Covariance errors - mobeets/nullSpaceControl GitHub Wiki

Source: "A simple procedure for the comparison of covariance matrices" Given: two samples D1, D2. We would like to find an error metric for the difference between cov(D1) and cov(D2).

S1 = S2 + S3

• S2 = error in covariance orientation
• S3 = error in covariance shape

High-level

• S1 (total): eigenvector from one sample should explain same amount of variance in both samples.
• S2 (orientation): one sample should be explained just as well by its ith eigenvector as by the other sample's ith eigenvector.
• S3 (shape): one sample should be as explained by its ith eigenvector as the other sample is explained by its ith eigenvector.

Low-level

Below, v_i12 is the variance in sample 1 explained by the ith eigenvector of sample 2. So v_11 and v_22, for example, these are the eigenvalues of D1 and D2, respectively.

Units

S1 can be normalized by the worst case scenario, which is when each data set has one axis of variance, and these axes are orthogonal. S1_max = 4*(sum(var(D1))^2 + sum(var(D2))^2). So you can normalize S1, S2, and S3 by this value to scale between 0 and 1.

Examples