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Stride-interval multifractality
Walking consists of a sequence of steps and the corresponding time series consists of
the time intervals for these steps, say, the time from one sequential striking of the right
heel to the ground to the next. In normal relaxed walking it is often assumed that the
stride interval is constant. However, it has been known for over a century that there is
variation in the stride interval of approximately 3%-4%. This random variability is so
small that the biomedical community has historically considered these fluctuations to be
an uncorrelated random process. In practice this means that the fluctuations in gait were
thought to convey no useful information about the underlying motor-control web. On
the other hand, Hausdorff et al .[ 8 ] demonstrated that stride-interval time series exhibit
long-time correlations and suggested that walking is a self-similar, fractal activity. In
their analysis of stride-interval data Scafetta et al .[ 25 ] concluded that walking is in fact
multifractal.
We apply the partition-function measure to a stride-interval time series [ 7 ] and numer-
ically evaluate the mass exponent using ( 5.89 ). In this analysis the partition function was
evaluated by constructing a random-walk sequence at a given level of resolution [ 32 ].
The results of this analysis are depicted in Figure 5.3 (a). Rigorously, the expression
for the mass exponent requires
0, but we cannot do that with the data, so there is
some error in the results depicted. In Figure 5.3 (a) we show the mass exponent averaged
over ten subjects, since their data individually do not look too different from the curve
shown. It is clear from the figure that the mass exponent is not linear in the moment
index q . In Table 5.1 we record the fitting coefficients for each of the ten time series
using the quadratic polynomial in the interval
δ
3
q
3,
a 2 q 2
τ(
q
) =
a 0 +
a 1 q
+
.
(5.100)
The fit to the data using ( 5.100 ) is indicated by the solid curve in Figure 5.3 (a).
Table 5.1. The fitting parameters for the scaling exponent τ( q ) are
listed for the stride-interval time series. The column − a 1 is the
fractal dimension for the time series. In each case these numbers
agree with those obtained elsewhere using a different method [ 7 ].
Wa lker
a 0
a 1
a 2
1
1 . 03
1 . 26
0 . 13
2
0 . 99
1 . 41
0 . 08
3
1 . 05
1 . 31
0 . 14
4
1 . 05
1 . 26
0 . 12
5
1 . 00
1 . 12
0 . 07
6
1 . 01
1 . 07
0 . 05
7
1 . 02
1 . 17
0 . 09
8
1 . 09
1 . 29
0 . 14
9
1 . 02
1 . 14
0 . 08
10
1 . 01
1 . 17
0 . 09
Average
1 . 03
1 . 19
0 . 10
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