Biomedical Engineering Reference
In-Depth Information
Multifractal spectrum
Estimated Turbulence
Estimated fBm
Theoretical fBm
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a
(q)
Fig. 3. Multifractal Spectra for mono- (dashed line) and multi-fractal (solid line) pro-
cesses. The dotted line indicates the theoretical slope of the spectrum for an fBm process
(mono-fractal) with a Hurst exponent of 1=3
in terms of Holder regularity indices 's.
This denition is also graphically presented in Figure 4. The deviation
from the monofractal could be fairly quantied using this Broadness mea-
sure since it posts a universal standard on the width spread. It is worth
to point out the threshold value 0.2 used in this denition could be ad-
justed empirically in the practice analysis to ensure that this measure is
well dened for all analyzed processes.
As mentioned earlier, the discreteness may produce diculties in the
computation. The problem is that it may be hard to nd the exact roots
of the equation f() + 0:2 = 0 among the discrete values of 's. To get
around this, we try to nd the minimum value ofjf() + 0:2jwith respect
of instead of solving the equation directly.
Applying our idea about extracting the spectral features to the PRB
measurements, we obtain the Broadness, spectral Mode (Hurst exponent),
and left Slope for each measurement. Table 2 summarizes the spectral char-
acteristics of the PRB datsets that we are using in our study. We will use
this result in Section 4.
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