Estimating the Hurst Exponent

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Estimating the Hurst Exponent

Contents

Why is the Hurst Exponent Interesting

White Noise and Long Memory

The Hurst Exponent and Fractional Brownian Motion

The Hurst Exponent and Finance

The Hurst Exponent, Long Memory Processes and Power Laws

So who is this guy Hurst?

The Hurst Exponent, Wavelets and the Rescaled Range Calculation

Understanding the Rescaled Range (R/S) Calculation

Reservoir Modeling

Estimating the Hurst Exponent from the Rescaled RangeBasic Variations on the Rescaled Range Algorithm

Applying the R/S Calculation to Financial Data

Estimating the Hurst Exponent using Wavelet Spectral Density

Wavelet Packets

Other Paths Not Taken

Retrospective: the Hurst Exponent and Financial Time Series

C++ Software Source

Books

Web ReferencesRelated Web pages on www.bearcave.com

Acknowledgments

Why is the Hurst Exponent Interesting?

The Hurst exponent occurs in several areas of applied mathematics, including fractalsand chaos theory, long memory processes and spectral analysis. Hurst exponentestimation has been applied in areas ranging from biophysics to computer networking.

Estimation of the Hurst exponent was originally developed in hydrology. However, themodern techniques for estimating the Hurst exponent comes from fractal mathematics.

The mathematics and images derived from fractal geometry exploded into the world the1970s and 1980s. It is difficult to think of an area of science that has not beeninfluenced by fractal geometry. Along with providing new insight in mathematics andscience, fractal geometry helped us see the world around us in a different way. Nature isfull of self-similar fractal shapes like the fern leaf. A self-similar shape is a shapecomposed of a basic pattern which is repeated at multiple (or infinite) scale. Anexample of an artificial self-similar shape is the Sierpinski pyramid shown in Figure 1.
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Figure 1, a self-similar four sided Sierpinski pyramid

(Click on the image for a larger version) From theSierpinski Pyramidweb page onbearcave.com.

More examples of self-similar fractal shapes, including the fern leaf, can be found ontheThe Dynamical Systems and Technology Projectweb page at Boston University.

The Hurst exponent is also directly related to the "fractal dimension", which gives ameasure of the roughness of a surface. The fractal dimension has been used to measurethe roughness of coastlines, for example. The relationship between the fractaldimension,D, and the Hurst exponent,H, is

Equation 1

D = 2 - H

There is also a form of self-similarity calledstatistical self-similarity. Assuming that wehad one of those imaginary infinite self-similar data sets, any section of the data setwould have the same statistical properties as any other. Statistical self-similarity occursin a surprising number of areas in engineering. Computer network traffic traces are self-similar (as shown in Figure 2)

Figure 2, a self-similar network traffic

This is an edited image that I borrowed from a page on network traffic simulation. I'vemisplaced the reference and I apologize to the author.

Self-similarity has also been found in memory reference traces. Congested networks,where TCP/IP buffers start to fill, can show self-similar chaotic behavior. The self-similar structure observed in real systems has made the measurement and simulation ofself-similar data an active topic in the last few years.
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Other examples of statistical self-similarity exist in cartography (the measurement ofcoast lines), computer graphics (the simulation of mountains and hills), biology(measurement of the boundary of a mold colony) and medicine (measurement ofneuronal growth).

White Noise and Long Memory

Figure 3, A White Noise Process

Estimating the Hurst exponent for a data set provides a measure of whether the data is apure white noise random process or has underlying trends. Another way to state this isthat a random process with an underlying trend has some degree of autocorrelation(more on this below). When the autocorrelation has a very long (or mathematicallyinfinite) decay this kind of Gaussian process is sometimes referred to as a long memory

process.

Processes that we might naively assume are purely white noise sometimes turn out toexhibit Hurst exponent statistics for long memory processes (they are "colored" noise).One example is seen in computer network traffic. We might expect that network trafficwould be best simulated by having some number of random sources send random sized

packets into the network. Following this line of thinking, the distribution might bePoisson (an example of Poisson distribution is the number of people randomly arriving

at a resturant for a given time period). As it turns out, the naive model for networktraffic seems to be wrong. Network traffic is best modeled by a process which displaysa non-random Hurst exponent.

The Hurst Exponent and Fractional Brownian Motion

Brownian walks can be generated from a defined Hurst exponent. If the Hurst exponentis 0.5 < H < 1.0, the random process will be a long memory process. Data sets like thisare sometimes referred to as fractional Brownian motion (abbreviated fBm). FractionalBrownian motion can be generated by a variety of methods, including spectral synthesisusing either the Fourier tranform or the wavelet transform. Here the spectral density is

proportional to Equation 2 (at least for the Fourier transform):


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Equation 2

Fractional Brownian motion is sometimes referred to as 1/f noise. Since these randomprocesses are generated from Gaussian random variables (sets of numbers), they arealso referred to as fractional Gaussian noise (or fGn).

The fractal dimension provides an indication of how rough a surface is. As Equation 1shows, the fractal dimension is directly related to the Hurst exponent for a statisticallyself-similar data set. A small Hurst exponent has a higher fractal dimension and arougher surface. A larger Hurst exponent has a smaller fractional dimension and asmoother surface. This is shown in Figure 4.

Figure 4, Fractional Brownian Motion and the Hurst exponent

FromAlgorithms for random fractals, by Dietmar Saupe, Chapter 2 of The Science ofFractal Imagesby Barnsley et al, Springer-Verlag, 1988

The Hurst Exponent and Finance

Stock Prices and Returns

Random Walks and Stock Prices

A simplified view of the way stock prices evolve over time is that they follow a randomwalk.

A one dimensional random walk can be generated by starting at zero and selecting aGaussian random number.In the next step (in this case 1), add the Gaussian random
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number to the previous value (0). Then select another Gaussian random number and inthe next time step (2) add it to the previous position, etc...

step value

0 0

1 0 + R01 R0+ R1

2 R0+ R1+ R2

etc...

This model, that asset prices follow a random walk or Gaussian Brownian Motion,underlies the Black-Scholes model for pricing stock options (see Chapter 14, WienerProcesses and Ito's Lemma, Options, Futures and Other Derivatives, Eighth Edition,John C. Hull, 2012).

Stock Returns

One way to calculate stock returns is to use continuously compounded returns: r t=log(P t) - log(P t-1). If the prices that the return is calculated from follow GaussianBrownian Motion, the the returns will be normally distributed. Returns that are derivedfrom prices that follow Gaussian Brownian Motion will have a Hurst exponent of zero.

Stock Prices and Returns in the Wild

Actual stock prices do not follow a purely Gaussian Brownian Motion process. Theyhave dependence (autocorrelation) where the change at time t has some dependence onthe change at time t-1.

Actual stock returns, especially daily returns, usually do not have a normal distribution.The curve of the distribution will have fatter tails than a normal distribution. The curvewill also tend to be more "peaked" and be thinner in the middle (these differences aresometimes described as the "stylized facts of asset distribution").

