## 4 Results
### Machine Learning Algorithms  
Our proposed models are compared with traditional machine learning (ML) algorithms such as k-nearest neighbors (K-NN), random forest (RF) and support vector machine (SVM). These models do not lend themselves for processing raw signals, which means that we preprocessed the data into a set of features in both the time and frequency domain. The features used to feed the ML models are derived from various studies, e.g., [60, 61] and are described in detail in Table 3. All of these features were used in the ML models and are calculated across the sequences (with length 2048). Features in the frequency-domain are calculated after transforming the time series to the frequency spectrum using the FFT algorithm that computes the one-dimensional discrete Fourier transform (DFT) with backward normalization, which is known as a potent baseline [10, 45, 59]. In total, the feature set consists of nine different features for every one of the separate time series.  
For all ML models, we used grid search optimization with five-fold cross-validation derived from the Skaern package to assess which model performs best for every experiment. The parameter settings of the best performing model vary throughout the different experiments and datasets used. In Table 4 the properties of the grid search optimization are described. In addition, we compared the models with the standard WDCNN models as proposed in [10]. The results are described per dataset since these sets have different properties and classification conditions. Further, we distinguished between different percentages of training data to see how well the ML models perform under varying circumstances regarding data availability.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Bearing Condition** & **Nr. of Classes** & **Samples per Class** \\ \hline Healthy & 6 & 10.000 \\ Real damage & 14 & 10.000 \\ Artificial damage & 12 & \(\approx\)10.000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of data samples (sequences) for every bearing condition, with their respective number of classes and samples per class.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Features** & **Formula** & **Description** \\ \hline
**Time-Domain** & & \\ Mean & \(\mu_{x}=\frac{1}{2}\big{(}\sum_{i-1}x_{i}\big{)}\) & The average value of sequence \(x=(x_{i}),i=0\ldots t-1\). \\ Standard Deviation & \(\sigma_{x}=\frac{1}{2}\sum_{i-1}\sum_{i-1}^{t-1}(x_{i}-\mu)^{2}\) & The standard deviation of sequence \(x=(x_{i}),i=0\ldots t-1\). \\ Variance & \(\sum_{i-1}\sum_{i-1}^{t-1}\sum_{i-1}^{t-1}\) & Square of standard deviation. \\ Median & Median \(x_{i}=x_{(i+1)/2}\) & Median value of a sequence \(x\) given by \(x_{i}\) as above. \\ Minimum & Min\(x_{i}\) & Minimum value of sequence \(x\) given by \(x_{i}\) as above. \\ Maximum & Max\(x_{i}\) & Maximum value of sequence \(x\) given by \(x_{i}\) as above. \\ Range & Range \(=\) Max\(x_{i}-\) Min\(x_{i}\) & Difference between the maximum and minimum value. \\ \hline
**Frequency-Domain** & & \\ Signal Energy & Peng \(=\)\(\sum(\text{fft}\,x_{i})^{2}\) & Energy of a signal calculated using FFT. \\ Signal Power & \(\text{Fsig}=\sum\frac{(\text{fft}\,x_{i})^{2}}{\sum_{i}^{t}}\) & Power of a signal calculated using FFT. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Set of features used for the ML algorithms with their formula and description.