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2018 ITU Kaleidoscope Academic Conference
units. Figure 3b illustrates the complete construction of the 3.2 Extension of the variance estimator and
time series for Channel 1 with a length of more than 3500 statistical moments
packages. Finally, the same process is repeated for all
channels, leading to the chart in Figure 3c which shows the By extending the estimator in Eq. (1) for statistical moments
behavior of the entire radioelectric spectrum. to q ≠ 2 and considering positive and negative values of q in
a set of real numbers, the analysis of the stochastic tool
3. MULTIFRACTAL SPECTRUM CALCULATION discussed in the preview section can be broadened. By
denoting q as the order of the estimator, Equation (3) can be
3.1 Log-scale diagram and Hurst parameter derived as:
calculation
Ɋ ൌ σ ͳ ȁ ሺǡሻȁ ሺ͵ሻ
The newly obtained time series is analyzed with the help of ൌͳ
probabilistic tools. Firstly, the Log-scale Diagram (LD) is
used to analyze the time series at different scales. Therefore, In a self-similar processes with 0.5 < H < 1, the estimator
the series is split into logarithmic scales and a detail follows the power law and can be extended to Equation (4)
coefficient is estimated for each scale. A correlation between according to [20]
the detail coefficients ensures that the temporal estimators
have minimum variance. Additionally, it is estimated that the ሾȁ ሺǡሻȁ ሿ ൌ ʹ ൫ɃሺሻǦ ൗ൯
ʹ ሺͶሻ
variance of the detail coefficients dx(j, ) is given by Equation
(1) [18], [19].
where Cq is a function of q and (q) is a function that allows
the differentiation between monofractal and multifractal
ͳ
ʹ
Ɋ ൌ σ ȁ ሺǡሻȁ ሺͳሻ processes. Therefore, the scaling exponent of order q is
ൌͳ
estimated by [20] in the same method described with Eq. (2).
To create a representation of the behavior of such exponents
where nj is the number of detail coefficients in the octave j,
according to an order q, the Multi-scale linear Diagram (MD)
2
and j is the estimator of E[|dx(j, )| ]. Once the detail is used where H is projected according to the order q for
coefficients have been determined, it is noteworthy to
negative and positive values of it.
mention that the second statistical moment follows the power
law with an exponent of 2H-1 where H is the Hurst parameter
representing the self-similarity of the second stochastic
moment. Thus, the estimation of the Hurst parameter can be
described by Equation (2).
Ɋ ൌ ሺʹ Ǧͳሻ ሺʹሻ
ʹ
ʹ
The LD is the result of plotting the log2 j as a function of
octave j. Figure 4 shows the calculation of H for the time
series corresponding to Channel 1 as illustrated in Figure 3.
Furthermore, it performs a linear regression on the estimators
computed for each scale. Then, the slope of said regression
is calculated. A gradual estimate of H is obtained with
confidence intervals determined by the standard deviation of
H.
Figure 5 – Dissection of fluctuation traces for small
changes in negative moments and significant changes in
positive moments. Adapted from [21]
Figure 5 illustrates how the analysis is affected by the
changes in q as it goes from negative to positive values. On
the one hand, the analysis of negative values of q pertains to
the study of small fluctuations in multifractal traces. As q
decreases, the analysis of small fluctuations is more
Figure 4 – Hurst parameter estimation with LD noticeable in contrast with monofractal traces where
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