As discussed above, a bounded time series can be mapped to a self-similar process by integration. However, another challenge facing investigators applying this type of fractal analysis to physiologic data is that these time series are often highly non-stationary (Fig. 1a). A simplified and general definition characterizes a time series as stationary if the mean, standard deviation and higher moments, as well as the correlation functions are invariant under time translation. Signals that do not obey these conditions are nonstationary. The integration procedure will further exaggerate the non-stationarity of the original data.
To overcome this complication, we have introduced a modified root mean square analysis of a random walk(Click here for a hands-on experiment of random walk)--termed detrended fluctuation analysis (DFA; can be download online) [11, 12]--to the analysis of biological data. Advantages of DFA over conventional methods (e.g., spectral analysis and Hurst analysis) are that it permits the detection of intrinsic self-similarity embedded in a seemingly nonstationary time series, and also avoids the spurious detection of apparent self-similarity, which may be an artifact of extrinsic trends. This method has been successfully applied to a wide range of simulated and physiologic time series in recent years [11, 12, 13, 14, 15, 16].
Please note that the DFA algorithm works better for certain types of nonstationary time series (especially slowly varying trends). However, it is not designed to handle all possible nonstationarities in real-world data.
To illustrate the DFA algorithm, we use the interbeat time series
shown in Fig. 3a as an example. First, the interbeat interval
time series (of total length N) is integrated,
, where B(i) is the i-th
interbeat interval and is the average interbeat
interval. As discussed above, this integration step maps the original
time series to a self-similar process. Next we measure the
vertical characteristic scale of the integrated time series. To do so,
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) (Fig. 4).
The y coordinate of the straight line segments is denoted by
. Next we detrend the integrated time series, y(k), by
subtracting the local trend, , in each box. For a given box
size n, the characteristic size of fluctuation for this integrated
and detrended time series is calculated by
(This quantity F is similar but not identical to the quantity s measured in the previous section.)
Figure 4: The integrated time series: , where B(i) is the interbeat interval shown in Fig. 3(a). The vertical dotted lines indicate boxes of size n=100, and the solid straight line segments represent the ``trend'' estimated in each box by a linear least-squares fit.
This computation is repeated over all time scales (box sizes) to provide a relationship between F(n) and the box size n. Typically, F(n) will increase with box size n. A linear relationship on a double log graph indicates the presence of scaling (self-similarity)--the fluctuations in small boxes are related to the fluctuations in larger boxes in a power-law fashion. The slope of the line relating log F(n) to log n determines the scaling exponent (self-similarity parameter), , as discussed previously.