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Mathematica Wolfram Language Optimization Numerical Computation

Selecting parts of fitted model

Al Guy

Posted 11 years ago

here is a code to create a list and fit it with two gaussians: list = Join[ Table[{RandomReal[{0, 1}], RandomReal[{0, 10}]}, {i, 100}], Table[{RandomReal[{1, 3}], RandomReal[{0, 30}]}, {i, 100}]]; fit = NonlinearModelFit[list, A1PDF[NormalDistribution[\[Mu]1, \[Sigma]1], t] + A2PDF[NormalDistribution[\[Mu]2, \[Sigma]2], t], { {A1, 15}, {\[Sigma]1, 0.5}, {\[Mu]1, 0.5}, {\[Mu]2, 1.5}, {A2, 40}, {\[Sigma]2, 1.5}}, t] Show[ListPlot[list], Plot[fit[t], {t, -1, 5}, PlotStyle -> Red]] I can select the first part of the fitted formula by: Normal[fit][[1]] But I can't plot or do numerical manipulation: Plot[Normal[fit][[1]], {t, 0, 3}] I know I can use `/.t-> d` construct , but it is a numerically inefficient way to do when dealing with large formulas. Any ideas?

here is a code to create a list and fit it with two gaussians:

list = Join[
   Table[{RandomReal[{0, 1}], RandomReal[{0, 10}]}, {i, 100}], 
   Table[{RandomReal[{1, 3}], RandomReal[{0, 30}]}, {i, 100}]];
fit = NonlinearModelFit[list, 
  A1*PDF[NormalDistribution[\[Mu]1, \[Sigma]1], t] + 
   A2*PDF[NormalDistribution[\[Mu]2, \[Sigma]2], t], { {A1, 
    15}, {\[Sigma]1, 0.5}, {\[Mu]1, 0.5}, {\[Mu]2, 1.5}, {A2, 
    40}, {\[Sigma]2, 1.5}}, t]
Show[ListPlot[list], Plot[fit[t], {t, -1, 5}, PlotStyle -> Red]]

I can select the first part of the fitted formula by:

Normal[fit][[1]]

But I can't plot or do numerical manipulation:

Plot[Normal[fit][[1]], {t, 0, 3}]

I know I can use /.t-> d construct , but it is a numerically inefficient way to do when dealing with large formulas.

Any ideas?

POSTED BY: Al Guy

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Jim Baldwin

Jim Baldwin, Retired

Posted 11 years ago

I think that the "properties" of the fit will provide what you want: fit["BestFit"][[1]] fit["BestFit"][[2]] Plot[Evaluate[fit["BestFit"][[1]]], {t, 0, 4}]

POSTED BY: Jim Baldwin

David Keith

Posted 11 years ago

But this will work: Plot[Normal[fit][[1]] // Evaluate, {t, 0, 3}] And you're very welcome!

POSTED BY: David Keith

Al Guy

Posted 11 years ago

You are right, I see what was the problem, I can not do it in one step like Plot[Normal[fit][[1]], {t,0,4}], i need to define additional variable. Seems legit. Thanks, man.

POSTED BY: Al Guy

David Keith

Posted 11 years ago

term1 plots OK for me. And in the above, In[25] defines a function t1 which is described by the term1 expression. I attach a notebook. (It might be helpful to work with a fixed data set rather than random.) Best, David Attachments:

POSTED BY: David Keith

Al Guy

Posted 11 years ago

`Plot[term1 , {t, 0, 4}]` doesn't work. How to make it work in a numerically sound manner? How to define and compile function `f[t_]:= term1`?

POSTED BY: Al Guy

David Keith

Posted 11 years ago

We get different values each time because the fit is generated from random data, but I think you can get the expression as follows, and can use it to define a new function as needed. Yes? In[23]:= exp = Normal[fit] Out[23]= -20.981 E^(-50.6429 (-3.00695 + t)^2) + 18.0241 E^(-0.257059 (-2.65573 + t)^2) In[24]:= term1 = exp[[1]] Out[24]= -20.981 E^(-50.6429 (-3.00695 + t)^2) In[25]:= t1[t_] = term1 Out[25]= -20.981 E^(-50.6429 (-3.00695 + t)^2)

We get different values each time because the fit is generated from random data, but I think you can get the expression as follows, and can use it to define a new function as needed. Yes?

In[23]:= exp = Normal[fit]

Out[23]= -20.981 E^(-50.6429 (-3.00695 + t)^2) + 
 18.0241 E^(-0.257059 (-2.65573 + t)^2)

In[24]:= term1 = exp[[1]]

Out[24]= -20.981 E^(-50.6429 (-3.00695 + t)^2)

In[25]:= t1[t_] = term1

Out[25]= -20.981 E^(-50.6429 (-3.00695 + t)^2)

POSTED BY: David Keith

David Keith

Posted 11 years ago

I'm not sure I understand. What is the first part of the sum?

POSTED BY: David Keith

Al Guy

Posted 11 years ago

The Normal[fit] consists of two summands: `-137.812 E^(-18.4523 (-11.7337 + t)^2)` and `15.2716 E^(-0.525651 (-1.92061 + t)^2)` I need the first of them: Normal[fit][[1]], this outputs the first part, but I can't plot it or do numerical computations on it as I posted above.

POSTED BY: Al Guy

Al Guy

Posted 11 years ago

No, I need to select the first part of the sum.

POSTED BY: Al Guy

David Keith

Posted 11 years ago

This will use the fitted model to define a function, but it is not clear there is an advantage over using fit[t], as you did in your plot. In[5]:= f[t_] = Normal[fit] Out[5]= -137.812 E^(-18.4523 (-11.7337 + t)^2) + 15.2716 E^(-0.525651 (-1.92061 + t)^2)

POSTED BY: David Keith

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