# Can Self-Similarity Processes Be Reflected by the Power-Law Dependencies?
## 2 Confirmation of the SSP for the Weierstrass-Mandelbrot Function and in Other Models
### Detection of the Hidden Self-Similarity  
Here and below the measured data are denoted \"Signal-1 (S1)\" and \"Signal-2 (S2)\", correspondingly. Each signal has approximately 8 \(\times\) 10\({}^{5}\) data points. Signal 1 is shown in Figure 3. How can the number of data points be reduced and the hidden self-similar behavior found? The desired algorithm can be divided into several successive steps. These steps are shown in an example of the processing of Signal 1. Figure 2: This figure demonstrates the similarity of the integral curve obtained from the TLS corresponding to Signal 2. These compressed curves are obtained in the same conditions (\(N=5000\) data points, a compression parameter \(b=100\) and a range between compressed curves of 0.0157). **S1. ** The initial noise band/tape having \(N_{b}\) data points is transformed to a rectangle matrix having \(N\) rows (=5000) \(\times\)\(M\) columns (=160) = \(N_{b}\) = 8 \(\times\) 10\({}^{5}\). **S2. ** Then for each column m having \(N\) rows, we find only three invariant points \(Yup(\text{m})\) = max(m), \(Yum(\text{m})\) = mean(m) and \(Ydn(\text{m})\) = min(m) that do not change their values relative to N! permutations in the fixed column. As the result of this action, we obtain three distributions over all the columns. They are shown in Figure 4. Then, in order to suppress the high-frequency fluctuations, we integrate these three distributions numerically by a trapezoid relative to its mean value  
\[ \begin{array}{l}J_{j}=J_{j-1}+\frac{1}{2}\left(x_{j}-x_{j-1}\right)\cdot \left(\Delta y_{j-1}+\Delta y_{j}\right),\\ \Delta y=y-mean(y)\equiv y-\frac{1}{N}\sum\limits_{j=1}^{N}y_{j}.\end{array}  \tag{40}\]  
These three integral curves \(Jup(X)\), \(Jmn(X)\) and \(Jdn(X)\) (\(0<X=m/M<1\)) are shown in Figure 5.  
**S3. ** Verification of the self-similarity principle. Integration of the initial trendless sequence depicted in Figure 3 with the help of expression (39) and comparison of this integral curve Jt(x) with the reduced integral curve Jmn(m). We should stress here again that the first curve contains 8 \(\times\) 10\({}^{5}\) data points, while the second one contains only 160 points coinciding with the number of columns. Their similarity is shown in Figure 6 for Signal S1. Figure 3: The initial Signal 1 forms a trendless sequence with 8 \(\times\) 10\({}^{5}\) data points. Figure 4: Distribution of three functions \(Yup(m)=\max(\text{m})\), \(Ymn(m)=\text{mean(m)}\) and \(Ydn(m)=\min(\text{m})\) over all the columns (\(M=160\)) that are obtained from the initial noise band depicted in Figure 3.  
Figure 5: Integral distributions that are obtained from the distributions depicted in Figure 4. The numerical integration plays the role of a specific filter helping to remove the high frequency fluctuations. It indicates that the three integral curves shown in the previous image are self-similar and are characterized by a linear combination of the power-law exponents. **S4. ** For fitting purposes, we use the functional Equation (9) that contains in solution (13) only two possible power-law terms and the term corresponding to the root \(r_{3}=1\). It is convenient to choose the normalized curve for all integral curves defined as  
\[Yn(m)=\frac{Y-mean(Y)}{Range(Y)},\;\;Range(Y)=\max(Y)-\min(Y) \tag{41}\]  
where the function \(Y(X)\) coincides with that of the integral curves \(Jup(X)\), \(Jmm(X)\) and \(Jdn(X)\). **S5. ** If the log-periodic corrections are removed from (13), we obtain the simplified function that enables the desired power-law exponents to be calculated  
\[F(z)\cong \begin{matrix}\sum\limits_{l=1}^{2}A_{l}^{(0)}z^{\
u_{l}}+A_{3}^{(0) },\;\;\
u_{l}=\frac{\ln(r_{l})}{\ln\lambda},\\ r^{3}=w_{2}r^{2}+w_{1}r+w_{0}.\end{matrix}  \tag{42}\]  
The approximate expression (41) can be transformed into the Euler equation  
\[D^{3}F(z)-w_{2}D^{2}F(z)-w_{1}DF(z)-w_{0}F(z)=0,\;D\equiv z\frac{d}{dz} \tag{43}\]  
That makes it possible to find the desired roots by the EC method [14].