函数法对股价波动的多重分形特性进行实证研究,结果发现多重分形谱参数与指数的变化趋势及对数收益率具有明显的关联性,并且在股价发生大幅波动的情况下,多重分形谱参数α0,αmax,αmin,W和C均具有较明显的变化特征,这为研究股票市场的运行规律提供了有力的依据。
3.中国股票市场的价格预测研究
基于多重分形理论,对上证指数股指价格进行方向预测及非线性预测,得到了较好的预测效果：
(1)基于日收盘指数价差(ΔI)的符号序列方法与基于日多重分形谱参数(Δf)的符号序列方法都能以一定的概率预测指数的涨落,而且将两种符号序列方法结合可以更好地预测指数的大涨落；在引入股价指数大涨落的阈值和条件平均增益后,发现大幅涨落时,条件与指数变化的关联性比小的涨落要强得多。
(2)提出基于多重分形谱的神经网络模型,并将其应用于股票价格的短期预测,发现该模型对于模拟股市的短期走势具有较好的预测效果,对防范和控制风险具有现实意义。
Stock price fluctuation in stock markets is a very important issue in financial researches. The right description to the evolution law of stock markets and to the running mechanism of stock prices is related to assets pricing、risk controlling、markets supervision and price prediction. After systemically analyzed the multifractal theory and deepened the methods of it, an empirical research on the stock markets in China is presented. The main research work of this paper is as follows:
1. The abnormal characteristics of stock markets in China
Through studying the basic statistics、long-term memory、clustering property、multi-scaling property of the stock price index returns and predictability of stock price index, some conclusions can be drawn:
(1) The returns series of Shanghai and Shenzhen stock markets are disobedient normally distribution but has obvious peak kurtosis and fat tails; And the returns series of Shanghai stock market has peaker kurtosis and fatter tail than those of Shenzhen stock market.
(2) The stock markets of China are not efficient markets, and the returns series had obvious long-term memory, indicating that the stock market did not reached the soft efficiency.
(3) The clustering structure and multi-scaling behavior of the time series were revealed. It indicates that a single scale exponent is insufficient to describe the scaling properties of the returns series and a multifractal model may be more suitable for describing the time series.
(4) The variation of the stock price in China is not totally random, but predictable, or at least predictable in a short period of time.
2. The multifractal characteristics of stock markets in China
Using MF-DFA、multi-affine、qth-order moment structure partition function and a quadratic function fitting to study the returns series and the stock price index series in China, some conclusions can be drawn:
(1) The returns series of stock markets in China showed pronounced multifractal characteristics. Furthermore, the sources of multifractality are analyzed. It is found that there are two different types of sources for multifractality in time series, namely, fat-tailed probability distributions and nonlinear temporal correlations. Most multifractality of the series is due to different long-term correlations, but the fat-tailed probability distributions also contribute to the multifractal behavior of the time series.
(2) The returns series showed a crossover time scale. This crossover divides the whole time scale into two different scale domains. There are different multifractal characteristics and scale exponents for these two different domains. The strength of multifractality in time scales s
(3)The stock markets showed multi-affine phenomenon. In addition, it is found that the generalized Hurst exponents H(1) and H(2) show remarkable differences between developed and emerging markets:Emerging markets are associated with high value of H(1) and developed markets are associated with low values of H(1). Besides, it is found that all the emerging markets have H(2)>0.5 whereas all the developed have H(2)≤0.5. The stock markets in China belong to the category of emerging financial markets.
(4) There are pronounced statistical correlations between the parameters of the multifractal spectra and the variation of the closing index, the logarithmic return and the gain probability. And when the stock price index fluctuates sharply, a strong variability is clearly characterized by the multifractal parameters (α0,αmax,αmin, W and C).
4. Forecasting research of stock price in China
Through directional prediction and nonlinear prediction to the stock price in China, some conclusions can be drawn:
(1) Both of two sign sequence methods have the forecasting capacity. One is the symbol sequence of difference of daily closing quotation indexes; the other is the symbol sequence of multifractal spectrum parameter of the high frequency data. The relation between the condition of different symbol sequences and changes of index in larger fluctuating of stock price index is stronger than the relation in smaller fluctuating of stock price index when threshold values are introduced. And the combining of these two methods has remarkable forecasting capacity.
(2) A neural network model based on the multifractal spectrum is advanced and is applied to the stock price forecasting. The test results indicated that the model can simulate stock market trends in a short time. It is useful to prevent and control risks.
引文
[1]Bachelier L. The Random Character of Stock Market Price[M], Cambridge:M.I.T press,1964.
[2]Kendall M. The Analysis of Economic Time Series, Part I[J], Journal of the Royal Statistical Society,1953,96:11-25.
[3]Fama E. Efficient Capital Markets:A Review of Theory and Empirical Work[J], Journal of Finance,1970,25(2):383-417.
[4]Haugen R A. The inefficient stock market[M], New Jersey:Prentice hall,1999:1-98.
[5]Kahneman D, Tversky A. Prospect theory:An analysis of decision under risk[J], Econometrica,1979,47(2):263-291.
[6]魏宇,黄登仕.经济物理学研究评述[J],经济学动态,2002,7：74-78.
[7]Mandelbrot B B. The variation of certain speculative prices[J], The Journal of Business, 1963,36(4):394-419.
[8]Mandelbrot, B B. How long is the coast of Britain? Statistical self-similarity and fractal dimension[J], Science,1967,156:636-638.
[9]Peters E E. Chaos and order in the Capital Market[M], New York:Wiley,1991.
[10]Peters E E. Fractal market analysis[M], New York:Wiley,1994.
[11]Lo A W. Long-term memory in stock market prices[J], Econometrica,1991,59(5):1279-1313.
[12]Mantegna R N, Stanley H E. Scaling behavior in the dynamics of an economic index [J], Nature,1995,376(6):46-49.
[13]Mandelbrot B B. Fractals and scaling in finance:discontinuity, concentration, risk [M], New York: Springer Verlag,1997:146-197.
[14]Liu Y, Cizeau P, Meyer M, et al. Quantification of correlations in economic time series[J], Physica A, 1997,245:437-440.
[15]Vandewalle N, Ausloos M. Coherent and random sequences in financial fluctuations[J], Physica A, 1997,246(3):454-459.
[16]Scalas E. Scaling in the market of futures [J], Physica A,1998,253(1～4):394～402.
[17]Stanley H E, Amaral L A N, Canning D, et al. Econophysics:Can physicists contribute to the science of economics?[J], Physica A,1999,269(1):156-169.
