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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 

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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 

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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 

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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



In article <7Dcvg.62598$R26.24864@tornado.southeast.rr.com>,
"Cheng Cosine" <acosine@spamfree.com> writes:
Quote: 
"Han de Bruijn" <Han.deBruijn@DTO.TUDelft.NL> 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
peter 

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