🚀 go-pugleaf

Anthony,

Your problem is of wide interest to many people, not
sure I can help, but I have been working on a related
problem...below is the context of this problem, something
I have been trying to sort out wrt. to my own work.  Perhaps
others could contribute...perhaps I whistling something 
other that Dixie?

Problem:  INTERPOLATION

Approach: Non-stochastic vs. Stochastic

  Non-Stochastic techniques
     polynomial fitting
     splines

  Stochastic techniques
     moving average
     kriging

Using polynomial fitting one does not preserve the original
data points --- therefore it is very useful as a global or
smoothing technique. This is also true of moving average
techniques.

Splines do preserve the original data points and spline 
fitting is a huge area (literature wise).  Generally splines
are fit locally so that changing or removing points does not
affect the entire data set.

IMHO the stochastic methods are most interesting, the class
of techniques called BLUE (best linear unbiased estimators) 
particularly those called kriging (or Gandin optimal methods
in meteorology) allow you to derive error variance estimates
so that you can where your curve is likely to be less 
reliable.


I am not sure if it is important in your case, but remember
that if you are fitting polynomials you are implicitly imposing
a certain structure on your data (true also in the case of
sin and cos functions).  

I have found the following to be extremely helpful in all
areas of interpolation, but particularly the two dimensional
cases:

Davis, J.C. 1986. Statistics and Data Analysis in Geology,
2nd Ed., John Wiley and Sons, New York.
-it also comes with software and sample data sets

Cheers,
Chris