## 2 Materials and Methods
### Methodology
#### 2.2.4 Classification Models  
In this study, the performance of the selected condition indicators (CIs) was evaluated using two classification models: Random Forest (RF) and K-nearest Neighbours (K-NN), both widely recognised and proven to be effective in fault diagnosis tasks [4; 14; 20; 21; 22]. RF was used as the primary model, and K-NN was applied as a comparative technique. These algorithms were selected due to their robustness, interpretability, simplicity, low  
Figure 5: Feature selection process. computational demand, reduced sensitivity to noise, and ease of implementation--qualities that make them particularly suitable for real-time industrial applications [25; 26; 27]. Their use is therefore justified in scenarios where timely and reliable fault detection is prioritised over theoretical optimality [28; 29]. Although more complex models, such as deep neural networks or advanced ensemble architectures, are available, they typically require larger datasets, greater computational capacity, and often lack interpretability. In contrast, RF and K-NN offer optimal performance with minimal parameter tuning and allow better traceability of each CI's contribution to the classification outcome, which is essential for practical industrial monitoring applications. Moreover, these classifiers enable a direct validation of the selected subset of CIs, supporting the analysis of their relationship with varying fault severity levels. * Random forest (RF): RF is a classification model represented by Equation (1), composed of multiple tree-based classifiers. For each \(i\)th tree, an independent random vector \((V_{i})\) is generated. Each tree is trained on a subset of the data and votes for the most popular category in the input vector \((x)\). The classification error, described by Equation (2), depends on the margin \((mg)\), which measures the average number of votes received for the correct class, and on the probability distribution \(P_{X,Y}\) in the feature-label space [30]. \[RF=h(x,V_{i})_{i=1}^{N},\quad i=1,2,3...,N\] (1) \[E=P_{X,Y}(mg(X,Y))\] (2)
* k-nearest neighbours (K-NN): K-NN is a non-parametric algorithm used for classification tasks in which new instances are categorised based on their proximity to existing samples within the feature space. The method assigns weights according to distance and infers the class of an unknown observation through a majority voting mechanism [31]. K-NN requires only the selection of the parameter \(k\) to define the number of neighbours and the appropriate distance metric [32]. The K-NN classification algorithm works as follows: Given a training set \(D(x_{i},y_{i})_{i=1}^{N}\), where \(x_{i}\) is a training vector and \(y_{i}\) its class label, and a test instance \((x^{\prime},y^{\prime})\), the predicted class \(y^{\prime}\) is determined using Equation (3): \[y^{\prime}=\underbrace{\text{arg max}}_{i}\psi\sum(x^{\prime},y^{\prime})\in D ^{\prime}w_{i}\delta(\psi,y_{i})\] (3) Here, \(\psi\) is a candidate class label, \(y_{i}\) is the label of the \(i\)th nearest neighbour, \(\delta(\cdot)\) is the indicator function returning \(1\) if the labels match and \(0\) otherwise, and \(w_{i}\), as defined in Equation (4), represents a weighting coefficient derived from the distance \(d(x^{\prime},x_{i})\) between the query instance and its \(i\)th nearest neighbour. \[w_{i}=\frac{1}{(d(x^{\prime},x_{i}))^{2}}\] (4) The default distance metric is Euclidean, although alternatives such as Mahalanobis, Manhattan, and Minkowski distances can also be used [33]. A repeated hold-out resampling process was applied to evaluate model performance. During each iteration, the databases were partitioned in 70% for model training and 30% for model testing. The method enables the estimation of model variability across multiple subsets and yields robust performance metrics such as the accuracy rate and the area under the curve (AUC) associated with the receiver operating characteristic (ROC) analysis [19; 34].