# Risk Assessment for Linear Regression Models in Metrology
## 2 Materials and Methods
### Model Parameters
#### 2.2.2 Measurement Uncertainty
According to [49; 53], these measurement uncertainties can be calculated from the equation:  
\[\mathrm{u^{2}}\Big{(}\mathrm{y_{f}}\Big{)}=\mathrm{u^{2}}\big{(}\hat{\beta}_ {0}\big{)}+\mathrm{x_{j}^{2}u^{2}}\big{(}\hat{\beta}_{1}\big{)}+\hat{\beta}_ {1}^{2}\mathrm{\sigma_{x}}^{2}+2\mathrm{x_{j}u}\big{(}\hat{\beta}_{0},\hat{ \beta}_{1}\big{)},\mathrm{j}=1,2,\ldots,n_{2}, \tag{12}\]  
where the standard uncertainty associated with the input quantity \(\mathrm{X}\) is equal to  
\[\mathrm{\sigma_{x}}^{2}=\frac{1}{n-2}\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}\Bigg{(} \mathrm{x_{j}}-\frac{\mathrm{y_{ij}}-\hat{\beta}_{0}}{\hat{\beta}_{1}}\Bigg{)} ^{2}. \tag{13}\]  
The numerical calculation yields a value of \(\mathrm{\sigma_{x}}=0.029626~{}\mathrm{\upmu m}\) for the data from Table 1. According to [52], the standard uncertainty arising from the dependence of the parameters \(\hat{\beta}_{0}\) and \(\hat{\beta}_{1}\) equals:  
\[\mathrm{u}\big{(}\hat{\beta}_{0},\hat{\beta}_{1}\big{)}=-\overline{\mathrm{x }}\mathrm{u^{2}}\big{(}\hat{\beta}_{1}\big{)}=0. \tag{14}\]  
It is important to notice that the expression in equation (12) due to (14) only depends on the squares of the reference values of the moderate scale \(\mathrm{x_{j}^{2}},\mathrm{j}=1,2,\ldots,n_{2}\). Therefore, the graph of measurement uncertainties evaluated at the points of the moderate scale \(\mathbf{x}\) is a parabola with an upward opening that is axisymmetric concerning the direction \(\mathbf{x}=0\) (Figure 3). The measurement uncertainty calculated using the expression for the combined measurement uncertainty is the same for all points of the moderate scale \(\mathbf{x}\). In contrast, the measurement uncertainties calculated by the LRA method are different for different points of the scale. The vector of measurement uncertainties, calculated by the LRA, is denoted by \(\mathrm{\mathbf{u_{0}^{LRA}}}=\big{(}\mathrm{u_{0}^{L}},\mathrm{u_{0}^{L}}, \mathrm{u_{0}^{n_{2}}}\big{)}\). The point in the middle of the moderate scale has the lowest \begin{table}
\begin{tabular}{c c c} \hline \hline
**Source of Uncertainty \(\mathrm{L_{P}}\)** & **Probability Distribution** & **Standard Uncertainty/\(\mathrm{\upmu m}\)** \\ \hline ULM device & Normal & 0.1 \\ Linear error of the probe I & Rectangular & 0.000184462 \\ Linear error of the dial probe II & Rectangular & 0.065909762 \\ Resolution & Rectangular & 0.034641016 \\ \hline Combined measurement uncertainty of the probe \(\mathrm{u_{c}(L_{P})}\) & 0.1247 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Uncertainty budget for probe in the measuring range of \(\pm 30~{}\mathrm{\upmu m}\)[44]. measurement uncertainty, and the points on the edges have the highest measurement uncertainty. The values of measurement uncertainties for each point of the moderate scale **x** calculated by LRA are given in Table S2. In comparison with the value \(\mathrm{u}_{0}^{\mathrm{GUM}}\), the values \(\mathrm{u}_{0}^{\mathrm{LRA}}\) are underestimated. When calibrating devices, there are usually no historical data from previous measurements about the slope and intercept of the regression line or their standard errors. A well-performed calibration yields a slope close to one and an intercept close to zero. For the measure, it would be extremely difficult to determine, based on prior beliefs, what values can have a slope and an intercept, especially when it is taken into consideration that these values should be given to at least two decimal places. That kind of guessing would inevitably lead to an incorrect risk assessment. It is an even bigger problem if the data come from other non-metrology models, where the values for slope and intercept can be any. Therefore, the risk assessment problem for regression is reformulated so that the calibration data shown in Table 1 are used to determine the data \(\mathrm{y}_{0}\) and \(\mathrm{u}_{0}\) that are included in the expression for the prior distribution \(\mathrm{g}_{0}(\mathrm{n})\).