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Method
To illustrate the DFA method, we consider a noisy time series,
(
). We integrate the time series
,
![\begin{displaymath}
y(j) = \sum\limits_{i=1}^{j} (u(i) - <u>),
\end{displaymath}](img160.png) |
(1) |
where
![\begin{displaymath}
<u>=\frac{1}{N_{max}} \sum\limits_{j=1}^{N_{max}} u(i),
\end{displaymath}](img161.png) |
(2) |
and is divided into boxes of equal size,
. In each box, we
fit the integrated time series by using a polynomial function,
, which is called the local trend. For order-
DFA (DFA-1 if
, DFA-2 if
etc.),
order polynomial function
should be applied for the fitting. We detrend The integrated time
series,
by subtracting the local trend
in each box, and we calculate the detrended
fluctuation function
![\begin{displaymath}
Y(i) = y(i)-y_{fit}(i).
\end{displaymath}](img166.png) |
(3) |
For a given box size
, we calculate the root mean
square (rms) fluctuation
![\begin{displaymath}
F(n) =\sqrt{\frac{1}{N_{max}}\sum\limits_{i=1}^{N_{max}}\left [Y(i)\right ]^2}
\end{displaymath}](img167.png) |
(4) |
The above computation is repeated for box sizes
(different scales) to provide a relationship between
and
. A power-law relation between
and the box
size
indicates the presence of scaling:
. The parameter
,
called the scaling exponent or correlation exponent, represents the
correlation properties of the signal: if
, there
is no correlation and the signal is an uncorrelated signal (white
noise); if
, the signal is anticorrelated;
if
, there are positive
correlations in the signal.
Next: Noise with linear
trends Up: Effect
of Trends on Previous: Introduction
Zhi Chen 2002-08-28