add parameter correlation capabilities

[mmckerns]

add parameter correlation capabilities

[mmckerns]

Notes on MCMC and parameter correlation from Paul Kienzle:

pyMC is not an optimizer -- instead, it starts  
from the solution point and does a random walk according to the  
probability distribution defined by the problem.  You end up with a  
collection of points from you parameter space whose frequency is  
proportional to the probability of seeing that point.  You then  
histogram the results to see the frequencies.

Pymc determines the joint probability distribution of your parameter  
space near your solution.  Using the suggestions from the mailing  
list (increasing the number of generations or using adaptive  
metropolis) I was able to get much better results for correlated  
variables. If your parameters are independent, you can simply use the  
mean and standard deviation of the returned points as your  
uncertainty.  If they are correlated you will need to do something  
more clever both for analysis and reporting.

I added examples of non-linear correlations to:

  svn://danse.us/reflectometry/branches/pymc-exploration/

Potential uncertainty quantifiers:

1. Using the derivatives at the solution

   A Balsamo, G Mana and F Pennecchi
   The expression of uncertainty in non-linear parameter estimation
   Metrologia 43 (2006) 396–402
   http://www.iop.org/EJ/article/0026-1394/43/5/009/met6_5_009.pdf?request-id=488a34df-5809-45bb-b8f4-d12bda4b27f1

   Cramer-Rao
        sigma**2 inv(J'J)
   Hessian
        2 sigma**2 inv(H)
   Additive residuals
        4 sigma**2 inv(H) J'J inv(H)
   Exact
        sigma**2 inv(H) dJ'/dy dJ/dy inv(H)

2. Resampling and refitting, with the caveats on correlated  
parameters given above.

3. Methods from Dakota

   http://www.ne.ncsu.edu/research/ace/adams_ace.pdf

   - sampling from factorial design or latin hypercube
   - modeling response surface and measuring the model