# Development of a Prediction Model for Tractor Axle Torque during Tillage Operation
## 3 Results
### Prediction Model
#### 3.3.1 Development of the Regression Model
The results of the ANOVA regression analysis for each model are shown in Table 7. Df, SS, and MS were used to calculate the F-value. The F-value, expressing how much more useful prediction by the regression was than prediction by the average, was the highest in Model B (2413), followed by the values in Models E (1547), F (1215), H (1106), D (892), G (875), C (813), and A (71). In all the models, the significance probability was 0.000, and the explanatory variable included in each model significantly explains the axle torque at a significance level of 0.01. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Source** & **Regression Model** & **R\({}^{2}\)** & **R\({}^{2}\) adj** & **S.E** \\ \hline A & \(f_{1}\) & \(T_{a}=1.626T_{\text{e}}+0.138\text{Se}_{\text{e}}+252.210\) & 0.275 & 0.271 & 39.98 \\ \hline B & \(f_{2}\) & \(T_{a}=23.189D_{t}+689.467\) & 0.866 & 0.866 & 17.16 \\ \hline C & \(f_{3}\) & \(T_{a}=19.599s-16.860S_{t}+919.198\) & 0.814 & 0.813 & 20.26 \\ \hline D & \(f_{1}+f_{2}\) & \(T_{a}=0.419T_{\text{e}}+0.042\text{Se}_{\text{e}}+21.785D_{t}+484.555\) & 0.878 & 0.877 & 16.41 \\ \hline E & \(f_{1}+f_{3}\) & \(T_{a}=2.565T_{\text{e}}+30.302\text{Se}_{\text{e}}-107.497S_{t}+237.209\) & 0.926 & 0.925 & 12.79 \\ \hline F & \(f_{2}+f_{3}\) & \(T_{a}=22.340D_{t}-6.175S_{t}+741.218\) & 0.867 & 0.867 & 17.11 \\ \hline G & \(f_{1}+f_{2}+f_{3}\) & \(T_{a}=-0.047\text{Se}_{\text{e}}+20.179D_{t}-14.662S_{t}+935.057\) & 0.876 & 0.875 & 16.54 \\ \hline H & \(f_{1}+f_{2}+f_{3}\) & \(T_{a}=0.580T_{\text{e}}+14.461D_{t}-40.263S_{t}+897.125\) & 0.899 & 0.899 & 14.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Regression model for predicting the axle torque of the tractor. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Model** & **Df** & **SS** & **MS** & **F-Value** & **\(p\)-Value** \\ \hline A & Regression & 2 & 225,648 & 112,824 & 71 & 0.000 \\  & Residual & 372 & 594,473 & 1598 & & & \\  & Total & 374 & 820,120 & & & & \\ \hline B & Regression & 1 & 710,323 & 710,323 & 2413 & 0.000 \\  & Residual & 373 & 109,797 & 294 & & & \\  & Total & 374 & 820,120 & & & \\ \hline C & Regression & 2 & 667,470 & 333,735 & 813 & 0.000 \\  & Residual & 372 & 152,651 & 410 & & \\  & Total & 374 & 820,120 & & & \\ \hline D & Regression & 3 & 720,217 & 240,072 & 892 & 0.000 \\  & Residual & 371 & 99,903 & 269 & & \\  & Total & 374 & 820,120 & & & \\ \hline E & Regression & 3 & 759,415 & 253,138 & 1547 & 0.000 \\  & Residual & 371 & 60,706 & 164 & & \\  & Total & 374 & 820,120 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of the analysis of variance (ANOVA) for each prediction regression model. Table 8 shows the analysis results for the regression coefficients of each axle torque prediction model. The \(\beta\) value of the standardized coefficient is an indicator of the influence of the explanatory variable on the dependent variable in each model. The tillage depth was the most used variable, as it was used in five of eight models (Models B, D, F, G, and H). The \(\beta\) of the tillage depth in Models B, D, F, G, and H were 0.931, 0.874, 0.897, 0.810, and 0.580, respectively. In particular, in Models B, D, F, G, and H, the \(\beta\) of the tillage depth was higher than that of the other variables, and this means that the tillage depth had the highest influence on the axle torque prediction. The explanatory variables of Model E with the highest accuracy were engine torque (t = 37.306, \(p\) = 0.000), travel speed (t = -57.115, \(p\) = 0.000), and engine speed (t = 21.478, \(p\) = 0.000). In this model, according to \(\beta\), it was found that engine torque, travel speed, and engine speed had the high influence, in order.