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Re: [Q] Nonlinear Fitting in symbolic Integration ..?

• To: mathgroup at smc.vnet.net
• Subject: [mg82909] Re: [mg82842] [Q] Nonlinear Fitting in symbolic Integration ..?
• From: Darren Glosemeyer <darreng at wolfram.com>
• Date: Sat, 3 Nov 2007 03:22:56 -0500 (EST)
• References: <200711011012.FAA07488@smc.vnet.net>

hwoarang wrote:
>  Dear Mathgroup, 
>  
>  I have experiment data and am trying to fit data to convoluted equation. 
>  The equation contains UnitStep and symbolic Integrate function. 
>  I think that these functions have some compolited problems according to searching 
>  about related problems in archives. 
>  But still, I have no idea whether the problem is in UnitStep, symbolic Integrate or NonlinearRegress.. -_-. 
>  Does anybody out there have any ideas for this ? 
>  Any help would be appreciated. 
>  
>  I pasted math code.
>  
>  Sincerely, yours 
>  Hwoarang. 
>  
>  Math Code : 
>  ----------------------------------------------------------------------------- 
>  Remove["Global`*"]; 
>  << "Statistics`NonlinearFit`" 
>  k0 = 1; k1 =. ; k2 =. ; tau1 =. ; tau2 =. ; 
>  response[t_, x_] := (UnitStep[t - x]*{k2*(1 - E^(-((t - x)/tau2))) + 
>  (k1*(1 - E^(-((t - x)/tau1))))/E^((t - x)/tau2)})/E^(x^2/(2*0.1^2)) 
>  convol[t_] = Integrate[(k0*response[t, x])/(2*Pi*0.1^0.5), {x, -1, 0}, 
>  Assumptions -> {{t, k1, k2, tau1, tau2} ¡ô Reals && t > 0 && tau1 > 0 && 
>  tau2 > 0 && k1 > 0 && k2 > 0}, GenerateConditions -> False] + 
>  Integrate[(k0*response[t, x])/(2*Pi*0.1^0.5), {x, 0, 20}, 
>  Assumptions -> {{t, k1, k2, tau1, tau2} ¡ô Reals && t > 0 && tau1 > 0 && 
>  tau2 > 0 && k1 > 0 && k2 > 0}, GenerateConditions -> False] 
>  dat = Import["Reflsmooth.txt", "Table"]; 
>  ListPlot[dat]; 
>  NonlinearRegress[dat, convol, t, {{k1, 0.4, 0.35, 0.45}, 
>  {k2, 0.04, 0.035, 0.045}, {tau1, 0.35, 0.3, 0.4}, {tau2, 0.25, 0.2, 0.3}}, 
>  ShowProgress -> True] 
>
> -------------------------------------------------------------------------- 
> Jiwan Kim, Ph. D. Candidate,
> Dept. of Physics and Center for Nanospinics of Spintronic Materials, 
> KAIST 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea 
> Tel: +82-42-869-8163
> Cel: +82-16-870-7419
> Fax: +82-42-869-8162 
> E-mail: hwoarang at kaist.ac.kr 
>
>   

Without your data, I can't comment on exactly what you are seeing, but I 
can make a few comments on the code that might solve the problem.

In the definition of response there is a set of curly brackets {} that 
should be parentheses (). In the symbolic integrals it would be better 
to use exact numbers instead of floating point numbers.  In the 
NonlinearRegress call, convol should be convol[t] instead.

An example with some simulated data is below.


In[1]:= << "Statistics`NonlinearFit`"

In[2]:= k0 = 1;

In[3]:= response[t_, x_] := (UnitStep[t - x]*(k2*(1 -
                          E^(-((t - x)/tau2))) + (k1*(1 - E^(-((t - 
x)/tau1))))/
                      E^((t - x)/tau2)))/E^(x^2/(2*(1/10)^2))

In[4]:= convol[t_] =
            Integrate[(k0*response[t, x])/(2*Pi*(1/10)^(1/2)), {x, -1, 0},
                Assumptions -> {t > 0 && tau1 > 0 && tau2 > 0 && k1 > 0 
&& k2 > 0},
                GenerateConditions -> False] +
              Integrate[(k0*response[t, x])/(2*Pi*(1/10)^(1/2)), {x, 0, 
20},
                Assumptions -> {t > 0 && tau1 > 0 && tau2 > 0 && k1 > 0 
&& k2 > 0},
                GenerateConditions -> False];

In[5]:= dat = Table[
                Flatten[{t, convol[t]*(1 + Random[Real, {-.005, 
.005}])}], {t, 0,
                  1, .01}] /. {k1 -> .41, k2 -> .37, tau1 -> .31, tau2 
-> .28};

In[6]:= NonlinearRegress[dat,
          convol[t], t, {{k1, 0.4, 0.35, 0.45}, {k2, 0.04, 0.035, 
0.045}, {tau1, 0.35,
               0.3, 0.4}, {tau2, 0.25, 0.2, 0.3}}]

Out[6]= {BestFitParameters ->
 
 >     {k1 -> 0.416736, k2 -> 0.369518, tau1 -> 0.310294, tau2 -> 
0.285446},
 
 >    ParameterCITable ->        Estimate   Asymptotic SE   
CI                    ,
                         k1     0.416736   0.0774913       {0.262937, 
0.570535}

                         k2     0.369518   0.00137785      {0.366783, 
0.372252}

                         tau1   0.310294   0.250587        {-0.187052, 
0.807639}

                         tau2   0.285446   0.180313        {-0.0724248, 
0.643317}
 
                                   -8
 >    EstimatedVariance -> 1.1668 10  ,
 
 >    ANOVATable ->                     DF    SumOfSq        MeanSq     ,
                   Model               4     0.170563       0.0426407

                                                       -6            -8
                   Error               97    1.13179 10     1.1668 10

                   Uncorrected Total   101   0.170564

                   Corrected Total     100   0.0110745
 
 >    AsymptoticCorrelationMatrix -> 1.          0.951971    0.99865     
-0.997248,

                                    0.951971    1.          0.96587     
-0.970739

                                    0.99865     0.96587     1.          
-0.999742

                                    -0.997248   -0.970739   -0.999742   1.
 
 >    FitCurvatureTable ->                           Curvature}
                          Max Intrinsic             0.33271

                          Max Parameter-Effects     1132.36

                          95. % Confidence Region   0.636868


In several trials with different simulated data sets, I found that 
convergence is sometimes a problem. I think this is due, at least in 
part, to interdependency of the parameters in the model; note that the 
entries in the correlation matrix are all nearly 1 or -1.


Darren Glosemeyer
Wolfram Research

• References:
  • [Q] Nonlinear Fitting in symbolic Integration ..?
    • From: hwoarang <kjwan@kaist.ac.kr>