How to implement weighted Linear Regression

I am trying to reproduce the results of a paper. It is mentioned that they used weighted linear regression with three different weights.

Is Weighted least square regression is same as weighted linear regression?

How can i implement weighted linear regression in matlab?

Answer 1

[John D'Errico]

"Is Weighted least square regression is same as weighted linear regression?" Yes.

Given the problem

A*x = y

where x is the vector of unknowns, and a weight vector w. w must have the same number of elements as y. I'll assume that w and y are column vectors.

W = diag(W);
x = (W*A)\(w.*y);

If there are many data points, then creating W as a diagonal matrix (that is not sparse) and multiplying by W will be less efficient that you may want. If you are using R2016b (or later) then you need not create W at all.

x = (w.*A)\(w.*y);

Again, this presumes that w and y are column vectors, and that you have R2016b.

[Paul Safier]

Also, @John D'Errico, do you know why the formula for beta above differs from the one here (taken from documentation online as well as the WLS wiki page)?

A = [1;2;3;4]; Y = [1.1;2.3;2.9;10]; W = [1;1;1;0.4];

A1 = [ones(size(A)) A];

Beta1 = (W.*A1)\(W.*Y) % The one you show above.

Beta2 = (A1'*diag(W)*A1)\(A1'*diag(W)*Y) % Eqn 7 from: https://www.stat.cmu.edu/~cshalizi/mreg/15/lectures/24/lecture-24--25.pdf

Answer 2

[Paul Safier]

@John D'Errico do you know how to compute the R^2 value in this weighted case?