Robust least squares adjustment

The robust least squares adjustment used in GROOPS is based on a modified Huber estimator. It downweights observations with large otuliers iteratively.

The algorithm starts with a first solution with equal weights $\M P =\M I$ \begin{equation} \hat{\M x} = (\M A^T\M P\M A)^{-1}\M A^T\M P\M l. \end{equation}The solution is used to compute the residuals \begin{equation} \hat{e}_i = \left(\M l - \M A \hat{\M x}\right)_i \end{equation}and the redundancies of all observations \begin{equation} r_i = \left(\M I - \M A(\M A^T\M P\M A)^{-1}\M A^T\M P\right)_{ii}. \end{equation}For observations with large residuals a new standard deviation is assigned \begin{equation} \sigma_i = \begin{cases} 1 & \text{for } \left|\frac{\hat{e}_i}{r_i}\right| \le h\cdot\hat{\sigma} \\ \left|\frac{\hat{e}_i}{r_ih}\right|^p & \text{for } \left|\frac{\hat{e}_i}{r_i}\right| > h\cdot\hat{\sigma}, \end{cases} \end{equation}where $h$ is huber, $p$ is huberPower, and $\hat{\sigma}^2$ a robust overall variance factor computed from all residuals. The estimation is repeated huberMaxIteration times with a new weight matrix \begin{equation} \M P = \text{diag}\left(\frac{1}{\sigma_1^2}, \frac{1}{\sigma_2^2}, \ldots, \frac{1}{\sigma_n^2}\right) \end{equation} or until convergence is reached.