# Sensitivity Analysis for Gaussian-Associated Features
## 7 Sensitivity Matrix Associated with a Least-Squares Cylinder Fit to Data
### Analytical Approximation for Measuring a Cylindrical Patch  
For this section, it is convenient to work in cylindrical coordinates. Suppose data points \(\mathbf{x}_{i}=(r_{0}\cos\theta_{i},r_{0}\sin\theta_{i},z_{i})^{\top}\) are distributed approximately uniformly on a cylinder \(x^{2}+y^{2}=r_{0}^{2}\) with \(-\pi\leq-\alpha\leq\theta_{i}\leq\alpha\leq\pi\) and \(-a\leq z_{i}\leq a\). In words, the points on the cylindrical patch are limited to a circumferential arc of length \(2\alpha\). Let \(\mathbf{x}_{A}=(x_{A},y_{A},z_{A})^{\top}\) and \(\mathbf{v}_{A}=(u_{A},v_{A},w_{A})^{\top}\) specify the locating point and direction vector of the cylinder axis. Parametrizing the cylinder in terms of \(\mathbf{a}=(x_{A},y_{A},u_{A},v_{A},r_{0})^{\top}\) (holding \(z_{A}\) and \(w_{A}\) fixed), the associated \(m\times 5\) Jacobian matrix has \(i\)th row given by  
\[J(i,1:5)=-[\cos\theta_{i}\ \sin\theta_{i}\ -z_{i}\sin\theta_{i}\ z_{i}\cos \theta_{i}\ 1].\]  
Then, using the principle of Monte Carlo integration (15),  
\[\frac{1}{m}J^{\top}J\approx H_{aa},\]  
where the nonzero elements of \(H_{aa}\) are specified by  
\[H_{aa}(1,1) =\frac{1}{2\alpha}(\alpha+(\sin 2\alpha)/2),\] \[H_{aa}(1,5) =(\sin\alpha)/\alpha\] \[H_{aa}(2,2) =\frac{1}{2\alpha}(\alpha-(\sin 2\alpha)/2),\] \[H_{aa}(3,3) =\frac{a^{2}}{6\alpha}(\alpha-(\sin 2\alpha)/2),\] \[H_{aa}(4,4) =\frac{a^{2}}{6\alpha}(\alpha+(\sin 2\alpha)/2),\] \[H_{aa}(5,5) =1.\]  
For \(\alpha=\pi\), corresponding to data approximately uniformly distributed on the cylindrical surface, we have  
\[V_{a\pi}=H_{a\pi}^{-1}=\left[\begin{array}{cccc}2&0&0&0&0\\ 0&2&0&0&0\\ 0&0&6/a^{2}&0&0\\ 0&0&0&6/a^{2}&0\\ 0&0&0&0&1\end{array}\right],\]  
showing that the uncertainties associated with the direction vector \(\mathbf{v}_{A}\) scale with \(1/a\) (but are independent of the radius \(r_{0}\)). For general \(\alpha\), the elements in the first, second and fifth rows and columns of \(H_{aa}\) above are exactly the same as the elements of \(H_{a}\) in (16) associated with the analysis of measurements of an arc of a circle. In particular, the behavior for measurements of a section of a cylinder subtending a small angle can be derived from the analysis on an arc of a circle. Let \(V_{aa}=H_{aa}^{-1}\). The matrix \(V_{a}\) given by (20) is a submatrix of \(V_{aa}\). For points approximately uniformly distributed in \(-\pi\leq-a\leq\theta_{i}\leq a\leq\pi\), \(-a\leq z_{i}\leq a\) and if the point-cloud data are associated with variance matrix \(V_{X}=\sigma_{R}^{2}I\), then the variance matrix \(V_{A}\) associated with the fitted cylinder parameters \(\mathbf{a}\) is approximated by  
\[V_{A}\approx\frac{\sigma_{R}^{2}}{m}V_{aa}. \tag{25}\]  
Table 5 shows the square roots \(s(\mathbf{a})\) of the diagonal elements of \(V_{aa}\) as a function of \(a\) (the height of the cylinder is \(2a\)) and \(\alpha\). For \(V_{X}=\sigma_{R}^{2}\), \(u(\mathbf{a})=\sigma_{R}s(\mathbf{a})/\sqrt{m}\). For a small arc of a cylinder, \(\alpha\) near zero, the uncertainties in the \(x\)-coordinate of the axis locating point and the radius scale with \(1/\alpha^{2}\), the \(y\)-coordinate of the axis locating point and the angle of rotation about the \(x\)-axis scale with \(1/\alpha\) while the angle of rotation of about the \(y\)-axis is well defined. The uncertainties associated with the angles of rotation scale with \(1/a\).