## 2 Neo-Normal Mixture Model
### Neo-Normal Distribution  
The neo-normal distribution family consists of the adaptive distributions that has symmetrical properties, but it can also accommodate skewed or asymmetrical patterns; moreover, it can also be sharper or gentler [18]. These distributions slightly deviate from normal or Gaussian distributions but set their mode as the location parameter. Some of the distributions included in the neo-normal model are the exponential power distribution and Aazlini skew normal distribution. The exponential power distribution is similar to the Gaussian distribution, but it can be more pointed and has a thick tail [19]. The Aazlini skew normal distribution can accommodate symmetrical or skewed characteristics of a distribution, but this distribution's mode is not stable in its location parameters [20].  
[PERSON] and [PERSON] [15] overcame the weakness of the [PERSON]'s distribution by introducing another skew normal distribution. This distribution is able to maintain the stability of modes in its location parameters--whether it is symmetrical or skewed--by adding the transformation parameters. This distribution is called the FSSN. Another distribution that is also stable in mode is the MSNBurr by [PERSON] [16]. The advantage of using the MSNBurr is that the distribution can accommodate long-tail data, as in an MRI image's pattern. This study used the FSSN and MSNBurr distributions to build the mixture models. The cluster center was more robust since these distributions are stable in their modes.  
Supposing the random variable \(Y_{i}\) follows the FSSN distribution, [PERSON] et al. [17] provided the density as:  
\[f(y_{i}\big{|}\mu,\sigma,\gamma)=\frac{2}{\gamma+\frac{1}{\gamma}}\Bigg{(}\frac{ 1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{y_{i}-\mu}{\sigma})^{2}}\Bigg{(} \frac{1}{\gamma^{2}}I_{[\mu,255)}(y_{i})+\gamma^{2}I_{(0,\mu)}(y_{i})\Bigg{)} \Bigg{)}, \tag{3}\]  
where \(\mu\) is a location parameter, \(\sigma\) is a dispersion parameter, and \(\gamma\) is a skewness parameter. This distribution transforms the symmetrical normal distribution using inverse scale factors in the positive and negative axes. It loses its symmetry when \(\gamma\
eq 1\), skews to the left when \(\gamma<1\), and skews to the right when \(\gamma>1\). By substituting Equation (3) as \(f_{j}\big{(}y_{i}\big{|}\Theta_{j}\big{)}\) in Equation (1), we were able to obtain the FSSN mixture model.  
The MSNBurr distribution idea was designed for the skewed distribution, which ensures that the modes of the Burr II distribution for all values of its skewness parameter, \(\gamma\), remain at its location parameter. [PERSON] [16] defined the density function (p.d.f) for the MSBurr family as:  
\[f(y_{i}\big{|}\mu,\sigma,\gamma,\kappa)=\frac{\kappa}{\sigma}\exp\biggl{(}- \kappa\bigg{(}\frac{y_{i}-\mu}{\sigma}\bigg{)}\Bigg{)}\Bigg{(}1+\frac{\exp \biggl{(}-\kappa\bigg{(}\frac{y_{i}-\mu}{\sigma}\bigg{)}\bigg{)}}{\gamma} \Bigg{)}^{-(\gamma+1)}, \tag{4}\]  
where \(\mu\) and \(\sigma\) are location and dispersion parameters, respectively, and \(\kappa\) is a scale factor. The distribution in Equation (5) is called the MSNBurr \((\gamma,\mu,\sigma)\), with the value of \(\kappa\) is as Equation (5):  
\[\kappa=\frac{1}{\sqrt{2\pi}}\biggl{(}1+\frac{1}{\gamma}\biggr{)}^{(\gamma+1)},\ \kappa>0 \tag{5}\]  
The MSNBurr mixture model is built by substituting Equation (4) as \(f_{j}\big{(}y_{i}\big{|}\Theta_{j}\big{)}\) in Equations (1) and (2) with parameter \(\Theta_{j}=\big{(}\gamma_{j},\mu_{j},\sigma_{j},\kappa_{j}\big{)}\).