🚀 go-pugleaf

my original message:
>Suppose, I am dealing with the weighted least squares problem
>y=A*x
>(covariance matrix of y is C, weighting matrix W is inv(C))
>and instead of using the ordinary
>x=inv(A`*W*A)*A`*W*y
>I use
>x=inv(A`*W*A+alpha*E)*A`*W*y
>where E denotes unit matrix and alpha is a parameter for
>smoothing, stabilization, regularization or whatever you call it.
>Sometimes it looks even useful in order not to bother with
>singular matrices and pseudo inverses.
>
>Now, my questions is, how do I get the covariance matrix of x
>in this case? Is it equally simple as in the 1st case?
>...

Many thanks to all who have sent answers to me thus far.

In the message above, unit matrix E should better read identity matrix I
(badly translated german).

The point is: I know about transformations looking like B*C*B`
but I wonder if it is applicable in my case.
Suppose, the matrix operator A has smoothing properties and, thus, the
inversion of the equation y=A*x leads to strong amplification of
statistical noise present in y. In order to overcome this let me introduce
the additional information that the absolut value (actually the square) of the
solution x should be small. Formulating this leads to something like
(A*x-y)`*W*(A*x-y) + alpha*x`*x -> min
where alpha is some kind of Lagrangian multiplier.
Minimizing this expression leads to the result given above and, indeed,
I obtain reasonable looking results when alpha is suitably chosen.
Now, I would like to draw error bars and I doubt if it is as simple as B*C*B`.

To all who suggested looking at ridge regression: I have not heard about
it before, but I will have a look at it. Thanks.

Apparently, people figured out that I am not a statistician and if this is
trivial, sorry to bother, you but I am really interested in this question.

Chris

burger@dmrhrz11.bitnet