# Statistical Validation Framework for Automotive Vehicle Simulations Using Uncertainty Learning
## 5 Validation Domain
### Validation Uncertainty Learning  
To estimate the model uncertainty for the evaluation of a new application parameter configuration, an uncertainty learning method \(G_{\text{lea}}\) is needed to train a prediction model \(G_{\text{pred}}^{\text{va}}\) based on the measured regressor validation parameter configuration space \(\boldsymbol{\Theta}_{\text{r}}^{\text{v}}\) and the corresponding measured validation uncertainties \(\boldsymbol{u}^{\text{v}}\). A possible solution for this learning model  
\[G_{\text{lea}}:(\boldsymbol{\Theta}_{\text{r}}^{\text{v}},\boldsymbol{u}^{ \text{v}})\mapsto G_{\text{pred}}^{\text{va}} \tag{13}\]  
is discussed in this section. There are various possibilities to train such a prediction model. Since the aim of this framework is wide applicability, the requirements of the prediction model are general applicability, minimum amount of data learning points, no over-fitting, high extrapolation capability and the uncertainty quantification of the prediction reliability based on model fit error. Basic surrogate models using the Gaussian process (GP) need large datasets and the prediction capability outside the scope of the dataset is low because the model converges to the mean of the initial function [58]. A surrogate model using polynomial chaos expansion (PCE) also has the disadvantage of needing more data. Additionally, PCE does not deliver information about the quality of the fit, which disables uncertainty quantification of the prediction reliability [59]. As proposed by [PERSON] and [PERSON] [21] and [PERSON] [56], the framework consequently uses simple linear regression. Additionally, a prediction interval for a single future value is calculated that considers the error of the regression fit and thus the non-linearities of the prediction. Higher polynomials are also possible but they need more data and there is a greater chance of over-fitting. The prediction interval is based on the confidence band which was first solved by [PERSON] [60] for simple linear regression. It is the best solution for this purpose and has no competition in general, as ([PERSON] [61], p. 112) states. A linear regression forms the mean of the prediction interval. The width is calculated from the quality of the regression fit depending on the error between the data points and the response surface. A general multiple linear regression model with a normal distributed error is assumed:  
\[\mathbf{Y} =\mathbf{X}\beta+\mathbf{e} \tag{14}\] \[\mathbf{e} \sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\mathbf{I})\] \[\mathbf{X}_{n\times p} =[\mathbf{x}_{0,n\times 1},\mathbf{x}_{1,n\times 1},\ldots, \mathbf{x}_{p-1,n\times 1}]\text{ has rank }p\;(p<n),\]  
where \(\mathbf{x}_{0}\equiv 1\). \(\mathbf{Y}_{n\times 1}\) are the \(n\) observations of the \(n\) regressors \([x_{1},\ldots,x_{p-1}]_{n\times 1}\). \(p-1\) defines the dimension of the regressors and is the number variable inputs of the regression and the subsequent prediction. The regression model forms a \(p\) dimensional space. The estimators \(\hat{\beta}\) and \(s\) of the unknown parameters \(\beta\) and \(\sigma\) in the linear model can be approximated with:  
\[\hat{\beta} =\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{Y} \tag{15}\] \[s =\sqrt{\frac{\mathbf{Y}^{T}\mathbf{Y}-\hat{\beta}\,\mathbf{X}^{T} \mathbf{Y}}{n-p}} \tag{16}\]  
For a fixed \(\beta\), Equation (17) creates a surface over the \(p\) dimensional space. \(\mathbf{\tilde{Y}}_{1\times 1}\) is the mean prediction of the fitted surface at the predictor point \(\mathbf{\hat{x}}_{1\times p}=[\hat{x}_{0},\hat{x}_{1},\ldots,\hat{x}_{p-1}]\). \[\mathbf{\tilde{Y}}=\mathbf{\hat{x}}\hat{\beta} \tag{17}\]If \(k\) is the number of simultaneously predicted points, the simultaneous prediction intervals \(I_{\hat{Y}}(\hat{y}|\hat{\mathbf{x}})\) can be estimated for each of the \(k\) future values so that they all fall into their respective intervals with a total confidence of \(1-\alpha\).