[18]Cuniberti G, Raberto M, Scalas E. Correlation in the bound-future market [J], Physica A,1999, 269(1):90～97.
[19]Janosi I M, Janecsko B, Kondor I. Statistical analysis of 5s index data of the Budapest Stock exchange [J], Physica A,1999,269(1):111～124.
[20]Raberto M, Scalas E, Cunibert Q, et al. Volatilty in the Italian stock market:an empirical study[J], Physica A,1999,269(2):148-155.
[21]Gordon R R. The fractal structure of exchange rates:measurement and forecasting [J], Journal of International Financial Markets, Institutions and Money,2000,10(2):163-180.
[22]Masoliver J, Monter M, Porra J M.A dynamical model describing stock market price distributions[J], Physica A,2000,283(3-4):559-567.
[23]Pilar G C. Empirical evidence of long-range correlations in stock returns[J], Physica A,2000,287(3): 394-404.
[24]Vjushin D, Govindan R B, Monetti R A, et al. Scaling analysis of trend using DFA[J], Physica A, 2001,302(1-4):234-243.
[25]Panes E P. Estimating fractal dimension using stale distributions and exploring long memory through ARFIMS models in Athens stock exchange[J], Applied Financial Economics,2001,11(3):395-402.
[26]Wang B H, Hui P M. The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market[J], European Physical Journal B,2001,20(4):573-579.
[27]Tsonis A A, Heller F. A characteristic time scale in dollar yen exchange rates[J], Physica A,2001, 291(1-4):574-582.
[28]Ohira T, Sazuka N. Predictability of currency market exchange[J], Physica A,2001, 308(1-4):368-374.
[29]Gencay R. Scaling properties of foreigh exchange volatility[J], Physica A,2001,289(1-2):249-266.
[30]Olan T H. Long memory in stock returns:some international evidence [J], Applied Financial Economics,2002,12(10):725-729.
[31]Zhuang X T, Huang X Y, Sha Y L.Research on the fractal structure in the Chinese stock market[J], PhysicaA,2004,333(2):293-305.
[32]Mandelbrot B B, Fisher A, Calvert L. A multifractal model of asset returns[J], Yale University, Working paper,1997.
[33]Mandelbrot B B. A multifractal walk down Wall Street[J], Scientific American,1999,298(2):70-73.
[34]Ghashghaie S, Breymann W, Peinke J. Turbulent cascade in foreign exchange market[J], Nature, 1996,381(7):767-771.
[35]Mantega R N, Stanley H E.Modeling of financial data:Comparison of the truncated Levy flight and the ARCH(1) and GARCH(1,1) processes[J], Physica A,1998,254(1-2):77-84.
[36]Skjeltorp J A. Scaling in the Norwegian stock market[J], Physica A,2000,283 (3-4):486-528.
[37]Lilloa F, Mantegna R N. Dynamics of a financial market index after a crash [J],Physica A,2004, 338(1-2):125-134.
[38]Pasquini M, Serva M. Multiscaling and clustering of volatility [J], Physica A,1999,269(1):140-147.
[39]Pasquini M, Serva M. Multiscale behaviour of volatility autocorrelations in a financial market [J], Economics Letters,1999,65 (3):275-279.
[40]Mulligan R F, Lombardo G A. Maritime businesses:volatile stock prices and market valuation inefficiencies [J], The Quarterly Review of Economics and Finance,2004,44(2):321-336.
[41]Brachet M E, Taflin E, Tcheou J M. Scaling transformation and probability distributions for financial time series [J], Chaos, Solitons and Fractals,2000,11:2343-2348.
[42]Gorski A Z, Drozdz S, Speth J. Financial multifractality and its subtleties:anexample of DAX[J], Physica A,2002,316:496-510.
[43]Selcuk F. Financial earthquakes, aftershocks and scaling in emerging stock markets[J], Physica A, 2004,333:306-316.
[44]Ivanova K, Ausloos M. Low q-moment multifractal analysis of gold price, Dow-Jones industrial average and BGL-USD exchange rate[J], European Physical Journal B,1999,8(4):665-669.
[45]Schmitt F, Shertze D, Lovejoy S. Multifractal fluctuations in finance[J], International Journal of Theoretical and Applied Finance,2000,3(3):361-364.
[46]Andreadis J, Serletis A. Evidence of a random multifractal turbulent structure in the Dow Jones Industrial Average[J], Chaos, Solitons and Fractals,2002,13:1309-1315.
[47]Katsuragi H. Evidence of multi-affinity in the Japanese stock market[J], Physica A,2000, 278(1-2):275-281.
[48]Ausloos M, Ivanovab K. Multifractal nature of stock exchange prices[J], Computer Physics Communications,2002,147(1):582-585.
[49]Zhao X X, Ramazan G. Scaling, self-similarity and multifractality in FX markets [J], Physica A, 2003,323:578-590.
[50]Kantelhardt J W, Zschiegner S A, Koscielny-Bunde E, et al. Multifractal detrended fluctuation analysis of nonstationary time series[J], Physica A,2002,316(1-4):87-114.
[51]Kantelhardt J W, Koscielny-Bunde E, Rego H A, et al. Detecting long-range correlations with detrended fluctuation analysis [J], Physica A,2001,295(3-4):441-454.
[52]Koscielny-Bunde E, Kantelhardt J W, Braun P, et al. Multifractal of river runoff records:Detrended fluctuation studies [J], Journal of Hydrology,2006,322(1-4):120-137.
[53]Matia K, Ashkenazy Y. Multifractal properties of price fluctuations of stocks and commodities[J], Europhysics Letters,2003,61(3):422-428.
[54]Norouzzaden P, Rahmani B. A multifractal detrended fluctuation description of Iranian rial-US dollar exchange rate [J], Physica A,2006,367:328-336.
[55]Pavlov A N, Ebeling W, Molgedey L, et al. Scaling features of texts, images and time series[J], Physica A,2001(1-2),300:310-324.
[56]Gorski A Z, Drozdz S, Speth J. Financial multifractality and its subtleties:anexample of DAX[J], Physica A,2002,316(1-4):496-510.
[57]Budaev V P. Turbulence in magnetized plasmas and financial markets:comparative study of multifractal statistics[J], Physica A,2004,344(1-2):299-307.
[58]Fulco U L, Lyra M L, Petroni F, et al. A stochastic model for multifractal behavior of stock prices[J], International Journal of Modern Physics B,2004,18(4-5):681-689.
[59]Alvarez-Ramirez J, Cisneros M, Ibarra-Valdoz C, et al. Multifractal Hurst analysis of crude oil prices [J], Physica A,2002,313(3-4):651～670.
