From: osman@stargate.cinternet.net (Me)
Subject: Re: Curve fitting.
Date: 7 Mar 1999 10:12:45 -0500
Newsgroups:  sci.math.num-analysis
Keywords: standard regression

On Sun, 7 Mar 1999  Olivier Maës <olivier.maes@worldonline.fr> wrote:
>Hi all,
>
>I have few experimental points :
>
>      X               Y
>  9270          111
>  9411          107
>  9552            98
>  9693            88
>  9833            83
>  9974            82
>10115            87
>11242            96
>11383          111
>11524          121
>11665          117
>11805          104
>11946            96
>12087          103
>12228          121
>
>I know that they can be fitted with a curve like :
>
>  Y = aX + b + (cX + d)*sin(eX + f)
>
>How can I find a, b, c, d, e and f from my measurement points ?
>
>Thank you for help.
>
>
>

Just use gnuplot3.7 with its "fit" command. The only problem is that
you have quite a few unknowns and really not enough data. The results
of this fit will depend greatly on the initial values. Just keep the
coeff of x in the sin term small so it won't oscillate wildly.
In a first try I came up with:

fit f(x) 'foo.data'  via a,b,c,d,e,f

After 8 iterations the fit converged.
final sum of squares residuals    : 398.414
rel. change during last iteration : -8.57448e-06

Final set of parameters            68.3% confidence interval (one at a time)
=======================            =========================================

a               = 0.00849747       +/- 0.00135903      (15.9934%)
b               = 9.59555          +/- 14.6627         (152.807%)
c               = 0.0142024        +/- 0.00196512      (13.8365%)
d               = -151.585         +/- 21.1732         (13.9678%)
e               = 0.00732784       +/- 0.0001037       (1.41515%)
f               = 130.637          +/- 1.14767         (0.878521%)


correlation matrix of the fit parameters:

               a      b      c      d      e      f      
a               1.000 -0.996  0.320 -0.323 -0.194  0.173 
b              -0.996  1.000 -0.321  0.325  0.182 -0.162 
c               0.320 -0.321  1.000 -0.996 -0.178  0.169 
d              -0.323  0.325 -0.996  1.000  0.174 -0.164 
e              -0.194  0.182 -0.178  0.174  1.000 -0.994 
f               0.173 -0.162  0.169 -0.164 -0.994  1.000 

you can also plot the results with the initial data:
plot 'foo.data', f(x)


-- 
Osman Buyukisik
==============================================================================

From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Curve fitting.
Date: 8 Mar 1999 15:35:45 GMT
Newsgroups: sci.math.num-analysis

In article <7btifk$f0a$1@mars.worldonline.fr>,
 "Olivier Maës" <olivier.maes@worldonline.fr> writes:
|> Hi all,
|> 
|> I have few experimental points :
|> 
|>       X               Y
|>   9270          111
snip
|> I know that they can be fitted with a curve like :
|> 
|>   Y = aX + b + (cX + d)*sin(eX + f)
|> 
|> How can I find a, b, c, d, e and f from my measurement points 
this is a classical nonlinear least squares problem .
you may use any of the nonlinear least squares codes listed at
http://plato.la.asu.edu/guide.html 
using lmdif from minpack, 
renaming a,b,c,d by x[1],x[2],x[3],x[4],x{5],x[6] 
and X by (t-10700) i got
 sum of erros squared        =  .1713951591894696E+04
 loesungsvektor            =
 x[ 1]=  .5774884523611436D-02
 x[ 2]=  .1010204505122279D+03
 x[ 3]=  .4142059231591140D-02
 x[ 4]= -.6742606934928337D+00
 x[ 5]=  .9998152217705151D+00
 x[ 6]=  .1445380819587726D+01
hope this helps
peter