Fitting different datasets with different models but the same parameters

Question

I am looking for a method to fit two different datasets with different fit models but depending on the same parameters in Matlab. All solutions i found so far are either not for matlab or do not treat this exact problem.

Here is a rough example what my problem looks like: I have acquired two datasets that should take these forms:

Dataset 1: f(x) = a*exp(x)+b

Dataset 2: g(x) = atan(b*x/a)

The real formulas are more complicated but the principle should be the same.

When i try to fit them with a NonlinearLeastSquares independently, matlab obviously provides different values for the variables a and b for the two different datasets. I tried feeding one solution into the other fitting routine as a starting point but that rarely improves accuracy. Is there a way to tell Matlab to fit both datasets at the same time or at least find the variables that fit bost models the best?

Answer 1

In general you can use fmincon to do that. The idea is to define a function that takes both f(x) and g(x) in to account.

Lets do that.

function error2 = myFunction(betas,x)

lambda=0.5;

error2=0;

a=betas(1,1);
b=betas(2,1);

x1=x(:,1); %Assuming that both datasets have the same size. If they are not you can adjust it
y1=x(:,2);
x2=x(:,3);
y2=x(:,4);

n=size(x,1); 

for i=1:n
    f1=a*exp(x1(i,1))+b;
    f2=atan(b*x2(i,1)/a);
    error2=error2+lambda*(y1(i,1)-f1)^2 + (1-lambda)*(y2(i,1)-f2)^2;
end

Note that in "betas" I am keeping the parameters and "x" I am keeping the data. I had to introduce a new variable "lambda" in order to weight both functions f and g. This is good because varying lambda you are able to see how one of the functions affected the estimations of the other. You can actually start with lambda=0 and run several times this routine for values such as 0.1, 0.2,...,1.

Now you have to call this function using fmincon.

clear all
close all

    % Here you have to create your data x: Remember the structure I used for x=[x1,y1,x2,y2]

   x1=
   y1=
   x2=
   y2=

   x=[x1,y1,x2,y2];


    % you need to initiate the values of your parameters beta

a0=
b0=

beta0(1,1)=a0;
beta0(2,1)=b0;

beta = fmincon(@(beta)myFunction(beta,x), beta0);

This must work!

Answer 2

Off the wall, but how about try fitting: f(x)+g(x) = a*exp(x)+b + atan(b*x/a)

What do you think? Is that totally stupid?

I know it won't work (and probably little will) if the magnitudes of f(x) & g(x) don't match because then the errors (which your are minimizing) for the two different parts are going to be unequal and you'll end up fitting one function or the other anyway.