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>