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when the mean exists. The preceding discussion of the Zipf and Pareto distributions
concerned time-independent inverse power-law complex webs. It is simple to show that
these distributions scale as
b
−
μ
ψ(
ψ(
)
=
),
bW
W
(2.80)
with the parameter
referred to as the scaling index, the scaling parameter or any
of a number of other terms. We interpret this scaling as meaning that increasing the
resolution of the dynamic variable by a factor
b
μ
1 is equivalent to retaining the present
resolution and decreasing the amplitude of the distribution by
b
−
μ
<
>
1for
μ>
1
.
2.4
Scaling physiologic time series
The previous subsections have stressed the importance of scaling, so now let us look at
some data. The question is how to determine from data whether the web from which
the data have been taken has the requisite scaling behavior and is therefore a candidate
for this kind of modeling. One method of analysis that is relatively painless to apply is
determined by the ratio of the variance to the mean for a given level of resolution and
the ratio is called the relative dispersion. The relation of the relative dispersion to the
level of aggregation of the data from a given time series is readily obtained. In fact it is
possible to use the expression for the relative dispersion to write a relation between the
variance and the mean that was obtained empirically by Taylor [
78
] in his study of the
growth in the number of biological species, that being
b
Va r
Q
(
n
)
=
aQ
(
n
)
,
(2.81)
where
n
denotes the level of resolution.
Taylor sectioned off a large field into plots and in each plot sampled the soil for
the variety of beetles present. In this way he was able to determine the distribution
in the number of new species of beetle that was spatially distributed across the field.
From the empirical distribution he calculated the mean and variance in the number of
new species. Subsequently he partitioned the field into smaller plots and carried out his
procedure again, calculating a new mean and variance. After
n
such partitionings Taylor
plotted the variances against the means for the different levels of resolution and found
a relation of the form (
2.81
). In the ecological literature a graph of the logarithm of the
variance versus the logarithm of the average value is called a
power curve
, which is
linear in logarithms of the two variables and
b
is the slope of the curve:
log Var
Q
(
n
)
=
b
log
Q
(
n
)
.
log
a
+
(2.82)
Taylor [
78
] developed his ideas for spatial distributions in which the resolution is
increased between successive measurements. However, for fractal data sets it should not
matter whether we increase the resolution or decrease it; the scaling behavior should be
the same. Consequently we expect that aggregating the time-series data should allow us
to utilize the insight obtained by Taylor in his study of speciation.
