# The Predictive Model of Surface Texture Generated by Abrasive Water Jet for Austenitic Steels
## Materials and Methods
### Statistical Modelling by Regression Analysis  
In this subchapter, test and regression analysis are presented as methods to obtain a prediction model. This is a nonparametric alternative to [PERSON]'s analysis of variance (ANOVA). The Kruskal-Wallis test is a generalization of the nonparametric Mann-Whitney test for more than two groups being compared. Like the Mann-Whitney test, it does not test the consensus of specific parameters, but the consensus of the selected distribution functions of the compared files, with the key assumption being the independence of the compared values. If \(k\) is the number of compared selections, the null and alternative hypotheses of the Kruskal-Wallis test are expressed as (1), (2):  
\[H_{0}:F_{1}(x)=F_{2}(x)=\ldots=F_{k}(x) \tag{1}\]  
\[H_{1}:F_{1}(x)=F_{2}(x)=\ldots\
eq F_{k}(x) \tag{2}\]  
The alternative hypothesis \(H_{1}\) according to Equation (2) represents the fact that at least one \(Fi\) is different from the others. The main idea of the Kruskal-Wallis test is that under the validity of \(H_{0}\), the combined values from all samples are so well mixed that the average orders that correspond to the individual groups are similar [35; 36]. When calculating the test, all observations are sorted by order (as if they came from one selection), and we assign an order to the individual values (\(R_{ij}\) will indicate the order of the \(j\)-th value in the \(i\)-th group). Let \(k\) be the total number of groups and \(n\) the total number of observations and \(n_{1}\), \(n_{2}\),..., \(n_{k}\) the number of observations in individual groups (\(n=n_{1}+n_{2}+...+n_{k}\)). Next, let us denote \(T_{ij}\) as the sum of the orders in the \(i\)-th group (3):  
\[T_{i}=\sum_{j=1}^{n_{i}}R_{ij} \tag{3}\]then the test statistic of the Kruskal-Wallis test will take the form (4):  
\[Q=\frac{12}{n(n+1)}\sum_{i=1}^{k}\frac{T_{i}^{2}}{n_{i}}-3(n+1) \tag{4}\]  
It applies that the test statistic \(Q\) has a probability distribution with the parameter \(k-1\) under the validity of the null hypothesis \((H_{0})\)\(\chi^{2}\) (chi-square). We then reject the null hypothesis at the significance level \(\alpha\) if the realization of the test statistic \(Q\) is higher than the critical value (quantile) corresponding to the significance level \(\alpha\), i.e., if (5):  
\[Q\geq\chi_{k-1}^{2}(\alpha) \tag{5}\]  
Statistical modelling by regression analysis was applied to create a complex dependence of the experimentally investigated factors of the AWJ process on the surface roughness value \(Ra\). The basic results for the searched dependency in a general form can be expressed by Equation (6):  
\[\widehat{y}=b_{0}.x_{0}+\sum_{j=1}^{N}b_{j}.x_{j}+\sum_{ \begin{array}{c}u,j=1 \\ u\
eq j\end{array} }^{N}b_{uj}.x_{u}.x_{j}+\sum_{ \begin{array}{c}u,j=1\\ u\
eq j\end{array} }^{N}b_{uj}.x_{u}^{2}.x_{j}+\sum_{ \begin{array}{c}u,j=1\\ u\
eq j\end{array} }^{N}b_{uj}.x_{u}.x_{j}^{2}+\sum_{j=1}^{N}b_{jj}.x_{j}^{2} \tag{6}\]  
where \(Ra\), \(b_{0}\), \(b_{j}\), \(b_{uj}\), \(b_{jj}\) are the relevant regression coefficients, and \(x_{j}\) is the relevant independent variable; \(x_{1}\) represents nozzle-to-material distance [mm], \(x_{2}\) abrasive nozzle diameter [mm], \(x_{3}\) volume flow rate [l-min\({}^{-1}\)], \(x_{4}\) abrasive mass flow [g\(\cdot\)min\({}^{-1}\)], \(x_{5}\) cutting head feed rate [mm\(\cdot\)min\({}^{-1}\)], and \(x_{6}\) thickness of the material being cut [mm] (Table 7).