different solutions when applying ridge() or the standard formulation

Dear all,

Recently I have been working with a very simple ridge regression example. In particular, based on the data included bellow

y = Xr'·b + e

X = [3.4 50.3;6.2 60;8 67;9.2 72; 10.2 76;11.1 79;11.8 82.5];
Xr = [ones(7,1) X];
y = [0 3 6 9 12 15 18]';

I calculate the arising coefficients (b) using the ridge() function implemented in matlab:

br = ridge(y,X,0.05,0)

Note that the flag is equal to zero and, therefore, the obtained coefficients should be those according to the initial data. In addition, as suggested by the help of ridge(), The model does not include a constant term, and X should not contain a column of 1s.

br =

    -26.7914
      0.2816
      0.4800

However, so as to validate this result I also solve the original ridge formula, namely:

    b=inv(Xr'*Xr+0.05*eye(3))*Xr'*y

b =

     -2.1186
      2.8194
     -0.1868

Surprisingly, both results where completely different. The best solution should be that with the less value in the following penalized residual sum of squares:

PRSSr = sum((y-Xr*br).^2)+0.05*sum(br.^2)

PRSSr =

     48.1683
  PRSS = sum((y-Xr*b).^2)+0.05*sum(b.^2)

PRSS =

     16.8857

This is a little bit astonishing for me since a better solution was expected using the ridge() function.

Could any one put light on this please?

Many many thanks in advance!

Jon

[Matt Fig]

What is Xr?

[Jon]

Sorry, it slipped my mind! I edited my question.

[Gang Cheng]

I have the same confusion? Did you solved it?