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from limited amounts of data (
Newman, 2005; White et al., 2008
). To
overcome this problem, the use of logarithmic bins has been suggested as
an alternative procedure for histogram construction. The method considers
making counts over bins of increasing width towards higher size classes, in an
effort to minimize the number of bins having zero-counts, therefore gaining
accuracy (
Newman, 2005
). The number of observations recorded in each size
class is then standardized dividing counts by the interval width size, in order
to transform frequency into a density estimate per unit interval (
Newman,
2005
). In spite of these improvements, binned methods were designed to
summarize data and reduce pattern complexity by collapsing large amounts
of data into a single measure of frequency. In turn, this has led to a poor use
of available information when trying to make precise estimations of distribu-
tional parameters or detect slope regime shifts (see
Clauset et al., 2009;
Edwards, 2008; Newman, 2005; White et al., 2008
). In addition, comparisons
among methods and their performance have shown that, in spite of their
widespread use, binned approaches can be biased and imprecise even when
considering large sample sizes (see
Clauset et al., 2009; Edwards, 2008; White
et al., 2008
). Recent advances in the estimation of parameters are based on
the analysis of the inverse cumulative distribution and on maximum likeli-
hood (ML;
Clauset et al., 2009; Edwards, 2008; White et al., 2008
).
Most ecological studies on power-law-like distributions used binned meth-
ods (
White et al., 2008
). However, the poor performance of binned
approaches in comparison to unbinned methods suggests that several
reported patterns could be biased (see
Clauset et al., 2009; Edwards, 2008;
White et al., 2008
; and the sections below). Unbinned methods make a much
more efficient use of available data for the estimation of parameters. The
optimum use of information and the development of ML estimations proba-
bly account for their accurate performance with unbiased and precise esti-
mation, even with small sample sizes (
Edwards, 2008; White et al., 2008
). The
cumulative distribution function estimates the probability that an individual
has a body mass greater than or equal to a reference size (see
Figure 4
A).
Importantly, if the original data follow a power-law distribution P(M)
M
a
M
(
a
1)
with an exponent biased in one unit with respect to the non-cumulative value
(
Newman, 2005
). It should be highlighted that the value P(M) can be esti-
mated for each one of the body masses measured, so no binning of data needs
to be involved and therefore every single measure of body size can be used to
estimate the exponent parameter. This represents a substantial improvement
if considering that data binning usually involves the aggregation of hundreds
or even thousands of measurements into a single frequency interval.
The cumulative function can also be plotted to visualize the relationship,
which is called the rank/frequency plot (
Newman, 2005
). Each individual
is ranked according to its mass in descending order, placing its mass
then its cumulative distribution also follows a power-law P(M)
