Hi Edward.
I thing the estimation for of r2eff errors via Co-variance matrix
should be considered complete.
The facts:
We get the same co-variance as reported by scipy.minimise.leastsq.
Using the direct exponential Jacobian, seems to give double error on
R2eff, but after fitting,
the extracted parameters of 'r1', 'r2', 'pA', 'dw', 'kex' are very similar.
The chi2 values seems 4x as small, but that agrees with the R2eff
errors being 2x as big.
This should complete the analysis.
I have not found any "true" reference to which Jacobian to use.
I think it is my imagination, to use the chi2 function Jacobian.
It seems from various tutorials also refers to the direct Jacobian:
http://www.orbitals.com/self/least/least.htm
This is an experimental feature, and that is pointed out in the user function.
The last point here is, that it seems that just 50 Monte-Carlo
simulations is enough for the
estimation of R2eff errors.
After your implementation in C-code of the Jacobian, and the Hessian,
which give the possibility to use "Newton" as minimisation technique,
Monte-Carlo simulations are now so super-fast, that if one is in a
HURRY, at least 50 MC simulations will give a better, and more safe,
result than estimating from the Co-Variance matrix.
But now the option is there, giving freedom to the user to try
different possibilities.
I think the code should reside where it is, since it is still so experimental.
When the usability has been verified, it could be split up.
If the usability is low, then one can quickly delete it.
If minfx was extended to use constraints in BFGS, it should be quite
easy to make the Jacobian of the different relaxation dispersion
models, and implement those.
That will speed up the analysis, and also make it possible to extract
the estimated errors from the Jacobian.
But this is distant future.
Best
Troels
The estimation for R2eff errors via Co-variance matrix.