The self-similarity parameter of an integrated time series is related
to the more familiar auto-correlation function, 
, of the
original (non-integrated) signal. Briefly:
[7, 17]. The auto-correlation function, ,
is 0 for any 
(time-lag) not equal to 0.
, 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.
greater than 0.5 and less than or equal to 1.0
indicates persistent long-range power-law correlations, i.e.,
. The relation between and
is 
. Note also that the power spectrum,
S(f), of the original (non-integrated) signal is also of a power-law
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
years--it corresponds to 1/f noise (
).
, power-law anti-correlations are present
such that large values are more likely to be followed by small values
and vice versa [10].
, correlations exist but cease to be
of a power-law 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 ``trade-off'' 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.