## 2 The Proposed LPC-NN Method
### Feature Extraction  
In order to understand the structure of the vibration signal from rolling bearing, we first examined their frequency components by applying FFT. Figure 2 shows some examples of spectrums for signals from the CWRU dataset that have different conditions. A detailed description of the CWRU dataset is provided in Section 3.1. Note here that these data were taken from an SK6205 bearing located at the drive end, and the acquisition frequency was 12 kHz. There are three fault types in the CWRU dataset (inner ring, outer ring, and ball) with three fault sizes. Figure 2 shows the spectra of the normal signal and signals with defects of the inner ring with different fault sizes and with a rotation speed of 1797 RPM. It is clear that signals from different classes have different spectrums and fundamental components. It can also be observed that the spectrum of these signals resembles a speech-like spectrum; the number of spikes (formants) in the spectrum is not high and they are well spaced in each signal. Like a speech signal, the energy of a vibration signal is concentrated in narrow frequency bands. Linear prediction for speech signals models a mechanism that introduces correlation into the signal, the main source of which is the vocal tract [39]. For vibration signals, there is also a correlation caused by periodic impacts due to the presence of a defect. These properties suggest that LPC spectrum estimation works sufficiently well to track the formants of the signal, as is the case in speech processing. The number of coefficients should correspond to the number of formants in the signal. Figure 3 shows the estimated spectrum of a healthy signal with zero loads on the bearing from the CWRU dataset using 12, 50, and 100 LPC coefficients. Using only 12 coefficients was not sufficient to obtain an accurate estimate, while using 100 coefficients did not significantly improve the estimate using 50 parameters. Therefore, using 50 parameters was suitable for achieving a smooth estimate. Figure 4 shows the LPC spectrum for normal and defective signals from Figure 2a,b, with the number of LPC coefficients equal to 50. It can be seen that the compressed spectrum captures the main behavior in the original spectrum and, in addition, the spectra of signals of different types are significantly different. Figure 2: Spectrum of different signals; (**a**): (healthy case); (**b**): (fault type: inner, fault diameter: 0.007′′); (**c**) (fault type: inner, fault diameter: 0.014′′); (**d**) (fault type: inner, fault diameter: 0.021′′). As a method for choosing the number of LPC coefficients, we can also propose an analog of the \"elbow\" method, which is a common heuristic in mathematical optimization used to identify a point where diminishing returns are no longer worth the additional cost. Figure 5 shows the dependence of the mean squared error  
\[MSE=\frac{1}{L}\sum_{i=1}^{L}\left(\left|X(f)\right|_{i}-X_{i}^{est}\right)^{2} \tag{9}\]  
on the number of coefficients. Here, \(\left|X(f)\right|-\) represents the power spectral density of a signal, and \(X^{est}-\) denotes its estimate using a specific number of LPC coefficients. The idea is that after a certain number of coefficients, the spectrum estimate will already be quite good and further increases in the number of coefficients will not significantly improve it. Figure 4: LPC spectra for normal and defective signals. Figure 5: “Elbow” method to determine the number of coefficients. Figure 3: Estimated spectrum of a healthy signal (no load) from the CWRU dataset. In Figure 5, for some signals of the CWRU dataset, we see a sharp elbow: a rapid decrease is replaced by a smoother one. Not all signals will exhibit such an elbow in the same location, so it is worth choosing the maximum number of coefficients. Another way to select the model order is to use the criteria for autoregression models, the most famous of which are _FPE_, _AIC_, and _MDL_. These methods use a loss function which is minimized.