I am trying to use SVD to fit a particle track through a detector and getting the direction vector from the right singular matrix. The matrix $A$ is defined by the positions $\vec{p}_i$ of the signals minus the average $\vec{p}_0$. $$ A = \begin{pmatrix} \vdots & \vdots & \vdots\\ x_i & y_i & z_i \\ \vdots & \vdots & \vdots \end{pmatrix} - \begin{pmatrix} \overline{x_i} & \overline{y_i} & \overline{z_i} \end{pmatrix}$$ $$ A = U \Sigma V^T $$ The first row of $V^T$ should then be the direction vector $\vec{v}$ of the track. So the particle track should be described by the parametric equation $$\vec{p}_0 + t \cdot \vec{v} $$
So my question is - obviously the positions of the signals do have an uncertainty (pixel size in $x$ and $y$ and time resolution in $z$). What can I do to have an uncertainty estimate for the direction vector? My goal is to extend the track to $z=0$ and get the $(x,y)$ position on that plane where I would like to know the uncertainty of the position.