given a set of 3d points and their covariance matrices finding the mean point

Question

Say I have a set of observations of a 3d object in space. Knowing only the location of the observations my best guess for the location of the object would be the mean point. But let's say I have the covariance matrix for each point based on some noise model. How can I calculate the most likely location of the object using this new information?

Answer 1

Given covariance $\Lambda_i$ per observation $y_i$ we have:

$P(\{y_{1}\cdots y_{n}\}|y_{mean})\sim e^{-\frac{1}{2}\sum(y_{i}-y_{mean})^{T}\Lambda_{i}^{-1}(y_{i}-y_{mean})}$

the log prob is: $\log P \sim -{1\over 2}\sum(y_{i}-y_{mean})^{T}\Lambda_{i}^{-1}(y_{i}-y_{mean})$

Extremizing the log probability according to the mean (Max likelihood):

$\sum\Lambda_{i}^{-1}y_{i}=\left(\sum\Lambda_{i}^{-1}\right)y_{_{mean}}^{estimated}$

So that the maximum likelihood estimate of the center is

$y_{mean}^{estimated}=\left(\sum\Lambda_{i}^{-1}\right)^{-1}\sum\Lambda_{i}^{-1}y_{i}=\Lambda^{tot}\sum\Lambda_{i}^{-1}y_{i}$

If we plug that in to the probability expression we find that:

$P(\{y_{1}\cdots y_{n}\}|y_{mean})\sim e^{-\frac{1}{2}(y_{mean}-y_{mean}^{estimated})^{T}(\Lambda^{tot})^{-1}(y_{mean}-y_{mean}^{estimated})}$

So we can interpret $\Lambda^{tot}$ as the covariance our guess for the location itself !