## 2 Materials and Methods
### Data Obtained by Experiment and Their Formal Interpretation
#### 2.2.2 Regression Analysis of Experimentally Obtained Results  
The entire regression analysis procedure can be divided into three points:  
* Selection of the appropriate regression model;
* Determination or estimation of relevant coefficients;
* Description of the applied regression model by examining the residuals. Therefore, the first task is the choice of a suitable regression model. Basically, it is a proposal of the mutual behaviour of the investigated quantities. Through the points of the ordered pairs, it is necessary to lead a certain curve, analytically represented by a certain mathematical function, which will represent them as best as possible. The simplest regression model is a linear model, represented by a linear function, a polynomial of the first degree. In more complex cases of interdependence, one of the non-linear models can be applied; it can be a quadratic (parabolic) model prescribed by a second-degree polynomial or, in more complex cases of dependence, a cubic model prescribed by a third  
Figure 10: Values of pressures \(p\) [MPa] with a display coefficient of 0.08 in the pipeline in a time range of approximately 9 h were obtained from the pipeline operator control station. degree polynomial. Other nonlinear models used are hyperbolic, logarithmic, exponential, power, etc., prescribed by relevant functional dependencies. Based on the knowledge and experience acquired, the solver then chooses one or some of them. The degree of correctness is subsequently verified by means of appropriate tests and means of correlation analysis. In this case, it was a matter of determining three regression dependencies: \(p=f(t)\) (RD1), \(\Delta\sigma_{t}=f(\Delta p)\) (RD2) and \(\Delta\sigma_{m}=f(\Delta p)\) (RD3). In searching for them, a design procedure of several models was chosen, and the most appropriate one was selected based on their evaluation. For each of them, three models were chosen: linear, quadratic and cubic, represented by polynomials of the first degree, second degree, and third degree. For example, for RD1, the proposed cubic regression model (regression curve, line) has the analytical form:  
\[p=\beta_{0}t^{0}+\beta_{1}t^{1}+\beta_{2}t^{2}+\beta_{3}t^{3}, \tag{1}\]  
where \(\beta_{j}\) for \(j=0\), \(1\), \(\ldots\)\(s\), (where s is the degree of the polynomial) are the unknown values of the parameters of the regression curve. Nevertheless, during the calculation, the so-called smoothing regression line will be applied, expressed in the form:  
\[\hat{p}=\hat{\beta}_{0}t^{0}+\hat{\beta}_{1}t^{1}+\hat{\beta}_{2}t^{2}+\hat{ \beta}_{3}t^{3}, \tag{2}\]  
where \(\hat{p}\) is the balanced (theoretical, calculated) value of the real variable and \(\hat{\beta}_{j}\) for \(j=0\), \(1\), \(\ldots\)\(s\), \(j=0\), \(1\), \(\ldots\)\(s\), are point estimates (parameters) that should approximate as accurately as possible, i.e., approximately replace unknown parameters (regression coefficients) \(\beta_{j}\). A point estimate means an estimate by a single number, as opposed to an interval estimate, which is an estimate by a number interval. The second step of regression analysis is parameter estimation. Point estimates \(\hat{\beta}_{j}\) of the unknown parameters \(\beta_{j}\) of the regression line are most often obtained from the analysed data by the method of least squares (MLS). The purpose of applying this method is to minimize errors arising as a result of differences between the calculated, i.e., theoretical, and established (empirical) values, e.g., measured values of the dependent variable; these differences can acquire both positive and negative values. Therefore, they are squared, and their sums are expressed, referred to as the residual error sum of squares or the sum of square errors \(SSE\). \(SSE\) represents the sum of squared deviations of empirical and theoretical values of the dependent variable. One such considered square is graphically marked in Figure 11 for one of the values of the third degree polynomial for RD1.