[60]Alvarez-Ramirez J. Characteristic time scales in the American dollar-Mexican peso exchange currency market[J], Physica A,2002,309(1-2):157-170.
[61]Kim K, Yoon S M. Multifractal features of financial markets[J], Physica A,2004,344(1-2):272～278.
[62]Lee J W, Lee K E, Rikvold P A. Multifractal behavior of the Korean stock-market index KOSPI[J], Physica A,2006,364:355-361.
[63]Jiang J, Ma K, Cai X. Non-linear characteristics and long-range correlations in Asian stock markets[J], Physica A,2007,378(2):399-407.
[64]Bacry E, Delour J, Muzy J F. Modeling financial time series using multifractal random walks[J], Physica A,2001,299(1-2):84-92.
[65]Los C A, Yalamova R. Multifractal Spectral Analysis of the 1987 Stock Market Crash. Kent State University, Working Paper,2004.
[66]Owicimka P, Kwapien J, Drozdz S. Multifractality in the stock market:price increments versus waiting times[J], Physica A,2005,347:626-638.
[67]Selcuk F, Gencay R. Intraday dynamics of stock market returns and volatility [J], Physica A,2006, 367:375-387.
[68]Muzy J F, Delour J, Bacry E. Modeling fluctuation of financial time series:from cascade process to stochastic volatility model[J], The European Physical Journal B,2000,17:537-548.
[69]Jiang Z Q, Zhou W X. Scale invariant distribution and multifractality of volatility multipliers in stock markets[J], Physica A,2007,381:343-350.
[70]Matteo T D, Aste T, Dacorogna M M. Scaling behaviors in differently developed markets[J], Physica A,2003,324(1-2):183-188.
[71]Norouzzadeh P, Jafari G R. Application of multifractal measures to Tehran price index[J], Physica A, 2005,356(2-4):609-627.
[72]Cajueiro D O, Tabak B M. Long-range dependence and multifractality in the term structure of LIBOR interest rates [J], PhysicaA,2007,373:603-614.
[73]Lux T. The multifractal model of asset returns:its estimation via GMM and its use for volatility forecasting. University of Kiel, Working Paper,2003.
[74]Muzy J F, Sornette D, Delour J, et al. Multifractal returns and hierarchical portfolio theory[J], Quantitative Finance,2001,1(1):131-148.
[75]Liu R P, Matteo T D, Lux T. True and apparent scaling:The proximity of the Markov-switching multifractal model to long-range dependence.[J], PhysicaA,2007,383(1):35-42.
[76]黄登仕.金融市场的标度理论[J],管理科学学报,2000,3(2)：27-33.
[77]胡雪明,吴坤.国外金融资产价格变化行为研究进展述评[J],经济师,2003,9：87-88.
[78]黄诒蓉,罗奕.资本市场分形结构的理论与方法[J],当代财经,2006,3：54-59.
[79]张永东,毕秋香.中国股票市场多标度行为的实证分析[J],预测,2002,21(4)：56-59.
[80]何建敏,朱林,常松.中国股票市场价格波动的尺度特性[J],中国管理科学,2003,1：1-5.
[81]朱林,常松,何建敏.我国股票市场多仿射特性研究[J],管理工程学报,2002,16(3)：86-90.
[82]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,2004,6：51-55.
[83]胡学明,宋学锋.深沪股票市场的多重分形分析[J],数量经济技术经济研究,2003,8：124-127.
[84]施锡铨,艾克凤.股票市场风险的多重分形分析[J],统计研究,2004,9：33-36.
[85]Ho D S, Lee C K. Scaling characteristics in the Taiwan stock market [J], Physica A,2003,332(2): 448-460.
[86]黄诒蓉.中国股票市场多重分形结构的实证研究[J],当代财经,2004,11：53-56.
[87]王德佳,朱海英,于海,朱伟勇.证券市场中的混沌分形现象与多分形走动[J],东北大学学报(自然科学版),2001,22(5)：501-504.
[88]于建玲,臧保将,商朋见.股市时间序列的多重分形分析[J],北京交通大学学报,2006,30(6)：69-72.
[89]舒建平,谭德庆,吴炯.中国股票市场分形结构探索[J],西南交通大学学报,2003,38(2)：212-215.
[90]吴文锋,黄登仕,吴炯.中国股票市场的多标度特征[J],数量经济技术经济研究,2003,9：112-115.
[91]魏宇,黄登仕.中国股票市场多标度特征的实证研究[J],系统工程,2003,21(3)：7-12.
[92]苑莹,庄新田.上海股票市场的标度特征[J],东北大学学报(自然科学版),2006,27(10)：1177-1180.
[93]Sun, X, Chen H P, Wu Z Q, et al. Multifractal analysis of Hang Seng index in Hong Kong stock market[J], Physica A,2001,291 (1-4):553-562.
[94]Sun X, Chen H P, Wu Z Q et al. Predictability of multifractal analysis of Hang Seng stock index in HongKong[J], Physica A,2001,301:473-482.
[95]施锡铨,艾克凤.股票市场风险的多重分形分析[J],统计研究,2004,9：33-36.
[96]周孝华,宋坤.高频金融时间序列的异象特征分析及应用[J],财经研究,2005,31(7)：123-131.
[97]周孝华,宋坤,杨秀苔.股票价格持续大幅波动前后多重分形谱的异常及分析[J],管理工程学报,2006,20(2)：92-96.
[98]赵巍,何建敏.多重分形的统计物理方法在证券市场中的应用[J],数理统计与管理,2007,26(3)：535-542.
[99]Du G X, Ning X X. Multifractal properties of Chinese stock market in Shanghai[J], Physica A,2008, 387(1):261-269.
[100]何建敏,常松.中国股票市场多重分形游走及其预测[J],中国管理科学,2002,10(3)：11-17.
[101]Chen H P, Sun X, Wu Z Q, et al. Enlightenment from various conditional probabilities about Hang Seng index in Hong Kong stock market[J], Physica A,2004,335(1-2):183-196.
[102]Wei Y, Huang D S. Multifractal analysis of SSEC in Chinese stock market:A different empirical result from Hang Seng index[J], Physica A,2005,355(2-4):497-508.
[103]魏宇,黄登仕.基于多标度分形理论的金融风险测度指标研究[J],管理科学学报,2005,8(4)：50-59.
[104]黄超,吴列清,武忠,朱扬勇.基于方差波动多重分形特征的金融时间序列聚类[J],系统工程,2006,24(6)：100-103.
