## 2 Tracking Algorithms Overview
### Covariance Matrix to Ellipsoid  
It is crucial to understand the influence of a single term of the covariance matrix on the size and orientation of the ellipsoid. The principle component analysis (PCA) framework [11; 53] describes the relationship between the covariance matrix Q and the rotated ellipsoid as follows:  
\[\text{Q}=\begin{bmatrix}\sigma_{\text{x}}^{2}&\text{\rho}_{\text{xy}}\text{ \sigma}_{\text{x}}\text{\sigma}_{\text{y}}&\text{\rho}_{\text{x}}\text{ \sigma}_{\text{x}}\text{\sigma}_{\text{z}}\\ \text{\rho}_{\text{xy}}\text{\sigma}_{\text{x}}\text{\sigma}_{\text{y}}&\text{ \sigma}_{\text{y}}^{2}&\text{\rho}_{\text{yz}}\text{\sigma}  
A convenient way to express the ellipsoid equation is to use the covariance matrix Q:  
\[\overset{\rightarrow}{\mathrm{x}}^{\mathrm{T}}\mathrm{Q}^{-1}\overset{ \rightarrow}{\mathrm{x}}=1 \tag{16}\]  
where \(\overset{\rightarrow}{\mathrm{x}}=\left\{\mathrm{xyz}\right\}^{\mathrm{T}}\) is a vector expressed in Cartesian coordinates. \(\mathrm{Q}^{-1}\) can be written as:  
\[\mathrm{Q}^{-1}=\frac{1}{\rho} \begin{bmatrix}\frac{1-\rho_{\mathrm{yz}}^{2}}{ \sigma_{\mathrm{x}}^{2}}&\frac{-\rho_{\mathrm{xy}}+\rho_{\mathrm{yz}}\rho_{ \mathrm{yz}}}{\sigma_{\mathrm{x}}\sigma_{\mathrm{y}}}&\frac{-\rho_{\mathrm{zx} }+\rho_{\mathrm{yz}}\rho_{\mathrm{yz}}}{\sigma_{\mathrm{x}}\sigma_{\mathrm{z}} }\\ \frac{-\rho_{\mathrm{xy}}+\rho_{\mathrm{yz}}\rho_{\mathrm{yz}}}{\sigma_{\mathrm{x }}\sigma_{\mathrm{y}}}&\frac{1-\rho_{\mathrm{zx}}^{2}}{\sigma_{\mathrm{y}}^{2} }&\frac{-\rho_{\mathrm{yz}}+\rho_{\mathrm{xy}}\rho_{\mathrm{xz}}}{\sigma_{ \mathrm{y}}\sigma_{\mathrm{z}}}\\ \frac{-\rho_{\mathrm{yz}}+\rho_{\mathrm{yz}}\rho_{\mathrm{yz}}}{\sigma_{ \mathrm{x}}\sigma_{\mathrm{z}}}&\frac{-\rho_{\mathrm{yz}}+\rho_{\mathrm{xy}} \rho_{\mathrm{xz}}}{\sigma_{\mathrm{y}}\sigma_{\mathrm{z}}}&\frac{1-\rho_{ \mathrm{xy}}^{2}}{\sigma_{\mathrm{z}}^{2}}\end{bmatrix}  \tag{17}\]  
where:  
\[\rho=\left(1+2\rho_{\mathrm{xy}}\rho_{\mathrm{xz}}\rho_{\mathrm{yz}}-\rho_{ \mathrm{yz}}{}^{2}-\rho_{\mathrm{xx}}{}^{2}-\rho_{\mathrm{xy}}{}^{2}\right) \tag{18}\]  
Hence, the rotated ellipsoid equation can be written as:  
\[ \begin{split}\frac{\left(1-\rho_{\mathrm{yz}}^{2}\right)}{\sigma_ {\mathrm{x}}^{2}}\chi^{2}+\frac{\left(1-\rho_{\mathrm{yz}}^{2}\right)}{\sigma_ {\mathrm{y}}^{2}}\chi^{2}+\frac{\left(1-\rho_{\mathrm{xy}}^{2}\right)}{\sigma _{\mathrm{z}}^{2}}\chi^{2}-\frac{\rho_{\mathrm{xy}}-\rho_{\mathrm{xz}}\rho_{ \mathrm{xz}}}{\sigma_{\mathrm{x}}\sigma_{\mathrm{y}}}2\chi\mathrm{y}\\ -\frac{\rho_{\mathrm{xx}}-\rho_{\mathrm{yz}}\rho_{\mathrm{xy}}}{\sigma_{ \mathrm{x}}\sigma_{\mathrm{z}}}2\chi\mathrm{z}-\frac{\rho_{\mathrm{yz}}-\rho _{\mathrm{xy}}\rho_{\mathrm{yz}}}{\sigma_{\mathrm{y}}\sigma_{\mathrm{z}}}2 \chi\mathrm{z}=\rho\end{split}  \tag{19}\]  
Equation (20) expresses the contribution of the individual terms of the covariance matrix to the overall ellipsoid shape and size:  
\[\mathrm{Q}=\mathrm{U}\mathrm{A}\mathrm{U}^{\mathrm{T}} \tag{20}\]  
where:  
\[\Lambda= \begin{bmatrix}\lambda_{1}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\lambda_{\mathrm{m}}\end{bmatrix}  \tag{21}\]  
Hence, the ellipsoid equation in (17) can be modified as:  
\[\left(\overset{\rightarrow}{\mathrm{x}}-\overset{\rightarrow}{\mathrm{c}} \right)^{\mathrm{T}}\mathrm{R}\mathrm{A}^{-1}\mathrm{R}^{\mathrm{T}}\left( \overset{\rightarrow}{\mathrm{x}}-\overset{\rightarrow}{\mathrm{c}} \right)=1 \tag{22}\]  
where:  
\(\overset{\rightarrow}{\mathrm{x}}\) is a vector in the Cartesian coordinates about the nominal position (origin), \(\overset{\rightarrow}{\mathrm{c}}\) of the ellipsoid. R = rotation matrix,  
A = diagonal eigenvalues matrix respectively, which are derived from Q. \[\overset{\rightarrow}{\mathrm{x}}= \begin{bmatrix}\mathrm{x}_{\mathrm{ECI}}\\ \mathrm{y}_{\mathrm{ECI}}\\ \mathrm{z}_{\mathrm{ECI}}\end{bmatrix}  \tag{23a}\] \[\overset{\rightarrow}{\mathrm{c}}= \begin{bmatrix}\mathrm{x}_{\mathrm{EC} \mathrm{N}_{\mathrm{n}}}\\ \mathrm{y}_{\mathrm{EC}\mathrm{N}_{\mathrm{n}}}\\ \mathrm{z}_{\mathrm{EC}\mathrm{N}_{\mathrm{n}}}\end{bmatrix}