My interest in the Hurst exponent was motivated by financial data sets (time series) likethe daily close price or the 5-day return for a stock. I originally delved into Hurstexponent estimation because I experimenting with wavelet compression as a method forestimating predictability in a financial time series (seeWavelet compression,

determinism and time series forecasting).

My view of financial time series, at the time, was noise mixed with predictability. I readabout the Hurst exponent and it seemed to provide some estimate of the amount of

predictability in a noisy data set (e.g., a random process). If the estimation of the Hurstexponent confirmed the wavelet compression result, then there might be some reason to

believe that the wavelet compression technique was reliable.

I also read that the Hurst exponent could be calculated using a wavelet scalogram (e.g.,a plot of the frequency spectrum). I knew how to use wavelets for spectral analysis,so Ithough that the Hurst exponent calculation would be easy. I could simply reuse the

wavelet code I had developed for spectrual analysis.
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Sadly things frequently are not as simple as they seem. Looking back, there are anumber of things that I did not understand:

1. The Hurst exponent is not so much calculated as estimated. A variety oftechniques exist for doing this and the accuracy of the estimation can be a

complicated issue.2. Testing software to estimate the Hurst exponent can be difficult. The best way to

test algorithms to estimate the Hurst exponent is to use a data set that has aknown Hurst exponent value. Such a data set is frequently referred to as

fractional brownian motion(or fractal gaussian noise). As I learned, generatingfractional brownian motion data sets is a complex issue. At least as complex asestimating the Hurst exponent.

3.

The evidence that financial time series are examples of long memory processesis mixed. When the hurst exponent is estimated, does the result reflect a longmemory process or a short memory process, like autocorrelation? Sinceautocorrelation is related to the Hurst exponent (see Equation 3, below), is this

really an issue or not?

I found that I was not alone in thinking that the Hurst exponent might provideinteresting results when applied to financial data. The intuitively fractal nature offinancial data (for example, the 5-day return time series in Figure 5) has lead a numberof people to apply the mathematics of fractals and chaos analyzing these time series.

Figure 5

Before I started working on Hurst exponent software I read a few papers on theapplication of Hurst exponent calculation to financial time series, I did not realize howmuch work had been done in this area. A few references are listed below.

Benoit Mandelbrot, who later became famous for his work on fractals, wrote theearly papers on the application of the Hurst exponent to financial time series.


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Many of these papers are collected in Mandelbrot's bookFractals and Scaling inFinance, Springer Verlag, 1997.

Edgar Peters' bookChaos and Order in the Capital markets, Second Editionspends two chapters discussing the Hurst exponent and its calculation using thethe rescaled range (RS) technique. Unfortunately, Peters only applies Hurst

exponent estimation to a few time series and provides little solid detail on theaccuracy of Hurst exponent calculation for data sets of various sizes.

Long-Term Memory in Stock Market Prices, Chapter 6 inA Non-Random WalkDown Wall Streetby Andrew W. Lo and A. Craig MacKinlay, PrincetonUniversity Press, 1999.

This chapter provides a detailed discussion of some statistical techniques toestimate the Hurst exponent (long-term memory is another name for a longmemory process). Lo and MacKinlay do not find long-term memory in stockmarket return data sets they examined.

In the paperEvidence of Predictability in Hedge Fund Returns and Multi-StyleMulti-Class Tactical Style Allocation Decisionsby Amenc, El Bied and Martelli,April 2002) the authors use the Hurst exponent as one method to analyze the

predictability of hedge funds returns. John Conover applies the Hurst exponent (along with other statistical

techniques) to the analysis of corporate profits. SeeNotes on the FractalAnalysis of Various Market Segments in the North American ElectronicsIndustry (PDF format) by John Conover, August 12, 2002.This is an 804 page(!) missive on fractal analysis of financial data, including the application of theHurst exponent (R/S analysis).

John Conover has an associated root web pageSoftware for Industrial MarketMetricswhich has links toNotes on the Fractal Analysis...and associatedsoftware source code (along with documentation for the source code).

The Hurst Exponent, Long Memory Processes and Power Laws

That economic time series can exhibit long-range dependence has been a hypothesis ofmany early theories of the trade and business cycles. Such theories were often motivated

by the distinct but nonperiodic cyclical patterns, some that seem nearly as long as theentire span of the sample. In the frequency domain such time series are said to have

power at low frequencies. So common was this particular feature of the data thatGranger (1966) considered it the "typical spectral shape of an economic variable." It hasalso been called the "Joseph Effect" by Mandelbrot and Wallis (1968), a playful but notinappropriate biblical reference to the Old Testament prophet who foretold of the sevenyears of plenty followed by the seven years of famine that Egypt was to experience.Indeed, Nature's predilection toward long-range dependence has been well-documentedin hydrology, meteorology, and geophysics...

Introduction to Chapter 6,A Non-Random Walk Down Wall Streetby Andrew W. Loand A. Craig MacKinlay, Princeton University Press, 1999.

Here long-range dependence is the same as a long memory process.
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Modern economics relies heavily on statistics. Most of statistics is based on the normalGaussian distribution. This kind of distribution does occur in financial data. Forexample, the 1-day return on stocks does closely conform to a Gaussian curve. In thiscase, the return yesterday has nothing to do with the return today. However, as thereturn period moves out, to 5-day, 10-day and 20-day returns, the distribution changes

to a log-normal distribution. Here the "tails" of the curve follow a power law. Theselonger return time series have some amount of autocorrelation and a non-random Hurstexponent. This has suggested to many people that these longer return time series arelong memory processes.

I have had a hard time finding an intuitive definition for the term long memory process,so I'll give my definition: a long memory process is a process with a randomcomponent, where a past event has a decaying effect on future events. The process hassome memory of past events, which is "forgotten" as time moves forward. For example,large trades in a market will move the market price (e.g., a large purchase order willtend to move the price up, a large sell order will tend to move the price downward).

This effect is referred to as market impact(seeThe Market Impact Modelby NicoloTorre, BARRA Newsletter, Winter 1998). When an order has measurable marketimpact, the market price does not immediately rebound to the previous price after theorder is filled. The market acts as if it has some "memory" of what took place and theeffect of the order decays over time. Similar processes allow momentum trading to havesome value.

The mathematical definition of long memory processes is given in terms ofautocorrelation. When a data set exhibits autocorrelation, a value x iat time tiiscorrelated with a value xi+dat time ti+d, where dis some time increment in the future. Ina long memory process autocorrelation decays over time and the decay follows apowerlaw.A time series constructed from 30-day returns of stock prices tends to show this

behavior.

In a long memory process the decay of the autocorrelation function for a time series is apower law:

Equation 3

In Equation 3, Cis a constant andp(k)is the autocorrelation function with lag k. The

Hurst exponent is related to the exponent alpha in the equation by

Equation 4

The values of the Hurst exponent range between 0 and 1. A value of 0.5 indicates a truerandom process (a Brownian time series). In a random process there is no correlation

between any element and a future element. A Hurst exponent valueH, 0.5


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autocorrelation). Here an increase will tend to be followed by a decrease. Or a decreasewill be followed by an increase. This behavior is sometimes called "mean reversion".

So who is this guy Hurst?

