# Risk Assessment for Linear Regression Models in Metrology
## 2 Materials and Methods
### Risk Calculation  
This paper observes the behavior of the global producers' risk \(\mathrm{R_{P}}\) and global consumers' risk \(\mathrm{R_{C}}\) at points defined along the guard band axis and along the moderate scale. As a result of the analysis, according to the procedure described below, risk surfaces were obtained. It is assumed that the maximal length of the guard band at each point of the moderate scale \(\mathbf{x}\) is equal to  
\[\mathrm{w_{max_{j}}}=0.2\left(\mathrm{T_{Uj}}-\mathrm{T_{Lj}}\right),j=1,2, \ldots,\mathrm{n_{2}}, \tag{18}\]  
where the value of the upper limit of the tolerance interval at the points of the moderate scale \(\mathbf{x}\) is given by the following equation:  
\[\mathrm{T_{Uj}}=\mathrm{y_{j}}+\frac{\Delta\mathrm{T}}{2},j=1,2,\ldots, \mathrm{n_{2}}, \tag{19}\]  
and the value of the lower limit of the tolerance interval at points of the moderate scale \(\mathbf{x}\) is given by the following equation:  
\[\mathrm{T_{Lj}}=\mathrm{y_{j}}-\frac{\Delta\mathrm{T}}{2},j=1,2,\ldots, \mathrm{n_{2}}. \tag{20}\]  
In equations (19) and (20), the values for \(\Delta\mathrm{T}\) from Table 3 are included. In addition, it is valid that the points \(\mathrm{y_{j}}=\mathrm{y}\left(\mathrm{x_{j}}\right)=\mathrm{x_{j}}\) are the points of the line \(\mathrm{y}=\mathrm{x}\), which corresponds to the perfect measurement. In other words, from equation (18), it follows that the acceptance interval occupies \(80\%\) of the tolerance interval. The \(10\%\) of the length of the guard band  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Uncertainty** & **Tolerance Interval** \\ \hline M1 & \(\mathrm{u_{0}}=\mathrm{u_{0}^{GUM}}\) & \(\Delta\mathrm{T}=0.6\ \mathrm{\SIUnit  
occupies the area between the acceptance line (curve) \(A_{L}\) and the tolerance line (curve) \(T_{L}\), below the line \(y=x\), and \(10\%\) of the length of the guard band occupies the area between the acceptance line (curve) \(A_{U}\) and the tolerance line (curve) \(T_{U}\), above the line \(y=x\) (Figure 4). The maximal length of the guard band \(w_{\max_{i}},j=1,2,\ldots,n_{2}\) in linearized models is the same for all points of the moderate scale \(x\). For the non-linearized model M3, the maximal length of the guard band \(w_{\max_{j}}\) has the smallest value at the middle point of the moderate scale, for \(x_{\gamma}=0\). The \(w_{\max_{j}}\) in non-linearized models has the highest value at the edges of the moderate scale. The lower and upper limits of the acceptance interval are defined by introducing a multiplicative factor \(r\in[-1,1]\). An equidistant subdivision of the interval \([-1,1]\) with a subdivision rate of \(0.1\) results in 21 subdivision nodes [18]. These subdivision nodes have the form \(r_{k}=-1+0.1\cdot(k-1),k=1,2,\ldots,n_{3}\), where \(n_{3}=21\). For nodes of the subdivision \(r_{k}\), the lower and upper limits of the acceptance interval at the points of the reference scale \(x\) are calculated from the following formulas:  
\[A_{L_{ijk}}=T_{L_{j}}+\frac{r_{k}\cdot w_{\max_{j}}}{2},j=1,2,\ldots,n_{2},k=1,2,\ldots,n_{3}, \tag{21}\]  
and  
\[A_{U_{ijk}}=T_{U_{j}}-\frac{r_{k}\cdot w_{\max_{j}}}{2},j=1,2,\ldots,n_{2},k=1,2,\ldots,n_{3}. \tag{22}\]  
For \(k=1\), it is valid that \(r_{k}=-1\), and in that case, \(\Delta A_{j}=1.2\Big{(}T_{U_{j}}-T_{L_{j}}\Big{)},j=1,2,\ldots,n_{2}\), i.e., the tolerance interval is within the acceptance interval (Figure 1a). For \(k=21\), it is valid that \(r_{k}=1\). In this case, \(\Delta A_{j}=0.8\Big{(}T_{U_{j}}-T_{L_{j}}\Big{)},j=1,2,\ldots,n_{2}\), and the acceptance interval is within the tolerance interval (Figure 1c). For \(k=11\), it is valid that \(r_{k}=0\) and that \(\Delta A_{j}=\Big{(}T_{U_{j}}-T_{L_{j}}\Big{)}=\Delta T,j=1,2,\ldots,n_{2}\). In this case, it is a shared risk model (Figure 1b). It should be noted that for all models, the tolerance lines (curves) calculated from equations (19) and (20) are fixed.