Re: Question: Fitting numerical solutions of ord. diff. equ.

*To*: mathgroup at smc.vnet.net*Subject*: [mg23381] Re: Question: Fitting numerical solutions of ord. diff. equ.*From*: "Atul Sharma" <atulksharma at yahoo.com>*Date*: Fri, 5 May 2000 02:07:14 -0400 (EDT)*References*: <8er9r7$hj2@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

I've found the timing to be quite variable depending on the quality of the initial estimates, how many equations, how well behaved they are etc. The largest system I've used this way contained 13 coupled equations with 4 unknowns, but "actual mileage may vary". I include an excerpt from the relevant FAQ on the WRI web site. This example uses FindMinimum, but I agree with you that there is an advantage to using NonlinearFit to simplify the goodness of fit evaluation. Once you've defined the error function, it may be used by NonlinearFit in an analogous fashion. There is also a FAQ on using penalty functions to constrain the parameter values, which you may find helpful. A. Sharma ********************************************************** How can I fit data to a model that is defined using NDSolve? ---------------------------------------------------------------------------- ---- You can in principle fit any data to any model by constructing an expression for the difference between the data and the model, and using FindMinimum to minimize that expression with respect to the parameters of the model. The example below shows one way to do this when the model is computed by NDSolve, and the difference to be minimized is the sum of the squared differences between the data and the model at each data point (least-squares fitting). If the model is given by a function f, the sum of the squared differences between the data and the model can be constructed using In[1]:= data = {{0.5, 6.2}, {1.1, 8.1}, {1.5, 8.8}, {2.1, 8.3}, {2.3, 6.8}, {3.1, 2.9}} ; In[2]:= Plus @@ Apply[(f[#1] - #2)^2 &, data, {1}] 2 2 2 Out[2]= (-6.2 + f[0.5]) + (-8.1 + f[1.1]) + (-8.8 + f[1.5]) + 2 2 2 > (-8.3 + f[2.1]) + (-6.8 + f[2.3]) + (-2.9 + f[3.1]) In this example the function f is the solution of a differential equation {f'[x] == If[x < p, 10 - f[x], -f[x]], f[0] == q} where p and q are parameters of the solution. A function sse[p, q] to compute the sum of the squared differences between the data and the model in this example can be written as In[3]:= sse[p_?NumberQ, q_?NumberQ] := Block[{sol, f}, sol = NDSolve[{f'[x] == If[x < p, 10 - f[x], -f[x]], f[0] == q}, f, {x, 0, 5}][[1]]; Plus @@ Apply[(f[#1] - #2)^2 &, data, {1}] /. sol ] This expression can be minimized using FindMinimum to obtain the best fit values for p and q. In[4]:= FindMinimum[sse[p, q], {p, 2, 3}, {q, 4, 5}] Out[4]= {0.0743759, {p -> 1.99611, q -> 3.93593}} These calculations can be computationally and mathematically quite demanding. In addition to all of the usual difficulties with non-linear fitting, solving a differential equation is a non-trivial task, and when the model is defined by a differential equation, that differential equation must be solved (with new parameters) at each step in the minimization process. "Marc Datz" <datz at ise.fhg.de> wrote in message news:8er9r7$hj2 at smc.vnet.net... > > Hi, > for my diploma thesis I have to optimize the parameters of a only > numerical solvable System of ordinary differential equations, so that it > fits optimal to my measured curve. > I have 6 implicit ordinary differential eqautions with derivatives to > time which I have to solve numerically. The parameters of the equations > have to be optimized in a way, that the chi^2 value of the calculated > NF(t) and my measured data is minimized (see atached File). > > [get the file by contacting the author ---moderator] > > I work on a solution in Pascal, but I heard it would be easy to solve > this problem with Mathematica. Unfortunately I dont know anybody who > could help me. The advantage would be that I could get the standard > errors of the parameters (alpha, beta...) and see very fast which are > significant. > I never worked with Mathematica before and I want to know, how I have to > proceed and how much time you think it will take. > Many thanks > Marc > - -------------------------------------------------- Atul Sharma MD, FRCP(C) Pediatric Nephrologist, McGill University/Montreal Children's Hospital email: atulksharma at yahoo.com

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**Question: Fitting numerical solutions of ord. diff. equ.**

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