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quiet standing” by Collins and De Lucca [
14
]. Subsequently it was determined that
postural sway is really chaotic [
11
], so one might expect that there exists a relatively
simple dynamical model for balance regulation that can be used in medical diagnosis.
Here again the fractal dynamics can be determined from the scaling properties of
postural-sway time series and it has been established that a decrease of postural stability
is accompanied by an increase of fractal dimension.
2.5
Some afterthoughts
In this chapter we have discussed a number of interesting ideas, but the emphasis has
been on power-law scaling and the relation of that scaling to fractal processes both
theoretical and observational. It is useful to state explicitly the different ways in which
the power law has been used.
First of all, the power law refers to the power spectrum for random time series that
have fractal dimensions, namely those whose second moment may diverge. The autocor-
relation function for such processes increases algebraically with time so that its Fourier
transform, the power spectral density, decreases as an inverse power law with frequency.
The scaling index in this case characterizes the degree of influence present events have
on future events. The greater the index the more short term is the influence.
Second of all, the term inverse power law has also been applied to probability densi-
ties that asymptotically behave as inverse power laws in the web variable. These were
also called hyperbolic distributions and the variates are hyperbolic random variables.
The statistical behavior of such variables is very different from normal random pro-
cesses. In the “normal” case the data being acquired appear with increasing frequency
in the vicinity of the average value with very few data points falling very far away. In
many of the hyperbolic cases we mentioned the initial data are acquired locally with
a width determined by the parameter
a
. Every so often, however, a value of size
b
is
realized and then a cluster of data points of size
a
in that region is accumulated. Even
more rarely a data point of size
b
2
occurs and a cluster of values of size
a
in that
region is accumulated. In this way an ever-expanding set of data is accumulated with
holes between regions of activity and whose overall size is determined by the inverse
power-law index
The values of the web variable out in the tail of the
distribution dominate the behavior of such processes.
In the spatial domain the inverse power law is manifest through data points that
resemble clumps of sheep grazing in a meadow as opposed to the regular spacing of
trees in an orchard. In the time domain the heavy-tailed phenomena give rise to a pattern
of intermittency. However, even in a region of activity, magnification reveals irregularly
spaced gaps between times of activity having the same intermittent behavior down to
smaller and smaller time scales.
We closed this chapter with a review of how the notions of complexity reveal the
underlying structure in physiologic time series such as the time intervals between
heart beats, as well as those between breaths, and the variability of strides during
μ
=
log
a
/
log
b
.
