NICE sample-error correction

[odunbar]

Purpose

Implements the NICE sample-error correction (Morzfeld, Vishny, et al 2024)

Content

• added the Localizer: SECNice() with a couple of constructors SECNice(), SECNice(delta_ug, delta_gg) and SECNice(n_samples, delta_ug, delta_gg) n_samples are the number of samples to approximate the standard deviation through Fisher transformation and deltas are "fudge factors" on the discrepancy principle tolerance
• added this to the example Localization/localization_example_lorenz96.jl
• docstrings and API for variable parameters
• Added L96 example to docs page, prettified plots and add recommendation for SECNice here
• unit tests
• Improved implementation efficiency for calc correlations, based on creating an interpolant on first call. NB this requires adding Interpolations into the Project toml (though we could do without if necessary)
• added a branch for N_ens<=6 for a crude sampling scheme, (still performant on the lorenz example even with 5 particles)

Examples

Implement Vishny, Morzfeld, 2024 localization. Using Fisher transformation to avoid lookup tables,
Instead we create a linear interpolation of the std of correlation space on first call, which is reused in future calls.

There is a gentle decline in performance as the iteration is decreased down to ~20 samples

Example: learning a 200dim state x(t) from a future time of x(t+T) where T=1, using 20 ensemble members.
SECNice(n_samples, delta_ug, delta_gg)

Result - SECNice(5000,1.0,1.0) and SECNice(5000,0.8,1.0)

Result - SECNice(1000,1.0,1.0) and SECNice(1000,0.8,1.0)

Results for even smaller samples = 200,50,20

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• I have read and checked the items on the review checklist.

[eviatarbach]

This is great, thank you for all the work on this Ollie! I suggested some changes as comments. The most substantive are:

1. What should we do for N_ens < 6? Right now a warning is issued, but it seems to me that it may be more useful to hard-code a look-up table for this case.
2. Shouldn't corr_psd = corr_tmp .^ α_min_exceeded[1] be corr_psd = corr_tmp .^ (α_min_exceeded[1] + 1), and similar for corr_psd_prev? Please check if I got it wrong.

[eviatarbach]

Looks good, thank you for making these changes! I only made two minor additional comments.

[odunbar]

Offline changes were approved