Page 69 - Proceedings of the 2018 ITU Kaleidoscope

Machine learning for a 5G future

fluctuations are scarce, and the trend is the same for all q. On quite intense. The result of this transformation is a long arc
the other hand, for positive values of q, the analysis requires where the difference between the maximum and the
studying large fluctuations related to the average fluctuations minimum values of H(q) corresponds to the width of the
of the traces. As the value of q increases, the analysis of multifractal spectrum [21]. The MS form can be
larger variations is more noticeable, and the trend of approximated to a second order polynomial function and its
monofractal traces remains the same as in negative values of width can be measured with the zero-crossing operation
q. D(q)=0.

3.3 Legendre transformation and multifractal 4. BEHAVIOR OF THE RADIOELECTRIC
spectrum calculation SPECTRUM IN BOGOTÁ

It is then proceeded to calculate the multifractal spectrum 4.1 Calculation of the Hurst exponent for the
using the Legendre transformation. This method measures radioelectric spectrum in Bogotá
the singularity dimension of order q denoted as D(q) and the
resolution function is named H(q) [22]. D(q) is a linear The next step involves calculating the Hurst parameters for
transformation that converts scales into statistical moments each channel based on the time series seen in Figure 3. Using
since the mapping function of sampling scales into the detail coefficients, the variance of the estimator is
individual statistical moments is non-linear [23]. Hence, the calculated, and the slope is derived from the estimation
Legendre transformation is computationally more efficient around the octaves for each channel Eq (1). Figure 7
than other methods used to calculate the multifractal corresponds to a plot of the variable H for all channels of the
spectrum [8]. This version of the multifractal spectrum radioelectric spectrum. The channels with a value of H > 0.5
calculation was implemented hereby. are denoted in light green, stating a persistent behavior in the
trend and short-range dependence. The channels with H > 1
To calculate the singularity dimension, τ(q) serves as an are denoted in dark green, indicating persistence against the
intermediate variable in the form of Equation (5). trend-related behavior and long-range dependence. Out of
the 461 channels, 25 channels showed a value of H < 0.5,
߬ሺݍሻ ൌ ݍܪሺݍሻ െ ͳ ሺͷሻ two channels had a value of H > 1 and the remaining 434
channels were in the range 0.5 < H < 1.
The calculation of the singularity dimension D of order q in
Equation (6) [24] can be determined once τ(q) has been
estimated.
ఛሺ௤ሻ ௤ுሺ௤ሻିଵ
ܦሺݍሻ ؠ ൌ ሺ͸ሻ
௤ିଵ ௤ିଵ

Figure 7 – Hurst parameter estimation for all channels

4.2 Sampling correction in the extensor of the
variance estimator

Moving on, the Multi-scale linear Diagram (MD) is
Figure 6 – Multifractal spectrum for a monofractal and
computed for all channels of the spectrum. When calculating
multifractal time series the MD, the channels presented irregularities in the shape
and distribution of H(q). In Figure 8a, the green curve
Plotting D(q) as a function of H(q) results in the Multifractal highlights the example of a channel with irregularities in the
Spectrum shown in Figure 6 where the shape of the spectrum sampling process of the diagram. The appropriate selection
for monofractal and multifractal time series can be of H(q) is carried out by a hierarchical decision tree that
appreciated. As expected, the shape of the monofractal chooses the maximum value of the function in the vicinity of
spectrum is not as broad as for the multifractal spectrum. q=0 [25] that can be compared based on a curve similar to a
Furthermore, the monofractal behavior is not as prolific sigmoidal function. To correctly deermine the width of the
during fluctuations of q, which is the opposite case for the multifractal spectrum, as shown in Figure 8b, the green curve
multifractal scenario where the activity surrounding q is

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