## 2 The GUM Methodology and CMM Point Clouds
### Summary of the Gum Methodology
The GUM methodology involves an input-output model in which the measurand(s) \(x\) are described as having a functional relationship \(\mathbf{x}=\mathbf{f}(\mathbf{b})\) on a set of inputs or _influence factors_\(\mathbf{b}\). Any statistical characterisation of the influence factors \(\mathbf{b}\) defines a corresponding statistical characterisation of the outputs \(x\). In particular, if \(\mathbf{b}\) is associated with a (multivariate) probability distribution with mean \(\hat{\mathbf{b}}\) and variance matrix \(V_{B}\), the mean \(\hat{\mathbf{x}}\) and variance matrix \(V_{X}\) associated with \(x\) are completely defined by the functional relationship \(\mathbf{x}=\mathbf{f}(\mathbf{b})\). If \(\mathbf{f}\) is a nonlinear function of \(\mathbf{b}\), the mean and variance associated with \(x\) may be difficult to compute exactly but can be approximated by linearising \(\mathbf{f}\) about \(\hat{\mathbf{b}}\). If \(G_{X|B}\) is the _sensitivity matrix_ of \(\mathbf{x}\) with respect to \(\mathbf{b}\),  
\[G_{X|B}(i,j)=\frac{f_{i}}{b_{j}} \tag{1}\]  
then the law of propagation of uncertainty (LPU, [24]) states that \(\hat{\mathbf{x}}\) and \(V_{X}\) are approximated by  
\[\hat{\mathbf{x}}\approx\mathbf{f}(\hat{\mathbf{b}}),\quad V_{X}\approx G_{X|B}V_{B}G_{X|B }^{\top}, \tag{2}\]  
a multivariate version of the well-known formula used in the GUM. The standard uncertainties \(u(\mathbf{x})\) associated with \(\hat{\mathbf{x}}\) are given by the square roots of the diagonal elements of \(V_{X}\). If the inputs \(\mathbf{b}\) are associated with a multivariate Gaussian distribution \(\mathbf{b}\sim\mathcal{N}(\hat{\mathbf{b}},V_{B})\), then the distribution associated with \(\mathbf{x}\) is approximated by \(\mathcal{N}(\hat{\mathbf{x}},V_{X})\). Here and elsewhere, the symbol \(\sim\) can be read as 'is distributed according to'. The LPU is exact for linear functions \(\mathbf{f}\) and, for linear \(\mathbf{f}\), if \(\mathbf{b}\) is associated with a Gaussian distribution, then \(\mathbf{x}\) is also associated with the given Gaussian distribution and no approximation is involved. For measurements that are associated with a number of influence factors, the distribution \(\mathcal{N}(\hat{\mathbf{x}},V_{X})\) is usually a suitable approximation to the true distribution. The Monte Carlo method (MCM, [9]) for uncertainty evaluation can be used as an alternative to the the LPU approach. The Monte Carlo approach involves assigning distributions to the influence factors \(\mathbf{b}\), sampling at random \(\mathbf{b}_{q}\), \(q=1,\ldots,M\), from these distributions and then evaluating \(\mathbf{x}_{q}=\mathbf{f}(\mathbf{b}_{q})\). Then, \(\mathbf{x}\) is estimated by the mean of the \(\mathbf{x}_{q}\) and the associated standard uncertainty is given by standard deviation of the sampled \(\mathbf{x}_{q}\). The Monte Carlo method is particularly appropriate if input-output function \(\mathbf{f}\) is significantly nonlinear. In coordinate metrology, relative accuracies are of the order of 1 part in \(10^{5}\) so that second order effects are of the order of 1 part in \(10^{10}\) and can be ignored in almost all applications. This means that the linearisation of \(\mathbf{f}\) in Equation (2) used to propagate the uncertainty information introduces no significant approximation error. The fact that CMM measurement is subject to a large number of influence factors also implies, through the central limit theorem, that the distributions associated with the measurands are approximated well by Gaussian distributions. Since Gaussian distributions are determined solely by their means and variances, the LPU method provides the essential information about the measurands.