Integration, Minimization and complex numbers Problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg45860] Integration, Minimization and complex numbers Problem*From*: mfific <mfific at indiana.edu>*Date*: Tue, 27 Jan 2004 04:51:21 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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