Hurst spent a lifetime studying the Nile and the problems related to water storage. Heinvented a new statistical method -- the rescaled range analysis(R/S analysis) -- whichhe described in detail in an interesting book,Long-Term Storage: An ExperimentalStudy(Hurst et al., 1965).

Fractalsby Jens Feder, Plenum, 1988

One of the problems Hurst studied was the size of reservoir construction. If a perfectreservoir is constructed, it will store enough water during the dry season so that it neverruns out. The amount of water that flows into and out of the reservoir is a random

process. In the case of inflow, the random process is driven by rainfall. In the case of

outflow, the process is driven by demand for water.

The Hurst Exponent, Wavelets and the Rescaled Range Calculation

As I've mentionedelsewhere,when it comes to wavelets, I'm the guy with a hammer towhom all problems are a nail. One of the fascinating things about the wavelet transformis the number of areas where it can be used. As it turns out, one of these applicationsincludes estimating the Hurst exponent. (I've referred to the calculation of the Hurstexponent as an estimate because value of the Hurst exponent cannot be exactlycalculated, since it is a measurement applied to a data set. This measurement will

always have a certain error.)

One of the first references I used wasWavelet Packet Computation of the HurstExponentby C.L. Jones, C.T. Lonergan and D.E. Mainwaring (originally published inthe Journal of Physics A: Math. Gen., 29(1996) 2509-2527). Since I had alreadydeveloped wavelet packet software, I thought that it would be a simple task to calculatethe Hurst exponent. However, I did not fully understand the method used in this paper,so I decided to implement software to calculate Hurst's classic rescaled range. I couldthen use the rescaled range calculation to verify the result returned by my waveletsoftware.

Understanding the Rescaled Range (R/S) Calculation

I had no problem finding references on the rescaled range statistic. AGooglesearch for

"rescaled range" hurstreturned over 700 references. What I had difficulty findingwere references that were correct and that I could understand well enough to implementthe rescaled range algorithm.

One of the sources I turned to was the bookChaos and Order in the Capital Markets,Second Edition, Edgar E. Peters, (1996). This book provided a good high leveloverview of the Hurst exponent, but did not provide enough detail to allow me toimplement the algorithm. Peters bases his discussion of the Hurst exponent on Chapters8 and 9 of the bookFractalsby Jens Feder (1988). Using Feder's book, and a little extra
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illumination from some research papers, I was able to implement software for theclassic rescaled range algorithm proposed by Hurst.

The description of the rescaled range statistic on this web page borrows heavily fromJens Feder's bookFractals. Plenum Press, the publisher, seems to have been purchased

by Kluwer, a press notorious for their high prices.Fractalsis listed on Amazon for $86.For a book published in 1988, which does not include later work on fractional brownianmotion and long memory processes, this is a pretty steep price (I was fortunate to

purchase a used copy onABE Books). Since the price and the age of this book makesProf. Feder's material relatively inaccessable, I've done more than simply summarize theequations and included some of Feder's diagrams. If by some chance Prof. Federstumbles on this web page, I hope that he will forgive this stretch of the "fair use"doctrine.

Reservoir Modeling

The rescaled range calculation makes the most sense to me in the context in which itwas developed: reservoir modeling. This description and the equations mirror JensFeder'sFractals. Of course any errors in this translation are mine.

Water from the California Sierras runs through hundreds of miles of pipes to the CrystalSprings reservoir, about thirty miles south of San Francisco. Equation 5 shows the

average (mean) inflow of water through those pipes over a time period .

Equation 5, average (mean) inflow over time

The water in the Crystal Springs reservior comes from the Sierra snow pack. Someyears there is a heavy snow pack and lots of water flows into the Crystal Springsreservoir. Other years, the snow pack is thin and the inflow to Crystal Springs does notmatch the water use by the thirsty communities of Silicon Valley area and San

Francisco. We can represent the water inflow for one year as . The deviation from

the mean for that year is (the inflow for year uminus the mean). Note that

the mean is calculated over some multi-year period . Equation 6 is a running sum of

the accululated deviation from the mean, for years 1 to .

Equation 6

I don't find this notation all that clear. So I've re-expressed equations 5 and 6 in pseudo-code below:
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Figure 6 shows a plot of the points X(t).

Figure 6

Graph by Liv Feder fromFractalsby Jens Feder, 1988

The range, is the difference between the maximum value X(tb) and the minimum

value of X(ta), over the time period . This is summarized in equation 7.

Equation 7

Figure 7 shows Xmaxthe maximum of the sum of the deviation from the mean, and X min,the minimum of the sum of the deviation from the mean. X(t) is the sum of the

deviation from the mean at time t.


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Figure 7

Illustration by Liv Feder fromFractalsby Jens Feder, 1988

The rescaled range is calculated by dividing the range by the standard deviation:

Equation 8, rescaled range

Equation 9 shows the calculation of the standard deviation.

Equation 9, standard deviation over the range 1 to

Estimating the Hurst Exponent from the Rescaled Range

The Hurst exponent is estimated by calculating the average rescaled range over multipleregions of the data. In statistics, the average (mean) of a data set Xis sometimes writtenas the expected value, E[X]. Using this notation, the expected value of R/S, calculatedover a set of regions (starting with a region size of 8 or 10) converges on the Hurstexponent power function, shown in Equation 10.

Equation 10

If the data set is a random process, the expected value will be described by a powerfunction with an exponent of 0.5.


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Equation 11

I have sometimes seen Equation 11 referred to as "short range dependence". This seemsincorrect to me. Short range dependence should have some autocorrelation (indicatingsome dependence between value xiand value xi+1). If there is a Hurst exponent of 0.5, itis a white noise random process and there is no autocorrelation and no dependence

between sequential values.

A linear regression line through a set of points, composed of the log of n(the size of theareas on which the average rescaled range is calculated) and the log of the averagerescaled range over a set of revions of size n, is calculated. The slope of regression lineis the estimate of the Hurst exponent. This method for estimating the Hurst exponentwas developed and analyzed by Benoit Mandelbrot and his co-authors in papers

published between 1968 and 1979.

The Hurst exponent applies to data sets that are statistically self-similar. Statisticallyself-similar means that the statistical properties for the entire data set are the same forsub-sections of the data set. For example, the two halves of the data set have the samestatistical properties as the entire data set. This is applied to estimating the Hurstexponent, where the rescaled range is estimated over sections of different size.

As shown in Figure 8, the rescaled range is calculated for the entire data set (hereRSave0= RS0). Then the rescaled range is calculated for the two halves of the data set,resulting in RS0and RS1. These two values are averaged, resulting in RSave1. In this

case the process continues by dividing each of the previous sections in half andcalculating the rescaled range for each new section. The rescaled range values for eachsection are then averaged. At some point the subdivision stops, since the regions get toosmall. Usually regions will have at least 8 data points.


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Figure 8, Estimating the Hurst Exponent

To estimate the Hurst exponent using the rescaled range algorithm, a vector of points iscreated, where xiis the log2of the size of the data region used to calculate RSaveiand yiis the log2of the RSaveivalue. This is shown in Table 1.

Table 1

region size RS ave. Xi: log2(region size) Yi: log2(RS ave.)

