How to fit experimental data on only one ordinary differential equations (out of multiple equations) with multiple unknown parametrs

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Learner am 3 Jun. 2014

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Kommentiert: Johan Sebastian Diaz Tovar am 21 Nov. 2019

Hi,

I am trying to estimate 3 unknown parameters that exist in 3 ordinary differential equations (ODEs) by fitting it to experimental data of only one ODE.

My equations are:

dy1/dt = (c(1)*y1.^2) + (c(2)*(y2)) - (c(3)*y3);

dy2/dt = (c(3)*y3) + (c(4)*y1);

dy3/dt = (c(1)*y1.^2) - (c(5)*y2) + (c(3)*y3);

My experimental 'x-axis data (time axis)' and 'y-axis data' for dy3/dt is as follows:

xdata = [0 1 2 3 4 5 6 7 8 9];

ydata = [3 17 9 4 8 2 1 9 2 4].

Starting guess of unknown parameters: c(1), c(2), c(3) i.e.

c0 = [1, 1, 1]; where c(4) and c(5) are known = 2 and 1.5.

Thanks in advance.

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Learner am 4 Jun. 2014

Please anyone help me.

Is it possible to solve this type of problems?

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Star Strider am 4 Jun. 2014

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Definitely possible! The easiest way to describe this is to quote Monod kinetics and curve fitting.

The strategy is to put your differential equation and the ODE solver call inside the objective function you want to fit. It integrates the equations with the parameters passed to it, and returns the solved equation as the output of the objective functions. You can specify that c(4) and c(5) are known inside the objective function file, and only fit c(1)...c(3). Specify the initial conditions inside the objective function, or you can also fit them as parameters if you wish. (I’ve done all those.)

With a vector output, you can use nlinfit, lsqcurvefit, or with an extra step, fminsearch to do the fit.

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Star Strider am 5 Jun. 2014

My pleasure!

Probably the easiest way is to save the estimated parameters (a save command works well here), copy your ODE and solver system and statements to your main script workspace, then with the estimated parameters in the workspace, solve your differential equation at the values of your independent variable, xdata. Your y3 variable should now be in your workspace, so plot it against xdata. (I suggest you plot your ydata as points rather than a line.)
The easiest way is to print them from inside your objective function. So the first line (just above your y0 assignment) might simply to put one line as c, because that will print the values of c to the Command Window. To be a bit more formal, use fprintf(1,'\\tc(1) = %.6f\ c(2) = %.6f\ c(3) = %.6f\',c) I don’t suggest plotting during the fitting process because it slows the process significantly. If you want to write the c values serially to a mat-file, you can load the file and write code to see the fitting process later without slowing the actual fitting process itself. There may also be a display option for lsqcurvefit, so you can use the optimset function and set 'Display', 'iter-detailed' to see everything. This wouldn’t save anything, just prints to the Command Window.

StarSign1997 am 11 Apr. 2019

I copied and pasted the code here but an error occurs saying function value and YDATA are not equal in size. I am confused.

Johan Sebastian Diaz Tovar am 21 Nov. 2019

StarSign1997 when copy the code make sure to take the transpose of Sdata, if you don't do that, you'll get that error. So: Sdata = Sdata'; Doing this you solve that.

regards.

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