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 yn(k). Next, we detrend the integrated time series, y(k), by subtracting the local trend, yn(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.
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