## Appendix A Wavelet Processing Toolkit  
Wavelets have been introduced few decades ago in mathematical theory, although their discovery was related to applications in petroleum extraction [12; 13; 14; 15; 16; 34]. A wavelet is simply a short wave function oscillating such as the Fourier modes, but with high frequency and small support, which we call in wavelet theory localization in time-frequency and/or time-space. To analyze a statistical (time, financial, etc.) series, we have to compute the so-called wavelet transform, a two-parameter quantity evaluated by the correlation type (a convolution) of the analyzed series with translated-dilated copies of one fixed wavelet known as the mother wavelet, which is a special function that should satisfy at least an admissibility assumption as  
\[\mathcal{A}_{\psi}=\int_{\mathbb{R}}\frac{|\widehat{\psi}(\xi)|^{2}}{|\xi|}d \xi<\infty. \tag{1}\]  
The wavelet analysis appeared originally in the theoretical form as a refinement and also terminology to Fourier and harmonic analysis is general. Therefore, it associates to the analyzed function a type of transform known as the wavelet transform or exactly the continuous wavelet transform obtained by a convolution product of the analyzed function with special copies of a source function known as the mother wavelet. More precisely, denote for \(s>0\) and \(u\in\mathbb{R}\) fixed  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]  
For a function \(F\in L^{2}(\mathbb{R})\), its continuous wavelet transform at the scale \(s\) and the position \(u\) is defined by  
\[\mathcal{WT}_{s,u}(F)=\int_{-\infty}^{\infty}F(t)\psi_{s,u}(t)dt. \tag{3}\]  
The parameters \(s\) and \(u\) have many nominations according to the context of use of the wavelet transform. \(s\) is known as a frequency, scale, or a dilation or compression parameter. \(u\) is known also as the translation parameter. When covering all the real values of \(s\) and \(u\), we obtain the so-called time-frequency or time-space domain. This transform is calledcontinuous because of the nature of the parameters \(s\) and \(u\) that may operate on all the space \((0,\infty)\times\mathbb{R}\). It holds in wavelet theory that the function \(F\) may be reproduced by means of its continuous wavelet transform in an \(L^{2}\)-sense and analogously as in Fourier analysis via the \(L^{2}\)-equality  
\[F(t)=\frac{1}{\mathcal{A}_{\psi}}\int\int_{\mathbb{R}}\mathcal{WT}_{u,s}(F)\psi \left(\frac{x-u}{s}\right)\frac{dsdu}{s^{2}}. \tag{10}\]  
We conclude that the wavelet transform operates according two parameters: the parameter \(s\) permitting to compress or to dilate the graph of \(\psi\), which allows in turn to reach the high/low magnitude fluctuations in the signal, and the parameter \(u\), which permits translating the graph of \(\psi\) to localize or to approach the local fluctuations. The discrete wavelet transform is a variant of the continuous one evaluated on a discrete grid for the parameters \(s\) and \(u\). The most known one is the dyadic, while there is no essential difference between the discrete grids. In the dyadic case, we restrict on the set \(\{(s,u)=(2^{-j},k2^{-j})\;;\;j,k\in\mathbb{Z}\}\). In this case, the translated-dilated copies are defined by  
\[\psi_{j,k}(t)=2^{j/2}\psi(2^{j}t-k). \tag{11}\]  
The continuous wavelet transform \(\mathcal{WT}_{s,u}\) will be called the discrete wavelet transform, which is denoted usually by \(d_{j,k}\). The cosntruction formula (10) will be evaluated via a discrete representation as  
\[F(t)=\sum_{j,k}d_{j,k}(F)\psi_{j,k}(t), \tag{12}\]  
known as the wavelet series of the function \(F\). In the discrete case such as statistical series and discrete signals, the wavelet transform of a series \(X(t)\) (known also as the discrete wavelet transform (DWT)) is obtained by correlation-type (discrete convolution)  
\[d_{j,k}(X)=\sum_{n}X(n)\psi_{j,k}(n), \tag{13}\]  
known also as the wavelet coefficient or detail coefficient at the level or the scale \(j\) and the position \(k\).