## 3 Material and Methods
### Experimental Dataset Formation and Preprocessing
The formula used to calculate the internal quality criterion for clustering in this case is given by:  
\[QC_{int}=\frac{ADW}{K\cdot ADB} \tag{1}\]  
where \(K\) is the number of clusters that form the cluster structure; \(ADW\) (average distance within) is the average distance between gene expression profiles and the medians of the clusters in which these profiles are located:  
\[ADW=\frac{1}{m}\sqrt{\sum_{k=1}^{K}\sum_{i=1}^{m_{k}}MI(e_{i},M_{K})^{2}} \tag{2}\]  
where \(m\) is the total number of gene expression profiles; \(m_{k}\) is the number of gene expression profiles in the \(k\)-th cluster; \(M_{k}\) is the median of the \(k\)-th cluster; \(e_{i}\) is the vector of expression values for the \(i\)-th gene; and \(MI(\cdot)\) is the value of mutual information (a higher value of mutual information corresponds to a smaller distance between gene expression profiles). The second component of the criterion (1) is calculated as the average distance between all pairs of cluster medians that make up the cluster structure:  
\[ADB=\frac{2}{K\cdot(K-1)}\sqrt{\sum_{i=1}^{K-1}\sum_{j=i+1}^{K}MI(M_{i},M_{j}) ^{2}} \tag{3}\]  
It is worth mentioning that a smaller value of criterion (1) indicates a better clustering based on this particular quality criterion. The methodology presented in [25] was used to calculate the external and balance criteria. The methodology for implementing the clustering algorithm within the inductive objective clustering framework, as presented in [9; 25], involves the formation of two equivalent subsets of gene expression profiles using a corresponding distance metric. In this study, a modified distance metric, based on a complex application of different methods for maximizing mutual information, was used. The optimal cluster structure was formed by simultaneously applying the spectral clustering algorithm to equivalent subsets of gene expression profiles. The next stage involved forming the cluster structures, with the number of clusters ranging from 2 to 7. In each step, the internal quality criteria for clustering were calculated using Formulas (1)-(3), as mentioned earlier. The external quality criterion was calculated as the normalized difference between the internal criteria at each step of this procedure. \[QC_{ext}=\frac{|QC_{int}^{A}-QC_{int}^{B}|}{QC_{int}^{A}+QC_{int}^{B}} \tag{4}\]  
The reproducibility error that occurs during the process of forming the cluster structure was minimized with a smaller value of criterion (4). The balance criterion, which combines both internal and external quality criteria, was evaluated using [PERSON]'s desirability function, based on Algorithm 1 presented in [9]. ```
Initialization: Calculate the vectors of \(QC_{int}\) and \(QC_{ext}\) for each obtained clustering within the range of the clustering algorithm parameters; create the empty vector of \(QC_{bal}\); set the iteration counter of \(t\) and the iteration counter of the used criteria to \(m=1\); determine the length of the vectors \(QC_{int}\) or \(QC_{ext}\) as \(n\); transform the scales of both \(QC_{int}\) and \(QC_{ext}\) into an index scale \(Y\) using the appropriate dependence type, which is given by: \[ \begin{cases}Y=a+b\cdot QC_{int};\\ Y=a-b\cdot QC_{ext}.\end{cases} \] (5) while\(t\leq n\)do while\(m\leq 3\)do  Calculation of \(Y_{m}\) value for each of the used criteria by the Equation (5);  Calculation of the private desirabilities for each of the criteria: \[d_{m}=exp(-exp(-Y_{m}))\] (6) \[m=m+1;\]  end while  Calculation of the \(QC_{bal}\) as geometric average of all partial desirabilities: \[QC_{bal_{t}}=\sqrt[3]{\prod_{m=1}^{3}d_{m}}\] (7) \[\text{t}=\text{t}+1;\]  end while Return the vector of \(QC_{bal}\) criterion values. ```  
**Algorithm 1**Balance quality criterion value calculation  
Figure 2 displays the simulation outcomes related to the identification of the most suitable cluster configuration.