## 2 The GUM Methodology and CMM Point Clouds
### Propagation of Variances  
The law of propagation of uncertainty, the basis of the GUM [3], in its multivariate setting [10] describes how uncertainties associated with the measured coordinates in Equation (4) can be evaluated on the basis of uncertainties associated with the systematic and random effects. Suppose effects \(\mathbf{e}_{i,B}=\mathbf{e}_{i}(\mathbf{b})\), \(i=1,\dots,m\) are specified by \(n_{B}\) parameters \(\mathbf{b}=(b_{1},\dots,b_{n_{B}})^{\top}\) and that a statistical model for \(\mathbf{b}\) specifies the \(n_{B}\times n_{B}\) variance matrix \(V_{B}\) associated with \(\mathbf{b}\). If \(G_{X|B}\) is the \(3m\times n_{B}\) sensitivity matrix of \(\mathbf{x}_{1:m}\) with respect to \(\mathbf{b}\) constructed from \(3\times n_{B}\) matrices  
\[G_{X|B,i}=\frac{\partial\mathbf{x}_{i}}{\partial\mathbf{b}^{\top}},\]  
then  
\[V_{X|B}=G_{X|B}V_{B}\sigma_{X|B}^{\top}.\]  
If \(V_{B}\) can be factored using an eigenvalue decomposition or Cholesky decomposition [25], for example, as \(V_{B}=K_{B}K_{B}^{\top}\) where \(K_{B}\) is an \(n_{P}\times p_{B}\) matrix (usually \(p_{B}=n_{B}\)), then \(V_{X|B}\) can be factored as  
\[V_{X|B}=K_{X|B}K_{X|B}^{\top},\quad K_{X|B}=G_{X|B}K_{B}.\]  
The role of the sensitivity matrix \(G_{X|B}\) can be explained as follows. If the parameters \(\mathbf{b}\) describing the systematic effects are perturbed by \(\Delta\mathbf{b}\), then the resulting perturbation on \(\mathbf{e}_{1:m}\), and hence, \(\mathbf{x}_{1:m}\), is given by \(\Delta\mathbf{x}_{1:m}=G_{X|B}\Delta\mathbf{b}\) to the first order. Often we are interested in quantities derived from a set of point coordinates. As a consequence of the chain rule in calculus, if \(\mathbf{a}=(a_{1},\dots,a_{n_{A}})^{\top}\) depends on \(\mathbf{x}_{1:m}\), and \(G_{A,X}\) is the \(n_{A}\times 3m\) sensitivity matrix of \(a\) with respect to \(x_{1:m}\); then, the \(n_{A}\times n_{B}\) sensitivity matrix \(G_{A|B}\) of \(A\) with respect to influence factors \(b\) is given by  
\[G_{A|B}=G_{A|X}G_{X|B}, \tag{5}\]  
and the \(n_{A}\times n_{A}\) variance matrix \(V_{A|B}\) describing the variance contribution to \(a\) arising from factors \(b\) is given by  
\[V_{A|B}=G_{A|B}V_{B}G_{A|B}^{\top}=K_{A|B}K_{A|B}^{\top},\quad K_{A|B}=G_{A|B}K _{B}.\]  
If the systematic effects \(b\) are perturbed by \(\Delta b\), then the derived parameters \(a\) are perturbed by \(\Delta a=G_{A|B}\Delta b\) to the first order.  
This paper considers the following CMM influence factors:  
* Repeatability effects (R)
* Probe qualification/location effects (PQ)
* Scale and squareness effects (S)
* Kinematic/geometrical errors: straightness errors (ET)
* Kinematic/geometrical errors: angular/rotation errors (ER)
* Probing effects: probe radius, errors depending on probing direction (P)  
The labels in brackets are used consistently in this paper to denote the corresponding influence factor. Temperature effects are assumed to arise via changes in scale and machine geometry. Influence factors arising from software used in coordinate metrology are also extremely important [26] but are not considered further here.  
In this paper, we concentrate on the evaluation point cloud uncertainties involving sensitivity matrices \(G_{X|B}\) and the uncertainties associated with the distance between pairs of points, a simple but important derived feature. The case of Gaussian-associated features is considered in [27] and shows how the sensitivity matrices \(G_{A|X}\) for parameters associated with geometric elements with respect to point cloud data can be calculated. Taken together with the sensitivity matrices \(G_{X|B}\) derived in this paper, the sensitivity matrices \(G_{A|B}\) can be evaluated as in Equation (5), allowing the uncertainty contribution to the geometric element parameters from each of the influence factors to be evaluated, e.g., the uncertainty associated with the radius of a Gaussian-associated cylinder to squareness errors.