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Re: Data fitting from coupled differential equations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg93343] Re: Data fitting from coupled differential equations
*From*: dh <dh at metrohm.ch>
*Date*: Tue, 4 Nov 2008 06:19:13 -0500 (EST)
*References*: <gemjhd$4s9$1@smc.vnet.net>
Hi,
draw your example solution. Is an exponential growning example really
what you want? Your measured data will probably not be of the order of
10^50.
hope this helps, Daniel
margeorias at gmail.com wrote:
> Hi,
>
> I'm working on a predator-prey model for 2-species:
> n1'[t] == a1 n1[t] - b1 n1[t]^2 + c1 n1[t] n2[t]
> n2'[t] == a2 n2[t] - b2 n2[t]^2 + c2 n1[t] n2[t]
> n1[0] == n10,
> n2[0] == n20
>
> I have real data for (t,n1[t],n2[t]), i.e. data={t,n1[t],n2[t]} and my
> objective is to estimate the coefficients a1,a2,b1,b2,c1,c2.
>
> To do this, I generate synthetic data by solving the above equations
> using {n10,n20,a1,a2,b1,b2,c1,c2}={0.01,0.001,2,3,1,1,1,2} and sample
> the solutions,i.e.
> data = Table[{t, n1[t], n2[t]} /. sol, {t, 0, 20, 0.25}];
>
> I then set-up the function below:
>
> sse[a1_?NumericQ,a2_?NumericQ,b1_?NumericQ,b2_?NumericQ,c1_?
> NumericQ,c2_?NumericQ]:=
> Block[{sol,n1,n2},
> sol=NDSolve[{
> n1'[t]==a1 n1[t]-b1 n1[t]-c1 n1[t] n2[t],
> n2'[t]==a2 n2[t]-b2 n2[t]-c2 n1[t] n2[t],
> n1[0]==0.01,n2[0]==0.001},
> {n1,n2},{t,0,20}][[1]];
> Plus@@Apply[((n1[#1]-#2)^2+(n2[#1]-#3)^2)&,data,{1}]/.sol]
>
> and use FindMinimum to retrieve the coefficients:
> FindMinimum[sse[a1,a2,b1,b2,c1,c2],{{a1,1.},{a2,1.},{b1,1.},{b2,1.},
> {c1,1.},{c2,1.}},AccuracyGoal->20,PrecisionGoal->18,WorkingPrecision-
>> 40]
>
> My problem however is that the resulting coefficients are not the same
> as the original one, ie:
> {a1->-9.49992942438124754467310,a2->-9.50000018840483650972573,b1-
>> -9.50007117410767898402213,b2->-9.50000041008883222559689,c1-
>> -9.50000036535088621114653,c2->-9.50000030036113407572174}
>
> I can improve the results by making starting points very close to the
> actual value of the coefficients i.e. {a1,2}. Well this somehow
> defeats the purpose and in reality, I don't know whats the actual
> value of the coefficients - they range from -Infinity to Infinity.How
> do I go about obtaining the correct coefficients? Is there something
> wrong with the function above?
>
> Another thing, wouldn't NMinimize be appropriate for this as well
> since I want to find the global minimum of sse? I have tried it but it
> gave the wrong values for the coefficients as well :(
>
> I'm trying to make FindFit work on the coupled differential equations
> but no luck yet. Any guidance on this?
>
> Appreciate any feedback you have.
>
>
> Thanks again.
>
--
Daniel Huber
Metrohm Ltd.
Oberdorfstr. 68
CH-9100 Herisau
Tel. +41 71 353 8585, Fax +41 71 353 8907
E-Mail:<mailto:dh at metrohm.com>
Internet:<http://www.metrohm.com>
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