2.1. Hierarchical Cluster Analysis MethodsHierarchical cluster analysis methods included of the analyzed objects into a hierarchical system of clusters. This system is defined as a system of mutually distinct non-empty subsets of the original set of objects. The main characteristic of hierarchical methods of cluster analysis is creating a decomposition of the original set of objects, in which each of the partial decomposition refines next or previous decomposition.
According to the way of creating decompositions (Figure 2) the hierarchical clustering methods are divided into several groups:
• Agglomerative clustering—at the beginning of clustering are considered individual objects as separate
clusters. The next steps will then be the most similar clusters combine into larger clusters until the specified criteria of quality decomposition is fulfilled.
• Divisional clustering—at the beginning of the clustering process all objects are in one cluster. This cluster is then divided into smaller clusters.
Agglomerative hierarchical clustering methods assign to set of objects O the sequence of its decomposition 
to clusters and hereby the real nonnegative number 
is assigned to each cluster
.
1) The decomposition of the set of objects 
are its individual objects, i.e., single element clusters whereby the number 
for 
belongs to each single element cluster
.
2) There is a decomposition 
and the numbers 
for 
are assigned to clusters. A pair of cluster which has the minimal dissimilarity of coefficient D is chosen, it means, they are the most similar. These clusters are combined to form one cluster. Other clusters stay unchanged and they pass to next decomposition.
2.1.1. Simple Linkage MethodThe simple linkage method can be defined as follows: if D is a random coefficient of dissimilarity, symbols C1, C2 are two different clusters, Ai object belongs to a cluster C1 and object Aj belongs to cluster C2 then
determines the distance of clusters for the Simple linkage method [3].
2.1.2. Complete Linkage MethodThe complete linkage method is a dual method to the simple linkage method its principle is following [3]:
If D is a random coefficient of dissimilarity, symbols C1, C2 are two different clusters, Ai object belongs to a cluster C1 and object Aj belongs to cluster C2 then
determines the distance of clusters for the complete linkage method.
2.1.3. Average Linkage MethodThe distance between the clusters for the average linkage method is defined as follows [3]:
If D is a random coefficient of dissimilarity, symbols C1, C2 are two different clusters, Ai object belongs to a cluster C1 and object Aj belongs to cluster C2 then
determines the distance of clusters for the average linkage method, where n1 and n2 are the number of objects in clusters C1 and C2.
2.1.4. Centroid’s MethodIn Centroid’s method the dissimilarity of 2 clusters is expressed as the distance of centroids of these clusters. Each cluster is represented by the average of its elements, which is called the centroid. The distance between clusters is determined by the Lance-William correlation:
where n1, n2 and n3 are the number of objects in clusters C1, C2 and C3.
2.1.5. The Median MethodIf the size of the clusters is different, the centroid of new cluster may lie within a larger cluster or near the larger cluster. The median method tries to reduce this deficiency in that way that it does not reflect the size of clusters, but it reflects its average. The distance between newly-formed clusters and other clusters is calculated by equation [3]:
2.1.6. Ward’s MethodWard’s method is also marked as a method of minimizing the increases of errors of sum squares. It is based on optimizing the homogeneity of clusters according to certain criteria, which is minimizing the increase of errors of sum squares of deviation points from centroid. This is the reason why this method is different from previous methods of hierarchical clustering, which are based on optimization of the distance between clusters [4].
The loss of information is determined at each level of clustering, which is expressed as the increase of total sum of aberrance square of each cluster point from the average ESS value. Then it comes to a connection of clusters where there is a minimal increase in the errors of sum of squares [5].
The accruement of ESS function is calculated according to [5]:
where
.
, [
comprises all the space.
and any dissimilarity coefficient of objects D, then a cluster is called a subset of p sets of objects o to which it applies [
and
. This means that the maximum distance of objects belonging to the cluster must always be less than the minimum distance any object from the cluster and object outside cluster.
and
. Therefore
.