### Detrended Fluctuation Analysis (DFA)

The method of detrended fluctuation analysis has proven useful in
revealing the extent of long-range correlations in time series.
Briefly, the time series to be analyzed (with *N* samples) is
first integrated. Next, the integrated time series is divided into
boxes of equal length, *n*. In each box of length *n*, a
least squares line is fit to the data (representing the *trend*
in that box). The *y* coordinate of the straight line segments
is denoted by *y*_{n}(k).
Next, we detrend the integrated time
series, *y*(*k*), by subtracting the local trend,
*y*_{n}(k),
in each box. The root-mean-square fluctuation of this integrated and
detrended time series is calculated by

This computation is repeated over all time scales (box sizes) to
characterize the relationship between *F*(*n*), the average
fluctuation, as a function of box size. Typically, *F*(*n*)
will increase with box size *n*. A linear relationship on a
log-log plot indicates the presence of power law (fractal)
scaling. Under such conditions, the fluctuations can be characterized
by a scaling exponent , the slope of the line relating log *F(n)* to
log *n*.

Learn more about DFA

Click here to
download the detrended fluctuation analysis(DFA) software package
discussed here.