## 3 Results and Discussion
### Optimization of UHRE Conditions
Analysis of variance (ANOVA) for second-order polynomial models is shown in Table 3. The statistical significance and adequacy of the regression model are evaluated by F-value and \(p\)-value. As can be seen from Table 3, greater F-values (F \(>\) 7.84) and low \(p\)-values (\(p<\) 0.01) for model terms displayed that the four fitting models were statistically significant and indicated that the yields of four flavonoids by UHRE could be well described with those models. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Source**} & **Sum of** & **Degrees of** & **Mean** & \multirow{2}{*}{**F-value**} & \multirow{2}{*}{**\(p\)-Value**} & \multirow{2}{*}{**Significance a**} \\  & **Squares** & **Freedom** & & & & \\ \hline \multicolumn{7}{c}{Yield of quercetin (QU)} \\ Model & \(3.68\times 10^{-3}\) & 9 & \(4.09\times 10^{-4}\) & 32.80 & 0.000 & **\(\ast\ast\)** \\ \(X_{1}\) & \(3.66\times 10^{-6}\) & 1 & \(3.66\times 10^{-6}\) & 0.29 & 0.605 & ns \\ \(X_{2}\) & \(6.03\times 10^{-6}\) & 1 & \(6.03\times 10^{-6}\) & 0.48 & 0.509 & ns \\ \(X_{3}\) & \(1.64\times 10^{-3}\) & 1 & \(1.64\times 10^{-3}\) & 131.40 & 0.000 & **\(\ast\ast\)** \\ \(X_{1}X_{2}\) & \(6.59\times 10^{-5}\) & 1 & \(6.59\times 10^{-5}\) & 5.29 & 0.055 & ns \\ \(X_{1}X_{3}\) & \(1.14\times 10^{-4}\) & 1 & \(1.14\times 10^{-4}\) & 9.15 & 0.019 & \(\ast\) \\ \(X_{2}X_{3}\) & \(4.49\times 10^{-5}\) & 1 & \(4.49\times 10^{-5}\) & 3.61 & 0.099 & ns \\ \(X_{1}{}^{2}\) & \(2.21\times 10^{-4}\) & 1 & \(2.21\times 10^{-4}\) & 17.76 & 0.004 & **\(\ast\ast\)** \\ \(X_{2}{}^{2}\) & \(6.57\times 10^{-4}\) & 1 & \(6.57\times 10^{-4}\) & 52.73 & 0.000 & **\(\ast\ast\)** \\ \(X_{3}{}^{2}\) & \(7.52\times 10^{-4}\) & 1 & \(7.52\times 10^{-4}\) & 60.37 & 0.000 & **\(\ast\ast\)** \\ Residual & \(8.72\times 10^{-5}\) & 7 & \(1.25\times 10^{-5}\) & & & \\ Lack of fit & \(8.72\times 10^{-5}\) & 3 & \(2.91\times 10^{-5}\) & & & \\ Pure error & 0.00 & 4 & 0.00 & & & \\ \hline \multicolumn{7}{c}{Yield of kaempfer (KA)} \\ Model & \(4.07\times 10^{-2}\) & 9 & \(4.52\times 10^{-3}\) & 14.36 & 0.001 & **\(\ast\ast\)** \\ \(X_{1}\) & \(2.00\times 10^{-4}\) & 1 & \(2.00\times 10^{-4}\) & 0.64 & 0.451 & ns \\ \(X_{2}\) & \(1.18\times 10^{-4}\) & 1 & \(1.18\times 10^{-4}\) & 0.37 & 0.560 & ns \\ \(X_{3}\) & \(9.19\times 10^{-3}\) & 1 & \(9.19\times 10^{-3}\) & 29.21 & 0.001 & **\(\ast\)** \\ \(X_{1}X_{2}\) & \(1.04\times 10^{-3}\) & 1 & \(1.04\times 10^{-3}\) & 3.29 & 0.112 & ns \\ \(X_{2}X_{3}\) & \(8.88\times 10^{-4}\) & 1 & \(8.88\times 10^{-4}\) & 2.82 & 0.137 & ns \\ \(X_{2}X_{3}\) & \(3.91\times 10^{-3}\) & 1 & \(3.91\times 10^{-3}\) & 12.42 & 0.010 & \(\ast\) \\ \(X_{1}{}^{2}\) & \(1.84\times 10^{-3}\) & 1 & \(1.84\times 10^{-3}\) & 5.86 & 0.046 & \(\ast\) \\ \(X_{2}{}^{2}\) & \(1.06\times 10^{-2}\) & 1 & \(1.06\times 10^{-2}\) & 33.60 & 0.001 & **\(\ast\)** \\ \(X_{3}{}^{2}\) & \(1.06\times 10^{-2}\) & 1 & \(1.06\times 10^{-2}\) & 33.74 & 0.001 & **\(\ast\)** \\ Residual & \(2.20\times 10^{-3}\) & 7 & \(3.15\times 10^{-4}\) & & & \\ Lack of fit & \(2.20\times 10^{-3}\)A 3D surface plot visualizes the correlation between responses and two independent variables, by viewing a three-dimensional surface of the predicted responses. The effects of the extraction time, ultrasonic power and the liquid-solid ratio on the flavonoids yield and their interactions are seen in Figure 5.  
Figure 5: Response surface diagrams indicating the effects of variables (ultrasonic power, W; extraction time, min; liquid–solid ratio, mL/g) on the yields of QU, KA, GI and SC.