MATLAB: Using lscov for weighted least squares; why do the results differ for the same weights

Question

Hello,

Using the weighted least squares example on the matlab lscov documentation page…

x1 = [.2 .5 .6 .8 1.0 1.1]';

x2 = [.1 .3 .4 .9 1.1 1.4]';

X = [ones(size(x1)) x1 x2];

y = [.17 .26 .28 .23 .27 .34]';

w = [1 1 1 1 1 .1]';

(1) A call to lscov with the weights as a vector produces…

>> lscov(X,y,w)

ans =

0.1046

0.4614

-0.2621

(2) However, if I represent the weight vector as a diagonal matrix (should be okay) I get very different results, why?

>> lscov(X,y,diag(w))

ans =

0.1310

0.2380

-0.0423

(3) Also, if I instead augment X and y with the weights and run it through ordinary least squares I get similar results to the first try, but still different, why?

>> Xp = diag(w)*X;

>> yp = diag(w)*y;

>> lscov(Xp,yp)

ans =

0.1005

0.4956

-0.2957

Which of these three is "best"?

Answer 1

The correct is

>> lscov(X,y,w)
ans =
    0.1046
    0.4614
   -0.2621

It is equivalent to

>> lscov(X,y,inv(diag(w)))
ans =
    0.1046
    0.4614
   -0.2621

Following is incorrect

Xp = diag(w)*X;
yp = diag(w)*y;
lscov(Xp,yp)

'w' is weight for errors not the values itself.