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| The '''Hurst exponent''' is used as a measure of [[Long-range dependency|long-term memory]] of [[time series]]. It relates to the [[autocorrelation]]s of the time series, and the rate at which these decrease as the lag between pairs of values increases. | | The writer is known as Irwin Wunder but it's not the most masucline title out there. I used to be unemployed but now I am a librarian and the wage has been truly fulfilling. To play baseball is the hobby he will by no means stop doing. California is exactly where her std testing at home ([http://www.hard-ass-porn.com/blog/111007 previous]) is but she requirements to move simply because of her family. |
| Studies involving the Hurst exponent were originally developed in [[hydrology]] for the practical matter of determining optimum dam sizing for the [[Nile river]]'s volatile rain and drought conditions that had been observed over a long period of time.<ref>{{cite journal | first1 = H.E. | last1 = Hurst | journal = Trans. Am. Soc. Civ. Eng. | volume = 116 | page = 770 | year = 1951 }}</ref><ref>{{cite book | last1 = Hurst | first1 = H.E. | last2 = Black | first2 = R.P. | last3 = Simaika | first3 = Y.M. | year = 1965 | title = Long-term storage: an experimental study | publisher = Constable | location = London }}</ref> The name "Hurst exponent", or "Hurst coefficient", derives from [[Harold Edwin Hurst]] (1880–1978), who was the lead researcher in these studies; the use of the standard notation ''H'' for the coefficient relates to his name also.
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| In [[fractal geometry]], the '''generalized Hurst exponent''' has been denoted by [[H (disambiguation)|''H'']] or ''H<sub>q</sub>'' in honor of both Harold Edwin Hurst and [[Otto Ludwig Holder|Ludwig Otto Hölder]] (1859–1937) by [[Benoît Mandelbrot]] (1924–2010).<ref>{{cite journal | first1 = B.B. | last1 = Mandelbrot | first2 = J.R. | last2 = Wallis | journal = Water Resour. Res. | volume = 4 | page = 909 | year = 1969 }}</ref> ''H'' is directly related to [[fractal dimension]], ''D'', and is a measure of a data series' "mild" or "wild" randomness.<ref>{{cite paper | title = The (Mis)Behavior of Markets | first = Benoît B. | last = Mandelbrot | page = 187 }}</ref>
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| The Hurst exponent is referred to as the "index of dependence" or "index of long-range dependence". It quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.<ref>Torsten Kleinow (2002)[http://edoc.hu-berlin.de/dissertationen/kleinow-torsten-2002-07-04/PDF/Kleinow.pdf Testing Continuous Time Models in Financial Markets], Doctoral thesis, Berlin {{Page needed|date=September 2010}}</ref> A value ''H'' in the range 0.5–1 indicates a time series with long-term positive autocorrelation, meaning both that a high value in the series will probably be followed by another high value and that the values a long time into the future will also tend to be high. A value in the range 0 – 0.5 indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the value after that will tend to be high, with this tendency to switch between high and low values lasting a long time into the future. A value of ''H''=0.5 can indicate a completely uncorrelated series, but in fact it is the value applicable to series for which the autocorrelations at small time lags can be positive or negative but where the absolute values of the autocorrelations decay exponentially quickly to zero. This in contrast to the typically [[power law]] decay for the 0.5 < ''H'' < 1 and 0 < ''H'' < 0.5 cases.
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| ==Definition==
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| The Hurst exponent, ''H'', is defined in terms of the asymptotic behaviour of the [[rescaled range]] as a function of the time span of a time series as follows;<ref name="Qian">{{cite conference | first1 = Bo | last1 = Qian | first2 = Khaled | last2 = Rasheed | id = {{citeseerx|10.1.1.137.207}} | title = HURST EXPONENT AND FINANCIAL MARKET PREDICTABILITY | conference = IASTED conference on Financial Engineering and Applications (FEA 2004) | pages = 203–209 | year = 2004 }}</ref><ref name="Feder">{{cite book |title=Fractals |last=Feder |first=Jens |year=1988 |publisher=Plenum Press |location=New York |isbn=0-306-42851-2}}</ref>
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| :<math>\operatorname{E} \left [ \frac{R(n)}{S(n)} \right ]=C n^H \text{ as } n \to \infty \, ,</math>
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| where;
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| * <math>R(n)</math> is the [[range]] of the first <math>n</math> values, and <math>S(n)</math> is their [[standard deviation]]
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| * <math>\operatorname{E} \left [x \right ] \,</math> is the [[expected value]]
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| * <math>n</math> is the time span of the observation (number of data points in a time series)
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| * <math>C</math> is a constant.
