Re: Any ideas how to linearize (and evaluate) this nonlinear equation?
- From: spellucci@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Peter Spellucci)
- Date: Thu, 16 Jun 2005 18:35:09 +0000 (UTC)
In article <1118937725.338809.21090@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Znarf" <znarf_akfak@xxxxxxxxxxx> writes:
>Hi,
>
>The equation is:
>
>Y = alpha0.X0 . [1 + alpha1.X1 + alpha1.X2] / [1 + beta1.Z1 + beta2.Z2]
>
>X1, X2 and Z1, Z2 are amounts of different products that each contain a
>nutrient, X0 is the total amount of the nutrient contained in all
>products, and Y is the amount of the nutrient that is absorbed. The X
>products are thought to improve aborption, and the Z products to
>degrade aborption.
>
>It is also possible that the nutrient could be given in pure form, in
>which case, the Xs and Zs are zero, and alpha0 should take on a known
>value.
>
>All the variables Y, X and Z can take on real values. There are no
>required bounds upon the alphas and betas, though the situation in
>which they are all constrained to be positive is also of interest.
>
>I've found nonlinear solutions using the Gauss-Newton method, which
>converge rapidly.
>
>I wonder though if anyone can see a way to linearize this equation? I
>suppose then I could at least use the linear solution as a starting
>point for the nonlinear one.
>
>Also, any advice or references too look up on evaluating nonlinear
>solutions would be appreciated. For example, if I had done a linear
>regression, I could use residuals to evaluate the model. But what
>diagnostics are there for nonlinear solutions?
>
>TIA,
>
>Znarf
>
it is a statistical crime, but for getting an initial guess it is o.k.:
multiply by the denominator , rewrite
gamma1=aplpah1/alpha0 gamma2=alpha2/alpha0
then in matlab notation
[ Y.*Z1 , Y.*Z2 , -X0 , -X0.*X1 , -X0.*X2 ] * [beta1, beta2 , alpha0 ,
gamma1 , gamma2]' = -Y
to be solved in the least squares sense (.* is componentwise multiplication
of vectors).
having got the nonlinear least sqaures solution of the original problem,
there are also possibilities to compute intervals of confidence for
the parameters:
from my annotations
\begin{citation}
%-------------------
@book { Bate88,
author = "D. M. Bates and D. W. Watts",
title = "Nonlinear Regression Analysis and Its Applications",
publisher = "Wiley",
address = "New York",
year = "1988"}
There are _five_ different kinds of confidence intervals you might be
interested in: (a) The 'approximate marginal' confidence interval (which
is the one you are asking about) is equation 1.12 on page 6. (b) The
'approximate confidence band' is equation 1.13 on page 6. (c) The
'likelihood region' is equation 1.14 on page 6. (d) The 'Bayesian
inference region' is equation 1.17 on page 7.
These four kinds of confidence intervals (including type (a) which you
asked about) have one very serious disadvantage: they assume that the
fitting model is sufficiently _linear_ in parameters. That is, the
model is assumed to be approximately linear within the confidence
interval for the parameters. [Yes, somewhat circular logic.]
A kind of confidence interval that is very much preferable to all four
types above is (e) likelihood region determined by a systematic _search_
of the least-squares surface in the parameter space. It is still based
on a _linear_ statistical theory, but much less so than (a) through
(d).
%------------------
Seber and Wild. (1988). NonLinear Regression. Wiley.
%-----------------------
hth
peter
.
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