[curve fitting] dependence between coefficients

Dear all -

I need to fit my experimental data (x_data, y_data) with a biexponential decay model:

      % define fit options
      fo_ = fitoptions('method','NonlinearLeastSquares','Lower',lower,'Upper',upper);
      % define fittype
      ft_ = fittype('offset+a*exp(-(x-x0)/b)+c*exp(-(x-x0)/d)',...
          'dependent',{'y'},'independent',{'x'},...
          'coefficients',{ 'offset', 'x0', 'a', 'b', 'c', 'd'});
      % perform fit
      [cf_, gof, output] = fit(x_data,y_data,ft_,fo_);

    offset = y-offset

    x0 = x-offset

a and c = amplitudes (weighing factors)

b and d = decay constants

As my experimental data are normalised, i.e. the decay occurs from 1 to 0, I would like to implement the following condition in my fitting routine: a + c = 1

How can I do this?

I appreciate your help!

Sebastian

Answer 1

[Matt J]

Replace occurrences of c in your model with 1-a. Then fit the remaining parameters.

      ft_ = fittype('offset+a*exp(-(x-x0)/b)+(1-a)*exp(-(x-x0)/d)',...
          'dependent',{'y'},'independent',{'x'},...
          'coefficients',{ 'offset', 'x0', 'a', 'b', 'd'});

[Sebastian König]

Dear Matt,

thank you for your reply! This would be an elegant solution to this particular problem.

However, how should I proceed if I was to perform a triexponential fit and the same condition should apply (amp1 + amp2 + amp3 = 1)?

Cheers, Sebastian

[Matt J]

You could do the same thing.

   amp1 = 1 - amp2 - amp3

The equation always allows one variable to be eliminated, no matter how many terms you have.

[Sebastian König]

That is in fact true. :-)

Thanks for your help!