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| ==Estimating the exponent==
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| To estimate the Hurst exponent, one must first estimate the dependence of the [[rescaled range]] on the time span ''n'' of observation.<ref name="Feder"/> A time series of full length ''N'' is divided into a number of shorter time series of length ''n'' = ''N'', ''N''/2, ''N''/4, ... The average rescaled range is then calculated for each value of ''n''. | |
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| For a (partial) time series of length <math>n</math>, <math>X=X_1,X_2,\dots, X_n \, </math>, the rescaled range is calculated as follows:<ref name="Qian"/><ref name="Feder"/>
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| 1. Calculate the [[mean]];
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| :<math>m=\frac{1}{n} \sum_{i=1}^{n} X_i \,.</math>
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| 2. Create a mean-adjusted series;
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| :<math>Y_t=X_{t}-m \quad \text{ for } t=1,2, \dots ,n \,. </math>
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| 3. Calculate the cumulative deviate series <math>Z</math>;
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| :<math>Z_t= \sum_{i=1}^{t} Y_{i} \quad \text{ for } t=1,2, \dots ,n \,. </math>
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| 4. Compute the range <math>R</math>;
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| :<math> R(n) =\operatorname{max}\left (Z_1, Z_2, \dots, Z_n \right )-
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| \operatorname{min}\left (Z_1, Z_2, \dots, Z_n \right ). </math>
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| 5. Compute the [[standard deviation]] <math>S</math>;
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| :<math>S(n)= \sqrt{\frac{1}{n} \sum_{i=1}^{n}\left ( X_{i} - m \right )^{2}}. </math>
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| 6. Calculate the rescaled range <math>R(n)/S(n)</math> and average over all the partial time series of length <math>n.</math>
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| The Hurst exponent is estimated by fitting the [[power law]] <math>\operatorname{E} \left [ \frac{R(n)}{S(n)} \right ]=C n^H</math> to the data.
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| This can be done by plotting the logarithm of <math>\operatorname{E} \left [ \frac{R(n)}{S(n)} \right ]</math> as a function of <math>\log n</math>, and fitting a straight line; the slope of the line gives <math>H</math>. Such a graph is called a pox plot. However, this approach is known to produce biased estimates of the power-law exponent. A more principled approach fits the power law in a maximum-likelihood fashion.<ref>{{cite journal |author=Aaron Clauset, Cosma Rohilla Shalizi, M. E. J. Newman |year=2009 |title=Power-law distributions in empirical data |journal=SIAM Review |volume=51 |pages=661–703 |arxiv=0706.1062 |doi=10.1137/070710111 }}</ref>
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| ==Generalized exponent==
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| The basic Hurst exponent can be related to the expected size of changes, as a function of the lag between observations, as measured by E(|''X<sub>t+τ</sub>-X<sub>t</sub>''|<sup>2</sup>). For the generalized form of the coefficient, the exponent here is replaced by a more general term, denoted by ''q''.
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| There are a variety of techniques that exist for estimating ''H'', however assessing the accuracy of the estimation can be a complicated issue. Mathematically, in one technique, the Hurst exponent can be estimated such that:<ref>[http://www.iop.org/EJ/abstract/1367-2630/11/9/093024/ Preis, T. et al. (2009) Accelerated fluctuation analysis by graphic cards and complex pattern formation in financial markets, New J. Phys. '''11''' 093024.]</ref><ref>
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| A.Z. Gorski et al. (2002) "Financial multifractality and its subtleties: an example of DAX", ''Physica'', 316 496 –510</ref>
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| :''H''<sub>''q''</sub> = ''H''(''q''),
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| for a time series
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| :''g''(''t'') (''t'' = 1, 2,...)
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| may be defined by the scaling properties of its [[Algebraic structure|structure]] functions ''S<sub>q</sub>''(<math>\tau</math>):
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| :<math>S_q = \langle |g(t + \tau) - g(t)|^q \rangle_t \sim \tau^{qH(q)}, \, </math>
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| where ''q'' > 0, <math>\tau</math> is the time lag and averaging is over the time window
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| :<math>t \gg \tau,\,</math>
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| usually the largest time scale of the system.
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| Practically, in nature, there is no limit to time, and thus ''H'' is non-deterministic as it may only be estimated based on the observed data; e.g., the most dramatic daily move upwards ever seen in a stock market index can always be exceeded during some subsequent day.<ref>[[Benoît Mandelbrot|Mandelbrot, Benoît B.]], ''The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward'' (Basic Books, 2004), pp. 186-195</ref>
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| ''H'' is directly related to [[fractal dimension]], ''D'', where 1 < ''D'' < 2, such that ''D'' = 2 - ''H''. The values of the Hurst exponent vary between 0 and 1, with higher values indicating a smoother trend, less volatility, and less roughness.
