Re: Least-Squares Fitting with asymetric errors

[Rodney Sparapani <rsparapa at post dot its dot mcw dot edu>]

>
>Hi,
>
>I was wondering if there was any advice on how to do least-squares
>fitting using GSL with asymmetric errors on my data.
>
>In the example for nonlinear least-squares fitting
>  http://www.gnu.org/software/gsl/manual/gsl-ref_36.html#SEC475
>the following snippet calculates the Jacobian of the function to be
>fitted...
>
>  /* 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);
>
>However, this assumes that the errors (AKA standard deviation 
>== sigma) is symmetric.
>
>What is the best way of allowing for asymmetric errors?  Is the way to
>do it redefine my "fi" function in both the function (to fit) and the
>calculation of its derivative to be a step function?
>i.e.   if ( yi <= Yi ) use the ErrorBar+ for sigma
>       if ( yi >  Yi ) use the ErrorBar- for sigma
>
>
>Thanks for any comments,
>
>S.

Steven:

If I understand your question correctly, it's simple:  that is the least squares 
criterion.  You could construct some other criterion, but usually it doesn't 
make much difference.  The reason that you have a non-linear model is because 
your errors are not mound-shaped and in this case they are not symmetric either.

Rodney Sparapani              Medical College of Wisconsin
Sr. Biostatistician           Patient Care & Outcomes Research
rsparapa@mcw.edu              http://www.mcw.edu/pcor
Was 'Name That Tune' rigged?  WWLD -- What Would Lombardi Do