Re: [Help-gsl] Fitting biexponential decay

[Totte Karlsson]

Hi,
Thanks for your suggestions. I have not had time to really get started 
on this until yesterday.
I'm not sure how to create a function of goodness of fit for my problem. 
If that is simple I could  use the multidimensional minimizer as you 
suggested. However, maybe using least squares (the nonlinear least 
squares fitting API) is what I want to use. At least thats what I 
started to work with since there is an example in the manual, which is a 
fit of a single exponential.

        y(t) = A1exp{-lambda_1*t} + b


I wonder if it is possible to use the example and extend it to a fit of 
a bi-exponential?

The part that i think would have to be modified is the creation of the 
Jacobian matrix. For the single exponential case
it is defined (from the manual)

     /* Jacobian matrix J(i,j) = dfi / dxj, */
     /* where fi = (Yi - yi)/sigma[i],      */
     /*       Yi = A * exp(-lambda * i) + b  */
     /* and the xj are the parameters (A,lambda,b) */
     double t = i;
     double s = sigma[i];
     double e = exp(-lambda * t);
     gsl_matrix_set (J, i, 0, e/s);
     gsl_matrix_set (J, i, 1, -t * A * e/s);
     gsl_matrix_set (J, i, 2, 1/s);

I'm not that mathematical and just started to use gsl, so it looks a 
little tough to proceed. Wonder if anyone can help to extend the above 
lines  to the bi-exponential case?

Also, what about independent gaussian errors, sigma_i? In my data each 
point  can be assumed to have the same error (same sigmas). How can I 
calculate sigma for an experimental data set? Is it the standard deviation?

regards
-totte
BTW I'm using the single exponential fitter and it works great.

Martin Jansche wrote:
> On Jan 22, 2008 9:09 AM, Totte Karlsson <address@hidden> wrote:
>
>   
> > Now I need to find a fitter for a bi exponential decay function, i.e.
> >
> > y(t) = A1exp{-lambda_1*t} + A2exp{-lambda_2*t}.
> >
> > In the equation, the A1 and A2 will typically be equal ,(A1 == A2) and
> > initially set to 1, so the fit will be for lambda_1 and lambda_2.
> >     
>
> You can use the multidimensional minimizer API
> (http://www.gnu.org/software/gsl/manual/html_node/Multidimensional-Minimization.html)
> to optimize the goodness of fit as a function of your two parameters.
> You'll have to implement a function which evaluates goodness of fit
> for your set of observation and choice of parameters, as well as the
> two partial derivatives in lambda_i.  Your decay function has simple
> partial derivatives; if your fit criterion does too, you're in very
> good shape.  If your fit criterion is least squares, you can also use
> the nonlinear least-squares fitting API
> (http://www.gnu.org/software/gsl/manual/html_node/Nonlinear-Least_002dSquares-Fitting.html).
>
> -- mj
>