MATLAB Answers

# lsqcurvefit to estimate parameters in ODE15s

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Antonella Treglia on 27 May 2021
Commented: Star Strider on 27 May 2021
Hello!
I am trying to solve a parametric ODE system. I woul like to estimate the parameters by fitting the calculated data with experimental data.
Model:
dydt(1) = - theta(2).*y(1).*y(2)-theta(3).*y(1).*(theta(4)-y(3))-theta(5)*y(1).*y(2).^2;
dydt(2) = -theta(2).*y(1).*y(2)-theta(5)*y(1).*y(2).^2;
dydt(3) = theta(3).*y(1).*(theta(4)-y(3));
dydt(4) = 0;
it is a system of differential equations with 5 parameters (theta). These are constant parameters that affect the time evolution of the set of solutions y.
What I can empirically measure is the quantity
C=theta(2)*y(1)*y(2)/max(theta(2)*y(1)*y(2));
I would like to solve the system of ODE and fit C to the measured values to obtain the best guess for the set of parameters theta.
Here my code:
%parameters initial values
theta0 = [4e17 2e-11 7e-8 5e15 9.3e-30];
% Variables
n_abs =3.3e15;
%experimental values for time (t - in seconds) and evolution (c) are pasted at the end
%of the code
[theta,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(theta,t) kinetics(theta,t),theta0,t,c);
[Cfit,T] = kinetics(theta,t);%,precision);
fprintf(1,'\tRate Constants:\n')
for k1 = 1:length(theta)
fprintf(1, '\t\tTheta(%d) = %d\n', k1, theta(k1))
end
figure(1)
plot(t*1e9, c, 'p') %plot in ns
hold on
hlp = plot(T*1e9, Cfit); %plot in ns
hold off
grid
xlabel('Time')
ylabel('Concentration')
legend(hlp, 'C_1(t)', 'C_2(t)', 'C_3(t)', 'C_4(t)', 'Location','N')
set(gca,'yscale','log')
function [C,T] = kinetics(theta,t)
y0 = [3.3e15 3.3e15+theta(1) 0 0 ]; %initial conditions
[T,y] = ode15s(@odefcn, t, y0);
function dydt = odefcn(t,y)
fprintf( 'dentroOdefnc')
dydt = zeros(4,1);
dydt(1) = - theta(2).*y(1).*y(2)-theta(3).*y(1).*(theta(4)-y(3))-theta(5)*y(1).*y(2).^2;
dydt(2) = -theta(2).*y(1).*y(2)-theta(5)*y(1).*y(2).^2;
dydt(3) = theta(3).*y(1).*(theta(4)-y(3));
dydt(4) = 0;
end
C=theta(2)*y(:,1).*y(:,2)/max(theta(2)*y(:,1).*y(:,2));
end
The code is running but at the output i don't get any change in the values i initially set for theta0. Basically lsqcurvefit is not working properly.
This is what I get as output:
Local minimum possible.
lsqcurvefit stopped because the size of the current step is less than
the value of the step size tolerance.
Can you please help me?
