auto regression: add numpy-based 'predict' function

[mathause]

Extract the code to 'predict' auto regression and write a numpy-based function. This is used in two places:

• mesmer/mesmer/create_emulations/create_emus_lv.py

  Line 112 in 9def742

  def create_emus_lv_AR1_sci(emus_lv, params_lv, preds_lv, cfg):

• mesmer/mesmer/create_emulations/create_emus_gv.py

  Line 111 in 9def742

  def create_emus_gv_AR(params_gv, nr_emus_v, nr_ts_emus_v, seed):

The two approaches are slightly different - one is based on a multivariate gaussian and the other on a univariate. However, it should be possible to unify them. My idea would be to do both using multivariate_normal and then squeeze the 1D case.

import numpy as np

np.random.seed(1)
a = np.random.normal(loc=0, scale=1, size=3)

np.random.seed(1)
b = np.random.multivariate_normal([0], [[1]], size=3)

np.testing.assert_allclose(a, b.squeeze())

[mathause]

As far as I can see it is not possible to get rid of the for loop, but this computation could potentially be optimized using numba.jit.

[mathause]

Argh - the relation between normal and multivariate_normal is not exactly 1:1. normal needs the standard deviation while mulitvariate_normal wants the (co-) variance.

import numpy as np

c = 0.5

np.random.seed(1)
a = np.random.normal(loc=0, scale=c, size=3)

np.random.seed(1)
b = np.random.multivariate_normal([0], [[c]], size=3)

np.random.seed(1)
b = np.random.multivariate_normal([0], [[c ** 2]], size=3)


np.testing.assert_allclose(a, c.squeeze())

np.testing.assert_allclose(a, b.squeeze()) # failst