Merge #270

270: more robust empirical covariances r=odunbar a=odunbar

<!--- THESE LINES ARE COMMENTED -->
## Purpose 
<!--- One sentence to describe the purpose of this PR, refer to any linked issues:
#14 -- this will link to issue 14
Closes #2 -- this will automatically close issue 2 on PR merge
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Closes #269 

## To-do
<!---  list of proposed tasks in the PR, move to "Content" on completion 
- Proposed task
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## Content
<!---  specific tasks that are currently complete 
- Solution implemented
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- `cov(u,u) -> cov(u)`

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- followed the codebase contribution guide: https://clima.github.io/ClimateMachine.jl/latest/Contributing/
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In the Content, I have included 
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Co-authored-by: odunbar <[email protected]>

src/EnsembleKalmanInversion.jl

@@ -93,7 +93,7 @@ function eki_update(
     obs_noise_cov::Union{AbstractMatrix{CT}, UniformScaling{CT}},
 ) where {FT <: Real, IT, CT <: Real}
 
-    cov_est = cov([u; g], [u; g], dims = 2, corrected = false) # [(N_par + N_obs)×(N_par + N_obs)]
+    cov_est = cov([u; g], dims = 2, corrected = false) # [(N_par + N_obs)×(N_par + N_obs)]
 
     # Localization
     cov_localized = ekp.localizer.localize(cov_est)

src/EnsembleKalmanProcess.jl

@@ -235,7 +235,7 @@ Returns the unconstrained parameter sample covariance at the given iteration.
 """
 function get_u_cov(ekp::EnsembleKalmanProcess, iteration::IT) where {IT <: Integer}
     u = get_data(ekp.u[iteration])
-    return cov(u, u, dims = 2)
+    return cov(u, dims = 2)
 end
 
 """
@@ -244,7 +244,7 @@ end
 Returns the unconstrained parameter sample covariance for the initial ensemble.
 """
 function get_u_cov_prior(ekp::EnsembleKalmanProcess)
-    return cov(get_u_prior(ekp), get_u_prior(ekp), dims = 2)
+    return cov(get_u_prior(ekp), dims = 2)
 end
 
 """
@@ -421,7 +421,7 @@ function sample_empirical_gaussian(
     n::IT;
     inflation::Union{FT, Nothing} = nothing,
 ) where {FT <: Real, IT <: Int}
-    cov_u_new = Symmetric(cov(u, u, dims = 2))
+    cov_u_new = Symmetric(cov(u, dims = 2))
     if !isposdef(cov_u_new)
         @warn string("Sample covariance matrix over ensemble is singular.", "\n Appplying variance inflation.")
         if isnothing(inflation)

test/Localizers/runtests.jl

@@ -60,7 +60,7 @@ const EKP = EnsembleKalmanProcesses
         # Test Schur product theorem
         u_final = get_u_final(ekiobj)
         g_final = get_g_final(ekiobj)
-        cov_est = cov([u_final; g_final], [u_final; g_final], dims = 2, corrected = false)
+        cov_est = cov([u_final; g_final], dims = 2, corrected = false)
         cov_localized = ekiobj.localizer.localize(cov_est)
         @test rank(cov_est) < rank(cov_localized)
         # Test localization getter method