[105]Mandelbrot B B. Fractals:Form, Chance and Dimension[M], San Francisco:Freeman,1977.
[106]Mandelbrot B B. The Fractal Geometry of Nature[M], New York:Freeman,1982.
[107]Mandelbrot B B. Fractaional brownian motions, fractional noises and applications[J], SIAM Review, 1968,10(4):422-437.
[108]樊智,张世英.金融市场的效率与分形市场理论[J],系统工程理论与实践,2002,3：13-19.
[109]Fisher A, Calvet L, Mandelbrot B B. Multifractality of Deutschmark/US dollar exchange rates[J], Yale University, Working Paper,1997.
[110]Mandelbrot B B, Fisher A, Calvet L. A multifractal model of asset returns[J], Yale University, Working Paper,1997.
[111]Calvet L, Fisher A, Mandelbrot B B. Large deviations and the distribution of price changes[J], Yale University, Working Paper,1997.
[112]Halsey T C, Jensen M H, Kadanoff L P. Fractal measures and their singularities:the characterization of strange sets[J],Phys Rev A,1986,33(2):1141-1150.
[113]孙霞,傅竹西,吴自勤.薄膜生长的多重分形谱的计算[J],计算物理,2001,18(3)：247-252.
[114]黄诒蓉.中国股市分形结构的理论研究与实证分析[D],厦门厦门大学,2004.
[115]Dickey D A, Fuller W A. Distribution of the estimators for autoregressive time series with a unit root[J], Journal of the American Statistical Association,1979,74:427-431.
[116]Brock W, Dechert D, Scheinkman L B. A test for independence based on the correlation[J], Econometric Reviews,1996,15:197-235.
[117]徐龙炳,陆蓉.R/S分析探索中国股票市场的非线性[J],预测,1999,2：59-62.
[118]史永东.中国证券市场股票收益持久性的经验分析[J],世界经济,2000,11：29-33.
[119]张维,黄兴.沪深股市的R/S实证分析[J],系统工程,2001,19(1)：1-5.
[120]叶中行,曹奕剑.Hurst指数在股票市场有效性分析中的应用[J],系统工程,2001(3)：21-24.
[121]伍海华,李道叶,高锐.论证证券市场的分形与混沌[J],世界经济,2001,7：32-37.
[122]伍海华.论证券市场的复杂性：以上海股票市场为例[J],系统仿真学报,2002,14(11)：1486-1516.
[123]庄新田,黄小原.证券市场的标度理论及实证研究[J],系统工程理论与实践,2003,3：1-8.
[124]王新宇,宋学锋,吴瑞明.中国证券市场的分形分析[J],管理科学学报,2004,7(5)：68-74.
[125]王明涛.基于R/S分析中国股票市场的非线性特征[J],预测,2002,3：42-45.
[126]邹新月.深圳综合指数长程相关性特征分析及其预测[J],中国管理科学,2002,10(2)：96-100.
[127]李亚静,何跃,朱宏泉.中国股市收益率与波动性长期记忆性的实证研究[J],系统工程理论与实践,2003,23(1)：9-15.
[128]王春峰,王庆翠.中国股票市场收益的长记忆性研究[J],系统工程,2003,21(1)：22-28.
[129]陈梦根.中国股市长记忆效应的实证研究[J],经济研究,2003,3：70-78.
[130]陈梦根.股票价格分形特征的实证研究,修正R/S分析[J],统计研究,2003,4：57-61.
[131]李亚静,何跃,朱宏泉.中国股市收益率与波动性长记忆性的实证研究[J],系统工程理论与实践,2003,23(1)：9-15.
[132]赵桂芹,曾振宇.证券市场长期记忆特征的实证分析[J],管理科学,2003,16(2)：40-44.
[133]胡彦梅,张卫国,陈建忠.中国股市长记忆的修正R/S分析[J],数理统计与管理,2006,25(1)：73-77.
[134]Lo W. Long-term memory in stock market prices[J], Econometrica,1991,59(5):1279-1313.
[135]Jan W K, Eva K B, Henio H A, et al. Detecting long-range correlations with detrended fluctuation analysis[J], Physica A,2001,295(3-4):441-454.
[136]Lee C K, Juang L C, Wang C C, et al. Scaling characteristics in ozone concentration time series(OTCS)[J], Chemosphere,2006,62(6):934-946.
[137]范龙振,胡畏.深圳股市价格运动可预测性的研究[J],系统工程学报,2001,16(6)：58-66.
[138]邹新月.深圳综合指数长程相关性特征分析及其预测[J],中国管理科学,2002,2：23-28.
[139]黄少军.中国股票市场可预测性研究[J],华南师范大学学报(社会科学版),2004,2：9-14.
[140]魏宇,黄登仕.中国股票市场价格波动特征及其可预测性研究[J],管理工程学报,2004,18(4)：117-120.
[141]陈春晖,曾德明.我国股市可预测性的统计研究[J],统计与决策,2006,8：6-7.
[142]Ausloos M, Vandewalle N, Boveroux P H, et al. Application of statistical physics to economic and financial topics[J], Physica A,1999,274(1-2):229-240.
[143]Bouchaud J P, Potters M, Meyer M. Apparent multifractality in financial time series[J], European Physical Jornal of B,2000,13:595-599.
[144]Muzy J F, Delour J, Bacry E. Modeling fluctuation of financial time series:from cascade process to stochastic volatility model[J], European Physical Jornal of B,2000,17:537-548.
[145]Bandiera L, Cester A, Paccagnella A, et al. Detrended fluctuation analysis of the soft breakdown current[J], Microelectronic Engineering,2001,59(1):49-53.
[146]Govindan R B, Vjushin D, Brenner S, et al. Long-range correlations and trend in global climate models:Comparison with real data[J], Physica A,2001,294(1-2):239-248.
[147]Peng C K, Havlin S, Stanley H E, et al. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series[J], Chaos,1995,5(1):82-87.
[148]Mandelbrot B B. Fractals and scaling in finance:discontinuity, concentration, risk[M], New York: Springer Verlag,1997,146-197.
[149]Shimizu Y, Thurner S, Ehrenberger K. Multifractal spectra as a measure of complexity in human posture[J], Fractals,2002,10(1):103-116.
[150]Harvey C R. Predictable risk and returns in emerging markets[J], Review of financial studies,1995, 8(3):773-816.
[151]Jung C, Boyd R. Forecasting U K stock prices[J], Applied Financial Economics,1996,6:279-286.
[152]刘国旗.非线性模型GARCH模型在中国股市波动预测中的应用研究[J],统计研究,2000,1：49-52.
