How does sklearn.linear_model.LinearRegression work with insufficient data? - scikit-learn

Question

To solve a 5 parameter model, I need at least 5 data points to get a unique solution. For x and y data below:
import numpy as np
x = np.array([[-0.24155831, 0.37083184, -1.69002708, 1.4578805 , 0.91790011,
0.31648635, -0.15957368],
[-0.37541846, -0.14572825, -2.19695883, 1.01136142, 0.57288752,
0.32080956, -0.82986857],
[ 0.33815532, 3.1123936 , -0.29317028, 3.01493602, 1.64978158,
0.56301755, 1.3958912 ],
[ 0.84486735, 4.74567324, 0.7982888 , 3.56604097, 1.47633894,
1.38743513, 3.0679506 ],
[-0.2752026 , 2.9110031 , 0.19218081, 2.0691105 , 0.49240373,
1.63213241, 2.4235483 ],
[ 0.89942508, 5.09052174, 1.26048572, 3.73477373, 1.4302902 ,
1.91907482, 3.70126468]])
y = np.array([-0.81388378, -1.59719762, -0.08256274, 0.61297275, 0.99359647,
1.11315445])
I used only 6 data to fit a 8 parameter model (7 slopes and 1 intercept).
lr = LinearRegression().fit(x, y)
print(lr.coef_)
array([-0.83916772, -0.57249998, 0.73025938, -0.02065629, 0.47637768,
-0.36962192, 0.99128474])
print(lr.intercept_)
0.2978781587718828
Clearly, it's using some kind of assignment to reduce the degrees of freedom. I tried to look into the source code but couldn't found anything about that. What method do they use to find the parameter of under specified model?

Answer 1

You don't need to reduce the degrees of freedom, it simply finds a solution to the least squares problem min sum_i (dot(beta,x_i)+beta_0-y_i)**2. For example, in the non-sparse case it uses the linalg.lstsq module from scipy. The default solver for this optimization problem is the gelsd LAPACK driver. If
A= np.concatenate((ones_v, X), axis=1)
is the augmented array with ones as its first column, then your solution is given by
x=numpy.linalg.pinv(A.T*A)*A.T*y
Where we use the pseudoinverse precisely because the matrix may not be of full rank. Of course, the solver doesn't actually use this formula but uses singular value Decomposition of A to reduce this formula.