## 3 Theoretical Background
It is defined by the integral Equation (18):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]7  
Since the signal generated by some aggregates of the drilling equipment is processed directly, a numerical method known as discrete Fourier transformation (DFT) was used. For the calculation (DFT) and to obtain the resulting amplitude spectrum, it was advisable to use the fast Fourier transform (FFT) algorithm. The DFT is defined by relation (19):  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]8  
The discrete value \(X(k)\) represents the amplitude. The values \(x(n)\) and \(X(k)\) have the same physical dimension. Subsequently, the moment of the vibration signal in the frequency domain was calculated from the amplitude spectra. The moment of signal spectra \(s_{13}\equiv\mathrm{m}_{\mathrm{x}}\)FFT as a vector symptom is a numerical characteristic that is calculated from the amplitude spectrum vibration signal. The moment symptom \(\mathrm{m}_{\mathrm{x}}\)FFT represents the center of gravity of the spectrum or the virtual center of the spectrum. When designing the signs for the drilling process, the moment is an essential numerical characteristic. The moment \(s_{13}\equiv\mathrm{m}_{\mathrm{x}}\)FFT of the signal represents the sum of products of all possible values of the spectrum with the frequencies of these spectra (20). \[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]9  
Since the result of the FFT is a vector of complex values, a suitable symptom is also the norm of the amplitude spectrum of the signal \(s_{14}\equiv\mathrm{L}_{2}\mathrm{x}\mathrm{FFT}\) and Equation (21) applies:  
\[s_{jk}^{\text{norm}}=\frac{s_{jk}-\min\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}}{ \max\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}-\min\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}}, \tag{2}\]0  
In industrial practice, power spectra are also calculated in addition to amplitude spectra. In complex industrial processes, to which the drilling process and equipment belong, power spectra have a stronger telling value about the complex state of the technological process. The Equation (22) was used to calculate the power spectrum:  
\[s_{jk}^{\text{norm}}=\frac{s_{jk}-\min\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}}{ \max\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}-\min\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}}, \tag{2}\]1  
where \(1\leq n\leq N\) is valid for a signal in the discrete form \(x(n)=x(n\Delta t)\) with a finite number of samples \(N\). On this basis, the moment of the power spectrum was calculated \(s_{15}\equiv\mathrm{m}_{\mathrm{x}}\mathrm{FFT}^{2}\) (23):\[s_{15}\equiv\mathrm{m_{x}S_{xx}}=\frac{\sum_{i=0}^{N}f_{i}|S_{xx}(\omega)|}{\sum_{i =0}^{N}f_{i}}\quad\text{ is valid for}\quad \begin{array}{l}f_{i}=\mathrm{i} \Delta f,\\ \Delta f=\frac{f_{s}}{N}.\end{array}  \tag{23}\]  
Subsequently, the norm of the power spectrum was calculated as \(s_{16}\equiv\mathrm{L_{2}XS_{xx}}\), as the last proposed symptom (24). \[s_{16}\equiv\mathrm{L_{2}XS_{xx}}\mathrm{=}\sqrt{\sum_{i=1}^{n}|S_{xx}(\omega )|^{2}}. \tag{24}\]  
After determining the symptoms of the aggregates, it is necessary to create clusters of \(\mathbf{C}\) objects. Clusters form a set of objects. The set of clusters for the investigated drilling stand can be written as \(\mathbf{C}=\{\mathrm{C_{1}},\ldots,\mathrm{C_{n}}\}\), i.e., for \(n=8\). In addition, each cluster contains at least one object. A typical symbolic representative was determined for the clusters created by the symptoms of the aggregates, which is the center of the symptom cluster. It was considered that if the objects represent points in the \(\mathbb{R}^{m}\) space, then it is possible to consider the center of gravity (centroids) of the objects in the feature cluster. Then, for a more effective assessment of the differentiability of the vibration signal from the drill stand aggregates using a defined symptom vector, centroids (centroids) were calculated (25). Centroids were calculated for each cluster of aggregates and symptoms.