constrained fitting using lsqcurvefit / fmincon

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

parameters x(4) and x(5) are used to define three fractions (as %), so their sum MUST be 1. To fit this curve 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 finally find my parameters. Many thanks for help!

Answer 1

[Alan Weiss]

fmincon can satisfy your conditions, which are, I believe:

 0 <= x(4) <= 1
 0 <= x(5) <= 1
 0 <= 1 - x(4) - x(5) <= 1, which, when combined with the previous, is the same as
 x(4) + x(5) <= 1

Create the following bounds and matrices:

 lb = zeros(5,1);
 ub = [inf;inf;inf;1;1];
 A = [0,0,0,1,1]; % for Ax <= b
 b = 1; % for Ax <= b

After you put xdata and ydata in your workspace, your objective function can be

 @(x)sum((model(x,xdata) - ydata).^2)

Use a reasonable fmincon algorithm, such as interior-point or sqp.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation