Minimisation Problem
- To: mathgroup at smc.vnet.net
- Subject: [mg40803] Minimisation Problem
- From: Mike Costa <run_mc2000 at yahoo.com>
- Date: Thu, 17 Apr 2003 23:18:50 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Dear All,
I have a little minimization problem. I'm essentially
trying to fit data points to a curve, which, in
particular, means minimising the chi-square function
in order to obtain the desired parameters. However,
the curve-fitting aspect is not important for now. The
main problem boils down to this: given f(x, p), the
theoretical function, with x being the simulated data
points and p being the parameter(s) of interest, and
y(x) being the actual obtained function value given
the simulated data set x, the usual chi-square method
of determining p consists of minimising the chi-square
function
ChiSq = Sigma[(f(x,p) - y(x))^2/y(x)]
where Sigma represents the sum over all the data
points(I realise that there are other definitions for
chi-square, but let's use this for now). The little
twist is this: I want instead to minimise the
ALTERNATIVE CHI-SQUARE
AltChiSq = Sigma[2(f-y)+(2y+1)Log[2y+1/2f+1]]
where again Sigma represents the sum over all the data
points. I want to minimise AltChiSq to get the
parameter p.
In my situation, the theoretical function f only has
one parameter that needs to be estimated
f(x,p) = p(0.4 + 3.8Exp[-|Cos[x]|^0.75]),
p being a kind of normalising factor. I would like to
know if there are any intrinsic functions in
Mathematica that can directly minimise a function like
AltChiSQ above. I recognise that the usual methods for
minimising functions like FindMinimum and such will
not work here due to the number of terms that Sigma
sums over (among other reasons). Is there any other
way to use FindMinimum in order to handle the function
AltChiSq? Or maybe, is there some way to somehow
change the default chi-square function that
Linear/NonLinearFit uses in order to instead minimise
AltChiSq? If these strategies lead to nowhere, can
anyone give a general strategy of how to tackle this
minimisation problem in Mathematica?
Any suggestions would be greatly appreciated. Thanks.
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