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| In the above mathematical estimation technique, the function ''H''(''q'') contains information about averaged generalized volatilities at scale <math>\tau</math> (only ''q'' = 1, 2 are used to define the volatility). In particular, the ''H''<sub>1</sub> exponent indicates persistent (''H''<sub>1</sub> > ½) or antipersistent (''H''<sub>1</sub> < ½) behavior of the trend.
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| For the BRW ([[brown noise]], 1/''f''²) one gets
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| :''H<sub>q</sub>'' = ½,
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| and for [[pink noise]] (1/''f'')
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| :''H<sub>q</sub>'' = 0.
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| The Hurst exponent for [[white noise]] is dimension dependant,<ref>{{cite journal |author=Alex Hansen, Jean Schmittbuhl, G. George Batrouni |year=2001 |title= Distinguishing fractional and white noise in one and two dimensions|journal=Phys. Rev. E |volume=63 |pages=062102 |arxiv=cond-mat/0007011 |doi=10.1103/PhysRevE.63.062102 }}</ref> and for 1D and 2D it is
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| :''H<sup>1D</sup><sub>q</sub>'' = -½ , ''H<sup>2D</sup><sub>q</sub>'' = -1.
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| For the popular [[Lévy stable process]]es and [[truncated Lévy process]]es with parameter α it has been found that
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| :''H<sub>q</sub>'' = ''q/α'' for ''q'' < ''α'' and ''H<sub>q</sub>'' = 1 for ''q'' ≥ α.
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| A method to estimate <math>H(q)</math> from non-stationary time series is called [[detrended fluctuation analysis]].<ref>{{cite journal|last=J.W. Kantelhardt|first=E. Koscielny-Bunde, H.A. Rego, S. Havlin, A. Bunde|title=Detecting long-range correlations with detrended fluctuation analysis|journal=Physica A: Statistical Mechanics and its Applications|year=2001|volume=295|pages=441–454|doi=10.1016/S0378-4371(01)00144-3|url=http://havlin.biu.ac.il/Publications.php?keyword=Detecting+long-range+correlations+with+detrended+fluctuation+analysis++&year=*&match=all}}</ref><ref>{{cite journal|last=J.W. Kantelhardt|first= S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H.E. Stanley|title=Multifractal detrended fluctuation analysis of nonstationary time series|journal=Physica A: Statistical Mechanics and its Applications|year=2001|volume=87|pages=316 |url=http://havlin.biu.ac.il/Publications.php?keyword=Multifractal+detrended+fluctuation+analysis+of+nonstationary+time+series&year=*&match=all}}</ref>
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| When <math>H(q)</math> is a non-linear function of q the time series in a [[multifractal system]].
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| ===Note===
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| In the above definition two separate requirements are mixed together as if they would be one.<ref>[http://www.uh.edu/~jmccaul2/ Joseph L McCauley], [http://complex.phys.uh.edu Kevin E Bassler], and Gemunu H. Gunaratne (2008) "Martingales, Detrending Data, and the Efficient Market Hypothesis", ''Physica'', A37, 202, Open access preprint: [http://arxiv.org/abs/0710.2583 arXiv:0710.2583]</ref> Here are the two independent requirements: (i) stationarity of the increments, x(t+T)-x(t)=x(T)-x(0) in distribution. this is the condition that yields longtime autocorrelations. (ii) [[Self-similarity]] of the stochastic process then yields variance scaling, but is not needed for longtime memory. E.g., both [[Markov process]]es (i.e., memory-free processes) and [[fractional Brownian motion]] scale at the level of 1-point densities (simple averages), but neither scales at the level of pair correlations or, correspondingly, the 2-point probability density.{{clarify|date=August 2011}}
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| An efficient market requires a martingale condition, and unless the variance is linear in the time this produces nonstationary increments, x(t+T)-x(t)≠x(T)-x(0). Martingales are Markovian at the level of pair correlations, meaning that pair correlations cannot be used to beat a martingale market. Stationary increments with nonlinear variance, on the other hand, induce the longtime pair memory of [[fractional Brownian motion]] that would make the market beatable at the level of pair correlations. Such a market would necessarily be far from "efficient".
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| ==See also==
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| * [[Long-range dependency]]
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| * [[Anomalous diffusion]]
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| * [[Rescaled range]]
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| * [[Detrended fluctuation analysis]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://www.scientio.com/Products/ChaosKit] Scientio's ChaosKit product calculates hurst exponents amongst other Chaotic measures. Access is provided online via a web service and Graphic user interface.
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| * [http://www.trusoft-international.com] TruSoft's Benoit - Fractal Analysis Software product calculates hurst exponents and fractal dimensions.
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| {{DEFAULTSORT:Hurst Exponent}}
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| [[Category:Stochastic processes]]
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| [[Category:Long-memory processes]]
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| [[Category:Fractals]]
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