## 4 Evaluation of the Soil Unit Weight Based on Laboratory Test Results
### Standard Regression  
The most important task of our analyses is to compare the results of standard regression to the neural network approach in the problem of soil unit mass identification. For this reason, a division of the entire set of 135 measurements into two groups (70%, 30%) was assumed. Ninety-four of them were used as basic data to calculate the parameters of the regression fit function. The remaining 41 were used to test the predictions of the soil unit mass from the regression model with measured values. To determine the statistical evaluation during the analysis, the basic data set was randomly selected 250 times. Four regression models of one variable were used: the two-parameter linear model (6), three-parameter polynomial model (7), two-parameter power model (8) and four-parameter power model (9). \[F1(x)=p_{1}+p_{2}x \tag{6}\]  
\[F2(x)=p_{1}+p_{2}x+p_{3}x^{2} \tag{7}\]  
\[F3(x)=p_{1}e^{p_{2}x} \tag{8}\]  
\[F4(x)=p_{1}e^{p_{2}x}+p_{3}e^{p_{4}x} \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline  & \(REy\) **(\(w\)) [\%]** & \(REy\) **(\(LOI_{T}\)) [\%]** & \(REy\) **(\(w\); \(LOI_{T}\)) [\%]** \\ \hline \(RE\)\({}_{MAX}\) & 21.76 & 121.21 & 22.45 \\ \(RE\)\({}_{MIN}\) & 0.07 & 1.19 & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 2: Relative error (RE) of the evaluated soil unit weight, based only on organic content, is not two-factor relationships. Figure 10: Comparison between measured values of soil unit weight and the values expected, based on regression analysis, including the model with (**a**) two independent variables (\(w\); \(LOI_{T}\)) and (**b**) one independent variable (\(w\)). Additionally, the next four regression models of the two variables were used: the three-parameter surface model (10), five-parameter surface model (11,12) and eight-parameter surface model (13). \[F5(x,y)=p_{1}+p_{2}x+p_{3}y \tag{10}\]  
\[F6(x,y)=p_{1}+p_{2}x+p_{3}y+p_{4}x^{2}+p_{5}x^{2}y \tag{11}\]  
\[F7(x,y)=p_{1}+p_{2}x+p_{3}y+p_{4}y^{2}+p_{5}xy^{2} \tag{12}\]  
\[F8(x,y)=p_{1}+p_{2}x+p_{3}y+p_{4}x^{2}+p_{5}y^{2}+p_{6}x^{2}y+p_{7}xy^{2}+p_{8 }x^{2}y^{2} \tag{13}\]  
The goodness of fit was checked using the coefficient of determination (14), mean relative error (15) and mean squared error (16):  
\[R^{2}=1-\frac{\sum_{p=1}^{n}\Bigl{(}d_{p}-y_{p}\Bigr{)}^{2}}{\sum_{p=1}^{n} \Bigl{(}d_{p}-\overline{d}_{p}\Bigr{)}^{2}}=\frac{\sum_{p=1}^{n}\Bigl{(}y_{p}- \overline{d}_{p}\Bigr{)}^{2}}{\sum_{p=1}^{n}\Bigl{(}d_{p}-\overline{d}_{p} \Bigr{)}^{2}} \tag{14}\]  
\[MRE=\frac{1}{n}\sum_{p=1}^{n}\Bigl{|}\frac{d_{p}-y_{p}}{d_{p}}\Bigr{|}100\% \tag{15}\]  
\[MSE=\frac{1}{n}\sum_{p=1}^{n}\Bigl{(}y_{p}-d_{p}\Bigr{)}^{2} \tag{16}\]  
where \(n\) is number of cases, \(d_{p}\) is the measured values, \(y_{p}\) is the fitted values (predicted) and \(\overline{d}_{p}\) is the mean of the measured values. The comparison of the median of the goodness of fit obtained for the regression of the one variable model is presented in Table 3. The median is the most resistant statistic datum and is of central importance in robust statistics. The best model obtained in the analyses was the power model F4, using natural water content. It has 1.83% of the median average relative error for testing the model. The comparison of the \(MRE\) models, included in Table 3 for laboratory data, is presented using box-and-whisker diagrams in Figure 10:0 On each box, the central mark is the median, the edges of the box are the 25 th and 75 th percentiles, the whiskers extend to the most extreme data points, which are not considered outliers, and outliers are plotted individually. Generally, we obtained better results using natural water content. Table 4 presents a comparison of the median of the goodness of fit obtained for the two variables regression models (F5-F8). The best result we obtained was for the F6 regression model.