[153]王培勋.非线性回归的弹性分析在股票投资与行情预测中的应用[J],系统工程理论与实践,2000,8：100-104.
[154]吴微,陈维强,刘波.用BP神经网络预测股票市场涨跌[J],大连理工大学学报,2001,41(1)：9-15.
[155]孙全,朱江.基于遗传神经网络的股票价格短期预测[J],计算机工程与应用,2002,5：237-239.
[156]Chen A S, Mark L T, Hazem D. Application of neural networks to an emerging financial market: forecasting and trading the Taiwan stock index[J], Computer&Operation Research,2003, 30(6):901-923.
[157]吕淑萍,赵咏梅.基于小波神经网络的时间序列预测方法应用[J],哈尔滨工程大学学报,2001,41(1)：9-15.
[158]彭岩,孙学惠,王庆余.基于混沌时序重构的股票价格预测研究[J],天津大学学报,2003,36(6)：65-69.
[159]常松.基于小波包和神经网络的股票价格预测模型[J],中国管理科学,2001,5：8-15.
[160]孙丹,张秀艳.基于人工神经网络的股市预测模型[J],吉林大学学报(信息科学版),2002,20(4)：68-70.
[161]汤凌冰,廖福元,罗键.模糊神经网络在股价预测中的应用[J],系统工程,2004,22(2)：107-109.
[162]李焕荣,王树明.一种改进的BP神经网络预测方法及其应用[J],系统工程,2000,5：76-79.
[163]杨成,程晓玲,殷旅江.基于人工神经网络方法的上市公司股价预测[J]统计与决策.2005,12：106-108.
[164]高振坤,熊正德.基于神经网络技术的股指预测模型及实证分析[J],财经理论与实践,2005(1)：56-61.
[165]陈兴,孟卫东,严太华.基于T-S模型的模糊神经网络在股市预测中的应用[J],系统工程理论与实践,2001(2)：66-72.
[166]郑丕谔,马艳华.基于RBF神经网络的股市建模与预测[J],天津大学学报,2000(7)：483-486.
[167]杨一文,刘贵忠,张宗平.基于嵌入理论和神经网络技术的混沌数据预测及其在股票市场中的应用[J],系统工程理论与实践,2001(6)：52-58.
[2]Kendall M. The Analysis of Economic Time Series, Part I[J], Journal of the Royal Statistical Society,1953,96:11-25.
[3]Fama E. Efficient Capital Markets:A Review of Theory and Empirical Work[J], Journal of Finance,1970,25(2):383-417.
[4]Haugen R A. The inefficient stock market[M], New Jersey:Prentice hall,1999:1-98.
[5]Kahneman D, Tversky A. Prospect theory:An analysis of decision under risk[J], Econometrica,1979,47(2):263-291.
[6]魏宇,黄登仕.经济物理学研究评述[J],经济学动态,2002,7：74-78.
[7]Mandelbrot B B. The variation of certain speculative prices[J], The Journal of Business, 1963,36(4):394-419.
[8]Mandelbrot, B B. How long is the coast of Britain? Statistical self-similarity and fractal dimension[J], Science,1967,156:636-638.
[9]Peters E E. Chaos and order in the Capital Market[M], New York:Wiley,1991.
[10]Peters E E. Fractal market analysis[M], New York:Wiley,1994.
[11]Lo A W. Long-term memory in stock market prices[J], Econometrica,1991,59(5):1279-1313.
[12]Mantegna R N, Stanley H E. Scaling behavior in the dynamics of an economic index [J], Nature,1995,376(6):46-49.
[13]Mandelbrot B B. Fractals and scaling in finance:discontinuity, concentration, risk [M], New York: Springer Verlag,1997:146-197.
[14]Liu Y, Cizeau P, Meyer M, et al. Quantification of correlations in economic time series[J], Physica A, 1997,245:437-440.
[15]Vandewalle N, Ausloos M. Coherent and random sequences in financial fluctuations[J], Physica A, 1997,246(3):454-459.
[16]Scalas E. Scaling in the market of futures [J], Physica A,1998,253(1～4):394～402.
[17]Stanley H E, Amaral L A N, Canning D, et al. Econophysics:Can physicists contribute to the science of economics?[J], Physica A,1999,269(1):156-169.
[18]Cuniberti G, Raberto M, Scalas E. Correlation in the bound-future market [J], Physica A,1999, 269(1):90～97.
[19]Janosi I M, Janecsko B, Kondor I. Statistical analysis of 5s index data of the Budapest Stock exchange [J], Physica A,1999,269(1):111～124.
[20]Raberto M, Scalas E, Cunibert Q, et al. Volatilty in the Italian stock market:an empirical study[J], Physica A,1999,269(2):148-155.
[21]Gordon R R. The fractal structure of exchange rates:measurement and forecasting [J], Journal of International Financial Markets, Institutions and Money,2000,10(2):163-180.
[22]Masoliver J, Monter M, Porra J M.A dynamical model describing stock market price distributions[J], Physica A,2000,283(3-4):559-567.
[23]Pilar G C. Empirical evidence of long-range correlations in stock returns[J], Physica A,2000,287(3): 394-404.
[24]Vjushin D, Govindan R B, Monetti R A, et al. Scaling analysis of trend using DFA[J], Physica A, 2001,302(1-4):234-243.
[25]Panes E P. Estimating fractal dimension using stale distributions and exploring long memory through ARFIMS models in Athens stock exchange[J], Applied Financial Economics,2001,11(3):395-402.
[26]Wang B H, Hui P M. The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market[J], European Physical Journal B,2001,20(4):573-579.
[27]Tsonis A A, Heller F. A characteristic time scale in dollar yen exchange rates[J], Physica A,2001, 291(1-4):574-582.
[28]Ohira T, Sazuka N. Predictability of currency market exchange[J], Physica A,2001, 308(1-4):368-374.
[29]Gencay R. Scaling properties of foreigh exchange volatility[J], Physica A,2001,289(1-2):249-266.
[30]Olan T H. Long memory in stock returns:some international evidence [J], Applied Financial Economics,2002,12(10):725-729.
[31]Zhuang X T, Huang X Y, Sha Y L.Research on the fractal structure in the Chinese stock market[J], PhysicaA,2004,333(2):293-305.
[32]Mandelbrot B B, Fisher A, Calvert L. A multifractal model of asset returns[J], Yale University, Working paper,1997.
[33]Mandelbrot B B. A multifractal walk down Wall Street[J], Scientific American,1999,298(2):70-73.
