# Statistical Validation Framework for Automotive Vehicle Simulations Using Uncertainty Learning
## 5 Validation Domain
### Validation Parameter Configurations  
The variations of all linear combinations of the parameter \(\theta\) form the possible parameter configuration space \(\mathbf{\Theta}\). Because measuring all parameter combinations is too time-consuming, ten linear combinations \(\theta^{\text{v}}\) that form the validation parameter configurations \(\mathbf{\Theta}^{\text{v}}\subseteq\mathbf{\Theta}\) are used. During the variation, only the three parameters \(\theta^{\text{v}}_{\text{r}}\) are varied. They are chosen to cover the three mayor driving roll, acceleration and air resistances of the longitudinal consumption cycle of the WLTP. They separate the parameter configuration \(\theta^{\text{v}}\) into fixed and regressor parameters defined as:  
\[\theta^{\text{v}} =(\theta^{\text{v}}_{\text{fix}},\theta^{\text{v}}_{\text{r}}) \tag{5}\] \[\theta^{\text{v}}_{\text{r}} =[M_{\text{veh}}\ c_{\text{d}}\ p]\subseteq\theta^{\text{v}} \tag{6}\]  
\(\theta^{\text{v}}_{\text{r}}\) is called regressor, because this vector will be used for validation uncertainty learning with regression. The minimum amount of validation experiments for the uncertainty learning method used is \(n_{\text{min}}=(p-1)+2=5\), while \((p-1)\) is the amount of regressor \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n_{i}\) & \(M_{\text{veh}}\) in kg & \(c_{\text{d}}\) & \(p_{\text{type}}\) in bar & \(c_{\text{rr}}\) in \% & \(r_{\text{dyn}}\) in mm & \(y^{\text{v}}_{\text{s},\text{1}-\text{3}}\) in Wh & \(u^{\text{v}}\) in Wh \\ \hline
1 & \([925,933]\) & 0.37 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1183,1182,1181)\) & 56.40 \\
2 & \([1025,1033]\) & 0.37 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1293,1280,1300)\) & 19.86 \\
3 & \([1125,1133]\) & 0.37 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1358,1344,1337)\) & 26.56 \\
4 & \([1225,1233]\) & 0.37 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1411,1406,1401)\) & 32.41 \\
5 & \([1025,1033]\) & 0.27 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1144,1139,1141)\) & 49.69 \\
6 & \([1025,1033]\) & 0.32 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1205,1200,1205)\) & 43.69 \\
7 & \([1025,1033]\) & 0.42 & 2.5 & \(\mathcal{N}(10.37,\,0.817^{2})\) & \(\mathcal{N}(275.2,\,0.22^{2})\) & \((1305,1307,1319)\) & 50.64 \\
8 & \([1025,1033]\) & 0.37 & 1.5 & \(\mathcal{N}(13.20,\,0.873^{2})\) & \(\mathcal{N}(273.8,\,0.26^{2})\) & \((1306,1339,1320)\) & 43.61 \\
9 & \([1025,1033]\) & 0.37 & 2.0 & \(\mathcal{N}(11.52,\,0.879^{2})\) & \(\mathcal{N}(274.6,\,0.16^{2})\) & \((1321,1311,1302)\) & 23.35 \\
10 & \([1025,1033]\) & 0.37 & 3.0 & \(\mathcal{N}(9.43,\,10.83^{2})\) & \(\mathcal{N}(275.8,\,0.21^{2})\) & \((1279,1281,1276)\) & 24.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Validation model uncertainty learning data set with the variation of 10 linear combinations of the regressor parameters \(\theta^{\text{v}}_{\text{r}}=[M_{\text{veh}}\ c_{\text{d}}\ p\text{type}]\) forming the input data set \(\mathbf{\Theta}^{\text{v}}_{\text{r}10\times 3}\in\mathbf{\Theta}^{\text{v}}\), with 10 corresponding results of the model uncertainty \(u^{\text{v}}\) forming the result data set \(\mathbf{\hat{a}}^{\text{a}}\). \(y_{\text{s},1-3}\) are three single measurements of the consumption in the WLTP cycle used for the calculation of \(u^{\text{v}}\). \(c_{\text{rr}}\) and \(r_{\text{wheel}}\) are additional data resulting from the changing type pressure \(p_{\text{type}}\). dimensions. One base configuration using the parameters of Table 1 is varied by the one factor at a time (OFAT) method to form the ten validation parameter configurations shown in Table 2. The ten parameter configurations \(\theta_{\rm r}^{\rm v}\) form the regressor validation parameter configurations \(\mathbf{\Theta}_{\rm E}^{\rm v}\subseteq\mathbf{\Theta}^{\rm v}\).