## 3 Materials and Methods
### Stable Distributions  
There are multiple applications of this family of distributions in many sectors. [PERSON]. [33] was the first to develop a stable distribution theory. Later, [PERSON]. [34] proposed a theory based on this distribution to solve the problem of price fluctuations. There are some issues in operating these distributions as they lack an analytical form. A random variable \(W\) has stable distribution \(X\sim S(\alpha,\ \beta,\ \gamma,\ \delta;1)\) with the characteristic function represented by Equation (11) [35].  
\[\mathrm{E}\ \exp(itW)=\left\{\begin{array}{l}\exp\left[-\gamma^{\alpha}|t|^{ \alpha}\left[1-i(sgn(t))\beta(tun\ \frac{\pi\alpha}{2})\right]+i\delta t\right],(\ \alpha\
eq 1)\\ \exp\left[-\gamma|t|\left[1+i(sgn(t))\beta\frac{2}{\pi}\ln\left(|t|\right) \right]+i\delta t\right],(\alpha=1)\end{array}\right\} \tag{11}\]  
where \(sgn(x)=\left\{\begin{array}{l}1\ if\ x>0\\ 0\ if\ x=0\\ -1\ if\ x<0\end{array}\right\}\) and \(\alpha\epsilon\left(0,\ 2\right]\) represents the impulsiveness of the random variable. The parameter \(\beta\in[-1,\ 1]\) is the symmetry of the distribution. \(\gamma>0\) is a scale parameter, and \(\delta\in R\) is the position parameter. Alpha-stable distribution with different parameters defines known distribution functions, Gaussian distribution in Equation (12), Cauchy distribution in Equation (13), and Levy distribution in Equation (14).  
\(f_{2,0}(.\left|\gamma\right|,\mu)\) is a Gaussian distribution with mean \(\delta\) and variance \(2\gamma^{2}\),  
\[N(x,\mu 2\gamma^{2})=\frac{1}{2\gamma\sqrt{\pi}}\exp\left\{\frac{\left(x-\mu \right)^{2}}{4\gamma^{2}}\right\} \tag{12}\]  
\(f_{1,0}(.\left|\gamma\right|,\mu)\) is a _Cauchy_ distribution with density,  
\[pCauchy(x\left|\mu,\gamma\right.)=p\frac{\gamma}{\pi(\left(x-\mu\right)^{2}+ \gamma^{2})} \tag{13}\]  
\(f_{1/2,1}(.\left|\gamma\right|,\mu)\) is a _Levy_ distribution with density,  
\[pLevy(x\left|\mu,\gamma\right.)=(\frac{\gamma}{2\pi})^{\frac{1}{2}}\frac{1}{ \left(x-\mu\right)^{\frac{3}{2}}}exp\left\{-\frac{\gamma}{2(x-\mu)}\right\} \tag{14}\]  
In this research, we use [PERSON]'s algorithm to estimate the alpha-stable distribution parameters. [PERSON]'s algorithm [36] is based on [PERSON]'s representation of the characteristic function [37]. The algorithm performs the numerical integration of a set of splits of the random variable domain. These splits are based on the sign change of the trigonometric functions (sine and cosine) that result from the transformation of [PERSON]'s [37] representation into the complex plane. There are other methodologies for estimating the parameters of the stable distribution, like those presented by [PERSON]. [38], proposing a combination of Gaussian and Laguerre's quadrature; or [PERSON] et al. [39] presenting an algorithm that applies the fast Fourier transform. Finally, [PERSON]. [40] used a Bergstrom series expansion on [PERSON]'s characteristic function representation for approximating a stable cumulative distribution. Nevertheless, [PERSON]'s algorithm [35; 41] has proven to be an easy method to implement, with good performance with high-frequency data, as is the case with the demand for electrical energy.