## 3 Theoretical Background
One symptom can acquire huge values, and the other can be very small. Standardizing the values of the characteristic vector using statistics is necessary to avoid distortion of results due to different scales of measurement parameters during cluster analysis of multivariate data. Equation (2) can be used to standardize the values of the characteristic:  
\[s_{jk}^{\text{norm}}=\frac{s_{jk}-\min\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}}{ \max\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}-\min\Bigl{\{}s_{jk}\Bigr{\}}_{k=1}^{N}}, \tag{2}\]  
where \(s_{jk}\) is the unstandardized \(j\)-th symptom, calculated from the \(k\)-th signal realization; \(N\) is the number of evaluated signal realizations; \(s_{jk}^{\text{norm}}\) is the normalized \(j\)-th symptom, calculated from the \(k\)-th realization of the signal. It was necessary to choose a suitable metric \(\rho\bigl{(}\mathbf{s}_{x},\mathbf{s}_{y}\bigr{)}\) for the feature space \(V_{m}(\mathbb{R})\), which determines the distance between a pair of feature vectors \(\mathbf{s}_{x},\mathbf{s}_{  
This obtained the metric feature space \(\left(V_{m}(\mathbb{R}),\rho_{\mathrm{e}}\right)\) of the object drilling stand and drilling process. In a broader context, it can be understood as a symptom space of the rock separation process by rotary drilling. A vibration signal was used as the basic information signal from which the proposed numerical indicators of the condition of the drilling stand were extracted. The essential partial task of experimental research was to design the symptom vector of the process under investigation. Laboratory experiments and measurements aimed to process the vibration signal of 16 symptoms as a probable symptom vector. An analysis of the numerical values of the symptoms depending on the drilling mode and aggregates of the horizontal experimental drilling stand was carried out. In addition, it was necessary to analyze the differentiability of vector signs concerning aggregate modes using metrics. The proposed symptoms were numerically calculated from one realization \(\left\{x_{i}\right\}_{i=1}^{n}\) of the realization length \(n=16\),384 samples. The signal samples were measured with a sampling frequency of \(f_{s}=18\),000 Hz. The calculation was repeated for \(N=30\) consecutive realizations of the signal. Subsequently, the symptom values calculated this way were normalized according to Equation (2). This procedure was used to analyze and compare accompanying vibration signals from the operation of individual aggregates and the drilling process. In the following text of the article, the calculation relations for the proposed symptoms are given. They were designed and calculated according to the area in which they were applied. In the time domain, vibration signal values generally change over time. As a result of the need to create a symptom vector, statistical characteristics were used as the basis. The first proposed feature \(s_{1}\equiv\bar{x}(t)\) was the average value of the vibration signal \(\bar{x}(t)\). It is the value of the amplitude in the waveform of the signal calculated according to relation (4):  
\[s_{1}\equiv\bar{x}(t)=\frac{1}{n}\sum_{i=1}^{n}x_{i}(t). \tag{4}\]  
The average value \(\bar{x}(t)\) is a symptom of the position around which the other values are more or less concentrated. The disadvantage is the high sensitivity to extreme values and the possible fictitious nature of the calculated value. The second significant feature of \(s_{2}\equiv\sigma_{x}^{2}\) is the variance. It expresses the degree of variability, which provides information on how the individual observed values in the analyzed signal are scattered or describes the variability of measured values. The dispersion symptom is the average value of the squares of the deviations of the measured values of the monitored symptom from their average value.