## 3 Predictive Models  
Multiple linear regression (MLR) models are built for predicting the MRR and TWR. The models are built using MATLAB(r) software and its dedicated function for linear regression. In a MLR model, linear functions are used to define the relationship between a response and a set of explanatory variables. The response can be explained as follows:  
\[\dot{y}\ =\ \ c_{0}+\sum_{i=\ 1}^{n_{v}}c_{i}*x_{i} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Run** & \multicolumn{5}{c}{**Processing Parameters**} \\ \hline
**\#** & \(\boldsymbol{T_{\text{on}}\) (\(\upmu\)s)} & \(\boldsymbol{U_{0}\) (V)} & \(\boldsymbol{F\) (kHz)} & \(\boldsymbol{K\) (-)} & \(\boldsymbol{U_{\text{e}}\) (V)} \\ \hline
1 & 7 & 130 & 83 & 62 & 62 \\
2 & 4 & 139 & 81 & 37 & 45 \\
3 & 3 & 131 & 83 & 27 & 31 \\
4 & 6 & 144 & 68 & 77 & 56 \\
5 & 3 & 117 & 61 & 65 & 55 \\
6 & 5 & 108 & 67 & 45 & 59 \\
7 & 7 & 141 & 95 & 53 & 50 \\
8 & 4 & 82 & 96 & 57 & 54 \\
9 & 5 & 132 & 57 & 53 & 39 \\
10 & 4 & 102 & 58 & 61 & 47 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Validation dataset: Applied processing parameters.  
where \(\hat{y}\) is the predicted value of the response, \(c_{0}\) is a constant, \(c_{i}\) is the regression coefficient for the \(i\)-th explanatory variable, \(x_{i}\) is the value of the \(i\)-th explanatory variable and \(n_{v}\) is the total number of explanatory variables.  
In this study, three regression models are built by using different explanatory variables to predict the same responses (MRR or TWR). Table 3 shows a summary of the explanatory variables, which are used in each model. The regression models are denoted as follows: PP, DD and PP + DD. In the PP model, the processing parameters are considered as explanatory variables. This model is used as a reference to analyse the effects of the data-driven approaches in terms of prediction accuracy. In the DD model, data collected through process monitoring are used for predicting the MRR and TWR. In particular, the average values of \(f_{\text{P}}\), \(v_{\text{f}}\) and \(u\) are used (\(F_{\text{P}}\), \(V_{\text{f}}\), and \(U\)). These average values are computed at the end of each experimental run from the recorded values of \(f_{\text{P}}\), \(v_{\text{f}}\) and \(u\), which are stored in the.csv files. The PP + DD model combines the explanatory variables used in the two other models. This model is built in order to understand whether combining the processing parameters with the information collected through process monitoring could be more advantageous.  
The normalised mean square error (NRMSE) is used to evaluate the prediction accuracy of the MLR models. The NRMSE is computed from the squared difference (SD) between the predicted (\(\hat{y_{j}}\)) and observed (\(y_{j}\)) values of the MRR and TWR for each \(j\)-th experimental run of the validation dataset. The average of the observed values (\(\overline{y}\)) is used for normalisation. In Equations (4) and (5), the formulas for computing the SD and NRMSE are shown respectively. Here \(n_{\text{r}}\) is the total number of experimental runs in the validation dataset (i.e., 10).  
\[\text{SD}_{j} = \left(y_{j}-\hat{y_{j}}\right)^{2} \tag{4}\] \[\text{NRMSE} = \frac{1}{\overline{y}}\sqrt{\frac{\sum_{j=1}^{n_{\text{r}}} \text{SD}_{j}}{n_{r}}} \tag{5}\]