## 3 Material and Methods
### Formation of Criteria for Assessing the Quality of the Cluster Structure  
The formation of clustering quality criteria for assessing the cluster structure of GEPs was conducted taking into account the principles of the objective clustering inductive technology (OCIT) [26; 27], the application of which involves the assessment of the cluster structure based on the use of both the internal and external clustering quality criteria. In this instance, the final decision regarding the optimal cluster structure formation is performed based on the analysis of the values of the balance criterion, which contains both internal and external criteria as the components. The application of OCIT involves the division of GEPs into two equivalent subsets applying hybrid profile proximity metrics based on the evaluation of both the mutual information using various methods of [PERSON]'s entropy evaluation and [PERSON]'s chi-squared test, which is proposed in [2]. The equivalent subsets contain an equal number of pairwise similar GEPs. The used hybrid proximity metric combines various methods of [PERSON] entropy calculation when the mutual information of gene expression profiles is evaluated and the [PERSON]'s chi-squared test. As has been shown in [2], the object classification results differ when various GEP proximity metrics are applied during the mutually correlated gene expression data formation. In order to increase the objectivity of the distance between GEP evaluation, in this research, the authors proposed the stepwise procedure of proximity metric formation. In the first step, the mutual information based on the use of various methods of [PERSON] entropy and Pearson's chi-squared criterion values are evaluated for appropriate pair of GEPs. Then, the Harrington desirability function is applied to form the general proximity metric. A larger value of the general proximity metric, in this case, corresponds to a greater proximity between this pair of GEPs. As a rule, the internal clustering quality criteria should take into account both the particularities of the gene expression profiles distribution within clusters relative to the respective cluster median (as the average value of all GEP is an abstraction and does not correspond to the real distribution of expression values in the profile) and the particularities of the clusters distribution (cluster medians) in the feature space. Within the framework of our research, the first component of the internal criterion was calculated as the square root of the sum of the squares of the distances from the GEPs to the median of the cluster where these profiles are allocated. Because within the framework of the research the distance is taken as the joint value of various types of mutual information [2], this component of the internal criterion can be presented as follows:  
\[CW=\sqrt{\sum_{k=1}^{K}\sum_{l=1}^{m}MI(e_{i}M_{k})^{2}} \tag{7}\]  
where \(MI\) is a hybrid metric based on joint application of the mutual information evaluation and [PERSON]'s chi-squared test (a larger value of this metric corresponds to a smaller distance between appropriate GEPs), \(e_{i}\) is a vector of \(i\)-th gene expression values, \(M_{k}\) is the median of the \(k\)-th cluster, \(m\) is the number of GEPs, and \(K\) is the number of clusters. It is obvious that a larger value of the internal criterion component (7) corresponds to a smaller distance from the gene expression profiles to the median of the respective cluster (or a larger density of the GEPs distribution within the clusters).