1024 96.4451 10.0 6.5916

512 55.7367 9.0 5.8006

256 30.2581 8.0 4.9193

128 20.9820 7.0 4.3911

64 12.6513 6.0 3.6612

32 7.2883 5.0 2.8656

16 4.4608 4.0 2.1573

8.0 2.7399 3.0 1.4541

The Hurst exponent is estimated by a linear regression line through these points. A line

has the form y = a + bx, where ais the y-intercept and bis the slope of the line. Alinear regression line calculated through the points in Table 1 results in a y-intercept of -0.7455 and a slope of 0.7270. This is shown in Figure 9, below. The slope is theestimate for the Hurst exponent. In this case, the Hurst exponent was calculated for asynthetic data set with a Hurst exponent of 0.72. This data set is from Chaos and Orderin the Capital markets, Second Edition, by Edgar Peters. The synthetic data set can bedownloaded here, as a C/C++ include filebrown72.h(e.g., it has commas after thenumbers).
http://www.bearcave.com/misl/misl_tech/wavelets/hurst/brown72.hhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/brown72.hhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/brown72.hhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/brown72.h


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Figure 9

Figure 10 shows an autocorrelation plot for this data set. The blue line is theapproximate power curve. The equations for calculating the autocorrelation function aresummarized on my web pageBasic Statistics
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Figure 10

Basic Variations on the Rescaled Range Algorithm

The algorithm outlined here uses non-overlapping data regions where the size of thedata set is a power of two. Each sub-region is a component power of two. Otherversions of the rescaled range algorithm use overlapping regions and are not limited todata sizes that are a power of two. In my tests I did not find that overlapping regions

produced a more accurate result. I chose a power of two data size algorithm because Iwanted to compare the rescaled range statistic to Hurst exponent calculation using thewavelet transform. The wavelet transform is limited to data sets where the size is a

power of two.

Applying the R/S Calculation to Financial Data

The rescaled range and other methods for estimating the Hurst exponent have beenapplied to data sets from a wide range of areas. While I am interested in a number ofthese areas, especially network traffic analysis and simulation, my original motivationwas inspired by my interest in financial modeling. When I first startedapplyingwavelets to financial time series,I applied the wavelet filters to the daily close pricetime series. As I discuss in a moment, this is not a good way to do financial modeling.In the case of the Hurst exponent estimation and the related autocorrelation function,using the close price does yield an interesting result. Figures 11 and 12 show the dailyclose price for Alcoa and the autocorrelation function applied to the close price timeseries.
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Figure 11, Daily Close Price for Alcoa (ticker: AA)

Figure 12, Autocorrelation function for Alcoa (ticker: AA) close price


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The autocorrelation function shows that the autocorrelation in the close price time seriestakes almost 120 days to decay below ten percent.

The Hurst exponent value for the daily close price also shows an extremely strong trend,where H= 0.9681 0.0123. For practical purposes, this is a Hurst exponent of 1.0.

I'm not sure what the "deeper meaning" is for this result, but I did not expect such a longdecay in the autocorrelation, nor did I expect a Hurst exponent estimate so close to 1.0.

Most financial models do not attempt to model close prices, but instead deal withreturns on the instrument (a stock, for example). This is heavily reflected in theliterature of finance and economics (see, for example,A Non-Random Walk Down WallStreet). The return is the profit or loss in buying a share of stock (or some other tradeditem), holding it for some period of time and then selling it. The most common way tocalculate a return is the log return. This is shown in Equation 12, which calculates thelog return for a share of stock that is purchased and held delta days and then sold. Here

Ptand Pt-deltaare the prices at time t(when we sell the stock) and t-delta(when wepurchased the stock, deltadays ago). Taking the log of the price removes most of themarket trend from the return calculation.

Equation 12

Another way to calculate the return is to use the percentage return, which is shown inEquation 13.

Equation 13

Equations 12 and 13 yield similar results. I use log return (Equation 12), since this isused in most of the economics and finance literature.

An n-day return time series is created by dividing the time series (daily close price for astock, for example) into a set of blocks and calculating the log return (Equation 12) forPt-delta, the price at the start of the block and P t, the price at the end of the block. The

simplest case is a 1-day return time series:

x0= log(P1) - log(P0)



...

xn= log(Pt) - log(Pt-1)

Table 2, below, shows the Hurst exponent estimate for the 1-day return time seriesgenerated for 11 stocks from various industries. A total of 1024 returns was calculatedfor each stock. In all cases the Hurst exponent is close to 0.5, meaning that there is nolong term dependence for the 1-day return time series. I have also verified that there isno autocorrelation for these time series either.


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Table 2, 1-day return, over 1024 trading days

Company Hurst Est. Est. Error

Alcoa 0.5363 0.0157

Applied Mat. 0.5307 0.0105

Boeing 0.5358 0.0077

Capital One 0.5177 0.0104

GE 0.5129 0.0096

General Mills 0.5183 0.0095

IBM Corp. 0.5202 0.0088

Intel 0.5212 0.0078

3M Corp. 0.5145 0.0081

Merck 0.5162 0.0075

Wal-Mart 0.5103 0.0078

The values of the Hurst exponent for the 1-day return time series suggest somethingclose to a random walk. The return for a given day has no relation to the return on thefollowing day. The distribution for this data set should be a normal Gaussian curve,which is characteristic of a random process.

Figure 13 plots the estimated Hurst exponent (on the y-axis) for a set of return timeseries, where the return period ranges from 1 to 30 trading days. These return timeseries were calculated for IBM, using 8192 close prices. Note that the Hurst exponentfor the 1-day return shown on this curve differs from the Hurst exponent calculated for

IBM in Table 2, since the return time series covers a much longer period.

Figure 13, Hurst exponent for 1 - 30 day returns for IBM close price


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Figure 13 shows that as the return period increases, the return time series have anincreasing long memory character. This may be a result of longer return periods pickingup trends in the stock price.

The distribution of values in the 1-day return time series should be close go a Gaussian

normal curve. Does the distribution change as the return period gets longer and the longmemory character of the return time series increases?

The standard deviation provides one measure for a Gaussian curve. In this case thestandard deviations cannot be compared, since the data sets have different scales (thereturn period) and the standard deviation will be relative to the data. One way tocompare Gaussian curves is to compare their probability density functions (PDF). ThePDF is the probabilty that a set of values will fall into a particular range. The probabilityis stated as a fraction between 0 and 1.

The probability density function can be calculated by calculating the histogram for the

data set. A histogram calculates the frequency of the data values that fall into a set ofevenly sized bins. To calculate the PDF these frequency values are divided by the totalnumber of values. The PDF is normalized, so that it has a mean of zero. The PDF forthe 4-day and 30-day return time series for IBM are shown in Figures 14 and 15.

Figure 14


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Figure 15

When expressed as PDFs, the distributions for the return time series are in the sameunits and can be meaningfully compared. Figure 16 shows a plot of the standarddeviations for the 1 through 30 day return time series.

Figure 16

As Figures 14, 15, and 16 show, as the return period increases and the time series gainslong memory character, the distributions move away from the Gaussian normal and get"fatter" (this is referred to as kurtosis).


