matlab: constrained fit with lsqcurvefit and fmincon

Question

I have a set of data points (data_x, data_y). I need to fit a model function into this data. Model is a function of 5 parameters, and I have defined it like that:

function F = model(x,xdata)

fraction1 = x(4);
fraction2 = x(5);
fraction3 = 1-x(4)-x(5);

F=1-(fraction1.*(exp(-(xdata)./x(1)))+(fraction2.*(exp(-(xdata)./x(2))))+(fraction3.*(exp(-(xdata)./x(3)))));

parameters x(4) and x(5) are used to define three fractions, so their sum MUST be 1. To fit this function I was using lsqcurvefit, like that:

%% initial conditions
a0 = [guess1 guess2 guess3 0.3 0.3];

%% bounds
lb = [0 0 0 0 0 ];
ub = [inf inf inf 1 1];

%% Fitting options
curvefitoptions = optimset( 'Display', 'iter' );

%% Fit
a = lsqcurvefit(@model,a0,x,y,lb,ub,curvefitoptions);

The thing is that don't know how to add constraints, to keep the sum of fractions = 1. I know that lsqcurvefit is not the best solution for this problem, but I have no idea how to feed fmincon with these data to find my parameters. Many thanks for help!

EDIT: just a note, the max value of F could be 1... I was trying to cheat i.e. by adding or even multiplying everything by something like 10^(1-fraction1-fraction2-fraction3), but then I was ending up with almost equal fractions (0.33), what has no sense at all, cause other parameters are screwed... When I was fitting the same data using Origin (with same model + constrains) it works perfectly... When I used fixed Origin output fraction parameters fits were also great, but... its not the way to do that having dozen of fits to do :(

Answer 1

If you use fmincon for this (and use another parameter as the third fraction) the constraints are quite simple. You may need to play around with the fmincon options to get good convergence.

function solution = my_fit_fun(xdata, ydata, a0)

lb = [0 0 0 0 0 0];
ub = [inf inf inf 1 1 1];

%Aeq and beq specify that the last three parameters add to 1
Aeq = [0 0 0 1 1 1];
beq = 1;

solution = fmincon(@objective,a0,[],[],Aeq,beq,lb,ub);

    function F = model(x)

        fraction1 = x(4);
        fraction2 = x(5);
        fraction3 = x(6);

        F=1-(fraction1.*(exp(-(xdata)./x(1)))+(fraction2.*(exp(-(xdata)./x(2))))+(fraction3.*(exp(-(xdata)./x(3)))));

    end

    function f = objective(x)

        yfit = model(x);
        f = sum((yfit - ydata) .^2);
    end

end

Answer 2

Unfortunately, MATLABs lsqcurvefit does not support constraints other than a lower and upper bound. Mathematically this would be possible - it is just not implemented. Use of fmincon is not advised to use for curve fitting.

In case of simple sum constrains the best solution is probably elimination of your variable x(5):

fraction1 = x(4);
fraction2 = 1-x(4);

This requires typically adjustment of the allowed range of x(4), so that x(5) = 1-x(4) will stay in the allowed range too. (In your case this is 0...1 for x(4) and x(5) so no adjustements have to be done).