# Anomaly Detection in IoT Communication Network Based on Spectral Analysis and Hurst Exponent
## 3 Self-Similarity and Multifractal Spectrum Dependences
### Multifractal Spectrum  
The analysis of multifractal dependencies of a given set of data allows for the determination of the characteristics of the tested data. Thanks to the application of multifractal decomposition, it is possible to analyze the processes taking place in small time scales. The examined processes are divided into sub-sets of points in such a way that their environment has similar geometrical features determined by Holder's exponent. Then, for the sub-sets obtained after this analysis, their Hausdorff dimensions are determined. In this way, a multifractal spectrum, which is a relation between the obtained Hausdorff dimension and the determined Holder's exponent, is obtained [25; 26]. [PERSON]'s interval exponent, determined for the probabilistic measure \(\mu\) in the specified range \(I\), is characterized by the use of Equation (5):  
\[a_{\mu}(I)=\frac{\log\mu(I)}{\log|I|} \tag{5}\]  
For this expression, \(|I|\) is the Lebesgue measure, which is defined for the range \(I\). Let \(x\) will be identified as a point of reference in the field of measurement \(\mu\), and \(\{l_{k}\}\) reflects the sequence of compartments:  
\[x\in I_{k}\,\ \lim_{k\rightarrow\infty}\lvert l_{k}\rvert=0 \tag{6}\][PERSON]'s exponent, which is a measure of \(\mu\) for a specific point \(x\), is described with the use of Equation (7):  
\[\mathrm{a}_{\mu}(\mathrm{x})=\lim_{\mathrm{k}\to\infty}\mathrm{a}_{\mu}(\mathrm{ I}_{\mathrm{k}})=\lim_{\mathrm{k}\to\infty}\frac{\log\mu(\mathrm{I}_{\mathrm{k}})}{ \log|\mathrm{I}_{\mathrm{k}}|}. \tag{7}\]  
The Hausdorff dimension determined for set \(F\) is expressed as a limit, Equation (8):  
\[\mathrm{dim}(\mathrm{F})=\lim_{\delta\to\infty}\frac{\log\mathrm{N}_{\delta}( \mathrm{A})}{-\log\delta}. \tag{8}\]  
for which:  
* \(F\) is a subset of the \(n\)-dimensional Euclidean space,
* \(A\) determines the set of \(n\)-dimensional spheres, where \(F\subseteq A\),
* \(\delta\) is the diameter of the coverage \(A\), which is the diameter of the largest of the spheres belonging to the coverage,
* \(N_{\delta}(A)\) means the minimum number of spheres that are part of the coverage with a diameter of \(\delta\). The multifractal spectrum, which is determined by multifractal decomposition, determines the relation between the Hausdorff dimension of a set of points of measurement with a fixed point dimension and the point dimension itself, Equation (9):  
\[f_{H}(a)=dim(K_{a})\,\ K_{a}=\{x:a(x)=a\}. \tag{9}\]  
The definition formulated in this way assumes that the spectrum is calculated for the probabilistic measure. In order to obtain a multifractal spectrum for the stochastic process, one must linearly rescale the values of the process so that the realization of the rescaled range is practically always a probabilistic measure. During the estimation of the multifractal spectrum concerning the realization of the processes of traffic intensity recorded in the measurements, it is necessary to linearly rescale the processes so that the values of these processes meet the condition of normalization. The generality of considerations is not reduced in any way by this type of rescaling because the multifractal spectrum does not depend on the mean value of the analyzed sample [27; 28; 29]. Multifractal decomposition divides the studied processes into sets of points, where each of them is defined as a set of Cantor. These sets of points are fractals, and the fractal dimension of each of them is different from unity [25; 29]. A multifractal spectrum can be determined in many different ways. The two primary methods are the Legendre transformation of the split function and determining spot metering histogram boundaries.