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Estimating the Hurst Exponent using Wavelet Spectral Density

As I've noted above, wavelet estimation of the Hurst exponent is what started me on myHurst exponent adventure. My initial attempts at using the wavelet transform tocalculate the Hurst exponent failed, for a variety of reasons. Now that the classical R/S

method has been covered, it is time to discuss the wavelet methods.

The Hurst exponent for a set of data is calculated from the wavelet spectral density,which is sometimes referred to as the scalogram. I have covered wavelet spectraltechniques on a related web pageSpectral Analysis and Filtering with the WaveletTransform.This web page describes the wavelet "octave" structure, which applies in theHurst exponent calculation. The wavelet spectral density plot is generated from thewavelet power spectrum. The equation for calculating the normalized power for octavejis shown in Equation 14.

Equation 14

Here, the power is calculated from the sum of the squares of the wavelet coefficients(the result of the forward wavelet transform) for octavej. A wavelet octave contains 2jwavelet coefficients. The sum of the squares is normalized by dividing by 2j, giving thenormalized power. In spectral analysis it is not always necessary to use the normalized

power. In practice the Hurst exponent calculation requries normalized power.

Figure 17 shows the normalized power density plot for for the wavelet transform of1024 data points with a Hurst exponent of 0.72. In this case theDaubechies D4 waveletwas used. Since this wavelet octave number is the log2of the number of waveletcoefficients, the x-axis is a log scale.
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Figure 17

The Hurst exponent is calculated from the wavelet spectral density by calculating alinear regression line through the a set of {xj, yj} points, where xjis the octave and yjisthe log2of the normalized power. The slope of this regression line is proportional to theestimate for the Hurst exponent. A regression plot for the Daubechies D4 powerspectrum in Figure 17 is shown in Figure 18.

Figure 18

When you want a real answer, rather than a feeling for how change in one thing causeschange in something else, the statement that something isproportionalto something


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else is not very useful. The equation for calculating the Hurst exponent from the slopeof the regression line through the normalized spectral density is shown in Equation 15.

Equation 15

Table 3 compares Hurst exponent estimation with three wavelet transform to the Hurstexponent estimated via the R/S method. The Daubechies D4 is an "energy normalized"wavelet transform. Energy normalized forms of the Haar and linear interpolationwavelet are used here as well. An energy normalized wavelet seems to be required inestimating the Hurst exponent.

Table 3Estimating the Hurst Exponent for 1024 point synthetic data sets.

Wavelet Function H=0.5 Error H=0.72 Error H=0.8 Error

Haar 0.5602 0.0339 0.6961 0.0650 0.6959 0.1079

Linear Interp. 0.5319 0.1396 0.8170 0.0449 0.9203 0.0587

Daubechies D4 0.5006 0.0510 0.7431 0.0379 0.8331 0.0745

R/S 0.5791 0.0193 0.7246 0.0149 0.5973 0.0170

The error of the linear regression varies a great deal for the various wavelet functions.Although the linear interpolation wavelet seems to be a good filter for sawtooth waveforms, like financial time series, it does not seem to perform well for estimating theHurst exponent. The Daubechies D4 wavelet is not as good a filter for sawtooth waveforms, but seems to give a more accurate estimate of the Hurst exponent.

I have not discovered a way to determine prioriwhether a given wavelet function willbe a good Hurst exponent estimator, although experimentally this can be determinedfrom the regression error. In some cases it also depends on the data set. In Table 4 theHurst exponent is estimated from the Haar spectral density plot and via the R/Stechnique. In this case the Haar wavelet had a lower regression error than theDaubechies D4 wavelet.

Table 4Normalized Haar Wavelet and R/S calculation for 1, 2, 4, 8, 16 and 32 day

returns for IBM, calculated from 8192 daily close prices (split adjusted), Sept. 25, 1970to March 11, 2003. Data fromfinance.yahoo.com.

Number of

return values

Return period

(days)Haar estimate Haar error R/S estimate R/S error

8192 1 0.4997 0.0255 0.5637 0.0039

4096 2 0.5021 0.0318 0.5747 0.0043

2048 4 0.5045 0.0409 0.5857 0.0057

1024 8 0.5100 0.0543 0.5972 0.0084

512 16 0.5268 0.0717 0.6129 0.0115

256 32 0.5597 0.0929 0.6395 0.0166
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Note that in all cases the R/S estimate for the Hurst exponent has a lower linearregression error than the wavelet estimate.

Figure 19 shows the plot of the normalized Haar estimated Hurst exponent and the R/Sestimated Hurst exponent.

Figure 19, Top line, R/S estimated Hurst exponent (magenta), bottom line, Haarwavelet estimated Hurst (blue), 1, 2, 4, 8, 16, 32 day returns for IBM

Wavelet Packets

Measuring the quality of the Hurst estimation is no easy task. A given method can beaccurate within a particular range and less accurate outside that range. Testing Hurstexponent estimation code can also be difficult, since the creation of synthetic data setswith a known accurate Hurst exponent value is a complicated topic by itself.

Wavelet methods give more accurate results than the R/S method in some cases, but notothers. At least for the wavelet functions I have tested here, the results are not radically

better. In many cases the regression error of the wavelet result is worse than the errorusing the R/S method, which does not lead to high confidence.

There are many wavelet functions, in particular wavelets which have longer filters (e.g.,Daubechies D8) than I have used here. So the conclustions that can be drawn from theseresults are limited.

Thewavelet packet algorithmapplies the standard wavelet function in a morecomplicated context. Viewed in terms of thewavelet lifting scheme,the result of thewavelet packet function can be a better fit of the wavelet function. The wavelet packetalgorithm produces better results forcompression algorithmsand a better estimate

might be produced for the Hurst exponent. Although I have written and published C++
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code for the wavelet packet algorithm, I ran out of time and did not perform anyexperiments with in Hurst estimation using this software.

Other Paths Not Taken

The literature on the estimation of the Hurst exponent for long memory processes, alsoreferred to as fractional Brownian motion (fBm) is amazingly large. I have used asimple recursively decomposable version of the R/S statistic. There are variations onthis algorithm that use overlapping blocks. Extensions to improve the accuracy of theclassical R/S calculation have been proposed by Andrew Lo (seeA Non-Random Walk).Lo's method has been critically analyzed inA critical look at Lo's modified R/S statistic

by V. Teverovsky, M. Taqqu and W. Willinger, May 1998 (postscript format)

Some techniques build on the wavelet transform and reportedly yield more accurateresults. For example, inFractal Estimaton from Noisy Data via Discrete FractionalGaussian Noise (DFGN) and the Haar Basis, by L.M. Kaplan and C.-C. Jay Kuo, IEEE

Trans. on Signal Proc. Vol. 41, No 12, Dec. 1993 developes a technique based on theHaar wavelet and maximum likelyhood estimation. The results reported in this paper aremore accurate than the Haar wavelet estimate I've used.

Another method for calculating the Hurst exponent is referred to as the Whittleestimator, which has some similarity to the method proposed by Kaplan and Kuo listedabove.

