Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = \mathbf{p} + \lambda \mathbf{u}$, which has a minimum the distance to all points, i.e.
$$\text{Var}\left\{\sum_{i=1}^n \big(\mathbf{x}_i - \mathbf{u}(\lambda_i)\big)\right\} \rightarrow \min$$
Nothing is known about $\lambda_i$.
I am interested in the following two cases:
- all $\mathbf{y}_i$ are stochastically independent.
- $\mathbf{y}_i$ are taken stochastically dependent. I.e., $\mathbf{y}_{i+1} = \mathbf{y}_i + \mathbf{v}_i + \mathbf{z}_i$, where $\mathbf{v}_i$ are pairwise independent meanfree Gaussians and $\mathbf{z}_i = \mathbf{p}^\text{real} + \lambda_i^\text{real} \mathbf{u}^\text{real}$ is the unknown deterministic part.
Particularly, only the direction vector $\mathbf{u}$ and not the point $\mathbf{p}$ needs to be determined. I am thankful for any pointers to feasible methods.