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kamek
kamek
2:14 PM
How do I correctly calculate the correlated (nondiagonal) covariance matrix?

Also, I thought of a situation where this approach will fail. Let's say $\theta_2=179$~deg and I have two measurements of $z_{12}$. After performing simplification through linear manipulation, it's possible that I'll have one measurement of $\theta_2$ that is (for example) 178 degrees, and another that is (for example) -178 degrees. When calculating $\theta_2$ by taking the weighted mean of these two measurements, I'll (incorrectly) get 0 degrees.
 
kamek
kamek
3:08 PM
Here is how I would calculate the covariance matrix: a set of linear operations on a MxN matrix A can be represented by pre-multiplying by the MxM matrix E. In other words, $Z = H\Theta$ in our example becomes $EZ = EH\Theta$, where Y = EZ. The covariance matrix of Y is then simply the transformation of the diagonal covariance matrix of Z by E (in other words $C_Y = E^T * C_Z * E$).
 
kamek
kamek
3:26 PM
I should also point out that I know how to calculate the mean of a set of angles to avoid the "(-179 + 179)/2 = 0$ problem. This can be avoided by calculating the mean of sin(z_i), and the mean of cos(z_i), then solving atan2(mean_of_sin, mean_of_cos). The problem now is turning that into a weighted mean with a nondiagonal covariance matrix.
 

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