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Integration, Minimization and complex numbers Problem

  • To: mathgroup at
  • Subject: [mg45860] Integration, Minimization and complex numbers Problem
  • From: mfific <mfific at>
  • Date: Tue, 27 Jan 2004 04:51:21 -0500 (EST)
  • Sender: owner-wri-mathgroup at

I have one integration and minimization problem. A function of interest 
is defined as follows:

model[l_,h_,scale_,shape_,tt]=Integrate[{den[l, h, 
t]*PDF[GammaDistribution[shape, scale], {tt - t}]}, {t, 0, tt}]

model[l_,h_,scale_,shape_,tt] is convolution function between first 
den[l, h, t] which is in my calculation cumulative distribution function 
and probability density function GammaDistribution[shape, scale]. So the 
convoluted function is strictly positive and should be continuous and 
differentiable at all points. Now it is NOT possible to get closed form 
for this convolution function, and as an output I get some long 
expression. When I had plot this function for some parameter values it 
turned out to be correct.

Now the problem I run into was when I try to use this 
model[l_,h_,scale_,shape_,tt] function to fit actual data. I used 
NMinimize on the function that is to be minimized (see bellow). This is 
standard routine for minimization. Dat2 file is a data file.

squares =Plus @@ Table[((model[l, h, scale, shape, tt] /.tt -> 
(dat2[[i,1]] -dat2[[i,2]]))^2, {i, Length[dat2]}];

nminsol=NMinimize[{squares}, {l, h, scale, shape},Method->{"NelderMead"}]

The problem was that it did not converge and I got report that the

"The function (145.080813 -(6.93321450269*10^-18 [maginaryI] did not 
evaluate to a number at the point (h 
->0.006370727396825097`,(l->0.128041641382088), (scale-> 
72.43025153),(shape-> 5.8088417556)?

(Note that I constrained parameters which is not presented in NMinimize 
function! above)

I tried different methods for minimization but it did not work out. When 
I paid close attention to the output of the table in squares above, 
I found some complex number (real + imaginary). After some further 
research I found that convoluted function model[l_,h_,scale_,shape_,tt] 
behaves chaotically for some parameter values, which was surprised for 
me, because both function that were used to construct it were regular 
probability functions. I guess these irregularities were because of 
mathematica used some numerical integration in order to approximate 
complicated convolution function.

Also, I used realonly package in order to turn-off imaginary part of 
the function output but it did not help. Obviously realonly will not 
prevent imaginary number when NMinimize is used.

I would appreciate any suggestion.

Mario Fific

Department of Cognitive Psychology

Indiana University

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