## 5 Conclusions
A hard cluster can be produced by assigning each observation to the cluster where it has the highest membership. FANNY clustering is based on the equation: \[C=\sum_{v=1}^{k}\frac{\sum\sum u_{iv}^{r}u_{ip}^{r}d(i,j)}{2\sum u_{jp}^{r}},\] (A4) where \(u_{ip}\) is the membership of element \(i\) in relation to group \(v\), \(n\) is the number of elements that form the data set, \(k\) is the number of groups to be formed, \(r\) corresponds to a pertinent exponent, and \(d(i,j)\) is the distance between elements \(i\) and \(j\). Machine learning algorithms possess the capability of improving their performance automatically through experience. To do so, they build a mathematical model based on sample data, known as \"training data\", in order to make predictions or decisions without being explicitly programmed to do so. Although machine learning approaches are commonly associated to modeling and prediction tasks, these tools can also be used to develop clustering essays [81]. In this work, only one machine learning clustering algorithm was used:  
5. SOM (Self-organizing maps [54]), an unsupervised learning technique based on neural networks that is popular among computational biologists and machine learning researchers. SOM is a concept of competition network that tries to find the most similar distance between the input vector and neuron with weight vector \(w_{i}\). SOM always consist of both input vector \(x\) and output vector \(y\). At the start of the learning, all the weights (\(w_{i}\)) are initialized to small random numbers. The set of weights forms a vector \(w_{i}=w_{ij}\), \(i=1,2,\ldots,k_{x}\), \(j=1,2\ldots,k_{y}\) where \(k_{x}\) is the row number and \(k_{y}\) is the column number. Euclidian distance \(d\) between the input vector \(x\) and the neuron with weight vector of the given neuron \(w_{c}\) is computed by: \[d(x,w)=|x(t)-w_{c}(t)|,\] (A5) where \(t\) is an integer. Next, SOM will search for the winner neuron using the minimum distance (best matching unit, BMU). BMU is calculated as follows: \[BMU=\text{argmin}\ |x(t)-w_{c}(t)|\] (A6)To increase the similarity with the input vector, weights are adjusted after obtaining the winning neuron. The rule for updating the weight vector is given by:  
\[w_{i}(t+1)=\frac{\sum h_{c(i)}(t)\times x_{j}}{\sum h_{c(i)}(t)}, \tag{10}\]  
where \(w_{i}(t+1)\) is the updated weight vector, \(x_{j}\) is the input record, \(h_{c}(i)(t)\) is the neighborhood function related to the winning unit \(c_{i}\) at step \(t\), and \(S\) is the number of input samples. The neighborhood function (usually assumed as Gaussian) determines the rate of change of the neighborhood around the winner neuron as in equation:  
\[h_{c(i)}(t)=e\frac{-\left|r_{c(i)}-r_{i}\right|^{2}}{2\sigma(t)^{2}}, \tag{11}\]  
where \(r_{c(i)}\) and \(r_{i}\) are, respectively, the positions on the map of the winning neuron and of the generic unit \(i\); \(\sigma(t)\) is the neighborhood radius at the iteration \(t\) of the training process and corresponds to the width of the neighborhood function at step \(t\). Initially, \(\sigma(t)\) can be as large as the size of the map and then, to guarantee convergence and stability, it decreases linearly with time till one during the process. Finally, model-based clustering algorithms are those that postulate a generative statistical model for the data and then use a likelihood (or posterior probability) derived from this model as the criterion to be optimized. Model-based clustering has recently gained widespread use both for continuous and discrete domains mainly because it allows one to identify clusters based on their shape and structure rather than on proximity between data points [82]. One model-based clustering algorithm was considered in this work:  
* MCL (Model-based clustering [53]) operates on the assumption that the analyzed data originate from a finite mixture of underlying probability distributions [83]. Each mixture component represents a cluster, and the mixture components and group memberships are estimated using maximum likelihood (EM algorithm).