# Modeling and Key Parameter Interaction Analysis for Ship Central Cooling Systems
## 5 Results and Analysis
### Model Evaluation  
(1) Coefficient of Determination \(R^{2}\)  
\[R^{2}=1-\frac{SS_{\rm E}}{SS_{T}} \tag{10}\]  
\[R^{2}{}_{\rm adj}=1-\frac{SS_{\rm E}/d_{\rm E}}{SS_{\rm T}/d_{\rm T}}=1-\frac{d_{ \rm T}}{d_{\rm E}}(1-R^{2}) \tag{11}\]  
\[R^{2}_{pred}=1-\frac{PRESS}{SS_{\rm T}} \tag{12}\]  
where \(SS_{\rm E}\) represents the sum of squares of error; \(SS_{\rm M}\) represents the model sum of squares; \(SS_{\rm T}\) represents the total sum of squares; \(d_{\rm E}\) represents the degrees of freedom for the sum of squares of error; \(d_{\rm T}\) represents the degrees of freedom for the total sum of squares; and PRESS represents the predicted residual sum of squares.  
The effective range for the values of the three evaluation metrics was between 0 and 1. A value closer to 1 for \(R^{2}\), \(R^{2}{}_{\rm adj}\), and \(R^{2}{}_{\rm pred}\) indicated a better model fit.  
From Table 5, it can be seen that in this regression, \(R^{2}=0.9688\), indicating that 96.88% of the variability in the dependent variable was explained by the model. \(R^{2}{}_{\rm adj}=0.9438\), indicating that the constructed response surface model can explain 94.38% of the variability in the response values, adjusted for the number of predictors. The difference between \(R^{2}\) and \(R^{2}{}_{\rm adj}\) was only 0.025 (0.9688-0.9438), which was small, suggesting a good model fit. \(R^{2}{}_{\rm pred}=0.8752\). The difference between \(R^{2}{}_{\rm pred}\) and \(R^{2}{}_{\rm adj}\) was 0.0686 (0.9438-0.8752), indicating that the model had good predictive ability. Therefore, the response surface model constructed in this experiment demonstrated good predictive performance and can be effectively used for predicting and analyzing the experimental data.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Source**} & **Sum of** & \multirow{2}{*}{**df**} & \multirow{2}{*}{**Mean Square**} & \multirow{2}{*}{**F-Value**} & \multirow{2}{*}{**\(p\)-Value**} \\  & **Squares** & & & & \\ \hline Model & 7647.29 & 20 & 382.36 & 10.94 & \textless{}0.0001 \\ \(X_{1}\) & 5581.58 & 1 & 5581.58 & 159.71 & \textless{}0.0001 \\ \(X_{2}\) & 326.25 & 1 & 326.25 & 9.34 & 0.0053 \\ \(X_{3}\) & 572.29 & 1 & 572.29 & 16.38 & 0.0004 \\ \(X_{4}\) & 162.31 & 1 & 162.31 & 4.64 & 0.0410 \\ \(X_{5}\) & 27.25 & 1 & 27.25 & 0.7797 & 0.3857 \\ \(X_{1}\) \(X_{2}\) & 57.61 & 1 & 57.61 & 1.65 & 0.2110 \\ \(X_{1}\) \(X_{3}\) & 151.17 & 1 & 151.17 & 4.33 & 0.0480 \\ \(X_{1}\) \(X_{4}\) & 5.22 & 1 & 5.22 & 0.1494 & 0.7024 \\ \(X_{1}\) \(X_{5}\) & 0.6400 & 1 & 0.6400 & 0.0183 & 0.8934 \\ \(X_{2}\) \(X_{3}\) & 4.00 & 1 & 4.00 & 0.1145 & 0.7380 \\ \(X_{2}\) \(X_{4}\) & 5.22 & 1 & 5.22 & 0.1494 & 0.7024 \\ \(X_{2}\) \(X_{5}\) & 9.095 \(\times 10^{-13}\) & 1 & 9.095 \(\times 10^{-13}\) & 2.602 \(\times 10^{-14}\) & 1.0000 \\ \(X_{3}\) \(X_{4}\) & 0.3025 & 1 & 0.3025 & 0.0087 & 0.9266 \\ \(X_{3}\) \(X_{5}\) & 4.45 & 1 & 4.45 & 0.1274 & 0.7241 \\ \(X_{4}\) \(X_{5}\) & 6.81 & 1 & 6.81 & 0.1949 & 0.6626 \\ \(X_{1}\)\({}^{2}\) & 486.20 & 1 & 486.20 & 13.91 & 0.0010 \\ \(X_{2}\)\({}^{2}\) & 8.07 & 1 & 8.07 & 0.2310 & 0.6349 \\ \(X_{3}\)\({}^{2}\) & 52.32 & 1 & 52.32 & 1.50 & 0.2325 \\ \(X_{4}\)\({}^{2}\) & 7.78 & 1 & 7.78 & 0.2225 & 0.6412 \\ \(X_{5}\)\({}^{2}\) & 5.51 & 1 & 5.51 & 0.1576 & 0.6948 \\ Residual & 873.71 & 25 & 34.95 & & \\ Lack of Fit & 306.60 & 20 & 15.33 & 0.1908 & 0.3465 \\ Pure Error & 304.96 & 5 & 60.99 & & \\ Cor Total & 8521.01 & 45 & & & \\ \hline \end{tabular}
\end{table}
Table 4: Significance test of regression coefficients.