Here the experimental data i am using for fitting
t=[4.88281749999997e-11
9.76562750000003e-11
1.46484375000000e-10
1.95312475000000e-10
2.44140675000000e-10
2.92968775000000e-10
3.41796875000000e-10
3.90624975000000e-10
4.39453175000000e-10
4.88281275000000e-10
5.37109375000000e-10
5.85937475000000e-10
6.34765675000000e-10
6.83593775000000e-10
7.32421875000000e-10
7.81249975000000e-10
8.30078175000000e-10
8.78906275000000e-10
9.27734375000000e-10
9.76562475000000e-10
1.02539067500000e-09
1.07421877500000e-09
1.12304687500000e-09
1.17187497500000e-09
1.22070317500000e-09
1.26953127500000e-09
1.31835937500000e-09
1.36718747500000e-09
1.41601567500000e-09
1.46484377500000e-09
1.51367187500000e-09
1.56249997500000e-09
1.61132817500000e-09
1.66015627500000e-09
1.70898437500000e-09
1.75781247500000e-09
1.80664067500000e-09
1.85546877500000e-09
1.90429687500000e-09
1.95312497500000e-09
2.00195317500000e-09
2.05078127500000e-09
2.09960937500000e-09
2.14843747500000e-09
2.19726567500000e-09
2.24609377500000e-09
2.29492187500000e-09
2.34374997500000e-09
2.39257817500000e-09
2.44140627500000e-09
2.49023437500000e-09
2.53906247500000e-09
2.58789067500000e-09
2.63671877500000e-09
2.68554687500000e-09
2.73437497500000e-09
2.78320317500000e-09
2.83203127500000e-09
2.88085937500000e-09
2.92968747500000e-09
2.97851567500000e-09
3.02734377500000e-09
3.07617187500000e-09
3.12499997500000e-09
3.17382817500000e-09
3.22265627500000e-09
3.27148437500000e-09
3.32031247500000e-09
3.36914067500000e-09
3.41796877500000e-09
3.46679687500000e-09
3.51562497500000e-09
3.56445317500000e-09
3.61328127500000e-09
3.66210937500000e-09
3.71093747500000e-09
3.75976567500000e-09
3.80859377500000e-09
3.85742187500000e-09
3.90624997500000e-09
3.95507817500000e-09
4.00390627500000e-09
4.05273437500000e-09
4.10156247500000e-09
4.15039067500000e-09
4.19921877500000e-09
4.24804687500000e-09
4.29687497500000e-09
4.34570317500000e-09
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4.44335937500000e-09
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4.63867187500000e-09
4.68749997500000e-09
4.73632817500000e-09
4.78515627500000e-09
4.83398437500000e-09
4.88281287500000e-09
4.93164087500000e-09
4.98046887500000e-09
5.02929687500000e-09
5.07812487500000e-09
5.12695287500000e-09
5.17578087500000e-09
5.22460987500000e-09
5.27343787500000e-09
5.32226587500000e-09
5.37109387500000e-09
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7.86132787500000e-09
7.91015587500000e-09
7.95898487500000e-09
8.00781287500000e-09
8.05664087500000e-09
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1.00097658750000e-08
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1.09863278750000e-08
1.10351558750000e-08
1.10839848750000e-08
1.11328128750000e-08
1.11816408750000e-08
1.12304688750000e-08
1.12792968750000e-08
1.13281248750000e-08
1.13769528750000e-08
1.14257808750000e-08
1.14746098750000e-08
1.15234378750000e-08
1.15722658750000e-08
1.16210938750000e-08
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1.18164058750000e-08
1.18652348750000e-08
1.19140628750000e-08
1.19628908750000e-08
1.20117188750000e-08
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1.21093748750000e-08
1.21582028750000e-08
1.22070308750000e-08
1.22558598750000e-08
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1.23535158750000e-08
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1.24511718750000e-08
1.24999998750000e-08
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1.38183598750000e-08
1.38671878750000e-08
1.39160158750000e-08
1.39648468750000e-08
1.40136718750000e-08
1.40624998750000e-08
1.41113278750000e-08
1.41601558750000e-08
1.42089848750000e-08
1.42578128750000e-08
1.43066408750000e-08
1.43554688750000e-08
1.44042968750000e-08
1.44531248750000e-08
1.45019528750000e-08
1.45507808750000e-08
1.45996098750000e-08
1.46484378750000e-08
1.46972658750000e-08
1.47460938750000e-08
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1.48437468750000e-08
1.48925778750000e-08
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1.49902348750000e-08
1.50390628750000e-08
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1.51855468750000e-08
1.52343748750000e-08
1.52832028750000e-08
1.53320308750000e-08
1.53808598750000e-08
1.54296878750000e-08
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1.55273468750000e-08
