least_absolute_regression

Least absolute error regression implemented using Linear Programming.

Below is an example using 5 samples with 2 independent variables that do not fit a plane. The mapping of independent variables to the dependent variable is

(1,1) --> 0
(1,-1) --> -2
(-1,-1) --> -4
(-1,1) --> -2
(0,0) --> -1.3

All but the last mapping fit the plane x + y - 2 = z.

from lae_regression import l1_fit
U = ([1, 1], [1, -1], [-1, -1], [-1, 1], [0, 0])
v = (0, -2, -4, -2, -1.3)
result = l1_fit(U, v)
result["m"], result["k"], result["residuals"], result["samples"], result["dimensions"]

The value of the last line gives:

(array([ 1.,  1.]),
 -2.000,
 array([ -4.441e-16,  -1.332e-15,  -4.441e-16,   4.441e-16,   7.000e-01]),
 5,
 2)

Here with m = (1,1) and k = -2 the regression finds the plane 1*x + 1*y - 2 = z and essentially rejects the last mapping as an outlier.

Background

Historically this code was set up to illustrate techniques for developing and publishing computational methods using methodologies associated with Jupyter notebooks.

Please see the ./presentation folder for detailed discussion of the methods used (as jupyter notebooks).

In particular the ./presentation/Notebooks as Notes.ipynb notebook introduces concepts of least absolute error regression and the ./presentation/1.1_Method Derivation.ipynb notebook discusses the derivation of the calculation.