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Figure 5.3.
(a) The mass exponent is calculated from stride-interval data using (
5.94
), indicated by the dots,
and the solid curve is the fit to the data obtained using (
5.100
). The singularity spectrum given in
(b) is determined from the data using (
5.93
) and is done independently of the fitting used in (a)
[
7
]. Reproduced with permission.
The singularity spectrum can now be determined using the Legendre transformation
by at least two different methods. One procedure is to use the fitting equation substituted
into (
5.93
). We do not do this here, but we note in passing that, if (
5.100
) is inserted into
(
5.92
), the fractal dimension is determined by the
q
=
0 moment to be
)
=−
τ
(
h
(
0
0
)
=−
a
1
.
(5.101)
The values of the parameter
a
1
listed in Table
5.1
agree with the fractal dimensions
obtained elsewhere by using a scaling argument for the same data [
7
].
A second method for determining the singularity spectrum is to numerically deter-
mine both
and its derivative. In this way we calculate the multifractal spectrum
directly from the data using (
5.93
). It is clear from Figure
5.3
(b) that we obtain the
canonical form of the spectrum; that is,
τ(
q
)
f
(
h
)
is a convex function of the scaling
