## 4 Criteria for Evaluating Sample Classification Quality  
In the present study, object classification was carried out by leveraging metrics that evaluate type 1 and type 2 errors [28]:  
* Classification accuracy is the overall percentage of correctly identified samples, calculated as \[ACC=\frac{TP+TN}{TP+FP+TN+FN}\] (2)
* F1-score is a metric that assesses the precision and recall of sample distribution across the relevant classes, computed as \[F1=\frac{2\cdot PR\cdot RC}{PR+RC}\] (3) where precision (_PR_), the probability of correct sample identification for a class, is \[PR=\frac{TP}{TP+FP}\] (4) and recall (_RC_), the probability of correctly identifying true positive cases for a class, is \[RC=\frac{TP}{TP+FN}\] (5) In these formulas, _TP_ (true positive) and _TN_ (true negative) indicate the count of objects correctly classified, while _FP_ (false positive) and _FN_ (false negative) represent those classified incorrectly. It is crucial to highlight that for a multi-class issue, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**No** & **Type of Cancer** & **Number of Samples** \\ \hline
1 & Adrenocortical carcinoma—ACC & 79 \\ \hline
2 & Glioblastoma multiforme—GBM & 169 \\ \hline
3 & Sarcoma—SARC & 263 \\ \hline
4 & Lung squamous cell carcinoma—LUSC & 502 \\ \hline
5 & Lung adenocarcinoma—LUAD & 541 \\ \hline
6 & Stomach adenocarcinoma—STAD & 415 \\ \hline
7 & Kidney renal clear cell carcinoma—KIRC & 542 \\ \hline
8 & Brain lower-grade glioma—LGG & 534 \\ \hline
9 & Normal & 224 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The classification of experimental gene expression data used in the modeling process. accuracy gauges the overall sample distribution accuracy among classes, while the _F1-score_ appraises the precision of sample distribution within each class independently. * Cross-entropy loss function (_Loss_)--computed during model validation to measure the disparity between predicted and actual probability distributions, in a multi-class classification task involving \(C\) classes: \[L(y,\hat{y})=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{C}y_{ij}log(\hat{y}_{ij})\] (6) where \(y_{ij}\) denotes a binary indicator of whether class \(j\) is the correct classification for observation \(i\), \(\hat{y}_{ij}\) is the predicted probability of observation \(i\) being of class \(j\), and \(N\) represents the total number of observations. Given the challenges of analyzing _F1-score_ values corresponding to multiple classes to select the optimal alternative from a hyperparameter list, an integrated _F1-score_ value was computed. This calculation utilized [PERSON]'s desirability method [29], a potent method for addressing multicriterion issues. The implementation algorithm of this procedure encompasses the following stages:  
1. Initialization. Transform the F1-score values in a matrix format, with rows representing classes and columns representing the hyperparameter values under examination in this phase. 2. Private desirability calculation. Identify the minimum and maximum values of the F1-score during the relevant neural network model operation phase (using the respective hyperparameter combination). Subsequently, transform the scale of F1-score values into a linear scale of the dimensionless indicator \(Y\), considering the boundary values of the F1-score identified in the prior step (the value of parameter \(Y\), according to the desirability method, varies from \(Y_{min}=-2\) to \(Y_{max}=5\)): \[Y_{min}=a+b\cdot F1_{min}\] (7) \[Y_{max}=a+b\cdot F1_{max}\] Transform the _F1_-score values into \(Y\) values: \[Y=a+b\cdot F1\] (8) Compute private desirability for each F1-score value: \[d=\exp(-\exp(-Y))\] (9)
3. Integrated _F1-score_ value calculation. For each column of the matrix obtained in step 2, compute the integrated _F1-score_ value as the geometric mean of all private desirability functions: \[F1_{int}^{j}=\left(\prod_{i=1}^{9}d_{ij}\right)^{\frac{1}{9}}\] (10)
4.