## 1 Introduction
### Measurement Uncertainties--Overview  
In metrology science, the true values (ideal quantities) may never be known, and all measurements could potentially have some degree of uncertainty, which is often a function of several variables (sources). The difference between the true and measured values is known as an error. Uncertainty can, as defined by the Guide to the Expression of Uncertainty in Measurement (GUM) [9], be considered as a _'parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand'_. Thus, the estimated value \(y\) of the measurand \(Y\) is generally calculated using the relationship presented in Equation (1):  
\[y=f(x_{1},x_{2},\ldots,x_{n}) \tag{1}\]  
where \(x_{i}\) is the estimation for each input variable \(X_{i}\) that could potentially have a significant influence on the measurement result \((y)\). The function \(f\) can be known and explicit. However, in some cases, the measurement function is unknown or very complex, and no analytic expression is available.  
If the function \(f\) is explicit and the input quantities are not correlated, the law of propagation of uncertainties given in [9] generally represents the combined standard uncertainty on the estimated value \(u(y)\) by:  
\[u(y)=\sqrt{\sum_{i=1}^{n}\left(\begin{array}{c}\partial f\\ \partial x_{i}\end{array}\right)^{2}\!\!u^{2}(x_{i})} \tag{2}\]  
Figure 1: The inspection process definition activity model.  
where \(u(x_{i})\) is the standard uncertainty of each input variable \(x_{i}\) In practice, the expanded uncertainty \(U(y)\) corresponds to the combined standard uncertainty multiplied by a coverage factor \(k\), where \(k\) is chosen, for a prior confidence interval (1-\(\alpha\)) to be the \(\mathrm{t}_{1-\alpha/2,\
u}\) critical value from the \(t\)-table, with \(\
u\) degrees of freedom (Section 6, [9]).  
Monte Carlo simulations are typically used to approximate the statistical behavior of the measured value in situations where the measurement function cannot be found directly. To determine the output, the input variables are generated randomly for each simulation within their respective uncertainty ranges. The output probability density function (PDF) is then used for evaluating the uncertainty [10]. Finally, in cases where the measurement function is very complex (or unknown), an empirical estimation can be established using certain assumptions and simplification hypotheses, as proposed in the Measurement System Analysis (MSA) guide from the Automotive Industry Action Group [11].  
The MSA consists of a specifically designed experiment aimed at determining the components of variation in the measurement (e.g., the reproducibility, repeatability, bias, etc.). Indeed, the process of obtaining measurements (and defect level estimation) may have variations and produce uncertainty. The analysis tools proposed by the MSA (e.g., the gage repeatability and reproducibility (R&R)) evaluate the uncertainty on a direct measure \(\langle f(x)=x\rangle\), such as the thickness measurement from a micrometer. The aim of the whole process is to guarantee the integrity of the data used for quality analysis and to consider the consequences of a measurement error for decisions taken on the product. The reader is referred to [11] for more details.