[34]Ghashghaie S, Breymann W, Peinke J. Turbulent cascade in foreign exchange market[J], Nature, 1996,381(7):767-771.
[35]Mantega R N, Stanley H E.Modeling of financial data:Comparison of the truncated Levy flight and the ARCH(1) and GARCH(1,1) processes[J], Physica A,1998,254(1-2):77-84.
[36]Skjeltorp J A. Scaling in the Norwegian stock market[J], Physica A,2000,283 (3-4):486-528.
[37]Lilloa F, Mantegna R N. Dynamics of a financial market index after a crash [J],Physica A,2004, 338(1-2):125-134.
[38]Pasquini M, Serva M. Multiscaling and clustering of volatility [J], Physica A,1999,269(1):140-147.
[39]Pasquini M, Serva M. Multiscale behaviour of volatility autocorrelations in a financial market [J], Economics Letters,1999,65 (3):275-279.
[40]Mulligan R F, Lombardo G A. Maritime businesses:volatile stock prices and market valuation inefficiencies [J], The Quarterly Review of Economics and Finance,2004,44(2):321-336.
[41]Brachet M E, Taflin E, Tcheou J M. Scaling transformation and probability distributions for financial time series [J], Chaos, Solitons and Fractals,2000,11:2343-2348.
[42]Gorski A Z, Drozdz S, Speth J. Financial multifractality and its subtleties:anexample of DAX[J], Physica A,2002,316:496-510.
[43]Selcuk F. Financial earthquakes, aftershocks and scaling in emerging stock markets[J], Physica A, 2004,333:306-316.
[44]Ivanova K, Ausloos M. Low q-moment multifractal analysis of gold price, Dow-Jones industrial average and BGL-USD exchange rate[J], European Physical Journal B,1999,8(4):665-669.
[45]Schmitt F, Shertze D, Lovejoy S. Multifractal fluctuations in finance[J], International Journal of Theoretical and Applied Finance,2000,3(3):361-364.
[46]Andreadis J, Serletis A. Evidence of a random multifractal turbulent structure in the Dow Jones Industrial Average[J], Chaos, Solitons and Fractals,2002,13:1309-1315.
[47]Katsuragi H. Evidence of multi-affinity in the Japanese stock market[J], Physica A,2000, 278(1-2):275-281.
[48]Ausloos M, Ivanovab K. Multifractal nature of stock exchange prices[J], Computer Physics Communications,2002,147(1):582-585.
[49]Zhao X X, Ramazan G. Scaling, self-similarity and multifractality in FX markets [J], Physica A, 2003,323:578-590.
[50]Kantelhardt J W, Zschiegner S A, Koscielny-Bunde E, et al. Multifractal detrended fluctuation analysis of nonstationary time series[J], Physica A,2002,316(1-4):87-114.
[51]Kantelhardt J W, Koscielny-Bunde E, Rego H A, et al. Detecting long-range correlations with detrended fluctuation analysis [J], Physica A,2001,295(3-4):441-454.
[52]Koscielny-Bunde E, Kantelhardt J W, Braun P, et al. Multifractal of river runoff records:Detrended fluctuation studies [J], Journal of Hydrology,2006,322(1-4):120-137.
[53]Matia K, Ashkenazy Y. Multifractal properties of price fluctuations of stocks and commodities[J], Europhysics Letters,2003,61(3):422-428.
[54]Norouzzaden P, Rahmani B. A multifractal detrended fluctuation description of Iranian rial-US dollar exchange rate [J], Physica A,2006,367:328-336.
[55]Pavlov A N, Ebeling W, Molgedey L, et al. Scaling features of texts, images and time series[J], Physica A,2001(1-2),300:310-324.
[56]Gorski A Z, Drozdz S, Speth J. Financial multifractality and its subtleties:anexample of DAX[J], Physica A,2002,316(1-4):496-510.
[57]Budaev V P. Turbulence in magnetized plasmas and financial markets:comparative study of multifractal statistics[J], Physica A,2004,344(1-2):299-307.
[58]Fulco U L, Lyra M L, Petroni F, et al. A stochastic model for multifractal behavior of stock prices[J], International Journal of Modern Physics B,2004,18(4-5):681-689.
[59]Alvarez-Ramirez J, Cisneros M, Ibarra-Valdoz C, et al. Multifractal Hurst analysis of crude oil prices [J], Physica A,2002,313(3-4):651～670.
[60]Alvarez-Ramirez J. Characteristic time scales in the American dollar-Mexican peso exchange currency market[J], Physica A,2002,309(1-2):157-170.
[61]Kim K, Yoon S M. Multifractal features of financial markets[J], Physica A,2004,344(1-2):272～278.
[62]Lee J W, Lee K E, Rikvold P A. Multifractal behavior of the Korean stock-market index KOSPI[J], Physica A,2006,364:355-361.
[63]Jiang J, Ma K, Cai X. Non-linear characteristics and long-range correlations in Asian stock markets[J], Physica A,2007,378(2):399-407.
[64]Bacry E, Delour J, Muzy J F. Modeling financial time series using multifractal random walks[J], Physica A,2001,299(1-2):84-92.
[65]Los C A, Yalamova R. Multifractal Spectral Analysis of the 1987 Stock Market Crash. Kent State University, Working Paper,2004.
[66]Owicimka P, Kwapien J, Drozdz S. Multifractality in the stock market:price increments versus waiting times[J], Physica A,2005,347:626-638.
[67]Selcuk F, Gencay R. Intraday dynamics of stock market returns and volatility [J], Physica A,2006, 367:375-387.
[68]Muzy J F, Delour J, Bacry E. Modeling fluctuation of financial time series:from cascade process to stochastic volatility model[J], The European Physical Journal B,2000,17:537-548.
[69]Jiang Z Q, Zhou W X. Scale invariant distribution and multifractality of volatility multipliers in stock markets[J], Physica A,2007,381:343-350.
[70]Matteo T D, Aste T, Dacorogna M M. Scaling behaviors in differently developed markets[J], Physica A,2003,324(1-2):183-188.
[71]Norouzzadeh P, Jafari G R. Application of multifractal measures to Tehran price index[J], Physica A, 2005,356(2-4):609-627.
[72]Cajueiro D O, Tabak B M. Long-range dependence and multifractality in the term structure of LIBOR interest rates [J], PhysicaA,2007,373:603-614.
[73]Lux T. The multifractal model of asset returns:its estimation via GMM and its use for volatility forecasting. University of Kiel, Working Paper,2003.