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1.56738278750000e-08
1.57226558750000e-08
1.57714848750000e-08
1.58203128750000e-08
1.58691408750000e-08
1.59179688750000e-08
1.59667968750000e-08
1.60156248750000e-08
1.60644528750000e-08
1.61132808750000e-08
1.61621098750000e-08
1.62109378750000e-08
1.62597658750000e-08
1.63085938750000e-08
1.63574218750000e-08
1.64062468750000e-08
1.64550778750000e-08
1.65039058750000e-08
1.65527348750000e-08
1.66015628750000e-08
1.66503908750000e-08
1.66992188750000e-08
1.67480468750000e-08
1.67968748750000e-08
1.68457028750000e-08
1.68945308750000e-08
1.69433598750000e-08
1.69921878750000e-08
1.70410158750000e-08
1.70898468750000e-08
1.71386718750000e-08
1.71874998750000e-08
1.72363278750000e-08
1.72851558750000e-08
1.73339848750000e-08
1.73828128750000e-08
1.74316408750000e-08
1.74804688750000e-08
1.75292968750000e-08
1.75781248750000e-08
1.76269528750000e-08
1.76757808750000e-08
1.77246098750000e-08
1.77734378750000e-08
1.78222658750000e-08
1.78710938750000e-08
1.79199218750000e-08
1.79687468750000e-08
1.80175778750000e-08
1.80664058750000e-08
1.81152348750000e-08
1.81640628750000e-08
1.82128908750000e-08
1.82617188750000e-08
1.83105468750000e-08
1.83593748750000e-08
1.84082028750000e-08
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1.87011718750000e-08
1.87499998750000e-08
1.87988278750000e-08
1.88476558750000e-08
1.88964848750000e-08
1.89453128750000e-08
1.89941408750000e-08
1.90429688750000e-08
1.90917968750000e-08
1.91406248750000e-08
1.91894528750000e-08
1.92382808750000e-08
1.92871098750000e-08
1.93359378750000e-08
1.93847658750000e-08
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1.94824218750000e-08
1.95312468750000e-08
1.95800778750000e-08
1.96289058750000e-08
1.96777348750000e-08
1.97265628750000e-08
1.97753908750000e-08
1.98242188750000e-08
1.98730468750000e-08
1.99218748750000e-08
1.99707028750000e-08
2.00195308750000e-08
2.00683598750000e-08
2.01171878750000e-08
2.01660158750000e-08
2.02148468750000e-08
2.02636718750000e-08
2.03124998750000e-08
2.03613278750000e-08
2.04101558750000e-08
2.04589848750000e-08
2.05078128750000e-08
2.05566408750000e-08
2.06054688750000e-08
2.06542968750000e-08
2.07031248750000e-08
2.07519528750000e-08
2.08007808750000e-08
2.08496098750000e-08
2.08984378750000e-08
2.09472658750000e-08
2.09960938750000e-08
2.10449218750000e-08
2.10937468750000e-08
2.11425778750000e-08
2.11914058750000e-08
2.12402348750000e-08
2.12890628750000e-08
2.13378908750000e-08
2.13867188750000e-08
2.14355468750000e-08
2.14843748750000e-08
2.15332028750000e-08
2.15820308750000e-08
2.16308598750000e-08
2.16796878750000e-08
2.17285158750000e-08
2.17773468750000e-08
2.18261718750000e-08
2.18749998750000e-08
2.19238278750000e-08
2.19726558750000e-08
2.20214848750000e-08
2.20703128750000e-08
2.21191408750000e-08
2.21679688750000e-08
2.22167968750000e-08
2.22656248750000e-08
2.23144528750000e-08
2.23632808750000e-08
2.24121098750000e-08
2.24609378750000e-08
2.25097658750000e-08
2.25585938750000e-08
2.26074218750000e-08
2.26562468750000e-08
2.27050778750000e-08
2.27539058750000e-08
2.28027348750000e-08
2.28515628750000e-08
2.29003908750000e-08
2.29492188750000e-08
2.29980468750000e-08
]; %experimental data (time axis)
c=[0.978949375536276
0.978377347697588
0.964744017542187
0.943006959672037
0.941386214129088
0.932043092763848
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0.00230717894937554]; %experimental data (yaxis)
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### Accepted Answer

Star Strider on 27 May 2021
The problem is in the definition of ‘theta0’ and not including the initial conditions in ‘theta0’.
Changing those produces —
T1 = readtable('Antonella_Treglia_Data.txt');
t = T1.t;
c = T1.c;
%parameters initial values
theta0 = [4e17 2e-11 7e-8 5e15 9.3e-30];
theta0 = [theta0 3.3e15 3.3e15+theta0(1) rand(1,2) ];
theta0 = rand(9,1);
% Variables
n_abs =3.3e15;
%experimental values for time (t - in seconds) and evolution (c) are pasted at the end
%of the code
[theta,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(theta,t) kinetics(theta,t),theta0,t,c,zeros(1,9));
dentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdent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efncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefnc...
