## Appendix A Wavelet Processing Toolkit
In wavelet theory, it is proved that any series \(X(t)\) may be decomposed in a series form as  
\[X(t)=\sum_{j,k}d_{j,k}(X)\psi_{j,k}(t), \tag{14}\]  
known as the wavelet series or the wavelet decomposition of \(X(t)\), and it guarantees a complete reconstruction formula of the original series \(X(t)\)[12; 13; 14; 15; 16; 17; 18; 19; 33; 34]. The greatest advantage of this decomposition is the fact that it allows splitting the data into different horizons known as levels. Each level is associated to a component of the series, and it makes itself a refinement of the preceding one, which we call in wavelet theory the concept of multi-resolution. Denote for \(j\in\mathbb{Z}\), \(W_{j}=spann(\psi_{j,k}\;;\;k\in\mathbb{Z})\) (known as the detail space at the level \(j\)), and \(V_{j}=\bigoplus\limits_{l\leq j}^{\perp}W_{l}\) (known as the approximation space at the level \(j\)). We get an orthogonal decomposition \(V_{j}=V_{j-1}\bigoplus W_{j-1}\). This permits splitting the wavelet decomposition above as  
\[X(t)=\sum_{j\leq\text{\tiny{Imin}},k}d_{j,k}(X)\psi_{j,k}(t)+\sum_{j>\text{ \tiny{Imin}},k}d_{j,k}(X)\psi_{j,k}(t), \tag{15}\]  
relative to a fixed integer \(J_{\text{\tiny{Imin}}}\in\mathbb{Z}\). For \(j\in\mathbb{Z}\), the component \[A_{j}(X(t))=\sum_{l\leq j,k}d_{l,k}(X)\psi_{l,k}(t) \tag{10}\]  
belongs to \(V_{j}\), and it is called the approximation of \(X(t)\) at the level \(j\). It describes the global behavior, the trend, or the shape of \(X(t)\). The component  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]0  
belongs to the space \(W_{j}\), and it is called the detail component of \(X(t)\) at the level \(j\). It reflects the higher frequency oscillations or the fine-scale deviations of the series near its trend. As a consequence, the wavelet decomposition of \(X(t)\) in (10) is a superposition as  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]1  
A second main advantage of wavelet theory is the reduction in computing the coefficients needed in the decomposition. Indeed, there exists a function \(\varphi\) (known as the scaling function or the father wavelet) characterized by the so-called two-scale relation  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]2  
and which is related to the function \(\psi\) by  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]3  
where  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]4  
It holds that \(V_{j}=spann(\varphi_{j,k}\ ;\ k\in\mathbb{Z})\), where the \(\varphi_{j,k}\) values are defined similarly to the \(\psi_{j,k}\) in (15). The component \(A_{j}(X(t))\) is therefore written as  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]5  
where the coefficients \(a_{j,k}(X)\) (known as the approximation or scaling coefficients of \(X(t)\)) are evaluated as the \(d_{j,k}(X)\) by replacing the function \(\psi\) by \(\varphi\). The relation (13) permits computing the level decomposition from each other as  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]6  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]7  
and  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]8  
The sequence \(H=(h_{k})_{k}\) is called the discrete wavelet low-pass filter, and the sequence \(G=(g_{k})_{k}\) is the discrete wavelet high-pass filter. The truncation of the last decomposition in (12) in a practical finite level \(J>J_{\min}\in\mathbb{Z}\) gives the so-called \(J\)-level finite wavelet decomposition of \(X(t)\) as  
\[\psi_{s,u}(x)=\frac{1}{\sqrt{s}}\psi(\frac{x-u}{s}). \tag{2}\]9The lower index \(J_{\rm min}\) is in fact more flexible, and it is usually chosen to be 0. The choice of \(J\) is always critical, and it is related to the eventual error estimates requested. See [12; 13; 14; 15; 16; 34] for more details.