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Is Huber Estimator IRLS Weight Wrong in Literature?

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aruzinsky

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Nov 28, 2009, 10:53:29 AM11/28/09
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The purpose Iteratively Reweighted Least Squares (IRLS) is to minimize
sum( f(ei) ) where ei are elements of

E = y - Ax

A typical expression for the Huber weight is given here
http://www.general-cathexis.com/images/Huber.png , but I derive a
different weight:

W(e) = 1/2 for |e| <= k

W(e) = k/|e| - ( (k/e)^2 )/2 for |e| > k

where e is an element of E.

My derivation is simply rho(e)/e^2 because, at convergence, et = et-1,
where t is iteration index,
therefore Wt*et^2 = (rho(et-1)/(et-1)^2)*et^2 = (rho(et)/et^2)*et^2 =
rho(et).

aruzinsky

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Dec 2, 2009, 11:23:24 AM12/2/09
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Prodigious spam justifies Bump Up.

Helmut Jarausch

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Dec 8, 2009, 2:18:41 PM12/8/09
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On 11/28/09 16:53, aruzinsky wrote:
> The purpose Iteratively Reweighted Least Squares (IRLS) is to minimize
> sum( f(ei) ) where ei are elements of
>
> E = y - Ax
>
> A typical expression for the Huber weight is given here
> http://www.general-cathexis.com/images/Huber.png , but I derive a
> different weight:

I can confirm the result quoted above.
If you are interested I can send you a pdf-file with my derivation.

Helmut.

>
> W(e) = 1/2 for |e|<= k
>
> W(e) = k/|e| - ( (k/e)^2 )/2 for |e|> k
>
> where e is an element of E.
>
> My derivation is simply rho(e)/e^2 because, at convergence, et = et-1,
> where t is iteration index,
> therefore Wt*et^2 = (rho(et-1)/(et-1)^2)*et^2 = (rho(et)/et^2)*et^2 =
> rho(et).


--
Helmut Jarausch

Lehrstuhl fuer Numerische Mathematik
RWTH - Aachen University
D 52056 Aachen, Germany

aruzinsky

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Dec 9, 2009, 10:21:47 AM12/9/09
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On Dec 8, 1:18 pm, Helmut Jarausch <jarau...@skynet.be> wrote:
> On 11/28/09 16:53, aruzinsky wrote:
>
> > The purpose Iteratively Reweighted Least Squares (IRLS) is to minimize
> > sum( f(ei) ) where ei are elements of
>
> > E = y - Ax
>
> > A typical expression for the Huber weight is given here
> >http://www.general-cathexis.com/images/Huber.png, but I derive a

> > different weight:
>
> I can confirm the result quoted above.
> If you are interested I can send you a pdf-file with my derivation.
>
> Helmut.
>
>
>
> > W(e) = 1/2                           for |e|<= k
>
> > W(e) = k/|e| - ( (k/e)^2 )/2      for |e|>  k
>
> > where e is an element of E.
>
> > My derivation is simply rho(e)/e^2 because, at convergence, et = et-1,
> > where t is iteration index,
> > therefore Wt*et^2 = (rho(et-1)/(et-1)^2)*et^2 = (rho(et)/et^2)*et^2 =
> > rho(et).
>
> --
> Helmut Jarausch
>
> Lehrstuhl fuer Numerische Mathematik
> RWTH - Aachen University
> D 52056 Aachen, Germany

It will suffice to tell me why my derivation is wrong in public.

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