# Modeling of Dry Clutch Wear for a Wide Range of Operating Parameters
## 7 Wear Rate Variability Model
### Model Validation  
The obtained 95% CIs are shown in Figure 16 along with recorded and expectation model-predicted wear rate values. An initial visual inspection reveals that 95% CIs of all variability models are similar, both for modeling (Figure 16a) and validation points (Figure 16b). The tight alignment of 95% CIs is reflected in similar cost function values in  
Figure 15: Flowchart of wear rate variability model parameter optimization procedure.  
Table 6. The richest, multi-input model (22) expectedly provides the lowest cost function. The 95% CIs in Figure 16 embrace almost all tested points, thus pointing to the variability model accuracy. A distinct exemption is validation point #13, which may be considered as an outlier, (as performed in Section 4). Figure 16 also provides an additional, visual confirmation of expectation model validity discussed in Section 4: recorded vs. expected wear rate points fall around the ideal identity curve.  
A joint analysis of model residuals and related variability model performance is hindered by the wear rate variability dependence on the operating point. The joint analysis is possible if the residuals are normalized with respect to corrected wear rate standard deviation:  
\[\varepsilon_{scaled,j}=\frac{w_{j}-w_{u,j}}{w^{\prime}_{\sigma,j}},\;j\in\{1, \;2,\;\ldots,\;N\}. \tag{23}\]  
Mean and standard deviation of the residuals (23) are further denoted as \(\varepsilon_{scaled,\mu}\) and \(\varepsilon_{scaled,\sigma}\), respectively. If the model predictions \(w_{u,j}\) and \(w^{\prime}_{\sigma,j}\) are accurate, the scaled residuals \(\varepsilon_{scaled,j}\) should distribute according to the standard normal distribution, with zero-mean (\(\varepsilon_{scaled,\mu}=0\)) and unit standard deviation (\(\varepsilon_{scaled,\sigma}=1\)) [32]. Thus, the analysis of scaled residuals and related normality tests are used as a basis for additional variability models' validation (note that the wear rate expectation model was already validated, see Table 3). For this purpose, Kolmogorov-Smirnov (KS) test is employed, which sets a null hypothesis that the collection of considered samples is drawn from the standard normal distribution to be true if related \(p\) value is larger than the threshold of 0.05. As shown in Table 6, all models satisfy the KS test, both for modeling and validation points, which is reflected in near-zero mean values \(\varepsilon_{scaled,\mu}\) and near-unit standard deviations \(\varepsilon_{scaled,\sigma}\). Also, the normal distribution fit curves corresponding to the normalized residual histogram data (obtained by Matlab function _hisfitfit._) align well with the standard zero-mean unit-variance distributions (rescaled to correspond to respective histograms, Figure 17). The normality of residuals may also be taken as an indicator that there are no considerable systematic errors of wear rate expectation modeling, i.e., that the related residuals captured by the variability model can be majorly attributed to inherent wear rate variability.  
Figure 16: 95% confidence intervals (CI) obtained by different wear rate variability models, shown along recorded vs. expectation model-predicted wear rate values for modeling (**a**) and validation points (**b**).