auto_regression: return co-variance instead of standard deviation

[mathause]

Our functions _fit_auto_regression_xr returns the "standard deviation of the residuals". But _draw_auto_regression_correlated_np requires the (co-)variance.

This was done because the two functions were extracted in separate PRs (#161 and #161) and the original code did the same. It's a bit wasteful if we constantly convert there and back. We should change this before we release v0.9.

mesmer/mesmer/stats/auto_regression.py

Line 233 in acd80e4

std = np.sqrt(AR_result.sigma2)

mesmer/mesmer/stats/auto_regression.py

Lines 82 to 83 in acd80e4

	def _draw_auto_regression_correlated_np(
	*, intercept, coefs, covariance, n_samples, n_ts, seed, buffer

mesmer/mesmer/calibrate_mesmer/train_gv.py

Line 214 in acd80e4

params_gv["AR_std_innovs"] = np.float64(params_scen.standard_deviation.values)

mesmer/mesmer/calibrate_mesmer/train_lv.py

Line 249 in acd80e4

params_lv["AR1_std_innovs"][targ_name] = params_scen.standard_deviation.values

[mathause]

What I want to say fit_auto_regression should return var but the old code path will keep using std - i.e. the place of the sqrt and **2 should change. Should not be too difficult but may (should) need some code test changes.

[mathause]

This is a bit less trivial than I thought - in train_gv_AR we take the average over the AR params - including the standard deviation. However, this is wrong: you have to average the covariances (or even take care of the size of the samples: https://stats.stackexchange.com/q/25848)

mesmer/mesmer/calibrate_mesmer/train_gv.py

Lines 195 to 208 in acd80e4

	params_scen = list()
	for scen in gv.keys():
	data = gv[scen]
	# create temporary DataArray
	data = xr.DataArray(data, dims=("run", "time"))
	params = _fit_auto_regression_xr(data, dim="time", lags=AR_order_sel)
	params = params.mean("run")
	params_scen.append(params)
	params_scen = xr.concat(params_scen, dim="scen")
	params_scen = params_scen.mean("scen")

These are used here:

mesmer/mesmer/create_emulations/create_emus_gv.py

Line 176 in acd80e4

covariance=AR_std_innovs**2, # pass the (co-)variance!