## 3 Theoretical Background  
Before the clustering process, it is necessary to determine all of the significant characteristics of the objects. Based on these data, it is then possible to start grouping the objects. The objects of the drilling stand equipment are its aggregates during operation. Therefore, it is possible to describe them using the form \(\mathbf{O=}\{\mathbf{o}_{1},\ldots,\mathbf{o}_{n}\}\) for \(n=8\). The investigated aggregates are the motor, pump, hydrogenerator, and their mutual combination during operation: a total of eight objects. The measured vibration signals define individual objects. This accompanying vibration signal is the basis for calculating the vector of symptoms of the drilling process and the operation of the aggregates. Given the empirical knowledge in this area, it can be concluded that the vibrational signal is deterministic with a strong random component. In the research papers [38; 39], the vibration signal was processed and presented using digital methods for technical diagnostics. Figure 3: Laboratory horizontal drilling stand: (**a**) 1—feed mechanism; 2—sheet metal cover of the working tool; 3—stand; (**b**) 4—slide; 5—control system for the drive spindle; 6—double acting hydraulic cylinder; (**c**) 7—core barrel; 8—drilling bit; 9—centering sled clamping mechanism. From a practical point of view, a single realization (i.e., sequence) of the length of the \(n\) samples is sufficient for its processing. It is possible to obtain a sufficiently strong informational value by processing it. This realization of the vibration signal can be described as a sequence of measured samples \(\{x_{i}\}_{i=1}^{n}\). The authors in [40; 41] designed a special acoustic sensor device for the qualitative research and monitoring of technological equipment. Realizations of signals with a length of \(n=16\),384 samples were processed to create a vector of symptoms for scientific purposes. The effort was to design and investigate the numerical signs of the drilling stand, which would sufficiently sensitived differentiate the generated vibration signals by creating clusters from the point of view of the working mode. The authors in [42; 43] present useful methods of processing vibration signals for the detection and diagnosis of mechanical systems. These symptoms can be calculated from one realization or a sequence of several consecutively measured realizations of the vibration signal. Extraction of the \(j\)-th numerical symptom \(s_{j}\in\mathbb{R}\) from the measured realization of the signal for which samples \(x_{i}\in X\) for \(i=1,2,\ldots,n\), is understood as the \(n\)-dimensional complex vector function:  
\[\mathbf{s}_{j}=f_{j}(x_{1},x_{2},\ldots,x_{n}), \tag{1}\]  
where \(\{x_{i}\}_{i=1}^{n}\) is the signal realization organized into a vector structure \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\). It is necessary to design and numerically calculate several symptoms to describe the state of individual aggregates. Subsequently, \(m\) numerical signs \(s_{j}\) where \(j=1,2,\ldots,m\) represents the \(m\)-component sign vector \(\mathbf{s}=(s_{1},s_{2},\ldots,s_{m})\), which presents the state of the process or its parts. Equation (1) describes the extraction of symptoms from the vibration signal. When applying mathematical abstraction, it is possible to imagine that the investigated process is located in an m-dimensional linear vector space \(V_{m}(\mathbb{R})\). It is assumed that \(\mathbf{s}_{j}\in\mathbb{R}\) for each \(j=1,2,\ldots,m\). It is an infinite and continuous \(m\)-dimensional vector space for which \(V_{m}(\mathbb{R})=\mathbb{R}^{m}\). In industrial conditions, where the vibration signal is measured and digitized by an AD converter with limited resolution, the designed vector space of symptoms is finite. Therefore, numerically calculated symptoms are components of a symptom vector and take values from finite sets of values. In general, the symptoms themselves have different physical dimensions and different ranges of values.