[74]Muzy J F, Sornette D, Delour J, et al. Multifractal returns and hierarchical portfolio theory[J], Quantitative Finance,2001,1(1):131-148.
[75]Liu R P, Matteo T D, Lux T. True and apparent scaling:The proximity of the Markov-switching multifractal model to long-range dependence.[J], PhysicaA,2007,383(1):35-42.
[76]黄登仕.金融市场的标度理论[J],管理科学学报,2000,3(2)：27-33.
[77]胡雪明,吴坤.国外金融资产价格变化行为研究进展述评[J],经济师,2003,9：87-88.
[78]黄诒蓉,罗奕.资本市场分形结构的理论与方法[J],当代财经,2006,3：54-59.
[79]张永东,毕秋香.中国股票市场多标度行为的实证分析[J],预测,2002,21(4)：56-59.
[80]何建敏,朱林,常松.中国股票市场价格波动的尺度特性[J],中国管理科学,2003,1：1-5.
[81]朱林,常松,何建敏.我国股票市场多仿射特性研究[J],管理工程学报,2002,16(3)：86-90.
[82]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,2004,6：51-55.
[83]胡学明,宋学锋.深沪股票市场的多重分形分析[J],数量经济技术经济研究,2003,8：124-127.
[84]施锡铨,艾克凤.股票市场风险的多重分形分析[J],统计研究,2004,9：33-36.
[85]Ho D S, Lee C K. Scaling characteristics in the Taiwan stock market [J], Physica A,2003,332(2): 448-460.
[86]黄诒蓉.中国股票市场多重分形结构的实证研究[J],当代财经,2004,11：53-56.
[87]王德佳,朱海英,于海,朱伟勇.证券市场中的混沌分形现象与多分形走动[J],东北大学学报(自然科学版),2001,22(5)：501-504.
[88]于建玲,臧保将,商朋见.股市时间序列的多重分形分析[J],北京交通大学学报,2006,30(6)：69-72.
[89]舒建平,谭德庆,吴炯.中国股票市场分形结构探索[J],西南交通大学学报,2003,38(2)：212-215.
[90]吴文锋,黄登仕,吴炯.中国股票市场的多标度特征[J],数量经济技术经济研究,2003,9：112-115.
[91]魏宇,黄登仕.中国股票市场多标度特征的实证研究[J],系统工程,2003,21(3)：7-12.
[92]苑莹,庄新田.上海股票市场的标度特征[J],东北大学学报(自然科学版),2006,27(10)：1177-1180.
[93]Sun, X, Chen H P, Wu Z Q, et al. Multifractal analysis of Hang Seng index in Hong Kong stock market[J], Physica A,2001,291 (1-4):553-562.
[94]Sun X, Chen H P, Wu Z Q et al. Predictability of multifractal analysis of Hang Seng stock index in HongKong[J], Physica A,2001,301:473-482.
[95]施锡铨,艾克凤.股票市场风险的多重分形分析[J],统计研究,2004,9：33-36.
[96]周孝华,宋坤.高频金融时间序列的异象特征分析及应用[J],财经研究,2005,31(7)：123-131.
[97]周孝华,宋坤,杨秀苔.股票价格持续大幅波动前后多重分形谱的异常及分析[J],管理工程学报,2006,20(2)：92-96.
[98]赵巍,何建敏.多重分形的统计物理方法在证券市场中的应用[J],数理统计与管理,2007,26(3)：535-542.
[99]Du G X, Ning X X. Multifractal properties of Chinese stock market in Shanghai[J], Physica A,2008, 387(1):261-269.
[100]何建敏,常松.中国股票市场多重分形游走及其预测[J],中国管理科学,2002,10(3)：11-17.
[101]Chen H P, Sun X, Wu Z Q, et al. Enlightenment from various conditional probabilities about Hang Seng index in Hong Kong stock market[J], Physica A,2004,335(1-2):183-196.
[102]Wei Y, Huang D S. Multifractal analysis of SSEC in Chinese stock market:A different empirical result from Hang Seng index[J], Physica A,2005,355(2-4):497-508.
[103]魏宇,黄登仕.基于多标度分形理论的金融风险测度指标研究[J],管理科学学报,2005,8(4)：50-59.
[104]黄超,吴列清,武忠,朱扬勇.基于方差波动多重分形特征的金融时间序列聚类[J],系统工程,2006,24(6)：100-103.
[105]Mandelbrot B B. Fractals:Form, Chance and Dimension[M], San Francisco:Freeman,1977.
[106]Mandelbrot B B. The Fractal Geometry of Nature[M], New York:Freeman,1982.
[107]Mandelbrot B B. Fractaional brownian motions, fractional noises and applications[J], SIAM Review, 1968,10(4):422-437.
[108]樊智,张世英.金融市场的效率与分形市场理论[J],系统工程理论与实践,2002,3：13-19.
[109]Fisher A, Calvet L, Mandelbrot B B. Multifractality of Deutschmark/US dollar exchange rates[J], Yale University, Working Paper,1997.
[110]Mandelbrot B B, Fisher A, Calvet L. A multifractal model of asset returns[J], Yale University, Working Paper,1997.
[111]Calvet L, Fisher A, Mandelbrot B B. Large deviations and the distribution of price changes[J], Yale University, Working Paper,1997.
[112]Halsey T C, Jensen M H, Kadanoff L P. Fractal measures and their singularities:the characterization of strange sets[J],Phys Rev A,1986,33(2):1141-1150.
[113]孙霞,傅竹西,吴自勤.薄膜生长的多重分形谱的计算[J],计算物理,2001,18(3)：247-252.
[114]黄诒蓉.中国股市分形结构的理论研究与实证分析[D],厦门厦门大学,2004.
[115]Dickey D A, Fuller W A. Distribution of the estimators for autoregressive time series with a unit root[J], Journal of the American Statistical Association,1979,74:427-431.
[116]Brock W, Dechert D, Scheinkman L B. A test for independence based on the correlation[J], Econometric Reviews,1996,15:197-235.
[117]徐龙炳,陆蓉.R/S分析探索中国股票市场的非线性[J],预测,1999,2：59-62.
[118]史永东.中国证券市场股票收益持久性的经验分析[J],世界经济,2000,11：29-33.
[119]张维,黄兴.沪深股市的R/S实证分析[J],系统工程,2001,19(1)：1-5.
[120]叶中行,曹奕剑.Hurst指数在股票市场有效性分析中的应用[J],系统工程,2001(3)：21-24.
[121]伍海华,李道叶,高锐.论证证券市场的分形与混沌[J],世界经济,2001,7：32-37.
