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Cheng Cosine
science forum Guru Wannabe

Joined: 26 May 2005
Posts: 168 Posted: Tue Jul 18, 2006 4:14 am    Post subject: ? types of LS Hi:

So far I saw ordinary LS, recursive LS, weighted LS, and total LS.
Whatelse?

For weighted LS, we look for min( norm(err,W) ), where norm(err,W) =

sqrt( err'*W*err ). W is positive definite. The easiest W is a diagonal
square

matrix with positive diagonal entries, W = D. Since Forbinous norm is
untinarily

invariant, we can extend W = U*D*U', where U is square unitary matrix. But
will

this have any effect on the LS soln we found when different U's are used?

For total LS, the problem is defined as min( norm( [A, a]-[AEst, aEst] ),
Forbinous ) under

condition that [AEst, aEst] belongs to R^Mx(N+1) and subject to aEst belongs

to range(AEst). A is MxN, x is Nx1, a is Mx1, M > N. [A, a] is argumented
matrix

in MatLab notation. That is, we know A*x = a, but we only have AEst and aEst
and

want to find some best approximted soln to A*x = a. How to solve problem
like this?

What if M <= N, and wnat to find its best approximted soln?

Thanks,
by Cheng Cosine
Jul/18/2k6 NC Han de Bruijn
science forum Guru

Joined: 18 May 2005
Posts: 1285 Posted: Tue Jul 18, 2006 9:07 am    Post subject: Re: ? types of LS Cheng Cosine wrote:

 Quote: So far I saw ordinary LS, recursive LS, weighted LS, and total LS. Whatelse?

Least Squares, perhaps?

Please explain your abbreviations let it be only _once_ in a poster.

Han de Bruijn Cheng Cosine
science forum Guru Wannabe

Joined: 26 May 2005
Posts: 168 Posted: Tue Jul 18, 2006 9:44 pm    Post subject: Re: ? types of LS "Han de Bruijn" <Han.deBruijn@DTO.TUDelft.NL> wrote in message
news:6708c\$44bca4c1\$82a1e228\$5632@news1.tudelft.nl...
 Quote: Cheng Cosine wrote: So far I saw ordinary LS, recursive LS, weighted LS, and total LS. Whatelse? Least Squares, perhaps? Please explain your abbreviations let it be only _once_ in a poster.

Aye, LS = Least Squares Peter Spellucci
science forum Guru

Joined: 29 Apr 2005
Posts: 702 Posted: Wed Jul 19, 2006 9:54 am    Post subject: Re: ? types of LS "Cheng Cosine" <acosine@spamfree.com> writes:
 Quote: "Han de Bruijn" wrote in message news:6708c\$44bca4c1\$82a1e228\$5632@news1.tudelft.nl... Cheng Cosine wrote: So far I saw ordinary LS, recursive LS, weighted LS, and total LS. Whatelse? Least Squares, perhaps? Please explain your abbreviations let it be only _once_ in a poster. Aye, LS = Least Squares

Cheng,
total least squares is simply done by the SVD of the matrix composed from
the given matrix and the right hand side (you imitate least squares solving
a homogeneous system in the least squares sense with the side condition
euclidean length of soution =1), realized by taking the right singular vector
corresponding to the (or a) smallest singular value and normalizing this later to
last component (correponding to the right hand side) =1
the orthogonal least squares is quite different, since here the errors in the
components of the matrix are not independent, but depend on the error in the
"independent" variable. you can translate orthogonal least squares into
ordinary least squares:
model y=f(x;a) a the model parameter
data (x(i),y(i))
minimize with respect to a and delta(i)

sum_i { y(i)-f(x(i)+delta(i);a) }^2 + sum_i delta(i)^2

this is a high dimensional nonlinear least squares problem which however can be
solved quite efficiently. see odrpack in http://www.netlib.org/opt
hth
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