# Some Aspects of the Modelling of Dried Red Beets Rehydration Process
## 3 Results and Discussion
We chose the structures of ANNs that were uncomplicated and simultaneously characterized by appropriately good statistics. The RI and WAC were predicted applying artificial neural networks. MLP 4-4-1 was adopted for RI (Figure 1a) and MLP 4-4-1 for WAC (Figure 2a), whereas MLP 4-4-2 was adopted for both indices RI and WAC together (Figure 3a) with _logsig_  
\[\mathrm{F(x)}=\frac{1}{1+exp^{-\beta x}} \tag{13}\]  
and _tagsig_  
\[\mathrm{F(x)}=\frac{2}{1+exp^{-\beta x}}-1 \tag{14}\]  
transfer functions in the hidden layer and the output layer, respectively. The _trainlm_ (training) and _learningdm_ (adaptation) functions were used. The ANN describing index RI gave the highest values of the correlation coefficient R for the validation set (R = 0.9662), for WAC, the discussed coefficient for the validation set equaled 0.9518, and for both indices RI and WAC together, the value of R = 0.9508. Table 11 presents the weights and biases (between the input and hidden layer and between the hidden and output layers) for indices RI, WAC, and both RI and WAC. Table 12 presents the results of statistical analyses on the modelling of rehydration indices. As far as the coefficient of determination R\({}^{2}\) is concerned, the highest values were obtained for the ANNs describing both indices RI and WAC, and the values were as follows: R\({}^{2}\) = 0.9348 for RI, R\({}^{2}\) = 0.9327 for WAC, and R\({}^{2}\) = 0.9347 for RI and WAC. The values of R\({}^{2}\) for the networks describing the discussed indices separately were lower, namely, 0.9207 for RI and 0.9078 for WAC. The same situation was observed for the values of RMSE. The lowest, so the best values, were achieved for the neural networks describing both indices RI and WAC: RMSE = 0.0529 for RI, RMSE = 0.5049 for WAC, and RMSE = 0.0539 for RI and WAC. The network adopted for RI gave the values of RMSE equal to 0.5084, and as far as the network adopted for WAC is concerned, RMSE = 0.2285. It resulted from the investigations that the ANNs adopted to characterize both indices RI and WAC together can be recommended as the most suitable for the prediction of the behavior of discussed  
In the backward stepwise (BS) method used in this work for the discussed rehydration indices, namely RI and WAC and both RI and WAC together, four models were generated, considering only three out of the four variables (parameters), namely, \(T_{d}\), \(v_{d}\), \(T_{r}\) and \(L\), as inputs. The omitted variable for which the generated models indicated the highest error (so the lowest value of R\({}^{2}\) and the highest value of RMSE) was regarded as the most important one. The BS results in which four models were generated by omitting one from the four considered variables are presented in Table 13. This table shows the values of R\({}^{2}\) and RMSE for each generated model. Figure 3: The ANN structure and details of ANN training. (**a**) The goodness of fit of ANN, (**b**) for both indices RI and WAC together (inputs: \(T_{d}\), \(v_{d}\), \(T_{r}\) and \(L\)). Figure 2: The ANN structure and details of ANN training. (**a**) The goodness of fit of ANN, (**b**) for index WAC (inputs: \(T_{d}\), \(v_{d}\), \(T_{r}\) and \(L\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rehydration Index** & **Omitted Parameter** & \(\mathbf{R^{2}}\) & **RMSE** \\ \hline \multirow{4}{*}{RI} & \(T_{d}\) & 0.3539 (2) *  & 0.2484 (2) \\  & \(v_{d}\) & 0.3539 (2) & 0.2844 (2) \\  & \(T_{r}\) & 0.8783 (3) & 0.0613 (3) \\  & \(L\) & 0.2139 (1) & 0.7067 (1) \\ \hline \multirow{4}{*}{WAC} & \(T_{d}\) & 0.4325 (2) & 0.2127 (4) \\  & \(v_{d}\) & 0.4325 (2) & 0.4811 (3) \\  & \(T_{r}\) & 0.7655 (3) & 0.5518 (2) \\  & \(L\) & 0.2295 (1) & 0.7068 (1) \\ \hline \multirow{4}{*}{RI + WAC} & \(T_{d}\) & 0.5077 (2) & 0.2013 (2) \\  & \(v_{d}\) & 0.5077 (2) & 0.201 (2) \\  & \(T_{r}\) & 0.8800 (3) & 0.0801 (3) \\  & \(L\) & 0.1536 (1) & 0.2510 (1) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Sensitivity analysis on the modelling of rehydration indices—BW method.