I am sure that there are other techniques that I have not listed. I've found this afascinating area to visit, but sadly I am not an academic with freedom to persue whatever avenues I please. It is time to put wavelets and related topics, like the estimation ofthe Hurst exponent, aside and return to compiler design and implementation.

Retrospective: the Hurst Exponent and Financial Time Series

This is one of the longest single web pages I've written. If you have followed me thisfar, it is only reasonable to look back and ask how useful the Hurst exponent is for theanalysis and modeling of financial data (the original motivation for wandering into this

particular swamp).

Looking back over the results, it can be seen that the Hurst exponent of 1-day returns is

very near 0.5, which indicates a white noise random process. This corresponds withresults reported in the finace literature (e.g., 1-day returns have an approximatelyGaussian normal random distribution). As the return period gets longer, the Hurstexponent moves toward 1.0, indicating increasing long memory character.

In his book Chaos and Order in the Capital Markets, Edgar Peters suggests that a hurstexponent value H (0.5 < H < 1.0) shows that the efficient market hypothesis is incorrect.Returns are not randomly distributed, there is some underlying predictability. Is thisconclusion necessarily true?

As the return period increases, the return values reflect longer trends in the time series

(even though I have used the log return). Perhaps the higher Hurst exponent value isactually showing the increasing upward or downward trends. This does not, by itself,
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show that the efficient market hypothesis is incorrect. Even the most fanatic theorist atthe University of Chicago will admit that there are market trends produced by economicexpansion or contraction.

Even if we accept the idea that a non-random Hurst exponent value does damage to the

efficient market hypothesis, estimation of the Hurst exponent seems of little use when itcomes to time series forecasting. At best, the Hurst exponent tells us that there is a longmemory process. The Hurst exponent does not provide the local information needed forforecasting. Nor can the Hurst exponent provide much of a tool for estimating periodsthat are less random, since a relatively large number of data points are needed toestimate the Hurst exponent.

The Hurst exponent is fascinating because it relates to a several different areas ofmathematics (e.g., fractals, the Fourier transform, autocorrelation, and wavelets, toname a few). I have to conclude that the practical value of the Hurst exponent is lesscompelling. At best the Hurst exponent provides a broad measure of whether a time

series has a long memory character or not. This has been useful in research on computernetwork traffic analysis and modeling. The application of the Hurst exponent to financeseems more tenuous.

Afterward to the Afterward

I get email every once-in-a-while from people who want to use the Hurst exponent forbuilding predictive trading models. Perhaps I have not been clear enough or directenough in what I've written above. I've included an edited version of a recent responseto one of these emails here.

Years ago I read a computer science paper on a topic that I now only vaguelyremember. As is common with computer science papers, the author included his emailaddress along with his name in the paper title. I wrote to him that his technique seemedimpractical because it would take so long to calculate relative to other techniques. Theauthor wrote back: "I never claimed it was fast". This disconnect can be summed up asthe difference between the theorist and the practitioner.

There was a popular theoretical view that was promoted strongly by the economistEugene Fama that financial markets are random walks. It is from this view that the book

by Burton G. Malkiel,A Random Walk on Wall Street, takes its title. Malkiel's book and

the work of the Fama school also inspired the title of Andrew Lo and Criag MacKinlay'sbookA Non-Random Walk Down Wall Streetwhich provides strong statistical evidencethat the market is not, in fact a Gaussian random walk. Peters' book Chaos and Order in

Financial Marketsmake this case as well. As does the paper by Amenc et al (Evidenceof Predictability in Hedge Fund Teturns and Multi-Style Multi-Class Tactical Style

Allocation Decisions). In fact, I'd say that the evidence against purely Gaussian marketbehavior is now so strong that fewer and fewer people hold this view. The non-Gaussianbehavior of markets is, in fact, a criticism of the Black-Scholes equation for optionspricing.

These are all academic arguments that that there is predictability in the markets, which

goes against what Fama wrote through most of his career. But there is a differencebetween saying that there is predictability and the ability to predict.
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The Hurst exponent, among other statistics, shows that there is predictability infinancial time series. In an applied sense it can be useful if you have some transform ona time series and you want to see if you've increased or decreased the amount of

predictive information. But the Hurst exponent is not useful for prediction in any directway. Any more than the statistics for the distribution of returns is useful for

predictability, although these statistics can be useful in analzying the behavior of marketmodels.

The problems of the Hurst exponent include accurate calculation. Andrew Lo spills a lotof ink describing problems in accurately calculating the Hurst exponent and then spillsmore ink describing a techique that he believes is better (a conclusion that other authorsquestioned). No one that I've read has ever been able to state how large a data set youneed to calculate the Hurst exponent the Hurst exponent accurately (although I did see a

paper on using "bootstrap" methods for calculating the Hurst exponent with smaller datasets).

Again, the Hurst exponent can be a useful part of the practitioners statistical toolbox.But I do not believe that the Hurst exponent is a useful predictor by itself.

C++ Software Source

The C++ source code to estimate the Hurst exponent and to generate the data that wasused for the plots on this Web page can be downloaded by clickinghere.This file is aGNU zipped (gzip) tar file. For wierd Windows reasons this file gets downloaded ashurst.tar.gz.tar. To unpack it you need to change its name to hurst.tar.gz, unzip it (with

the command gzip -d hurst.tar.gz) and then untar it (with the command tar xf

hurst.tar

).

TheDoxygengenerated documentation can be foundhere.

If you are using a Windows system and you don't have these tools you can down loadthem here. This code is courtesy ofCygnusand is free software.

gzip.exe tar.exe

The wavelet source code in the above tarfile was developed before I started working on

the Hurst exponent, as was the histogram code and the code for wavelet spectrumcalculation. The statistics code was developed to support the Hurst exponent estimationcode. The entire code base is a little over 4,000 lines. The actual code to estimate theHurst exponent is a fraction of this.

Looking back over the months of nights and weekends that went into developing thiscode, it is odd that the final code is so small. This shows one difference betweenmathematics codes and data structure based codes (e.g., compilers, computer graphics,or other non-mathematical applications). In the case of non-mathematical applicationslarge data structure based frameworks are constructed. In the case of a mathematicscode, the effort goes into understanding and testing the algorithm, which may in the end

not be that large.
http://www.bearcave.com/misl/misl_tech/wavelets/hurst/hurst.tar.gzhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/hurst.tar.gzhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/hurst.tar.gzhttp://www.doxygen.org/http://www.doxygen.org/http://www.doxygen.org/http://www.bearcave.com/misl/misl_tech/wavelets/hurst/doc/index.htmlhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/doc/index.htmlhttp://www.bearcave.com/misl/misl_tech/wavelets/hurst/doc/index.htmlhttp://www.cygnus.com/http://www.cygnus.com/http://www.cygnus.com/http://www.bearcave.com/software/btp/gzip.exehttp://www.bearcave.com/software/btp/gzip.exehttp://www.bearcave.com/software/btp/tar.exehttp://www.bearcave.com/software/btp/tar.exehttp://www.bearcave.com/software/btp/tar.exehttp://www.bearcave.com/software/btp/gzip.exehttp://www.cygnus.com/http://www.bearcave.com/misl/misl_tech/wavelets/hurst/doc/index.htmlhttp://www.doxygen.org/http://www.bearcave.com/misl/misl_tech/wavelets/hurst/hurst.tar.gz


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Books

Chaos and Order in the Capital Markets,Second Edition, Edgar E. Peters, JohnWiley and Sons, 1996

Fractalsby Jens Feder, Plenum Press, 1988.

