How to implement weighted Linear Regression

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Shagun Khare
Shagun Khare 2016-11-12
评论： Paul Safier 2024-2-6
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?

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回答（2 个）

John D'Errico
John D'Errico 2016-11-12
"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.
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Paul Safier
Paul Safier 2024-2-6
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

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Paul Safier
Paul Safier 2024-2-6
@John D'Errico do you know how to compute the R^2 value in this weighted case?

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