Why center E?
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In this line E is centered
This decreases the rank by 1.
Under equation 14 it says "E is the centered measurement-perturbation matrix whose columns are sampled from N(0, C_dd)".
Here "centered" could be interpreted as (1) that the mean is 0 in N(0, C_dd) or that (2) the rows in E should be centered.
If we think of equation (15), then the sample covariance does require us to define E as mean centered and divided by (N-1).
But don't we already know the true mean to be zero (ref. our model is eqn (7))?
So the best low-rank approximation to C_dd would be
# approach 1
a ~ N(0, C_dd)
C_dd_bar = a @ a.T / len(a)
and not
# approach 2
a ~ N(0, C_dd)
C_dd_bar = subtract_mean(a) @ subtract_mean(a).T / (len(a) - 1)
Again I appeal to the case when we draw a single sample.
Approach 1 would produce a rank 1 approximation to C_dd, while approach 2 would produce a rank 0 approximation to C_dd (C_dd_bar is identically zero).
If we have a run with a single ensemble member, then E will be identically zero, and there will be no permutation of the observations and no sampling. I would expect that:
- 1000 runs with 1 ensemble member (different seeds)
- 1 run with 1000 ensemble member
should both sample the posterior, but in our implementation the first option will always give the same answer is E is identically zero.
Thoughts on this @Blunde1 ? (we can discuss over chat. I just wanted to write it down here)
