How to fit one variable data into multiple ode equations

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Hi there
I have one set of experimental data for y(1) but I have 4-set of ODE equations as follows:
dy1/dt = F1(y1,y2,y3,y4,par1, par2, par3, par4,t)
dy2/dt = F2(y1,y2,y3,y4,par1, par2, par3, par4,t)
dy3/dt = F3(y1,y2,y3,y4,par1, par2, par3, par4,t)
dy4/dt = D4(y1,y2,y3,y4,par1, par2, par3, par4,t)
I have only experimental data for y(1), I need to fit all four equations to best parameters to best fit to one available data y(1).
Can MATLAB do this?

Accepted Answer

Thiago Henrique Gomes Lobato
Edited: Thiago Henrique Gomes Lobato on 5 Jul 2020
What you mean by y(1)? That you have data only for the first variable y1? Your equations seem to be coupled, so if you optimize all the parameters for y1, the other yX will also have a reasonable result based on your data. What you can do then is to have all parameters as optimization variables and define a cost function equals to the difference between the integration and your experimental data. A general code for this problem would be something like this:
function cost = opt(x,yexperimental)
% Integrate system
[t,y] = ode45(@(t,y)YourOdeFunction(t,y,par1,par2,par3,par4),tspan,y0)
% Calculate error
cost = rms(y-yexperimental)
%% Run optimizer
f = @(x)opt(x,yexperimental);
[par,fval] = fminsearch(f,x0)
Thiago Henrique Gomes Lobato
I'm glad it helped. There's some complicated and easy ways to control that. The easiest one, and also probably the best in this case, is to make sure your function use the absolute value of x, so, in the end, it will not matter if fminsearch finds a negative coefficient. As, for example:
function cost = opt(x,yexperimental)
x = abs(x); % x always positive in the function
[par,fval] = fminsearch(f,x0);
par = abs(par); % Since x always positive, negative values from fminsearch are irrelevant

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