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Re: RBF Toolbox?


From: Mike B.
Subject: Re: RBF Toolbox?
Date: Sun, 25 Apr 2010 18:42:59 -0700 (PDT)

--- On Fri, 23/4/10, Jordi Gutiérrez Hermoso <address@hidden> wrote:

> This sounds fairly simple to implement, except I don't know
> what
> cross-validation is. I thought picking the right parameters
> for the
> RBFs was still a dark art. Also, what's a linear RBF? Do
> you mean the
> polynomial/conical RBFs? Got a reference for me?

Selecting the RBF hyper-parameter is definitely not black-magic and is covered 
extensively under statistical model-selection (a few references floating 
around, no particular order):
* Rippa, An Algorithm for Selecting a Good Value for the Parameter c in Radial 
Basis Function Interpolation
* Golberg et al., Improved Multiquadric Approximation for Partial Differential 
Equations
* Tenne and Armfield, A Memetic Algorithm Assisted by an Adaptive Topology 
Artificial Neural Network and Variable Local Models for Expensive Optimization 
Problems
* Milroy et al., An Adaptive Radial Basis Function Approach to Modeling 
Scattered Data

The main methods (which I know of) are maximum likelihood, cross-validation 
(essentially an empirical ML) and bootstrap (computationally intensive). There 
is no `optimal' method and they all should provide `good' and similar estimates 
so small differences may not be crucial. A major consideration is how 
computationally-efficient is the method. You might want to allow the user to 
select their method of choice.

For reviews on RBFs:
* Powell, Radial Basis Function Methods for Interpolation of Functions of Many 
Variables
* Buhmann, Radial Basis Functions Theory and Implementations

Cheers.





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