Stock Market Data Analysis with Relative Dispersion Based Hurst Exponent

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Stock Market Data Analysis with Relative
Dispersion Based Hurst Exponent

• Matthew Selnekovic
• Advisor Soundararajan Ezekiel
• Computer Science Department

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Thanks to

• Professor James Wolfe, Chairman of Computer
  Science Department at IUP
• My Advisor Dr. Soundararajan Ezekiel
• My research group, especially Jason Gruber, Gary
  Greenwood, and others

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Introduction

• Fractal based methods is a new and promising
  approach for the analysis of non-stationary
  signals, such as stock market data.
• Traditional Fourier analysis methods assume that
  the signals are stationary in temporal windows.
  Such an assumption is inappropriate for stock
  market data because it changes constantly.
• Fractal based methods do not impose this
  assumption and therefore are better suited for
  this analysis.
• We present a fractal dimension based method to
  analyze stock market data. We use the Hurst
  exponent to calculate fractal dimensions and to
  present experimental results demonstrating their
  effectiveness.

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Introduction

• The results suggest that fractal based techniques
  can provide useful information that is not
  available from traditional methods.
• The basic idea is to calculate the Hurst exponent
  for a stock market data by using relative
  dispersion analysis. The Hurst exponent, H, has a
  broad applicability to signal processing because
  of its robustness. It has a few underlying
  assumptions about the signal and it can
  distinguish between random and nonrandom signals.
  If H equals 0.5, then the signal is random.

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• Further, we can measure the level of noise from
  H that is, how much lower H is from 0.5.
• There are three different cases of H
• i) 0ltHlt0.5 This represents anti-persistence.
  This means, if a signal is up in the last period
  then more likely it will go down in the next
  period.
• ii) H0.5 This represents randomness that
  implies the values are uncorrelated.
• iii) 0.5ltHlt1 This represents persistence in
  this case, if the signal is up in the last
  period, then it is more likely that in the next
  period the signal will continue going up.
• The fractal dimension is then derived from the
  relation 2 H.

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Fractal dimension

• Ordinary dimension
• A line is one dimensional.
• A smooth surface is two dimensional.
• A solid is three dimensional.
• Fractal dimension
• A really rough surface dimension more than two
• Fractal dimension says exactly how much more.
• For example we may find a dimension of 2.4.

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Methods

• Method for assessing the fractal characteristics
  of time varying biological signals
• heart rate -- neural pulse trains -- .
• Such signals which vary, apparently irregularly,
  have been considered to be driven by external
  influences, which are random, that is to say just
  noise

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Methods

• To distinguish truly random influence from other
  types of irregularity
• we will examine two methods to one-dimensional
  signals or signals which are recorded over the
  time
• Rescaled Range analysis(R/S)
• Relative Dispersional analysis(RD)
• These methods help us to understand the phenomena
  and of making estimate of characterizing
  parameters

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Hurst (Father of the Nile River)

• Hurst was a hydrologist- began to work on the
  Nile River Dam project in about 1907
• He spent next 40 years studying almost 800 years
  of records of Nile River
• Problem An ideal reservoir would never overflow
• Discharge certain amount of water each year
• However if the influx from the river were too low
  gt water level would become dangerously too low
• How much discharge could be set, such that the
  reservoir never overflowed or emptied?

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Hurst Exponent

• Hurst observed that the records of flow or
  levels at the Roda gauge, near Cairo, did not
  vary randomly, but showed series of low-flow and
  high flow years.
• Definition The Hurst exponent H gives a measure
  of the smoothness of a fractal objects, with
  0ltHlt1. A low H indicates high degree of
  roughness, so much that the object almost fills
  the next-higher dimension a high H indicate
  maximal smoothness so that the object intrudes
  very little into the next higher dimension.

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Relative Dispersion Algorithm

• Step 1. Define the signal. Consider the simple
  case of a signal measured at even intervals of
  the independent variable, for example function of
  positions along a line
• Start with group size m, with m1
• Step 2. Calculate the mean of the whole set of
  observations, f

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• Step 3. Calculate the standard deviation SD of
  the set of n observations
• where the denominator is n-1, using the sample SD
  rather than the population SD where n would be
  used.
• Step 4. For the second grouping aggregate
  adjacent samples into groups of two data points
  and calculate the mean for each group, the SD for
  the group means and the RD for the group means.

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• Step 5. Repeat Step 4 with increasing group size
  until the number of group is small.
• Step 6. Plot the relationship between the number
  of data points in each group and the observed
  RDs. That is plot log-log plot. A plot of log of
  RD(m) versus the log of the number of the data
  points in each group.
• Step 7. Calculate fractal dimension FD2-slope

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Sample Signal
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Log-Log Fit
H0.3723 and FD 1.6277.
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Note

• The analysis of the relative dispersion (standard
  deviation divided by the mean)as a function of
  the element size used to make the analysis
• The Hurst analysis of the range of the cumulative
  deviations normalized by the standard deviation
  as function of the length of the signal
• The fractal dimension FD of the signal, and the
  Hurst parameter H which quantify the degree of
  correlation, can be determined from each of these
  methods of analysis

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• To illustrate our proposed method we applied it
  to 49 sets of daily closing prices of various
  stocks listed from the New York Stock Exchange
  and the NASDAQ from January 25, 1988 to January
  24, 2003.
• We then calculated the H value for each data set.

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AMD Hurst 0.5413 FD 1.4587
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MMM Hurst 0.3817 FD 1.5326
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IBM Hurst 0.5025 FD 1.4975
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FOX Hurst 0.4395 FD 1.5605
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Conclusion

• Since the H values are differing from 0.5, it
  shows that the stock market is clearly fractal.
  Further, H also measures how jagged the signal
  is. Low H values represent higher noise, and
  more random-like, or volatile data, thus
  representing a higher risk. A higher H value, on
  the other hand, indicates lower noise levels and
  less randomness, representing a lower risk.
• In our observation, AMD had an H value of 0.5413,
  while 3M had an H value of 0.3817.
• However, further experimental analysis needs to
  be carried out with different data sources,
  including international markets, bonds and mutual
  funds, commodities, and currency exchanges.

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