# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Variable**} & \multicolumn{1}{c}{**SSE**} & \multicolumn{1}{c}{**R\({}^{2}\)**} & \multicolumn{1}{c}{**Adjusted R\({}^{2}\)**} & \multicolumn{1}{c}{**RMSE**} \\ \hline \(L^{\star}\) & \(0.0077\)–\(0.2359\) & \(0.6306\)–\(0.9889\) & \(0.6306\)–\(0.9889\) & \(0.0394\)–\(0.2068\) \\ \(b^{\star}\) & \(0.0009\)–\(0.5514\) & \(0.3516\)–\(0.9986\) & \(0.3516\)–\(0.9986\) & \(0.0177\)–\(0.3031\) \\ \(h^{\star}\) & \(0.0105\)–\(0.6388\) & \(0.2601\)–\(0.9865\) & \(0.2601\)–\(0.9865\) & \(0.0419\)–\(0.3574\) \\ C\({}^{\star}\) & \(0.0012\)–\(0.6660\) & \(0.3611\)–\(0.9980\) & \(0.3611\)–\(0.9980\) & \(0.0241\)–\(0.3085\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Summary results of statistical analyses on the modelling of lightness \(L^{\star}\), yellowness \(b^{\star}\), hue angle \(h^{\star}\), and chroma \(C^{\star}\) changes during the rehydration of dried apples. In the backward stepwise method used in the presented work, for the colour variations \(\Delta E\), five models were generated, taking only four 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 11 presents the backward stepwise results in which five models were generated by applying four of the discussed variables. The values of R, RMSE, R\({}^{2}\), adjusted R\({}^{2}\), and SSE of each generated model are shown in Table 11. The discussed table also shows the results of statistical analyses on the modelling of colour variations \(\Delta E\). It is shown from the conducted research that all of the ANNs applied in this work predicted the behaviour of colour variators quite acceptably. As can also be seen from the obtained results, the characteristic dimension \(L\) of apples had the greatest impact on the colour variations (only for RMSE did it occupy the second position). The drying air temperature took the second position. The drying air temperature took second place (with the exception of RMSE being in first place), and then, successively, drying air velocity held \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Statistical Test Method** & **Parameter** & **All Parameters** & **Omitted \(L\)** & **Omitted \(T_{d}\)** & **Omitted \(v_{d}\)** & **Omitted \(T_{r}\)** & **Omitted \(t\)** \\ \hline R & 0.9048 & 0.1395 & 0.4424 & 0.4482 & 0.4859 & 0.6721 \\ RMSE & 0.0567 & 0.2654 & 0.2953 & 0.1980 & 0.1246 & 0.1022 \\ R\({}^{2}\) & 0.8187 & 0.0195 & 0.1957 & 0.2009 & 0.2361 & 0.4517 \\ Adjusted R\({}^{2}\) & 0.8177 & 0.0143 & 0.1915 & 0.1967 & 0.2321 & 0.4488 \\ SSE & 0.0573 & 0.1332 & 0.1207 & 0.1203 & 0.1176 & 0.0996 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Sensitivity analysis for the colour variation \(\Delta E\)—stepwise method. Figure 5: The best ANN goodness of fit and correlation coefficients R between ANN and output (a), and window of neural network training (b) for the colour variation \(\Delta E\). the third position, and the rehydration temperature was in fourth place. Time (the duration of the rehydration process) had the least influence on colour variations. Due to complex biochemical reactions and water loss in the drying process come adverse changes in product quality, whereas these changes are dependent also on the drying regime. Due to the high content of water, sugars (glucose, fructose), and the presence of pectins and malic acid, the discussed product quality changes apply especially to apples [77]. The apple colour change could be associated with the synthesis of phenolic compounds, the non-enzymatic browning reactions [78], the decomposition of original pigments, the formation of brown pigments (by enzymatic and non-enzymatic browning reactions), and other undesirable pigments (pigments responsible for the original colour: chlorophyll (green), carotenoids, flavonoids (yellow), anthocyanins (red)) [79].