[Cfit,T] = kinetics(theta,t);%,precision);
dentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefnc
fprintf(1,'\tRate Constants:\n')
Rate Constants:
for k1 = 1:length(theta)
fprintf(1, '\t\tTheta(%d) = %10.5f\n', k1, theta(k1))
end
Theta(1) = 0.36428 Theta(2) = 61.16478 Theta(3) = 0.36475 Theta(4) = 1.75649 Theta(5) = 515.96813 Theta(6) = 35.60667 Theta(7) = 728.17186 Theta(8) = 1.73429 Theta(9) = 0.68600
fprintf('\nNOTE — The last four ‘theta’ values are the initial conditions.\n\n')
NOTE — The last four ‘theta’ values are the initial conditions.
figure(1)
plot(t*1e9, c, 'p') %plot in ns
hold on
hlp = plot(T*1e9, Cfit, 'LineWidth',2); %plot in ns
hold off
grid
xlabel('Time')
ylabel('Concentration')
legend(hlp, 'C_1(t)')%, 'C_2(t)', 'C_3(t)', 'C_4(t)', 'Location','N')
set(gca,'yscale','log')
function [C,T] = kinetics(theta,t)
% y0 = [3.3e15 3.3e15+theta(1) 0 0 ]; %initial conditions
y0 = theta(6:end);
[T,y] = ode15s(@odefcn, t, y0);
function dydt = odefcn(t,y)
fprintf( 'dentroOdefnc')
dydt = zeros(4,1);
dydt(1) = - theta(2).*y(1).*y(2)-theta(3).*y(1).*(theta(4)-y(3))-theta(5)*y(1).*y(2).^2;
dydt(2) = -theta(2).*y(1).*y(2)-theta(5)*y(1).*y(2).^2;
dydt(3) = theta(3).*y(1).*(theta(4)-y(3));
dydt(4) = 0;
end
C=theta(2)*y(:,1).*y(:,2)/max(theta(2)*y(:,1).*y(:,2));
end
One set of parameters that worked are:
% Parameters = [0.9; 22; -48; 15; 455]
% InitialConditions = [38; 776; -32; 0.7]
Constraining the parameters so they are all greater than 0 is an option with lsqcurvefit, and was added in this code.
Doing that produces:
% Parameters = [0.4; 61; 0.4; 1.8; 516]
% InitialConditions = [36; 728; 1.7; 0.7]
With an equally (visually) acceptable fit.
NOTE — I copied the posted data to a text file so that I could easily run the code here. The file is attached.
.
##### 6 CommentsShowHide 5 older comments
Star Strider on 27 May 2021
@Antonella Treglia — The parameters you chose are so far from the parameters that are appropriate for this model that they do not change because starting with them, lsqcurvefit is searching the parameter space for a gradient to descend toward a global minimum and not finding it.
This:
Warning: Failure at t=4.882817e-11. Unable to meet integration tolerances without reducing the step size below the smallest value allowed
indicates that the differential equation integration is enountering a singularity () and stopping. That also indicates a problem with the parameter scaling.
.

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### More Answers (1)

Torsten on 27 May 2021
function C = kinetics(theta,t)
instead of
function [C,T] = kinetics(theta,t)
Define stronger RelTol and AbsTol for ODE15S and lsqcurvefit (around 1e-8)
Can you include a plot in which we see the t-c-curve for the experiment together with the t-c_simulated curve with the starting parameters for theta ?
##### 3 CommentsShowHide 2 older comments
Antonella Treglia on 27 May 2021
I set:
options = optimoptions('lsqcurvefit','StepTolerance',1e-12,'MaxFunEvals',5e3,'MaxIterations',5e3);
Opt = odeset('RelTol',1e-8,'AbsTol',1e-8);
but continues to give me
Warning: Failure at t=4.882817e-11. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.033976e-25) at time t.
It stops at the first value of the time vector t

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