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In article <1992Oct11.153538.12821@zip.eecs.umich.edu> ayman@eecs.umich.edu (Ayman I. Kayssi) writes:
>I'm trying to fit data to a rational function. What is the transformation
>that I need to make to have a linear model?
>y = (a0 + a1*x + a2*x^2 + ... + am*x^m)/(1 + b1*x + b2*x^2 + ... + bn*x^n)
>means that:
>y + b1*y*x + b2*y*x^2 + ... + bn*y*x^n = a0 + a1*x + ... + am*x^m
>or:
>y = a0 + a1*x + ... + am*x^m + b1*(-y)*x + b2*(-y)*x^2 + ... + bn*(-y)*x^n

>I tried this and I'm not getting the right answers...is this the right 
>way to do it?

If you are trying a regression on your transformed equation, the estimates
will not even be consistent because of dependent variables on the right side.
There are ways that this can be handled (an INDICATION of how to do it is
to use higher powers of x to identify the situation, and to estimate it
as a single equation in a complete system), but this is very definitely
not optimal.  There is NO way to transform a non-linear model to a linear
model preserving the assumptions.  Figure out your assumptions, and then
use the appropriate statistical procedures.

A word of warning is appropriate.  Fitting a polynomial or a rational
function is very ill-conditioned, no matter how it is done.  You may
need not only the services of a good mathematical statistician, but
also an excellent numerical analyst.  You may very well need the numerical
analyst even if there are no random errors; crude programs are likely to
break down.

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)

hrubin@pop.stat.purdue.edu (Herman Rubin) wrote:
>In article <1992Oct11.153538.12821@zip.eecs.umich.edu> ayman@eecs.umich.edu (Ayman I. Kayssi) writes:
>>y = a0 + a1*x + ... + am*x^m + b1*(-y)*x + b2*(-y)*x^2 + ... + bn*(-y)*x^n
>
>If you are trying a regression on your transformed equation, the estimates
>will not even be consistent because of dependent variables on the right side.


TableCurve (from Jandel) lists the following form under *linear* equations:
    
    a + c x + e x^2 + ...
y = ---------------------
    1 + b x + d x^2 + ...

How do they (non-iteratively) calculate the parameters?
--
Ayman I. Kayssi                               |    ayman@eecs.umich.edu
EECS Department                               +------------------------
University of Michigan                        |          (313) 764-8033

In article <1992Oct11.192921.22075@zip.eecs.umich.edu> ayman@eecs.umich.edu (Ayman I. Kayssi) writes:
>hrubin@pop.stat.purdue.edu (Herman Rubin) wrote:
>>In article <1992Oct11.153538.12821@zip.eecs.umich.edu> ayman@eecs.umich.edu (Ayman I. Kayssi) writes:
>>>y = a0 + a1*x + ... + am*x^m + b1*(-y)*x + b2*(-y)*x^2 + ... + bn*(-y)*x^n

>>If you are trying a regression on your transformed equation, the estimates
>>will not even be consistent because of dependent variables on the right side.


>TableCurve (from Jandel) lists the following form under *linear* equations:

>    a + c x + e x^2 + ...
>y = ---------------------
>    1 + b x + d x^2 + ...

>How do they (non-iteratively) calculate the parameters?

I am not familiar with the work; I do not even know if data is being fitted,
or a theoretical function.  A standard numerical analysis situation is the
fitting of approximations of fixed type; as infinite series, the problem 
is hopelessly unidentified.  As I have stated before, there are quite 
substantial numerical problems, although these may not be too crucial for
the degree of fit.  If the fit is to observational data, it may even be a
good idea to use ridge regression or some other Bayesian, or approximate
Bayesian, approacth.

But I cannot see any good reason why it is important, or even desirable, to
calculate non-iteratively, unless this is to be a "field" situation, with
thousands or millions of cases to be computed.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)

>In article <1992Oct11.153538.12821@zip.eecs.umich.edu> ayman@eecs.umich.edu 
(Ayman I. Kayssi) writes:
>>I'm trying to fit data to a rational function. 

John Nelder wrote a paper in the '60's about inverse polynomial regression that
might give you the sort of fit you need.  I can't recall the exact reference,
but it is discussed in McCullagh & Nelder's "Generalized Linear Models".

>What is the transformation that I need to make to have a linear model?

Can't be done.  That's why you need a generalized linear model.


>>>>>>>>>>>>>>>>>>>bickis@math.usask.ca<<<<<<<<<<<<<<<<<<<<<<<<<<<
Mik Bickis       \,,,       .*#*.  \    University of Saskatchewan
Department of        \,,,  ^\\|//^  ```\   Saskatoon, Saskatchewan
Mathematics & Statistics \   )|(        ```\   Canada      S7N 0W0
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