## 3 Proposed ML Method
### Classification
#### 3.4.1 Support Vector Machines  
A support vector machine (SVM) is a supervised ML algorithm. The SVM algorithm is used to classify pre-labelled test cases (targets) by analysing the training cases (predictors) and finding a separator between classes. The use of SVM-classification algorithms has been recorded to achieve highly accurate results in high dimensional feature spaces [52; 53; 55; 72; 73]. Another key benefit that the SVM algorithm option offers is memory efficiency. Only a small subset of the training features, the support vectors, are required to calculate the location of optimised hyperplanes between wear-state classes.  
The pre-labelled training data, also referred to as the predictor features, is mapped to a higher-dimensional feature space, so that data points can be categorised as shown in Figure 6. This mapping process is often referred to as kernelling, as the transformation can be achieved through the application of various kernel functions. The predictor features are transformed in such a way that the separator can be formed as a hyperplane, which can be considered as a line function representing the largest separation, or margin, between classes (wear-states). The data points whose positions lie closest to the calculated hyperplane are identified as support vectors. To achieve the most accurate SVM prediction model, the hyperplane should be at the maximal distance possible from the nearest support vectors.  
This distance from a support vector to the hyperplane is identified as the margin. The classification of target instances is achieved by inputting the unseen, feature data  
Figure 5: Linear and non-linear wear-state class boundaries, \(\alpha_{i}\) and \(\beta_{i}\), respectively.  
values, without their corresponding wear state label, into the hyperplane function shown in Equation (7).  
\[w^{T}y+b=0 \tag{7}\]  
where \(w\) represents the weight vector, \(y\) is the input vector and \(b\) is the bias. The result determines whether the data point is an instance of the class above or below the hyperplane.  
This means that only a fraction of the overall predictor feature points are processed for calculation unlike other ML methods, such as Decision Trees, Logistic Regression, Naive Bayes and \(k\)-NN classification, which require all feature points to be included in each calculation. This significantly reduces the complexity of calculations while increasing the efficiency and speed of producing RUL estimations. The biggest challenge associated with the SVM classification algorithm as a prediction model is its tendency to over-fit data. Over-fitting would be most prevalent when the feature number (dimensionality) is high relative to the number of predictor instances. To counteract this, numerous bearing test-cases with large numbers of time-samples are used to train and test the performance of the SVM classification algorithms.  
The performance of six different kernel functions were investigated and compared in this study. Kernel options used to map the data into a higher dimensional feature space included Linear, Quadratic, Cubic, Fine Gaussian, Medium Gaussian and Coarse Gaussian functions.