# Modeling of Dry Clutch Wear for a Wide Range of Operating Parameters
## 6 Wear Rate Expectation Model
### Model Validation  
The finally selected model structure which maximizes \(R_{p}^{2}\) index (Figure 12d) has ten parameters (out of maximum 20 parameters) and it reads:  
\[\begin{array}{ll}\hat{w}=\beta_{0}+\beta_{1}d_{s}+&\beta_{2}T_{d}+\beta_{12 }d_{s}T_{d}+\beta_{23}T_{d}M_{z}+\beta_{33}M_{z}^{2}+\beta_{123}d_{s}T_{d}M_{z }\\ &+\beta_{12}d_{s}^{2}T_{d}+\beta_{113}d_{s}^{2}M_{z}+\beta_{223}T_{d}^{2}M_{z}.\end{array} \tag{15}\]  
The parameters of selected model (15) are finally re-calculated with respect to all modeling points (i.e., not with one point omitted). The model (15) reveals that the surrogate distance input influences the wear rate in a rather complex way, i.e., in several combinations with other inputs, with the highest influence found to be in combination with temperature \(T_{d}\) (the parameters \(\beta_{12}\), \(\beta_{112}\), and \(\beta_{123}\) take on high values; around 10 times larger than \(\beta_{0}\)). The temperature exhibits the highest individual influence and appears in several influential combinations with surrogate distance and clutch torque (the parameters \(\beta_{2}\) and \(\beta_{23}\) are also found to be about 10 times larger than \(\beta_{0}\)).  
The optimal model (15) is validated against the following two distinct models based on \(R^{2}\), \(R_{p}^{2}\), and root mean square error metrics RMSE and RMSE\({}_{p}\): (i) a simple linear model: \(\hat{w}=\beta_{0}+\beta_{1}t_{2}+\beta_{2}T_{d}+\beta_{3}M_{z}\) (four parameters), and (ii) a full cubic model involving all  
Figure 12: \(R^{2}\) (**a**) and \(R_{p}^{2}\) (**c**) performance indices versus number of model parameters, for all 1,048,575 submodels contained within full cubic model (10). Zoomed-in views shown in (**b**,**d**), respectively.  
features (20 parameters). The comparative performance indices are given in Table 2 (see also Figure 12 for a wider response of indices \(R^{2}\) and \(R_{p}^{2}\)). The selected, 10 th-order model has a favorable accuracy characterized by rather high \(R^{2}\) and \(R_{p}^{2}\) values (not far from the ideal value of 1). A small gap between \(R^{2}\) and \(R_{p}^{2}\) indicates low level of overfitting of the modeling data. The low-order linear model has significantly lower \(R^{2}\) and \(R_{p}^{2}\) values, being around 0.55, and significantly larger residuals with RMSE being 1.85 times higher than that of the selected model. The full cubic is only slightly better in terms of \(R^{2}\) index (0.884 vs. 0.875), regardless of its significantly higher flexibility to fit the modeling data (20 vs. 10 parameters). On the other hand, the higher flexibility of the full cubic model leads to overfitting, as indicated by lower \({R_{p}}^{2}\) value compared to the selected (optimal) model.  
Table 3 includes the performance metrics of wear rate expectation model (15) with respect to validation points from Figure 3 (20 points not used in model parameterization). The \(R^{2}\) value of 0.411 is significantly lower than the corresponding value 0.875 obtained for modeling points. When omitting one of the validation points that appears to be an outlier (Point 13 in Figure 16b), \(R^{2}\) increases to 0.721, which is significantly closer to the value 0.875 obtained for modeling points. Also, the mean and standard deviation values of validation points' residuals are rather comparable to those of modeling points (Table 3).