## 2 The Proposed LPC-NN Method
### Feature Extraction
The loss functions are as follows [40]:  
\[FPE(p)=\frac{N+(p+1)}{N-(p+1)}\hat{\sigma}^{2}, \tag{10}\]  
\[AIC(p)=N\ln\hat{\sigma}^{2}+2p, \tag{11}\]  
\[MDL(p)=N\ln\hat{\sigma}^{2}+p\ln N. \tag{12}\]  
Here, \(N\) is the number of samples, \(\hat{\sigma}^{2}\) is the estimation of the variance in the residual, and \(p\) is the order of the model. Note that sometimes other definitions can be found; for example, in the work [41], the loss function for \(AIC(p)\) is determined by the formula  
\[AIC(p)=\ln\hat{\sigma}^{2}+\frac{2(p+1)}{N} \tag{13}\]  
and differs from Function (11) by a multiplier and a constant term. This is not of fundamental importance, since we are looking for the minimum point of the function. We also applied these criteria to determining the model order. Figure 6 shows the corresponding dependencies on the number of model coefficients. These curves were plotted for a normal signal from the CWRU dataset. As can be seen, the curves for the _FPE_ and _AIC_ methods also resemble an elbow, and the _MDL_ method demonstrates a clear minimum at an order value of approximately 50. We will also use data from the SUSU dataset further (both datasets are described in detail below in Section 3). Figure 7 shows the dependences on the model order for a signal from a normal bearing from the SUSU dataset. It is clear that here, too, a model order of approximately 50 is also optimal. Therefore, throughout this article, we take this value of the model order. We applied UMAP dimension reduction [42] to visualize the 50-dimensional LPC features in a 3D space for all signal types in the CWRU dataset with zero load. Different colors in Figure 8 represent different conditions. More information about these conditions can be found in Section 3.1. The results suggest that the features chosen are effective in classifying the fault types, as the points are not randomly located, but exhibit a certain order. The Normal class \"0\" is well separated from the other defective classes. Figure 6: _FPE_, _AIC_, and _MDL_ methods to determine the number of coefficients (CWRU). Figure 9 shows a similar arrangement of points in the SUSU dataset after dimensionality reduction using UMAP. This dataset contains five signal classes and is described in more detail in Section 3.2. Again, we observe that the data form distinct clusters corresponding to different types of signals. \[MSE=\frac{1}{L}\sum_{i=1}^{L}\left(\left|X(f)\right|_{i}-X_{i}^{est}\right)^{2} \tag{9}\]0 UMAP visualization for the CWRU data (zero load) after feature extraction. \[MSE=\frac{1}{L}\sum_{i=1}^{L}\left(\left|X(f)\right|_{i}-X_{i}^{est}\right)^{2} \tag{9}\]1 _FPE, AIC, and MDL methods to determine the number of coefficients (SUSU)._  
\[MSE=\frac{1}{L}\sum_{i=1}^{L}\left(\left|X(f)\right|_{i}-X_{i}^{est}\right)^{2} \tag{9}\]2 UMAP visualization for the SUSU data after feature extraction. We also used the t-SNE dimensionality reduction method [43], and similar results obtained with it are presented in \[MSE=\frac{1}{L}\sum_{i=1}^{L}\left(\left|X(f)\right|_{i}-X_{i}^{est}\right)^{2} \tag{9}\]3 It can be seen that this method, when reducing dimensionality, splits the data into clear, separated clusters: 10 for CWRU dataset and 5 for SUSU dataset. In general, the t-SNE method confirms the results obtained by UMAP and also indicates the informativeness of the LPC coefficients and the possibility of successfully solving the classification problem. Figures 8-10 demonstrate that the feature extracted by LPC are informative and relevant. It is worth noting the small dimension of the feature vector: each signal frame can be represented using only 50 parameters, which, as will be shown below, are enough for successful classification.