MATLAB: Fitting global parameters using fminsearch with nested ode

fminsearchodeode45optimizationparameter estimation

Hello,

I have a problem fitting a parameter to experimental data. The model equations build a system of ODE´s. The simplified version looks like this (A and B are model parameters, C is a constant):

dx1_dt = -A*(x1-x2)
dx2_dt = A*(x1-x2) - B*(x2-C)

The task is to estimate the optimal value of A and B for both, x1(t) and x2(t).

Currently I use fminsearch with a nested ode45 in a loop. I calculate the RMSE for x1 and x2, then add them together to a global RMSE and give it back to fminsearch as quality criteria. It works quite well if you´re not in a hurry.

But I believe theres a better way to do it.

Please consider that I don´t have access to any toolboxes. I am aware of fminsearch´s drawbacks. More generally I am wondering how the algorithm should look like.

Thanks in advance.

mulm

Best Answer

I refer you to: Monod kinetics and curve fitting. I used lsqcurvefit here, but you can use fminsearch with slight modification of the code.

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MATLAB: Determining coefficients of an equation using your experimental data

What you refer to as ‘coefficients’ are actually called ‘parameters’. Estimating them is relatively straightforward. I am using fminsearch here, but the function ‘Yfit’ will work with the Statistics Toolbox nlinfit and the Optimization Toolbox lsqcurvefit functions. See the relevant documentation for details.

% Original Equation: Y = X * (a + (b*(Z)^c) * (F^d))
% Variable Matrix: 
Y = Y(:);
XZF = [X(:) Z(:) F(:)];
% Parameters: a = b(1),  b = b(2),  c = b(3),  d = b(4)
Yfcn = @(b,XZF) XZF(:,1) .* (b(1) + (b(2).*(XZF(:,2).^b(3)) .* XZF(:,3).^b(4)));
SSECF = @(b) sum((Y - Yfcn(b,XZF)).^2);
B0 = rand(4,1)*100;                         % Initial Parameter Estimates
[B,SSE] = fminsearch(SSECF, B0);

The ‘XZF’ matrix is necessary because most functions take only two parameters, the parameter vector and the independent variable. The ‘SSECF’ function is the sum-squared-error cost function for a least-squares fit. The estimated parameters are in the ‘B’ vector, and ‘SSE’ is the sum-squared-error when fminsearch ends.

This code works with created data, but fitting the parameters may be very difficult. It is important for your function here that your initial parameter estimates are as close to their actual values as you can estimate them to be. I cannot guarantee that fitting your function will give you the most appropriate parameters, in terms of their physical validity. You may have to experiment with several — perhaps hundreds — of different initial parameter estimates before you get a set that converges on physically appropriate estimates.

Because your data are actually 4-dimensional (with ‘Y’, ‘X’, ‘Z’, and ‘F’), you will not be able to plot it, or plot the fitted curve to it.

MATLAB: How to add a constraint to lsqcurvefit for estimated parameters

You CAN do this directly with lsqcurvefit, IF you reparameterize the model in to form of the parameters [a,a+b].

Then all you need to do is set a lower bound for b of zero.

Best Answer

Related Solutions

MATLAB: Determining coefficients of an equation using your experimental data

MATLAB: How to add a constraint to lsqcurvefit for estimated parameters

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