Non-linear Simultaneous Fitting/Solution

Hi,

   I am trying to model two vector data sets f1(x) and f2(x) using two non-linear equations with common variables.  

   For example:

   f1(x) = a0 + a1(k1,t)*exp(-l1*x) + a2(k1,t)*exp(-l2*x)
   f2(x) = b0 + b1(k1,t)*exp(-l1*x) + b2(k1,t)*exp(-l2*x) 

    I would like to simultaneously fit f1(x) and f2(x) to these two equations. Then graphically display the data and fits. The a1, a2, b1, and b2 are expressions which are different but contain the variables k1 and t which will be determined from the simultaneous fit. I would like to use matlab script to do this. 

    Is fsolve the way to do this?  If so, can somebody please give a little direction as to setting this up?

Edit: I have been trying fmincon, but need the minimization to output a global minimum for the vector data sets, not a minimum at each point. Is there a way around this in simple code?

    Thanks!

Answer 1

[Matt Tearle]

You can treat this as a least-squares problem with 6 parameters: a0, b0, k1, t, l1, and l2. Then make your objective function the total square error ((y1 - f1(x))^2 + (y2 - f2(x))^2). So something like

function err = myerrorfun(c,x1,y1,x2,y2)
f1 = c(1) + [function of c(3) and c(4)]*exp(-c(5)*x1) + ...;
f2 = c(2) + [function of c(3) and c(4)]*exp(-c(5)*x2) + ...;
err = (y1-f1).^2 + (y2-f2).^2;

Then in your main program, call a minimization routine like fmincon:

x1 = ... % enter/load
x2 = ... % all
y1 = ... % the
y2 = ... % data

% make a function handle of one variable (the parameters), with the data embedded
objective = @(c) myerrorfun(c,x1,y1,x2,y2);
% do the fitting
c_fit = fmincon(objective,...);

[Avigdor]

Thank you for the information and concise code!

Do you have any suggestions as to estimation of goodness of fit, or error in fit from such a method?

[Matt Tearle]

In this case, myerrorfun is actually returning the sum of the squared error. You can get the function value as a second output from fmincon (or whatever minimization function you used). But if you want more detailed info, you could make functions to evaluate f1 and f2; call these with c_fit and x1 or x2; take the difference with y1 and y2 and now you have the residuals. Apply whatever standard analysis you normally would to the residuals of a regression -- hist, normplot, scatter(resids(1:end-1),resids(2:end)), etc. Or, of course, apply your favorite goodness-of-fit formula (eg adjusted R^2).