# Development of a Prediction Model for Tractor Axle Torque during Tillage Operation
## 2 Materials and Methods
### Development of Multiple Regression Models
#### 2.5.3 Regression Model Development  
The dataset used for the regression model was obtained through a field experiment. The data for the selected explanatory variables in this study are not easy to collect under all conditions. The tractor may not be able to communicate with CAN or may not have a built-in potentiometer. Therefore, in this study, the regression model was developed for each function (\(f_{1}\), \(f_{2}\), and \(f_{3}\)) and function combination in order to consider the conditions of data collection and the usability of the model. Equation (6), as function of \(f_{1}\), can be used when the tractor is capable of CAN communication; Equation (7), as function of \(f_{2}\), can be used when the potentiometer is built into the tractor; and Equation (8), as function of \(f_{3}\), can be used when an additional sensor is installed to measure the travel speed and wheel speed of the tractor. \[T_{a}=f_{1}\left(T_{e},\ S_{e}\right) \tag{6}\]  
where \(T_{a}\) is the axle torque (Nm), \(f_{1}\) is the functions related to the engine parameters, \(T_{e}\) is the engine torque (Nm), and \(S_{e}\) is the engine speed (rpm). \[T_{a}=f_{2}\left(D_{t}\right) \tag{7}\]  
where \(f_{2}\) is the measured data from the built-in sensor in the tractor, and \(D_{t}\) is the tillage depth (cm). \[T_{a}\ =f_{3}\left(S_{t},\ s\right) \tag{8}\]  
where \(f_{3}\) is the data measured by the additional installed sensors, \(S_{t}\) is the travel speed (km/h), and \(s\) is the slip ratio (%). To develop the axle torque prediction model, a simple regression analysis was used for each function relationship (\(f_{2}\)) using the enter method, and a multiple regression analysis was performed for the combination of each functional relationship (\(f_{1}\), \(f_{3}\), \(f_{1}+f_{2}\), \(f_{1}+f_{3}\), \(f_{2}+f_{3}\), \(f_{1}+f_{2}+f_{3}\)) using the stepwise method. Equation (9) is the equation of the input and output variables of the prediction model [41]. Based on Equation (9), the prediction model was developed, with a total of six variable conditions, such as the engine torque, engine speed, travel speed, slip ratio, and tillage depth, using the SPSS 21 software (SPSS Inc., Chicago, IL, USA). \[T_{a}=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots a_{n}x_{n}+\varepsilon \tag{9}\]  
where \(a_{0}\), \(a_{1}\), \(a_{2}\cdots a_{n}\) are the coefficients of the multiple regression, \(x_{1}\), \(x_{2}\cdots x_{n}\) are the regression model variables, and \(\varepsilon\) is the error of the model. To use regression analysis (including multicollinearity, only when using multiple regression), the following basic assumptions must be satisfied:  
1. Linearity: The relationship between the dependent and explanatory variables should be linear. This can be confirmed through correlation analysis. 2. Normality: The residuals must have a normal distribution, regardless of the value of the independent variable. The mean of the regression standardized residuals is zero, and a normal distribution with a constant variance is assumed. The normality of the estimation error can be confirmed through the distribution of the regression-standardized residuals and the P-P plots of the regression-normalized residuals of the expected cumulative probability versus actual cumulative probability. 3. Independence: The residuals in the dependent variable measurements should not affect each other. A Durbin-Watson test (D.W) is conducted to confirm the independence of the residuals of each model. In general, D.W has a value of 0 to 4, and the closer this value is to 2, the more the residual independence is guaranteed [42]. 4. Homoscedasticity: Homoscedasticity indicates that the scatter of the explanatory variable should be the same, regardless of the value of the dependent variable. Homoscedasticity can be confirmed by a scatter plot of the standardized predicted values and standardized residuals. 5.