How radial basis interpolation works?

Question

Can any one explain to me how to understand this plot. This is 2D scatter data interpolation using Radial basis function.

http://www.alglib.net/interpolation/introductiontorbfs.php

In this link they shown this fig. but Don't understand what they trying to represent using this fig.

_{(source: alglib.net)}

Answer 1

These are the centers of your radial basis functions. Each circle corresponds to a Gaussian G(x;m,s) with mean m and (scalar) variance s. Here the mean varies from x=-5 to x=5 whereas |y|=2 is held fixed. The variance seems to be isotropic and identical across all Gaussians.

Using these functions, one can then go on and do some kind of regression. There are multiple alternatives for this: one can for example consider the simple linear superposition of those Gaussians and optimize the superposition coefficients (--that would be linear regression-style). On the other hand, one can also try to optimize the hyper-parameters, namely the means and variances, so that the fit becomes closest.

The basic problem, however, already becomes clear from this figure: the higher the dimension, the more functions you need in order to obtain a good fit (for example, think about interpolating something which is located around the point (-5,-5) -- it simply doesn't work). This problem is often termed the curse of dimensionality.