Weight setting?

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colorresearcher@gmail.com - 30 Oct 2008 12:28 GMT

Hi all,

I want to do a squares fitting:

minimize Sum_i{ W( i )*[G( i )-F( X,A(i) ) ] ^2}

subject to C(X,i=5)=0

where X is a 12-element column vector, W(i)( weight setting), G(i)
(reference), and A(i) are the functions of index i; Sum_i is the
notation of sum the thing inside {} when i runs from 0 to 200,
C(X,i=5) is the constraint. All these functions as well as
F( X,A(i) ), i=1...200 are nonlinear functions.

Is it necessary that I must set weightting W(i) such that
Sum_i{ W(i)}=1 or it is just application dependent issue (i.e in some
applications, it is necessary, it some others, it is not)?

Thanks,

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Peter Spellucci - 30 Oct 2008 13:48 GMT

>Hi all,
>
[quoted text clipped - 15 lines]
>
>Thanks,

if your least squares ansatz has some stochastic background
(i.e. the G(i) are values perturbed by independently normally
distributed (Gaussian) errors with mean zero) then you should
use as w(i) the reciprocal of the variance^2 of the G(i)
such that G(i)*w(i) all have variance one.
another reason for choosing w(i): if you want a fitting with
respect to the relative errors, you might take w(i)=1/G(i)^2.
you must be aware of the fact that least squares is quite
sensitive to outliers, hence minimizing the unscaled sum of squares
might have strange effects.

hth
peter

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Math Forum / Mathematics / Operations Research / October 2008

Weight setting?