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The selfsimilarity parameter of an integrated time series is related
to the more familiar autocorrelation function, , of the
original (nonintegrated) signal. Briefly:

For white noise where the value at one instant is completely
uncorrelated with any previous values, the integrated value, y(k),
corresponds to a random walk and therefore
[7, 17]. The autocorrelation function, ,
is 0 for any (timelag) not equal to 0.

Many natural phenomena are characterized by shortterm correlations
with a characteristic time scale, , and an autocorrelation
function, that decays exponentially, [i.e., ]. The initial slope of vs.
may be different from 0.5, but will approach 0.5 for
large window sizes.

An greater than 0.5 and less than or equal to 1.0
indicates persistent longrange powerlaw correlations, i.e.,
. The relation between and
is . Note also that the power spectrum,
S(f), of the original (nonintegrated) signal is also of a powerlaw
form, i.e., . Because the power spectrum
density is simply the Fourier transform of the autocorrelation
function, . The case of is a
special one which has interested physicists and biologists for many
yearsit corresponds to 1/f noise ().

When , powerlaw anticorrelations are present
such that large values are more likely to be followed by small values
and vice versa [10].

When , correlations exist but cease to be
of a powerlaw form; indicates brown noise, the
integration of white noise.
The exponent can also be viewed as an indicator of the
``roughness'' of the original time series: the larger the value of
, the smoother the time series. In this context, 1/f noise
can be interpreted as a compromise or ``tradeoff'' between the
complete unpredictability of white noise (very rough ``landscape'')
and the much smoother landscape of Brownian noise [18].
In the next sections, we apply these scaling analyses to the output of
two complex integrated neural control systems, namely those regulating human
heart rate and gait dynamics in health and disease.
Next: Fractal Dynamics of Human
Up: Fractal Analysis Methods
Previous: Detrended Fluctuation Analysis (DFA)