Re: Solve Ax=b with different weights for each x[i]
- From: "glenn.macgougan@xxxxxxxxx" <glenn.macgougan@xxxxxxxxx>
- Date: Thu, 3 Jan 2008 12:29:01 -0800 (PST)
On Jan 3, 12:31 am, spasmous <spasm...@xxxxxxxxx> wrote:
I'd like to try out a weighting approach that uses different weights
on each part of the solution. E.g. starting with Ax=b where x=x[1 to
n] and b=b[1 to n], then for computing each element x[i] I'd like to
weight b[i] more strongly.
Kind of like solving WAx=Wb with W=diag([2 1 1 ...]) and taking x[1]
from that solution, then W=diag([1 2 1 1...]) and taking x[2] from
that solution, etc. I know this is kind of "wrong" but is there a way
to do it by some clever weighting matrix that only means solving the
equation once?
Use
Weighted Least Squares
x = (A^T P A)^-1 * A^T P b
where P is your weight matrix (P is usually diagonal and each element
corresponds to 1/(\sigma_bi)^2, where (\sigma_bi)^2 is the variance of
element b[i]. P(i,i) = 1 is full weight, P(i,i) = 0.5 is half weight,
etc.
.
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