## 3 Feature Selection Using Agglomerative Hierarchical Clustering  
When the feature space exhibits collinearity, permuting one feature often has minimal impact on the overall performance of the estimator because the estimator can still extract similar information from other correlated features. This overlap can adversely affect the estimator's effectiveness. In other words, by selecting one representative feature from each cluster, the characteristics of the entire cluster are effectively captured and included in the assessment process of the machine's status, thus ensuring an efficient analysis while minimizing redundancy [30; 31; 32; 33]. The Spearman rank-order correlation coefficient, denoted by \(r_{s}\), is a Pearson correlation coefficient, denoted by \(\rho\), and is calculated using the ranked variables of the dataset. It quantifies the degree of connection between variables, features in this case, while considering their ranked values rather than raw data, providing a robust measure of monotonic relationships within the dataset (\(U\)). For the m-th feature \(U=[F_{1},\ F_{2},\ F_{3},\ldots,F_{m}]\), the \(r_{s}\) for two features of \(N\) observations is computed as follows [35; 36]:  
\[r_{s(i,j)}=\frac{12\sum_{n=1}^{N}\left\{\left[R(F_{i,n})-\left(\frac{N+1}{2} \right)\right]\left[R(F_{j,n})-\left(\frac{N+1}{2}\right)\right]\right\}}{N(N ^{2}-1)} \tag{6}\]  
Or  
\[r_{s(i,j)}=1-\frac{6\sum_{n=1}^{N}\left\{d_{(ij),n}^{2}\right\}}{N(N^{2}-1)} \tag{7}\]  
where \(d_{ij}=R(F_{i})-R\big{(}F_{j}\big{)}\), \(i,j=1,\ 2,\ 3,\ \ldots,\ m\), and \(R(F_{i})\) and \(R\big{(}F_{j}\big{)}\) signify the rank of samples in features \(F_{i}\) and \(F_{j}\). The Spearman correlation matrix for dataset \(Z\) can be formulated as follows:  
\[ \begin{bmatrix}1&1-\frac{6\sum_{n=1}^{N}\left\{R(F_{1})-R(F_{2})\right\}}{N(N ^{2}-1)}&\ldots&1-\frac{6\sum_{n=1}^{N}\left\{R(F_{1})-R(F_{m})\right\}}{N(N^ {2}-1)}\\ 1-\frac{6\sum_{n=1}^{N}\left\{R(F_{2})-R(F_{1})\right\}}{N(N^{2}-1)}&1&\ldots &1-\frac{6\sum_{n=1}^{N}\left\{R(F_{2})-R(F_{m})\right\}}{N(N^{2}-1)}\\ \vdots&\vdots&\ddots&\vdots\\ 1-\frac{6\sum_{n=1}^{N}\left\{R(F_{m})-R(F_{1})\right\}}{N(N^{2}-1)}&1-\frac{ 6\sum_{n=1}^{N}\left\{R(F_{m})-R(F_{2})\right\}}{N(N^{2}-1)}&\ldots&1\end{bmatrix}  \tag{8}\]  
Once the Spearman correlation coefficient matrix is obtained, the subsequent step involves applying the AHC algorithm, which is used to compute the distance linkage matrix. However, to obtain the linkage matrix in AHC, the choice of distance or dissimilarity measure in conjunction with the linkage criterion is essential for determining the similarity between data points or clusters. The most commonly used linkage methods include single linkage, complete linkage, Ward's method, and the average linkage method, each offering distinct insights into the structure of the data and influencing how clusters are defined [32]. As a matter of fact, linkage methods are specifically formulated to calculate the distance, indicated as \(D_{ij}\), between two clusters, \(i\) and \(j\). The general representation of this distance equation can be formulated as follows:  
\[D_{ij}=g\big{(}F_{i},F_{j}\big{)} \tag{9}\]  
In (9), \(D_{ij}\) represents the distance or dissimilarity between data points or clusters \(i\) and \(j\), where \(F_{i}\), and \(F_{j}\) are the feature vectors, and \(g\) is a distance or dissimilarity function that quantifies the dissimilarity between the data points, clusters or feature vectors in this study. However, considering the scope of this work, the Ward linkage method, utilizing the Euclidean distance, was employed to update the linkage matrix. For two data points \(x\) and \(y\) in the n-dimensional space, the Euclidean distance can be formulated as follows:  
\[d_{E}(x,y)=\sqrt{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}} \tag{10}\]Consider a scenario in which a cluster \(K\) emerges through the combination of original observations, namely \(K[0],\ K[1],\ \...,\ K[n_{K}-1]\), as a result of the fusion of clusters \(i\) and \(j\).