## 4 Evaluation of the Soil Unit Weight Based on Laboratory Test Results
### Artificial Neural Networks Analysis  
Next, we applied a neural regression model. In all the examples, standard multi-layer perception with one hidden layer was applied. In this case, the nets have only one element in the output vector (soil unit weight). The number of hidden neurons is obtained from a cross-correlation procedure. In the calculations, five to eight neurons were used in the hidden layers. Exactly the same pattern divided as that for the standard regression was used to learn and test the networks. The comparison of the median of the goodness of fit, obtained for a few architectures, is presented in Table 5.  
The presented results include the median obtained parameters for one element in the input vector. During the calculation of the values, 20 repetitions of the network training were taken into account for each of the 250 pattern divisions. In this way, 1000 results were finally taken into account. Better prediction was obtained, like that in the standard regression for natural water content, in the input vector. A comparison of the \(MRE\) of the nets, included in Table 5, is presented in Figure 14.  
In this particular case, the network architecture does not affect the accuracy of approximation. Obtained mean relative errors of testing using organic matter content were in the range of 2.73-2.76%. Respectively, the \(MRE\) of testing using natural water content were in the range of 1.65-1.67%. In Table 6, the comparison of the median result, obtained for the nets with two elements in the input vector, is presented. In that approach, we also obtained a better result, like that in the standard regression with two independent variables. The mean relative errors of testing using two parameters were in the range of 1.40-1.44%. Figure 13: Regression of the F6 model of soil unit weight (\(LOI_{T}\), \(w\)): (**a**) base data and (**b**) test data. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Soil Parameter**} & \multicolumn{3}{c}{**[1-5-1]**} & \multicolumn{3}{c}{**[1-6-1]**} & \multicolumn{3}{c}{**[1-7-1]**} & \multicolumn{3}{c}{**[1-8-1]**} \\ \cline{2-10}  & & **Learn** & **Test** & **Learn** & **Test** & **Learn** & **Test** & **Learn** & **Test** \\ \hline \multirow{3}{*}{\(\gamma_{t}\) (\(LOI_{T}\))} & \(R^{2}\) & 0.945 & 0.948 & 0.944 & 0.947 & 0.945 & 0.947 & 0.945 & 0.947 \\  & \(MRE\) [\%] & 2.70 & 2.73 & 2.69 & 2.76 & 2.68 & 2.76 & 2.64 & 2.75 \\  & \(MSE\) & 0.381 & 0.346 & 0.380 & 0.352 & 0.379 & 0.341 & 0.376 & 0.350 \\ \hline \multirow{3}{*}{\(\gamma_{t}\) (\(w\))} & \(R^{2}\) & 0.976 & 0.978 & 0.977 & 0.979 & 0.977 & 0.979 & 0.977 & 0.979 \\  & \(MRE\) [\%] & 1.63 & 1.65 & 1.62 & 1.66 & 1.60 & 1.67 & 1.56 & 1.65 \\  & \(MSE\) & 0.160 & 0.146 & 0.160 & 0.143 & 0.159 & 0.144 & 0.158 & 0.144 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The goodness of fit of the ANN models with one input. The comparison test results from the _MRE_ of models, included in Table 6 for two laboratory parameters (\(LOI_{T},w\)), are presented in Figure 15. There is no clear difference in the results. The obtained results for the two inputs are a little better, which is the opposite case for the nets with one input. The smallest median average relative error test of the ANN was equal to 1.40%. Figure 16 shows detailed results for the ANN's prediction of the soil unit weight using two variables (\(LOI_{T},w\)) in the input vector. Figure 16a shows learning data and Figure 16b shows the test data. The coefficients of determination are very similar.