## 3 Application of the Process Method
### Machine Learning-Based Polynomial Regression  
Regression models are built using linear regression modeling, where low order polynomials are combined into multilayer polynomial functions. Prediction selection begins with the selection of the predictor that has the greatest influence on the mutual information of the hydraulic system working area under investigation. Once the most relevant predictor is selected from all available variables, the next predictor that maximizes relevance must be sought. In previous research, the selection of predictors summarized a novel approach to consider GPR models built with different available predictors to check the minimum number of predictors required and to identify the predictors with the highest contribution while not compromising the accuracy of the prediction [42]. The semi-constant trend of the input variables \(F_{C}\), \(v_{C}\) on the response of the hydraulic press \(\Delta X_{C}\) was summarized. In such cases, the estimation of model parameters involves nonlinear optimization techniques, as opposed to the linear optimization used in linear regression. To examine the nonlinear behavior of the hydraulic press, our approach proposes the statistical interpretation of the actual behavior of the hydraulic press under various external disturbances, using regression modeling implemented in a MATLAB 2022b environment with the open-source toolbox \"Curve Fitting Toolbox\". The research field proves that the simplicity in creating regression models leads to very accurate results, and polynomial regression in particular has been shown to be accurate in modeling complex data patterns [43]. The process involves selecting the correct polynomial degree, fitting the model, and determining the correct coefficients that minimize the error in predictive accuracy. The polynomial regression approach models the relationship between a dependent variable, also referred to as the response of the regression model \(Y\), and one or more independent variables, referred to as the input variables \(Z\). The complexity of the regression model is determined by the training dataset \(D=(Z,\,Y)\) where the number of input variables consists of D-dimensional N-input vectors \(Z=\{F_{n},\,v_{n}\}_{n-1}^{N}\) or \(\{\,v_{n}\}_{n-1}^{N}\) for the input variables \(F_{C}\), \(v_{C}\) and the desired outputs \(Y=\{\,\Delta X_{n}\}_{n-1}^{N}\) for the hydraulic press response error \(\Delta X_{C}\). For a pair \((Z,\,Y)\) with \(n\) observations, a regression model is obtained as follows:  
\[\mathrm{Y}=f(Z,\,\beta)+\varepsilon=Z\beta+\varepsilon \tag{7}\]  
If the number of input variables in polynomial regression is more than one, it is interpreted as multivariate polynomial regression [44]. Given the matrix expression in Equation (8), the unknown parameters of the regression model \(\,\beta=\{\beta_{0},\,\beta_{1},\ldots,\,\beta_{s}\}\) denote the hyperparameters and determine the characteristics of the regression model. Here, \(\varepsilon\), \(i=\{1,\ldots,n\}\) denotes the uncorrelated error term with mean zero and variance \(\sigma^{2}\) in the normal distribution. In addition, the number of outputs \(n\) and the number of input variables \(s\) determine the size of the input vectors \(Z\) and thus the complexity of the polynomial regression model. \[Y= \begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{bmatrix} ,\beta= \begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \vdots\\ \beta_{s}\end{bmatrix} ,\,\varepsilon= \begin{bmatrix}e_{1}\\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{bmatrix} ,\,Z= \begin{bmatrix}1&z_{11}&\cdots&z_{s,1}\\ 1&z_{21}&\cdots&z_{s,2}\\ \vdots&\vdots&\ddots&\vdots\\ 1&z_{1,n}&\cdots&z_{s,n}\end{bmatrix}  \tag{8}\]  
The standard deviation of the residuals \(Y\) characterizes the variability around the fitted regression line, i.e., the smaller \(\sigma(y)\), the better the fit. To express the unbiased estimate of the variance, the number of degrees of freedom of the polynomial fit \(s\) in Equation (10) must be considered.