The Illustrated Wavelet Transform Handbook: Introductory Theory andApplications in Science, Engineering, Medicine and Finance, by Paul S.Addison, Institute of Physics Publishing, 2002

Chapter 7 of Prof. Addison's book discusses Hurst exponent calculation via thewavelet transform.

Self-similar Network Traffic and Performance Evaluationedited by Kihong Parkand Walter Willinger.

Research on computer network trafic and analysis has resulted in a blossoming

of techniques for Hurst exponent estimation and the simulation of long memoryprocesses (which resemble computer network traffic traces). In the area ofnetwork modeling this has been radical, overturning older queuing models,which have been used for almost twenty years. This book includes chapters

based on many of the core journal articles in this area. The book includes anintroductory chapter, providing background information. For anyone working onnetwork traffic modeling, this book is an important reference.

Web References

A Google search on the Hurst exponent will yield a large body of material. Here aresome references I found useful or interesting.

Hurst, the discoverer of the exponent that bears his name, studied power laws asthey related to Nile river floods. The Hurst exponent has been applied to avariety of other natural systems. A nicedefinition of the Hurst Exponentis

published in a very good mathematics glossary titledTerms Used in Time SeriesAnalysis of Cardiovascular Datafrom the Working Group on Blood Pressureand Heart Rate Variability, European Society of Hypertension, Centro DiBioingegneria, in Milan, Italy

Cameron L. Jones' Complex Systems Research Web Page

Cameron Jones seems to be a true polymath. He is an prolific researcher, whohas published on a range of topics from wearable computers to fractalmathematics. His work includes applications of fractal and wavelet mathematicsto a variety of areas in biology.

o Wavelet Packet computation of the Hurst exponent

From an implementation point of view, I found the description of thealgorithm to calculate the Hurst exponent in this paper somewhat murky.For a given wavelet and scaling function, the wavelet packet transform

provides a closer approximation of the data set, compared to the wavelettransform. The authors do not state whether this results in a more
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accurate estimation of the Hurst exponent, compared to the wavelettransform. Since the wavelet transform is faster and simpler to calculate,the wavelet packet transform should only be used if it yields asignificantly better result.

Scaling in the Market of Futuresby Enrico Scalas, November 25, 1997 Software for Programmed Trading of Equities Over the Internet

This is the root page for a discussion on using fractal analysis and othertechniques like classical information theory to analyze market information. JohnConover also publishes source code and the associated documentation.

Simply because one has an equation, or a mathematical technique, does notmean that the result is true and there are occasions where I find that JohnConover's conclusions are overly broad. For example, in his analysis of thesuccess ofLinux vs. Microsoftor in his analysis of thesuccess of "IT"

(software) projects.The factors that influence the success of Linux and thetimely completion of a software project are complex and cannot be described bysimple equations. For example, no simple equation can predict that IBM would(or would not) adopt Linux or that Dell would adopt Linux for their clustercomputers.

Fractal Analysis Programs of the National Simulation Resource

This site publishes code for generating a gaussian random walk series and forcalculating the Hurst exponent. These are Fortran codes.

Hurst's R/S-Analysis

A mathematical summary of Hurst exponent calculation

A Procedure to Estimate the Fractal Dimension of Waveformsby Carlos Sevcik,Complexity International, 1998

This paper outlines a method for calculating the Hurst exponent which is ratherunique. It is not the same as either the R/S method discussed in this paper or thespectral techniques (e.g., wavelets). I'm grateful to David Nicholls who pointed

me toward Carlos Sevcik's work.

The Hurst exponent seems to exert an attraction on people in a wide variety offields. Prof. Sevcik is a neuropharmacologist. His research interests includearthropod toxins (e.g.,. scorpion venom).

Self-similarity and long range dependence in Networks,a brief annotatedbibliography, byPaul BarfordandSally Floyd(Dr. Floyd is at theICIR NetworkResearch Center)

Long range order in network traffic dynamicsby J.H.B. Deane, C. Smythe andD.J. Jefferies, June 1996
http://citeseer.nj.nec.com/cache/papers/cs/20514/http:zSzzSzwww.ge.infm.itzSzeconophysicszSzpaperszSzgenoapaperszSzscalfutures.pdf/scaling-in-the-market.pdfhttp://citeseer.nj.nec.com/cache/papers/cs/20514/http:zSzzSzwww.ge.infm.itzSzeconophysicszSzpaperszSzgenoapaperszSzscalfutures.pdf/scaling-in-the-market.pdfhttp://www.johncon.com/ntropix/http://www.johncon.com/ntropix/http://www.johncon.com/john/correspondence/020217114704.27107.html#example-linuxhttp://www.johncon.com/john/correspondence/020217114704.27107.html#example-linuxhttp://www.johncon.com/john/correspondence/020217114704.27107.html#example-linuxhttp://www.johncon.com/john/correspondence/020218165107.2891.html#example-projecthttp://www.johncon.com/john/correspondence/020218165107.2891.html#example-projecthttp://www.johncon.com/john/correspondence/020218165107.2891.html#example-projecthttp://www.johncon.com/john/correspondence/020218165107.2891.html#example-projecthttp://nsr.bioeng.washington.edu/Software/NSR_SW_fractal.htmlhttp://nsr.bioeng.washington.edu/Software/NSR_SW_fractal.htmlhttp://www1.physik.tu-muenchen.de/~gammel/matpack/html/Mathematics/RSAnalysis.htmlhttp://www1.physik.tu-muenchen.de/~gammel/matpack/html/Mathematics/RSAnalysis.htmlhttp://journal-ci.csse.monash.edu.au/ci/vol05/sevcik/sevcik.htmlhttp://journal-ci.csse.monash.edu.au/ci/vol05/sevcik/sevcik.htmlhttp://www.cs.bu.edu/pub/barford/ss_lrd.htmlhttp://www.cs.bu.edu/pub/barford/ss_lrd.htmlhttp://www.cs.wisc.edu/~pb/http://www.cs.wisc.edu/~pb/http://www.cs.wisc.edu/~pb/http://www.icir.org/floyd/http://www.icir.org/floyd/http://www.icir.org/floyd/http://www.icir.org/http://www.icir.org/http://www.icir.org/http://www.icir.org/http://www.ee.surrey.ac.uk/Personal/D.Jefferies/Selfsim/htmlpaper.htmlhttp://www.ee.surrey.ac.uk/Personal/D.Jefferies/Selfsim/htmlpaper.htmlhttp://www.ee.surrey.ac.uk/Personal/D.Jefferies/Selfsim/htmlpaper.htmlhttp://www.icir.org/http://www.icir.org/http://www.icir.org/floyd/http://www.cs.wisc.edu/~pb/http://www.cs.bu.edu/pub/barford/ss_lrd.htmlhttp://journal-ci.csse.monash.edu.au/ci/vol05/sevcik/sevcik.htmlhttp://www1.physik.tu-muenchen.de/~gammel/matpack/html/Mathematics/RSAnalysis.htmlhttp://nsr.bioeng.washington.edu/Software/NSR_SW_fractal.htmlhttp://www.johncon.com/john/correspondence/020218165107.2891.html#example-projecthttp://www.johncon.com/john/correspondence/020218165107.2891.html#example-projecthttp://www.johncon.com/john/correspondence/020217114704.27107.html#example-linuxhttp://www.johncon.com/ntropix/http://citeseer.nj.nec.com/cache/papers/cs/20514/http:zSzzSzwww.ge.infm.itzSzeconophysicszSzpaperszSzgenoapaperszSzscalfutures.pdf/scaling-in-the-market.pdf


