# Modeling of Dry Clutch Wear for a Wide Range of Operating Parameters
## 6 Wear Rate Expectation Model
### Model Parametrization  
The model (10) is linear in parameters, and it can effectively be identified by using the least square method [29]. The parameters of each submodel contained within a vector \(\mathbf{\beta}\) are obtained by minimizing the following mean squared error (MSE) cost function:  
\[\mathbf{\beta}=\underset{\mathbf{\beta}}{argmin}\frac{1}{N}\sum_{j=1}^{N}\Big{(}w_{ \mu j}(\mathbf{\beta})-w_{j}\Big{)}^{2}, \tag{11}\]  
where \(N\) is the number of recorded points used for modeling (characterization and extrapolation points from Figure 3; \(N=101\)), \(w_{\mu j}\) is the model-predicted wear rate expectation, and \(w_{j}\) is the recorded wear rate for \(j\)th point of the modeling dataset. The parameters which minimize (11) are calculated as:  
\[\mathbf{\beta}=\Big{(}\mathbf{X}^{T}\mathbf{X}\Big{)}^{-1}\mathbf{X}^{T}\mathbf{w}, \tag{12}\]  
where \(\mathbf{w}\) is the column vector containing recorded wear rate points, and \(\mathbf{X}\) is the design matrix containing normalized features \(z_{j,i}\) for the related submodel. The matrix row and column indices \(j\) and \(i\), respectively, correspond to \(j\)th recorded point within the modeling dataset, \(j\in\{1,2,\ldots,N\}\), and \(i\)th feature, \(i\in\{1,2,\ldots,20\}\). The model features \(z_{j,i}\) are normalized to the interval [0, 1] as:  
\[z_{j,i,norm}=\frac{z_{j,i}-z_{i,min}}{z_{i,max}-z_{i,min}}, \tag{13}\]  
to obtain a well-conditioned design matrix \(\mathbf{X}\), where \(z_{i,min}\) and \(z_{i,max}\) correspond to minimum and maximum values of \(i\)th feature.  
Assessment of different submodels and selection of the optimal one are based on the coefficient of determination \(R^{2}\)[31]:  
\[R^{2}=1-\frac{\sum_{j=1}^{N}\Big{(}w_{\mu j}-w_{j}\Big{)}^{2}}{\sum_{j=1}^{N} \big{(}w_{j}-\overline{w}\big{)}^{2}}, \tag{14}\]  
where \(\overline{w}\) is the mean wear rate among \(N\) recorded modeling points. The index \(R^{2}\) takes values in the range \([0,1]\). It is often interpreted as a proportion of variation in the dependent response variable (here the wear rate) that is predictable from independent predictor variables. The maximum value \(R^{2}=1\) corresponds to the idealized, zero-error model case.  
By successively adding model terms to the model, i.e., by enriching it with additional degrees of freedom, the \(R^{2}\) index typically improves (increases) when calculated over modeling points, because of an increase in model fitting capacity (see Figure 12a,b). However, at some number of model parameters, the model starts to fit noise present in the data,while degrading target input-output dependencies, which is known as overfitting [29]. To avoid overfitting, a leave-one-out cross-validation is performed [29], providing a so-called predicted \(R^{2}\) value, denoted below as \(R_{p}^{2}\). The cross-validation procedure is as follows: (i) leave \(j\)th point from the modeling dataset and fit the model to remaining \(N-1\) points by using Equation (12), (ii) predict the wear rate by the model obtained in Point (i) for the removed \(j\)th point, \(w_{\mu j}\), and calculate the related model error/residual \(\varepsilon_{j}=w_{\mu j}-w_{j}\), (iii) repeat Steps (i)-(ii) until obtaining model residuals \(\varepsilon_{j}\) for all \(N\) points from the modeling dataset, and feed them into Equation (14) to obtain \(R_{p}^{2}\). The overfitting avoidance is demonstrated in Figure 12 through comparison of \(R^{2}\) and \(R_{p}^{2}\) indices. Figure 12 also indicates that the model accuracy strongly depends on combination of input features and not only on their number.