# Development of a Prediction Model for Tractor Axle Torque during Tillage Operation
## 3 Results
### Prediction Model  
#### 3.3.1 Development of the Regression Model  
Eight regression models were developed through enter and stepwise methods as a combination of each function source using the measured calibration dataset. Table 6 shows the developed regression \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Parameter** & **Mean** & **Max.** & **Min.** & **Range** & **SD** \\ \hline  & \(T_{a}\) (Nm) & 1079.60 & 1202.69 & 977.87 & 224.82 & 46.82 \\  & \(T_{e}\) (Nm) & 316.07 & 392.40 & 252.34 & 140.06 & 23.20 \\ Calibration & \(S_{e}\) (rpm) & 2265.09 & 2410.23 & 1725.71 & 684.53 & 112.58 \\ set (\(n=375\)) & \(D_{t}\) (cm) & 16.82 & 22.50 & 13.73 & 8.77 & 1.88 \\  & \(S_{t}\) (km/h) & 6.07 & 6.93 & 5.04 & 1.89 & 0.36 \\  & \(s\) (\%) & 13.40 & 18.00 & 9.50 & 8.50 & 1.93 \\ \hline  & \(T_{a}\) (Nm) & 1080.26 & 1170.26 & 987.65 & 182.61 & 42.38 \\  & \(T_{e}\) (Nm) & 314.65 & 372.62 & 259.05 & 113.57 & 21.14 \\ Validation & \(S_{e}\) (rpm) & 2276.65 & 2407.68 & 1889.99 & 517.69 & 94.25 \\ set (\(n=125\)) & \(D_{t}\) (cm) & 16.76 & 21.29 & 13.94 & 7.35 & 1.68 \\  & \(S_{t}\) (km/h) & 6.08 & 6.90 & 5.24 & 1.66 & 0.34 \\  & \(s\) (\%) & 13.40 & 17.50 & 9.88 & 7.63 & 1.81 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical description of the calibration and validation dataset for each variable used in this study. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \multicolumn{4}{c}{**Pearson Correlation Coefficient (\(p\)-Value)**} \\ \hline
**Variable** & \(T_{a}\) & \(T_{e}\) & \(S_{e}\) & \(D_{t}\) & \(S_{t}\) & \(s\) \\ \hline \(T_{a}\) & 1.000 & \begin{tabular}{c} 0.505 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} \(-\)0.396 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} 0.931 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} \(-\)0.688 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} 0.897 ** \\ (0.000) \\ \end{tabular} \\ \hline \(T_{e}\) & & 1.000 & \begin{tabular}{c} 0.904 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} 0.445 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} 0.132 ** \\ (0.005) \\ \end{tabular} & \begin{tabular}{c} \(-\)0.431 ** \\ (0.000) \\ \end{tabular} \\ \hline \(S_{e}\) & & & 1.000 & \begin{tabular}{c} \(-\)0.355 ** \\ (0.000) \\ \end{tabular} & \begin{tabular}{c} \(-\)0.033 \\ (0.261) \\ \end{tabular} & \begin{tabular}{c} \(-\)0.320 ** \\ (0.000) \\ \end{tabular} \\ \hline \(D_{t}\) & & & 1.000 & \begin{tabular}{c} \(-\)0.714 ** \\ (0.000) \\ \end{tabular} & \
\begin{tabular}{c} 0.984 ** \\ (0.000) \\ \end{tabular} \\ \hline \(S_{t}\) & & & 1.000 &
\begin{tabular}{c} \(-\)0.690 ** \\ (0.000) \\ \end{tabular} \\ \hline \(s\) & & & 1.000 \\ \hline \end{tabular}
\end{table} 
Table 5: Correlation matrix for variables for predicting axle torque. model, R\({}^{2}\), adjusted R\({}^{2}\), and standardized error (S.E) for each regression model. The developed eight regression models showed an adjusted R\({}^{2}\) range of 0.271-0.925, indicating that the actual axle torque can be predicted with an accuracy of about 27.1-92.5%. The adjusted R\({}^{2}\) of regression model A, which used the engine torque and engine speed as the main variables, was the lowest, at 0.271. This means that the prediction accuracy is somewhat low, because the adjusted R\({}^{2}\) is less than 0.3. On the other hand, the adjusted R\({}^{2}\) of Model B, with the tillage depth, and Model C, with the slip ratio, were 0.866 and 0.813, respectively. These results showed that they had a high accuracy compared to regression Model A. In addition, Model B showed the highest accuracy among the prediction models using a single input source. In case of using a combination of each source as a variable, adjusted R\({}^{2}\) in Models F and E showed lowest (0.867) and highest (0.925), respectively. The S.E was found to be approximately between 12.79 and 39.98 for all the regression models. An ANOVA analysis--including the degrees of freedom (Df), sum square (SS), mean square (MS), F-value, and \(p\)-value for each model--was performed.