9780521861724htl 1..2

The Ho ̈ lder exponent or singularity strength ( (^) i) of theith cell is defined as:
(^) i¼logiðlÞ
logl
( 8 : 12 )
A greater value of (^) iis related to smallerpivalues (concentrations of) and vice
versa. The function (q) is the Ho ̈ lder spectrum and is computed as a function of
q-moments following Chhabra & Jensen ( 1989 ) as:
ðqÞ¼liml! 0
NPðlÞ
i¼ 1
qðlÞlogpiðlÞ
logl (^8 :^13 )
where
NPðlÞ
i¼ 1
qðlÞis the partition function (Halseyet al., 1986).
For the purpose of a direct and mathematically precise evaluation of the
singularity spectrum,f( ), is defined as:
fð ðÞÞ¼q lim
l! 0
NPðlÞ
i¼ 1
qðlÞlogqðlÞ
logl
( 8 : 14 )
and takes its maximum value forq¼0, displaying a parabolic shape around this
point. The Ho ̈ lder and singularity spectrum for each size-frequency distribution
was calculated following Equations 8.12 to 8.14, respectively, for values
 10 q10 and 0.1 lag increments using least squares linear regressions. As a
test criterion for multifractality, the linearity between the log-partition function
(Eq. (8.7)) and theq-moments must be fulfilled for a range ofq-moments. Besides
visual inspection of these relationships (suggested by Evertsz & Mandelbrot,
1992 ), coefficients of determination (r^2 ) of log-linear regression fits were obtained
to characterize their scaling quality. Moments withr^2 values0.95 were consid-
ered to be significant, and, thus, attributing to multifractal patterns. In addition,
the validity of the results was tested by verifying that the tangent of the graphf( )
versus at ¼1 is the bisector defined bydf( )/d ¼qwith the intersection point
corresponding tof( (1))¼ (1)¼D 1 (Evertsz & Mandelbrot, 1992 ).
In Fig.8.4,theBSDisgivenover300sizeclassesinincrementsof100mm (total
size range30 mm), while the BSD represented in the insets encompasses the
same number of size classes over 10mm increments (size range3 mm). Thus,
similar patterns of irregularity can be observed across different scales of resolution
illustrating inherent self-similar size-frequency distributions. In the three streams,
the mean number of species found per unit area declined with increasing body
size with marked irregularities at both resolution scales particularly evident in the
Cwm Llwch (LL) (Fig.8.4). The BSD displayed a multifractal pattern in all com-
munities ranging over a wide range of scales (allq-moments,r^2 >0.99; Fig.8.5;
Table8.3).ThewiderangeofHo ̈ lder exponents covered by thef( )-spectrum
BODY SIZE AND SCALE INVARIANCE 155