Linear Regression Normal Equation

Fit ordinary least-squares linear regression with an intercept by solving the normal equations. Prepend a column of ones to X, then solve $({\tilde{X}}^{\mathrm{\top }}\tilde{X})\beta ={\tilde{X}}^{\mathrm{\top }}y(\backslash tilde\; X^\backslash top\backslash tilde\; X)\backslash beta=\backslash tilde\; X^\backslash top\; y$.

Implement solve(X, y).

Returns. Return a NumPy vector with the intercept first, followed by one coefficient per feature. The Moore-Penrose pseudo-inverse is used, so perfectly collinear features do not crash.

Examples

Example 1

Input:

solve([[1.0], [2.0], [3.0]], [2.0, 4.0, 6.0])

Output:

[0.0, 2.0]

Example 2

Input:

solve([[1.0, 0.0], [0.0, 1.0], [1.0, 1.0], [2.0, 1.0]], [1.0, 2.0, 3.0, 4.0])

Output:

[0.0, 1.0, 2.0]