MFDFA

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• ????
• ??????????? ?????? ?? ??????? ??????
• MFDFA
• ???????-?????? ???? ????????? ???????
• Grey forecasting ? ?????? ???????
• ???????????? ????

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Delay reconstruction ?? ???????? ???????????? ??
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???? ? ???????? ???? ???? Takens delay embedding
theorem (Sauer-Yorke-Casdagli version)
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MFDFA ???????? ????? ? ?????? ?? ???? ??? (R-S
analysis and the Hurst exponent)

• Once upon a time, a British government bureaucrat
  named Harold Edwin Hurst studied 800 years of
  records of the Nile's flooding. He noticed that
  there was a tendency for a high flood year to be
  followed by another high flood year, and for a
  low flood year to be followed by another low
  flood year.
• Was that accidental ... or was there really some
  correlation between levels? Did the height at
  year 5 have an effect on the height in year 6?

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??????????? ?? ?????

• To analyze, we might do something like this
• Note the heights of the n flood levels      
  h(1), h(2), ... h(n)
• Let m be the Mean of these levels       M
  (1/n) h(1)h(2)...h(n)
• Calculate the deviations from the mean      
  x(1) h(1) - M       x(2) h(2) - M       ...
        x(n) h(n) - M Note that the set of xs
  have zero mean. Positive x's indicate that the
  Nile level was above the average.
• Now calculate the Sums       Y(1) x(1)      
  Y(2) x(1) x(2)       ...       Y(n) x(1)
  x(2) ... x(n) Note that the set of partial
  sums, the Y's, are sums of zero-mean variables.
  They will be positive if there's a preponderance
  of positive x's. Note, too, that Y(k) Y(k-1)
  x(k).
• Let R(n) MAXY(k) - MINY(k) This difference
  between the maximum and minimum of the n values
  is called the Range
• Let s(n) be the standard deviation of the set of
  n h-values.

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??????????? ?? ????? (2)

• As it turns out, the probability theorist William
  Feller proved that if a series of random
  variables (like the x's) had finite standard
  deviation and were independent, then the
  so-called R/s statistic (formed over n
  observations) would increase in proportion to
  n1/2 (for large values of n).
• We now have       R(n) / s(n) kn1/2    ...
  where k is some constant
• If that were true, then we'd expect that
•       log(R/s ) log(k) (1/2) log(n)
• So, if we were to plot log(R/s ) vs log(n), we'd
  expect it to be approximately a straight line
  with slope (1/2).
• gtA logarithm to what base? ???? ???????? !

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????? ????? ?? ??????? 0.5 ? ???? ?????...

• Anyway, what Hurst apparently found, was that the
  plot had a slope closer to 0.7 (rather than 0.5).
  
• gtSo, what's that mean? I guess it means that the
  annual Nile levels weren't independent, but this
  year's level might be expected to affect next
  year's level. Indeed, if the slope of the log(R
  / s ) vs log(n) "best fit line" is H, then we'd
  expect       R / s knH
• That H is the Hurst Exponent, right?
• ??!

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The Hurst Exponent and Fractional Brownian Motion

• Brownian walks can be generated from a defined
  Hurst exponent. If the Hurst exponent is 0.5 lt H
  lt 1.0, the random walk will be a long memory
  process. Data sets like this are sometimes
  referred to as fractional Brownian motion
  (abbreviated fBm).
• Fractional Brownian motion is sometimes referred
  to as 1/f noise. Since these random walks are
  generated from Gaussian random variables (sets of
  numbers), they are also referred to as fractional
  Gaussian noise (or fGn).

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The Hurst Exponent and Fractional Brownian Motion
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???????????? ????????? ?? ?????, ?????????????
????????? ?? ??????? - MFDFA
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MFDFA (2)
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MFDFA (3)
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MFDFA (4)
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?????? ?? ?????????? ?? MFDFA
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?????????? ?? MFDFA (2)
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?????????? ?? MFDFA (3)
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?????????? ?? MFDFA (4)
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????? ?????? ???? ????????? ???????
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??? ????? ??????
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??????? ??? ?????????? ????????? Draupner
(01.01.1995)
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??????????????? ?? Weibull ? ??????? ???
??????????? Draupner

• ????????? ??? ? ????????????? ??????????? ????
  ?????????? ????????? ?? ????????-??????? ??
  ??????????? ?????? ??????? ? ?????-???? ???????
  (1000 ?????) ???????? ????

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?????? ?? ????????????? ? ???????? ??????????

• The clustering index T 1
• Estimation of T runs estimator, interexceedance
  times estimator
• Runs estimator

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Interexceedances times estimator of the
clustering index
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????? ? ??????????????? ?? ??????????? ?????????
quantile-quantile plots
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Quantile-quantile plots (2)
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Quantile-Quantile plots (3)

• Q standard exponential quantile
• Q time series quantile
• Presence of heavy tail

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??? ???? ????????????? ?? ??????????? ???????
Weibull, Gumbel, Frechet - Pareto
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Quantile-quantile plot calculation for Weibull
distribution
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Q-Q plots Gumbel vs. time series quantile

• ????????????? ?? ???????
• ?? !

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Q-Q plots Frechet-Pareto vs. time series quantile

• ????????????? ?? ????? ???????
• ?? !

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??? ?????? Weibull ?

• ??!

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Distribution of maxima of time series fitted by
the Weibull distribution

• F 1-exp(-? xr)
• ? 0.148
• r 1.468

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Probabilities for rogue waves at the Draupner
platform

• Height
• 18.45 m
• 19 m
• 21 m
• 23 m
• 25 m
• 27 m
• 29 m
• 31 m

• Probability
• 0.0000230
• 0.0000143
• 0.00000246
• 0.000000390
• 0.000000057
• 0.00000000776
• 0.000000000985
• 0.000000000117

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Grey forecasting ? ?????? ??????? (?????????
????? ?? ?????????????)

• ????? ?????? ?? ?? ????????? ?? ???? ?????????
  2010 ??

??????? ????????? ??????? ????? ??????? ? ???? ????????? ????
?????? ????????? 1082
?????? ?????? ????????? ????? ?????? 1562 (2 x 781)
?????? ?????? ????????? ????? ?????? 2042 (1562480)
????-???????????? ?? ??? 2522(2042480)
???????? 3002 (2522480)
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  ?????? ?? ???????? ?? ????? 2004 ? ?? ?????? 2009
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Grey forecasting method
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Grey forecasting method (2)
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????????? ????? ? ??????? GFM ?????
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?? ?????????

• Dark side of the Force (THE NONLINEAR TIME SERIES
  ANALYSIS) is pathway to many abilities some
  consider to be unnatural.

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??????? ?????? !
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??? ????? ? ?? ?????????????? ?????? ? wavelets !
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