# Sensitivity Analysis for Gaussian-Associated Features
## 8 Sensitivity Matrix Associated with a Least-Squares Cone Fit to Data  
The calculations associated with an axis given above in Section 2 and a cylinder fit can be extended to evaluate the sensitivity matrix associated with a least-squares cone fit to data. The calculations below are based on specifying the cone in terms of an axis locating point \(\mathbf{x}_{A}\), an axis unit direction vector \(\mathbf{v}_{A}\), cone radius \(r_{0}\), and cone angle \(\phi\). The radius parameter is the radius of the circle defined by the intersection of the cone with the plane passing through \(\mathbf{x}_{A}\) and orthogonal to \(\mathbf{v}_{A}\), i.e., the set of points \(\mathbf{x}\) on the cone satisfying \((\mathbf{x}-\mathbf{x}_{A})^{\top}\mathbf{v}_{A}=0\). The cone angle is the angle the cone generator makes with cone axis, i.e., half the vertex angle, with the convention that if \(\phi>0\), then the vertex of the cone lies at \(\mathbf{x}_{A}+\mathbf{v}_{A}\) with \(t>0\). Although it may be natural to use the cone vertex as the axis locating point, the parametrization in terms of a radius remains stable for cone angles near zero. The distance \(d\) from a point \(\mathbf{x}\) to a cone specified by \(\mathbf{b}^{\top}=(\mathbf{x}_{A}^{\top},\mathbf{v}_{A}^{\top},r_{0},\phi)^{\top}\) is given by  
\[d=d(\mathbf{x},\mathbf{b})=(\cos\phi)d_{\mathrm{C}}(\mathbf{x},\mathbf{b})+(\sin\phi)d_{ \mathrm{P}}(\mathbf{x},\mathbf{a}), \tag{26}\]  
where \(d_{\mathrm{C}}(\mathbf{x},\mathbf{b})\) is the distance of \(\mathbf{x}\) to the cylinder specified by \(\mathbf{x}_{A}\), \(\mathbf{v}_{A}\) and \(r_{0}\) and \(d_{\mathrm{P}}(\mathbf{x},\mathbf{b})\) is the distance of \(\mathbf{x}\) to the plane specified by \(\mathbf{x}_{A}\) and \(\mathbf{v}_{A}\). _Analytical Approximation for Measuring a Patch of a Cone_  
Suppose data points \(\mathbf{x}_{i}=(r_{i}\cos\theta_{i},r_{i}\sin\theta_{i},z_{i})^{\top}\), \(r_{i}=r_{0}-\tan\phi z_{i}\), are distributed approximately uniformly on a cone with \(\mathbf{x}_{A}=\mathbf{0}\), \(\mathbf{v}_{A}=(0,0,1)^{\top}\), with \(-\alpha\leq\theta_{i}\leq\alpha\leq\pi\) and \(-a\leq z_{i}\leq a\). Parametrizing the cone in terms of \(\mathbf{a}=(x_{A},y_{A},u_{A},v_{A},r_{0},\phi)^{\top}\) (holding \(z_{A}\) and \(w_{A}\) constant), the associated \(m\times 6\) Jacobian matrix has \(i\)th row given by  
\[J(i,1:6)=\left[ \begin{array}{c}-\cos\phi\cos\theta_{i}\\ -\cos\phi\sin\theta_{i}\\ w_{i}\sin\theta_{i}\\ -w_{i}\cos\theta_{i}\\ -\cos\phi\\ z_{i}/\cos\phi\end{array} \right],\quad w_{i}=z_{i}/\cos\phi-r_{0}\sin\phi.\]  
Then, using the principle of Monte Carlo integration (15),  
\[\frac{1}{m}J^{\top}J\approx H_{aa\phi},\]  
where the nonzero elements of \(H_{aa\phi}\) are given by the integrals of functions of \(\theta\), \(z\) and \(\phi\) determined from the form of the Jacobian matrix above. The integrals are somewhat more complicated than the other cases already considered but can be easily evaluated using one-dimensional quadrature routines [22]. Here we give some example calculations. Table 6 shows the square roots \(s(\mathbf{a})\) of the diagonal elements of \(V_{aa\phi}=H_{aa\phi}^{1}\) as a function of \(\phi\) for the case \(\alpha=\pi\), \(r_{0}=50\) and \(a=100\). For \(V_{X}=\sigma_{R}^{2}\), \(u(\mathbf{a})=\sigma_{R}s(\mathbf{a})/\sqrt{m}\). As \(\phi\) approaches 90 degrees, the uncertainties associated with \(x_{A}\), \(y_{A}\) and \(r_{0}\) increase markedly. Table 7 shows the square roots \(s(\mathbf{a})\) of the diagonal elements of \(V_{aa\phi}\) as a function of \(\alpha\). For \(V_{X}=\sigma_{R}^{2}\), \(u(\mathbf{a})=\sigma_{R}s(\mathbf{a})/\sqrt{m}\). For a small arc of a cone, \(\alpha\) near zero, the uncertainties associated with \(x_{A}\), \(u_{A}\), \(v_{A}\), \(r_{0}\) and \(\phi\) scale with \(1/\alpha^{2}\) while those associated with the \(y_{A}\) and \(u_{A}\) scale with \(1/\alpha\). No parameter is well defined.