Re: RBF Toolbox?

[Jordi Gutiérrez Hermoso]

2010/4/23 Jordi Gutiérrez Hermoso <address@hidden>:
> 2010/4/22 Mike B. <address@hidden>:
>
>> I am interested in interpolation from scattered data, dimension
>> (number of components per vector) can be as high as 200. Ideally,
>> the toolbox would allow to select from several types (multiquadric,
>> linear, Gaussian, inverse multiquadric etc.) and would calibrate any
>> hyper-parameters (such as the free coefficient in MQ and IMQ), say,
>> by cross-validation. The DACE toolbox calibrates the Kriging
>> hyper-parameters by maximum likelihood.
>
> 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?

So I looked around on my own. Cross validation looks like an
interesting statistical method, but I cannot figure out on my own how
to best implement it with RBFs. Randomly split the data into two sets?
Jackknife (leave one out) method? And on each split, how do you gauge
the goodness of fit? It's an interpolation method, not a regression
method, so the RBF parameter isn't estimated by the procedure, only
the RBF coefficients are computed "exactly".  So what do you do,
validate the training set on the target set and perform nonlinear
least squares to improve the RBF parameter without touching the RBF
coefficients?

What I've been doing in my own work is choosing the RBF parameter with
adhoc techniques based with the goal of making sure the shape of the
resulting RBFs approximately matches the separation distance of the
interpolation sites. This has worked more or less well for me since
for the PDE collocation problem I am free to choose more or less
uniform collocation points.