Implement eigh() fallback

[veni-vidi-vici-dormivi]

Implement eigh() as a fallback for loglikelihood estimation in covariance localization. Since physically, the covariance matrix can be positive semidefinite we should allow singular true. I also implemented some bug fixes mentioned in #413.

• Closes Reconsider try Cholesky approach #413
• Closes Raise LinAlg error in try Cholesky #420
• Tests added
• Fully documented, including CHANGELOG.rst

[veni-vidi-vici-dormivi]

We could cherry pick the rng commit to another PR, I just opened the branch for all thing covariance related as a tryout in the beginning.

[veni-vidi-vici-dormivi]

Before I fix the rest of the tests, could you look at it @mathause. I don't think it's pretty what I did to mesmer.utils.minimize_local_discrete but I couldn't think of a better way. maybe it would also be okay to check cholesky for every localization radius if it is rather likely that one where the matrix is pos def is the better one in which case we would only do the estimation with eigh once or max twice.

[mathause]

Thanks. I made a suggestion how we can potentially avoid changeable. It would be nice to have a separate PR about the random generator but I also see that they are intricately linked...

[veni-vidi-vici-dormivi]

Note: diff works better with ignore whitespace

[codecov]

Codecov Report

Attention: Patch coverage is 98.00000% with 1 line in your changes missing coverage. Please review.

Project coverage is 61.51%. Comparing base (e0ca738) to head (234890e).

Files with missing lines	Patch %	Lines
mesmer/stats/_localized_covariance.py	96.96%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #493      +/-   ##
==========================================
+ Coverage   61.36%   61.51%   +0.14%     
==========================================
  Files          50       50              
  Lines        3572     3588      +16     
==========================================
+ Hits         2192     2207      +15     
- Misses       1380     1381       +1     

Flag	Coverage Δ
unittests	61.51% <98.00%> (+0.14%)	⬆️

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[mathause]

Thanks that looks good and clean. One last thing we could do (and I leave it up to you): define _ecov_crossvalidation = _EcovCrossvalidation() and rename crossvalidate to __call__, then we can call _ecov_crossvalidation(...) again as before (but then this function is purely internal...)

[mathause]

Oh and some of the tests emit warnings - can you suppress them? E.g.:

mesmer/tests/integration/test_volc.py

Line 62 in 6e937ff

@pytest.mark.filterwarnings("ignore:The slope of")

or in a with pytest.warns(...): block,

[veni-vidi-vici-dormivi]

Stumbled upon #187. Just wanted to mention it here, but it seems like it does not necessarily speak against allowing singular true?

[mathause]

I think the problem in #187 was that certain folds did not contribute to the nll. We should make sure this doesn't happen. Need to take a look at the code...

[veni-vidi-vici-dormivi]

Another not so nice side effect about all of this is that if we select a singular covariance matrix and use eigh for drawing samples this could lead to slight differences in the emulations between machines...

[veni-vidi-vici-dormivi]

Maybe this is the solution to go for. Also because an ostrich looks very similar to an emu.

[veni-vidi-vici-dormivi]

Implement switch to set that per default, it will error if the matrix is singular

[mathause]

I've avoided reviewing this because I am unsure what's best (different results on different OSs is annoying but prob. unavoidable, however, if #187 happens it would be bad). Should/ can we wait for #521 so we can easily play with different models / scenarios before deciding here? (I think there were models with singular matrices for Lea, maybe only in CMIP5.)

[veni-vidi-vici-dormivi]

Okay