# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 3 Results and Discussion
The _trainlm_ training function (L-M) and _learningdm_ (gradient descent with momentum weight and bias learning) adaptation functions were used. The highest values of R were obtained for ANN describing RI\({}_{1}\) (0.9371 for validation set), whereas for RI\({}_{2}\), R = 0.9256, and for WAC, R = 0.9203. Weights and biases between input and hidden layer and between hidden and output layers for rehydration indices RI\({}_{1}\), RI\({}_{2}\), and WAC are provided in Table 6.  
Figure 1: The best ANN structure and window of ANN training (**a**); the ANN goodness of fit and correlation coefficients R between ANN and output (**b**) for index RI\({}_{1}\). Figure 3: The best ANN structure (**a**) and the ANN goodness of fit and correlation coefficients R between ANN and output (**b**) for index WAC. Figure 2: The best ANN structure (**a**) and the ANN goodness of fit and correlation coefficients R between ANN and output (**b**) for index \(\text{RI}_{2}\). Table 7 shows the results of statistical analyses on the modelling of rehydration indices considered. The values of correlation coefficient R were high enough. The lowest R value (for all data) was received for the networks describing index WAC (0.9270), whereas the highest value was obtained for index RI\({}_{1}\) (0.9423). The gained values of RMSE were within the range of 0.0756 to 0.0802, and the lowest value demonstrated the neural networks describing rehydration index RI\({}_{1}\), while the network-characterised index WAC showed the highest value. As far as \(\chi^{2}\) is concerned, its values were as follows: 0.0065 for the neural networks describing index RI\({}_{1}\) and 0.070 for the network-characterised indices RI\({}_{2}\) and WAC. The obtained results show that all of the ANNs applied in this work predicted the behaviour of the considered rehydration indices satisfactorily. In the backward stepwise method applied in the present research, for each considered index, namely, RI\({}_{1}\), RI\({}_{2}\), and WAC, four models were generated, taking only three of the variables (parameters) as inputs. The variable not taken into account for which the resulting models gave the highest value of error was considered to be the most important one. Table 8 shows the backward stepwise results in which four models were generated, applying three of the discussed variables. The values of R, RMSE, and \(\chi^{2}\) of each generated model are shown in Table 8.  
As can be seen from the obtained results, the characteristic dimension \(L\) of apples had the greatest impact on the two of considered rehydration indices, namely, RI\({}_{2}\) and WAC. As far as RI\({}_{1}\) is concerned, characteristic dimension took the first place for error of R, whereas for RMSE and \(\chi^{2}\), it occupied the second position. The air drying velocity had the least influence on all of the rehydration indices. The rehydration temperature influences considered indices to a lesser extent than the drying temperature (with the exception of R error for index RI\({}_{2}\)). The obtained results indicate that both drying and rehydration conditions influenced (although in different degree) the rehydration indices. The rehydration indices, the same as rehydration kinetics, indicated the structural changes that had arisen as a result of