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Re: I: NONLINEARMODELFIT

• To: mathgroup at smc.vnet.net
• Subject: [mg120314] Re: I: NONLINEARMODELFIT
• From: Ray Koopman <koopman at sfu.ca>
• Date: Mon, 18 Jul 2011 06:15:00 -0400 (EDT)
• References: <ivuc83$gta$1@smc.vnet.net>

On Jul 17, 3:06 am, maria giovanna dainotti
<mariagiovannadaino... at yahoo.it> wrote:
> Dear Mathematica Group,
>
> I am fitting a function of 3 parameters with the NonlinearModelFit
> and with the Marquardt Levemberg algorithm.  Since  this method gives
> the interval of the Parameters of the best fit considering the parameters
> not correlated, I was looking for a tool that takes into account the
> fact that the parameters are correlated and the  parameters interval
> should respect the following rule:
>
> The confidence interval for a given parameter should be computed by
> varying the parameter value until the chi^2 increases by a particular
> amount above the minimum, or best-fit value.  The amount that the chi
> square is allowed to increase (also referred to as the critical
> delta_chi^2) depends on the confidence level one requires, and on
> the number of parameters whose confidence space is being calculated.
> The critical delta_chi^2 for common cases are given in Avni, 1976:
>  For example in the case I am using for 3 parameters delta_chi^2=3.50.
>
> I would be very grateful if someone of you is acquainted with a such
> a tool in Mathematica or some of you has done a code to sort this
> problem out.  Thanks a lot in advance for your help.
>
> Best regards,
>
> Maria

https://groups.google.com/group/sci.stat.edu/browse_frm/thread/5eef0a33244e080f?hl=en#
discusses a similar problem. Here is the code I used.

In[1]:=
color = {"Blue","Brown","Green","Orange","Red","Yellow"};
{k,n} = {Length@#,Tr@#}&[f = {127, 63, 122, 147, 93, 76}];
e = Array[x,k];
init = Transpose@{e,f-1,f+1};
b[\[Alpha]_] = Tr[f^2/e] == n + 2 InverseGammaRegularized[(k-1)/2,
  \[Alpha]] && And@@Thread[e > 0] && Tr@e == n;

In[6]:=
"         95% Limits\nColor   Lower  Upper\n" <>
ToString@PaddedForm[TableForm[Table[{color[[j]],
First@NMinimize[{x[j],b[.05]},init]/n,
First@NMaximize[{x[j],b[.05]},init]/n},{j,k}],
TableSpacing->{0,1}],{4,3}]

Out[6]=
         95% Limits
Color   Lower  Upper
Blue    0.154  0.261
Brown   0.067  0.147
Green   0.147  0.252
Orange  0.183  0.295
Red     0.107  0.201
Yellow  0.084  0.171