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A definition for the autocorrelation function is remarkably difficult to find.Apparently it has primarily been used in econometrics. This 1996 paper providesa definition for the autocorrelation function, along with a set of basic statisticsfor calculating the Hurst exponent (e.g., the rescaled range, the Fouriertransform periodogram (a.k.a., scalogram)).

Empirical evidence of long-range correlations in stock returnsby Pilar Grau-Carles, Physica A 287 (2000) 396-404

Physics of fashion fluctuationsby R. Donangelo, A. Hansen, K. Sneppen andS.R. Souza, Physica A 287 (2000) 539-545.

There does not seem to be enough money to fund research work for all thosewho get Phd degrees in physics. So we see physicists doing work in anincreasingly broad variety of areas from thetheory of self-organizing networksto quantitative finance. I looked at this article because I liked its title so much.As it turns out, the topic applies directly to Hurst exponent estimation. The

authors apply Hurst exponent estimation to fashions. By fashion, the authors arenot directly referring to skirt length or whether the double breasted suit is "in"this year, but to what are sometimes referred to as "fads": fashions - goods that

become popular not due to any intrinsic value, but simply because "everybodywants it".

The problem with this article is that fads are known to follow a power law (x -a).Whether the authors were measuring a long memory process or the power lawnature of fads is not clear (perhaps fads dohave a long memory character).

TISEAN Nonlinear Time Series AnalysisRainer Hegger, Holger Kantz, ThomasSchreiber

Estimation of the Hurst exponent can be classified as a non-linear time seriestechnique. TISEAN is a software package for non-linear time series analysis andincludes software for a variety of fractal techniques. This software grew out ofHolger Kantz and Thomas Schreiber's bookNon-linear Time Series Analysis,Cambridge University Press, 1997. Considering the topic, this book is veryreadable. The authors attempt to cover a great deal of material and they glossover some topics that have considerably more depth than the space given them.

Volatility processes and volatility forecast with long memoryby GillesZumbach. This paper is available on theOlsen Research Library - Resources inhigh-frequency financeOlsen is a hedge fund manager and provider ofquantitative finance software and data. The founder of Olsen is Richard Olsen,who is the author of an interesting bookIntroduction to High Frequency

Finance

The approach to long memory processes described here is not directly related toHurst exponent estimation. The techniques are drawn from quantitative finance.

Math and Statistics published on the Rice UniversityConnexions Project
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The Connexions Project at Rice University provides a set tools and a web siterepository for publishing educational content, especially university courses. TheRice UniversitySignals and Systemscourse, taught by Richard Baraniuk, and

published on the Connexions site, includes material on a variety of usefulstatistics (written by Michael Haag):

o Random Processes: Mean and Varianceo Correlation and Covariance of a Random Signalo Autocorrelation of Random Processes

The definition for autocorrelation appears to be incorrect on this webpage.

Chaos and Fractals in Financial Markets,Part 7, by J. Orlin Grabbe.

This web page is the last of a readable multi-part collection titledChaos &

Fractals in Financial Marketsby Orlin Grabbe. Mr. Grabbe apparently is aperson of varied interests. Chaos & Fractals in Financial Marketsis a sub-section ofMr. Grabbe's web site(which is also mirrored in the Neatherlands onwww.xs4all.nl/~kalliste). The site includes links toThe Laissez Faire ElectronicTimes,which Mr. Grabbe publishes. After skimming some of the articles in this

publication, I'm not quite sure how to characterize it. It seems to have a definitelibertarian bent. Some of the articles are not particularly well written and can't

be compared to Mr. Grabbe's writing. Mr. Grabbe's web site also includes whatseem to be his own version of the British tabloid "page 3 girls". Given the styleof photography, these look like they are "borrowed" from a publication likePlayboy.

Whether someone is a crank or not is probably a matter of opinion. As far as I'mconcerned, one of the measures of "crankdom" is whether you believe (as theeditoral page writers at the Wall Street Journal did) that Vince Foster wasmurdered. Vince Foster was a White House Counsel during the early part of theClinton administration. In an interview with journalist Dan Moldea, who wrotethe bookA Washington Tragedy, Lori Leibovich writes inSalon:

On July 20, 1993, Vincent Foster was found dead at Ft. Marcy Park inNorthern Virginia with a .38-caliber revolver in his hand. An autopsy

revealed that it was a straight-ahead suicide -- Foster had placed the gunin his mouth and fired one shot that blasted through his head. End ofstory? Not by a long shot. Nearly five years later, the Foster suicide liveson in the hearts and minds of right-wing Clintonphobes and conspiracytheorists who believe that Foster, a close friend and advisor of the

president, was murdered because he knew too much.

...

In the resulting book, "A Washington Tragedy: How the Death ofVincent Foster Ignited a Political Firestorm," Moldea confirms -- again

that Foster's death was indeed a suicide and that a cabal of right-winggroups -- financed by banking heir Richard Mellon Scaife is responsible
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I am grateful to Eric Bennos who sent me a copy ofTime Series Prediction UsingSupervised Learning and Tools from Chaos Theory(PDF) byAndrew N. Edmonds,PhdThesis, Dec. 1996, University of Luton. It was this thesis that sparked my interest inusing the Hurst exponent to estimate the predictability of a time series. The journey has

been longer than I intended, but it has been interesting.

Credits

The plots displayed on this web page were generated with GnuPlot. Equations werecreated using MathType.

Ian KaplanMay 2003Revised: May 2013

Back toWavelets and Signal Processing
http://www.scientio.com/resources/thesis.pdfhttp://www.scientio.com/resources/thesis.pdfhttp://www.scientio.com/resources/thesis.pdfhttp://www.scientio.com/resources/thesis.pdfhttp://www.scientio.com/about.aspxhttp://www.scientio.com/about.aspxhttp://www.scientio.com/about.aspxhttp://www.bearcave.com/misl/misl_tech/wavelets/index.htmlhttp://www.bearcave.com/misl/misl_tech/wavelets/index.htmlhttp://www.bearcave.com/misl/misl_tech/wavelets/index.htmlmailto:[email protected]://www.bearcave.com/misl/misl_tech/wavelets/index.htmlhttp://www.scientio.com/about.aspxhttp://www.scientio.com/resources/thesis.pdfhttp://www.scientio.com/resources/thesis.pdf

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