## 5 Conclusions
Average linkage [51], mean distance between observations. 2. Complete linkage [51], maximum distance between observations. 3. Single linkage [51], minimum distance between observations. 4. Ward linkage [55], error sum of squares. For all cases, the manipulation of the distance function exerts an influence on the combination of any two groups to form a new one [10]. Manipulation of such a distance function can be achieved through the equation:  
\[D\big{(}G_{x},\big{(}G_{i},G_{j}\big{)}\big{)}=\alpha_{i}D(G_{x},G_{i})+\alpha_ {j}D\big{(}G_{x},G_{j}\big{)}+\beta D\big{(}G_{i},G_{j}\big{)}+\mu\big{|}D(G_{x },G_{i})-D\big{(}G_{x},G_{j}\big{)}\big{|}, \tag{1}\]  
where \(D\) is a distance function, \(\alpha_{i}\), \(\alpha_{j}\), \(\beta\) and \(\mu\) are coefficients that have their values determined according to the applied algorithm. A fifth hierarchical clustering approach was tested in this work:  
1. DINANA (Divisive analysis [52]) is an algorithm that initially starts with all observations in a single cluster, and successively divides the clusters until each one contains a single observation; thus, hierarchies are built in \(n-1\) steps. During each step, the cluster \(C\) with the largest diameter is selected based on the following equation: \[diam(C):=max_{i,j\in C}d(i,j)\] (2) Assuming \(diam(C)>0\), we then split up \(C\) into two clusters \(A\) and \(B\), according to a variant of the method of [PERSON] et al. [80]. At first \(A:=C\) and \(B:=\theta\), later one object is moved from \(A\) to \(B\) and then other objects are moved from \(A\) to \(B\). Partitioning algorithms divide a set of elements into \(k\) groups without constructing a hierarchical structure, following the principle that elements in a same group should be more similar than elements belonging to different groups [50]. These algorithms perform a division of the data to identify \(n\) natural groups into a certain number of disjoint groups, with a centroid for each group as a reference and employing a distance function. They can perform clustering automatically and seek to achieve the maximum similarity between the elements of the same group and the minimum similarity between different groups [10]. In our case, the following algorithms were used to implement partitioning clustering during the development of this work:  
2. PAM (Partitioning around medoids [52]), similar to \"k-means\", the number of clusters (i.e., \(k\)) is fixed in advance and an initial set of cluster centers (i.e., \"medoids\", in contrast to \"means\" used in k-means) is required to start the algorithm. PAM is considered more robust than k-means because it admits the use of other dissimilaritiesbesides Euclidean distance. The implementation of PAM clustering was based on the equation: \[TD:=\sum_{i=1}^{k}\sum_{x_{j}\in C_{i}}d\left(x_{j},m_{i}\right),\] (A3) where \(TD\) is the total deviation, defined as the sum of dissimilarities of each point \(X_{j}\in C_{1}\) to medioid \(m_{i}\) of its cluster. 3. CLARA (Clustering large applications [52]), a sampling-based algorithm that implements PAM on a number of sub-datasets, which allows for faster running times when a number of observations is relatively large. CLARA complies with the following algorithm: 1. Create randomly, from the original dataset, multiple subsets with fixed size (sampsize). 2. Compute PAM algorithm on each subset and choose the corresponding k representative objects (medioids). Assign each observation of the entire data set to the closest medioid. 3. Calculate the mean (or the sum) of the dissimilarities of the observations to their closest medioid. This is used as a measure of the goodness of the clustering. 4. Retain the sub-dataset for which the mean (or sum) is minimal. A further analysis is carried out on the final partition. 4. FANNY (Fuzzy analysis [52]), this algorithm performs fuzzy clustering, where each observation can have partial membership in each cluster. Thus, each observation has a vector that gives the partial membership to each of the clusters.