## 3 AI-Based Early Fault Detection
### One-Class Support Vector Machine (ML)  
SVM is a proven method for solving classification tasks [32]. One variant of this, which is particularly suitable for condition-based PdM, is OSVM. It requires only one data class for classification or for the detection of an anomaly. This is usually the class whose data are generated at the beginning of the inspection and are marked as \"normal\". To do this, it is necessary to identify in advance which features are suitable for differentiating between \"normal\" data and an anomaly. As previously outlined in the test procedure, the specific frequencies of the gears, including the tooth mesh \(f_{\mathrm{z}}\), the first side bands of the level gear \(f_{\mathrm{z}}\pm f_{\mathrm{1}}\), and the first and second side bands of the disk gear \(f_{\mathrm{z}}\pm f_{\mathrm{2}}\) and \(f_{\mathrm{z}}\pm 2f_{\mathrm{2}}\) can be used to make reasonable estimates about the condition of the differential. This is based on the frequencies  
\[f_{\mathrm{1}}=\frac{n}{60\;\mathrm{s}} \tag{1}\]  
and  
\[f_{\mathrm{2}}=\frac{n}{60\mathrm{s}}\cdot\frac{1}{i} \tag{2}\]  
at a specific speed n in rpm and transmission ratio \(i\). While damage to a single tooth (local) is more likely to be noticeable through an increase in the amplitude of the sidebands [33], damage to all teeth (circumferential), such as abrasive wear or scuffing, can be recognized by an increase in the amplitude of the tooth mesh frequency. This results in seven features that are extracted from the torque signal of the drive, which are explicitly defined in Table 1. MATLAB offers the option of doing this via the Diagnostic Feature Designer App. Before the values of the features are determined, the most frequently used operating point from the test run is selected for a comparable analysis. Here, this operating point has a constant torque of 500 Nm and a constant speed of 1584 rpm. This eliminates speed- and torque-dependent changes in gear-specific frequencies. In addition, the torque signal used is divided into sequences of equal length. This results in six sequences of a fixed length of 2000 data points per run in the test program used. The features values are extracted from these sequences and assigned to the test run together. The spectral power density is formed using [PERSON]'s method [34] to determine the feature value from the torque sequences. This method can be used to clearly determine all the named features from the torque signal. \begin{table}
\begin{tabular}{c c} \hline \hline
**Feature** & **Value (Hz)** \\ \hline \(f_{\mathrm{z}}\) & 396 \\ \(f_{\mathrm{z}}+f_{\mathrm{1}}\) & 422.4 \\ \(f_{\mathrm{z}}-f_{\mathrm{1}}\) & 369.6 \\ \(f_{\mathrm{z}}+f_{\mathrm{2}}\) & 408.8 \\ \(f_{\mathrm{z}}-f_{\mathrm{2}}\) & 383.2 \\ \(f_{\mathrm{z}}+2  
The basic idea of SVM is the identification of an optimal separating line or plane between two classes. In the context of a data set  
\[(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n}) \tag{3}\]  
comprising a feature vector \(x_{i}\)\(\epsilon\)\(\mathbb{R}^{d}\) of the \(i\)-th data point in dimension \(d\) and a class label \(y_{i}\)\(\epsilon\)\(\{-1,1\}\) the separating plane can be described by  
\[w\cdot x+b=0 \tag{4}\]  
where \(w\) represents the vector of weights and \(b\) denotes the bias, which indicates the distance from the origin to the separating plane. The optimization problem of the SVM  
\[min_{w,b}\frac{1}{2}\|w\|^{2} \tag{5}\]  
with the constraint  
\[y_{i}(w\cdot x_{i}+b)\geq 1,\ \forall_{i} \tag{6}\]  
is to identify the maximum distance of the separating plane to the nearest data points. While SVMs separate two classes, OSVM learns a decision function that describes the largest possible part of the trained class in a space. It identifies outliers from this as anomalies. Since no class labels are used for this, this method is unsupervised learning. This is sometimes also referred to as semi-supervised learning, since the class of the trained data is named.