## 3 Theoretical Background
The relation (5) applies to the dispersion:  
\[s_{2}\equiv\sigma_{x}^{2}=\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}(t)-\bar{x}(t) \right)^{2}. \tag{5}\]  
The skewness symptom \(s_{3}\equiv\gamma_{1}\) expresses the size of the asymmetry of the measured signal (6). \[s_{3}\equiv\gamma_{1}=\frac{\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}(t)-\bar{x}(t) \right)^{3}}{\sigma^{3}}. \tag{6}\]  
The kurtosis symptom \(s_{4}\equiv\gamma_{2}\) (7) informs whether the values of the investigated signal symptom are flatter or sharper than the values of the Gaussian curve. \[s_{4}\equiv\gamma_{2}=\frac{\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}(t)-\bar{x}(t )\right)^{4}}{\sigma^{4}}-3. \tag{7}\]  
Other selected characteristics were specific to the vibration signal in the time domain. The following features were proposed: the peak is the maximum distance of the wave's top from the reference value, which is usually the x-axis (8). \[s_{5}\equiv\max(x_{i}(t)). \tag{8}\]  
The peak-to-peak symptom \(s_{6}\equiv x_{p2p}(t)\), which represents the difference between the maximum and minimum value. The value is the maximum distance of opposite signal peaks (9):  
\[s_{6}\equiv x_{p2p}(t)=|\max(x_{i}(t))-\min(x_{i}(t))|. \tag{9}\]  
The symptom effective value \(s_{7}\equiv x_{\text{rms}}\) (i.e., root mean square level (rms)), where rms is an objective value used in diagnostic regulations, determined according to relation (10):  
\[s_{7}\equiv x_{\text{rms}}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}(t)}. \tag{10}\]  
The symptom \(s_{8}\equiv E_{s}\) is the energy of the signal. The energy of the accompanying vibration signal is given by the sum of the squares of the signal samples \(\{x_{i}\}_{i=1}^{n}\) (11):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]0  
The significant symptom \(s_{9}\equiv\text{L}_{2}\text{x}\) is the norm of the measured signal \(\text{L}_{2}\) from the time waveform of the signal; thus, Equation (12) applies:  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]1  
The design of the ninth symptom \(s_{9}\equiv\text{L}_{2}\text{x}\) was based on the idea of a vector whose components are the individual sample's realization of the signal \(\{x_{i}\}_{i=1}^{n}\). The \(\text{L}_{p}\) norm of the vector **x** was defined for the feature vector (13):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]2  
Specifically, the \(\text{L}_{2}\) norm of the vector \(\textbf{x}\)\((p=\textbf{2})\). The symptom \(s_{10}\equiv P_{s}\) represents the signal power; thus, Equation (14) applies:  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]3  
The entropy of the signal is the symptom \(s_{11}\equiv H_{s}\) (15):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]4  
An essential statistical characteristic in time analysis is the autocorrelation function \(R_{xx}(\tau)\) of the measured vibration signal \(x_{i}(t)\). The autocorrelation function \(R_{xx}(\tau)\) represents a generalization of the mean square value. It is defined by Equation (16):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]5  
It presents periodic parts of the vibration signal \(x_{i}(t)\), and nonperiodic ones disappear quickly. It provides information about the dependence of the values of the function \(x_{i}(t)\) at time \(t\) on the values at time \(t+\tau\). For the design of the vector symptom \(s_{12}\equiv\text{L}_{2}\text{x}R_{xx}\), the norm of the autocorrelation function \(R_{xx}(\tau)\) of the vibrational signal was calculated. It was determined based on Equation (17):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]6  
Amplitude and power spectra were calculated in the frequency domain. Their complex values are significant for the design of the feature vector in the frequency domain. Fourier transformation \(X(\mathrm{i}\omega)\) was used to calculate the amplitude spectrum of the vibration signal of the drilling stand.