# Experimental Research on Unit Operation for Fruit Crops' Bones Calibration
## 3 Results and Discussion
The adequacy check of the obtained models was carried out according to the Fisher criterion [31]. It is determined that the hypothesis about the adequacy of the description by the equation of experimental results is accepted with 95% probability:  
* for cherries \(Fp\) = 1.71 \(<Fcr\) = 2.19;
* for sweet cherries \(Fp\) = 1.03 \(<Fcr\) = 2.19;
* for plum cherries \(Fp\) = 2.01 \(<Fcr\) = 2.19;
* for apricots \(Fp\) = 1.80 \(<Fcr\) = 2.19;
* for almonds \(Fp\) = 1.92 \(<Fcr\) = 2.19. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Regression Statistics** & **Cherry** & **Sweet Cherry** & **Cherry Plum** & **Apricot** & **Almond** \\ \hline Adequacy variance, \(Da\) & 0.600 & 0.941 & 2.413 & 1.850 & 2.290 \\ \hline Average variance of the whole experiment, \(D_{m}\) & 0.200 & 0.314 & 0.800 & 0.617 & 0.765 \\ \hline Multiple correlation coefficient, \(R\) & 0.964 & 0.897 & 0.999 & 0.950 & 0.870 \\ \hline Determination coefficient, \(R^{2}\) & 0.930 & 0.804 & 0.997 & 0.900 & 0.750 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Regression statistics of mathematical models of the process of sizing the bones of fruit crops. The critical value of the Fisher criterion was determined by the number of degrees of freedom [31]: \(f_{1}=N-(k+1)=10;f_{2}=N\cdot(\gamma-1)=28\). To analyze the obtained Equations (4)-(8), the method of two-dimensional sections was used. Calculations to determine the optimal values of the optimization parameters in accordance with Equations (4)-(8) were carried out using the computer programs \"Mathematica 5\" and \"MathCad\". Consider the application of this method on the example of one crop--cherries. The resulting regression Equation (5) is reduced to the canonical form. To do this, we move the origin of coordinates to a new point in the factor space. Considering that each of the factors varies within:  
\[ \begin{array}{l}-1\leq X_{1}\leq 1;\\ -1\leq X_{2}\leq 1;\\ -1\leq X_{3}\leq 1;\end{array} \]  
the coordinates of the new centers in coded form for each crop will be:  
\[R=\left\{ \begin{array}{l}x_{1}\\ x_{2}\\ x_{3}\end{array} \right\}\]  
The resulting second-order regression equation is differentiated with respect to each independent variable, \(x_{1}\), \(x_{2}\), \(x_{3}\), and equated to zero. This gives:  
\[ \begin{array}{l}\frac{dy}{dx_{1}}=8.474+7.63\cdot x_{1}+8.695\cdot x_{2}+3. 169\cdot x_{3}=0\\ \frac{dy}{dx_{2}}=14.823+8.695\cdot x_{1}+2.758\cdot x_{3}=0\\ \frac{dy}{dx_{1}}=4.41+3.169\cdot x_{1}+2.758\cdot x_{2}+30.89\cdot x_{3}=0 \end{array}  \tag{9}\]  
Solving the system of equations, we obtain the coordinates of the new center:  
\[R_{\text{missweet cherry}}=\left\{ \begin{array}{l}0.053\\ -1\\ 0.059\end{array} \right\}\]  
Substituting these values into the regression Equation (5), we obtain the value of the criterion at the minimum point \(Y_{\text{min sweet cherry}}=9.61\). Thus, the analysis of Equations (4), (6)-(8) was performed and response surfaces for all crops were built (Figures 4-8), their appearance determined and an analysis carried out by the method of two-dimensional sections, depending on the factors of variation, fixing each of them at a certain level. Figure 4: The graph and contour curves of the dependence of the criteria \(Y\) on the factors \(x_{3}\), \(x_{2}\) for cherries, when \(x_{1}=0\). Figure 5: Graph and contour curves of the dependence of criteria \(Y\) on the factors \(x_{1}\), \(x_{3}\) for cherries, when \(x_{2}=0\). Figure 6: The graph and contour curves of the dependence of criteria \(Y\) on the factors \(x_{1}\), \(x_{3}\) for cherries, when \(x_{2}=0\). Figure 7: Graph and contour curves of the dependence factors of criterion \(Y\) on factors \(x_{2}\), \(x_{3}\) for apricot, when \(x_{1}=0\). The coordinates of the new centers for cherries, apricots, cherry plums and almonds are presented here:  
\[R_{cherry}=\left\{ \begin{array}{l}0.364\\ -1\\ -1\end{array} \right\}\text{; }R_{plum\ cherry}=\left\{ \begin{array}{l}0.193\\ -1\\ -1\end{array} \right\}\text{; }R_{apricot}\ =\left\{ \begin{array}{l}-1\\ -1\\ 0.037\end{array}