# Sensitivity Analysis for Gaussian-Associated Features
###### Abstract  
This paper is concerned with the evaluation of the uncertainties associated with Gaussian-associated features following the GUM methodology. We show how sensitivity matrices necessary for a GUM uncertainty evaluation can be calculated and how the variance matrices associated with the feature parameters can be estimated for a range of complete and partial features common in engineering. Example results are given in tables that allow practitioners to estimate, a priori, the uncertainties associated with fitted parameters, given a proposed measurement strategy for the case in which the point-cloud variance matrix is a multiple of the identity matrix  
\(\hat{\mathbf{a}}\). If \(G_{A|X}\) is the _sensitivity matrix_ (the symbol \(A|X\) can be read as '\(A\) given \(X\)') of \(\mathbf{a}\) with respect to \(\mathbf{x}\),  
\[G_{A|X}(i,j)=\frac{\partial f_{i}}{\partial x_{j}},\]  
then the law of propagation of uncertainty (LPU, [20,21]) states that \(\hat{\mathbf{a}}\) and \(V_{A}\) are approximated by  
\[\hat{\mathbf{a}}\approx f(\hat{\mathbf{x}}),\quad V_{A}\approx G_{A|X}V_{X}G_{A|X}^{ \top}, \tag{1}\]  
a multivariate version of the well-known formula used in the GUM. The standard uncertainties \(u(\mathbf{a})\) associated \(\hat{\mathbf{a}}\) are given by the square roots of the diagonal elements of \(V_{A}\). The key to implementing a GUM approach is the evaluation of the sensitivity matrices \(G_{A|X}\), the main subject of this paper. These sensitivity matrices depend on the measurement strategy as represented by the (nominal) points \(\mathbf{x}_{1:m}\) being measured and this paper discusses how \(G_{A|X}\) can be calculated for the geometric elements common in engineering. The uncertainties associated with the features depend mostly on the geometry of the patch of geometric surface being measured rather than on the precise details of where the measuring points are located on the patch, so long as the distribution of the points is representative of the patch. In this paper, we use tools borrowed from Monte Carlo integration [22] to estimate uncertainties under the assumption that the proposed measurement strategy is approximately equivalent to a strategy in which a finite set of points is dispersed evenly over the surface being sampled, enabling the variance matrix to be calculated analytically. This allows us to estimate uncertainties without knowing the measurement strategy, only the number of data points and geometry of the area being sampled. The approach is used to examine the effect of sampling only a portion of a geometric element, an arc of a circle, a cap of a sphere, etc., and provide asymptotic results for limiting cases, e.g., as the angle of arc of a circle measured tends to zero. Example uncertainty estimates for different partial features of circles, spheres, planes, cylinders and cones are given in tables. The methods described in this paper are all based on deriving the sensitivity matrix \(G_{A|X}\) that is used to construct the variance matrix \(V_{A}\) for the derived features \(\mathbf{a}\) from the variance matrix \(V_{X}\) associated with the point cloud \(\mathbf{x}_{1:m}\). For the features involved, the functional relationship \(\mathbf{a}=\mathbf{f}(\mathbf{x}_{1:m})\) of \(\mathbf{a}\) on the point cloud is smooth and almost linear so that the first order approximation of \(\mathbf{f}\) and the application of the law of propagation of uncertainty (1) is very effective in estimating uncertainties associated with the derived features. For features derived according to Chebyshev/minimum zone and related criteria, the functional relationship is not smooth and the first order approximation of \(\mathbf{f}\) might not be fit for purpose. An alternative approach is to use a Monte Carlo sampling approach [19] generating point-cloud data sets \(\mathbf{x}_{1:m,q}\) and derived features \(\mathbf{a}_{q}=\mathbf{a}_{q}(\mathbf{x}_{1:m,q})\) for each data set, \(q=1,\ldots,M\). The variance matrix associated with the sample \(\mathbf{a}_{1:M}\) is an approximation to \(V_{A}\).