[122]伍海华.论证券市场的复杂性：以上海股票市场为例[J],系统仿真学报,2002,14(11)：1486-1516.
[123]庄新田,黄小原.证券市场的标度理论及实证研究[J],系统工程理论与实践,2003,3：1-8.
[124]王新宇,宋学锋,吴瑞明.中国证券市场的分形分析[J],管理科学学报,2004,7(5)：68-74.
[125]王明涛.基于R/S分析中国股票市场的非线性特征[J],预测,2002,3：42-45.
[126]邹新月.深圳综合指数长程相关性特征分析及其预测[J],中国管理科学,2002,10(2)：96-100.
[127]李亚静,何跃,朱宏泉.中国股市收益率与波动性长期记忆性的实证研究[J],系统工程理论与实践,2003,23(1)：9-15.
[128]王春峰,王庆翠.中国股票市场收益的长记忆性研究[J],系统工程,2003,21(1)：22-28.
[129]陈梦根.中国股市长记忆效应的实证研究[J],经济研究,2003,3：70-78.
[130]陈梦根.股票价格分形特征的实证研究,修正R/S分析[J],统计研究,2003,4：57-61.
[131]李亚静,何跃,朱宏泉.中国股市收益率与波动性长记忆性的实证研究[J],系统工程理论与实践,2003,23(1)：9-15.
[132]赵桂芹,曾振宇.证券市场长期记忆特征的实证分析[J],管理科学,2003,16(2)：40-44.
[133]胡彦梅,张卫国,陈建忠.中国股市长记忆的修正R/S分析[J],数理统计与管理,2006,25(1)：73-77.
[134]Lo W. Long-term memory in stock market prices[J], Econometrica,1991,59(5):1279-1313.
[135]Jan W K, Eva K B, Henio H A, et al. Detecting long-range correlations with detrended fluctuation analysis[J], Physica A,2001,295(3-4):441-454.
[136]Lee C K, Juang L C, Wang C C, et al. Scaling characteristics in ozone concentration time series(OTCS)[J], Chemosphere,2006,62(6):934-946.
[137]范龙振,胡畏.深圳股市价格运动可预测性的研究[J],系统工程学报,2001,16(6)：58-66.
[138]邹新月.深圳综合指数长程相关性特征分析及其预测[J],中国管理科学,2002,2：23-28.
[139]黄少军.中国股票市场可预测性研究[J],华南师范大学学报(社会科学版),2004,2：9-14.
[140]魏宇,黄登仕.中国股票市场价格波动特征及其可预测性研究[J],管理工程学报,2004,18(4)：117-120.
[141]陈春晖,曾德明.我国股市可预测性的统计研究[J],统计与决策,2006,8：6-7.
[142]Ausloos M, Vandewalle N, Boveroux P H, et al. Application of statistical physics to economic and financial topics[J], Physica A,1999,274(1-2):229-240.
[143]Bouchaud J P, Potters M, Meyer M. Apparent multifractality in financial time series[J], European Physical Jornal of B,2000,13:595-599.
[144]Muzy J F, Delour J, Bacry E. Modeling fluctuation of financial time series:from cascade process to stochastic volatility model[J], European Physical Jornal of B,2000,17:537-548.
[145]Bandiera L, Cester A, Paccagnella A, et al. Detrended fluctuation analysis of the soft breakdown current[J], Microelectronic Engineering,2001,59(1):49-53.
[146]Govindan R B, Vjushin D, Brenner S, et al. Long-range correlations and trend in global climate models:Comparison with real data[J], Physica A,2001,294(1-2):239-248.
[147]Peng C K, Havlin S, Stanley H E, et al. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series[J], Chaos,1995,5(1):82-87.
[148]Mandelbrot B B. Fractals and scaling in finance:discontinuity, concentration, risk[M], New York: Springer Verlag,1997,146-197.
[149]Shimizu Y, Thurner S, Ehrenberger K. Multifractal spectra as a measure of complexity in human posture[J], Fractals,2002,10(1):103-116.
[150]Harvey C R. Predictable risk and returns in emerging markets[J], Review of financial studies,1995, 8(3):773-816.
[151]Jung C, Boyd R. Forecasting U K stock prices[J], Applied Financial Economics,1996,6:279-286.
[152]刘国旗.非线性模型GARCH模型在中国股市波动预测中的应用研究[J],统计研究,2000,1：49-52.
[153]王培勋.非线性回归的弹性分析在股票投资与行情预测中的应用[J],系统工程理论与实践,2000,8：100-104.
[154]吴微,陈维强,刘波.用BP神经网络预测股票市场涨跌[J],大连理工大学学报,2001,41(1)：9-15.
[155]孙全,朱江.基于遗传神经网络的股票价格短期预测[J],计算机工程与应用,2002,5：237-239.
[156]Chen A S, Mark L T, Hazem D. Application of neural networks to an emerging financial market: forecasting and trading the Taiwan stock index[J], Computer&Operation Research,2003, 30(6):901-923.
[157]吕淑萍,赵咏梅.基于小波神经网络的时间序列预测方法应用[J],哈尔滨工程大学学报,2001,41(1)：9-15.
[158]彭岩,孙学惠,王庆余.基于混沌时序重构的股票价格预测研究[J],天津大学学报,2003,36(6)：65-69.
[159]常松.基于小波包和神经网络的股票价格预测模型[J],中国管理科学,2001,5：8-15.
[160]孙丹,张秀艳.基于人工神经网络的股市预测模型[J],吉林大学学报(信息科学版),2002,20(4)：68-70.
[161]汤凌冰,廖福元,罗键.模糊神经网络在股价预测中的应用[J],系统工程,2004,22(2)：107-109.
[162]李焕荣,王树明.一种改进的BP神经网络预测方法及其应用[J],系统工程,2000,5：76-79.
[163]杨成,程晓玲,殷旅江.基于人工神经网络方法的上市公司股价预测[J]统计与决策.2005,12：106-108.
[164]高振坤,熊正德.基于神经网络技术的股指预测模型及实证分析[J],财经理论与实践,2005(1)：56-61.
[165]陈兴,孟卫东,严太华.基于T-S模型的模糊神经网络在股市预测中的应用[J],系统工程理论与实践,2001(2)：66-72.
[166]郑丕谔,马艳华.基于RBF神经网络的股市建模与预测[J],天津大学学报,2000(7)：483-486.
[167]杨一文,刘贵忠,张宗平.基于嵌入理论和神经网络技术的混沌数据预测及其在股票市场中的应用[J],系统工程理论与实践,2001(6)：52-58.