Geoscience Reference
In-Depth Information
A
B
2.2
0
−
−
2
2.0
1.8
−
−
6
1.6
1.4
−
4
−
2
02
−
5
−
3
−
1
1
log
2
(body size)
x
min
(log
2
body size)
C
α
All points -1.3
Without extremes -1.9
Without two extrems -2.9
Maximum likelihood -2.03
8
6
4
2
0
−
2
024
log
2
(body size)
Figure 4
Example of scaling exponent
a
estimated by cumulative probability distri-
bution (A), maximum likelihood (B), and binned histogram (C). (A) The cumulative
distribution estimates the probability that a randomly chosen individual has a body
size larger than or equal to a reference size. The parameter is estimated fitting an OLS
regression to the linear range of values selecting an x
min
value after which the power-
law regime occurs. (B) Maximum likelihood (ML) estimation also needs to identify an
x
min
value. A more objective way to find x
min
is to plot the estimated value of
a
as a
a
function of x
min
. In the range of values where
is stable (zero slope), the function
a
follows a power-law regime and the estimation of
is robust. To find this regime, we
calculated the absolute difference between consecutive estimations of
, fitted a
second order polynomial to this value as a function of x
min
, and obtained the x
min
at which a minimum is expected. The black horizontal line is the value of
a
a
estimated
with this procedure. (C) This panel shows estimates of
from a histogram; each point
is the log-transformed frequency of a size class. If all points are used,
a
is probably
underestimated because the smaller size classes show a different trend and the high
noise observed in the larger classes. When extreme classes are removed the estimated
a
approaches the value estimated by other regimens but the few points remaining lead
to a poor statistical estimation. The collapse of large amounts of information in a
single size class frequency, the use of extreme values with different distribution or high
noise could explain the discrepancy between binned and unbinned estimations of
a
.
a
