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I've been reading up recently on radial basis functions, and I've recently moved from Gaussian basis functions of the form $\phi(x)=\exp(-\alpha||x-x_i||)$ to multiquadratic basis functions. My source is this paper, which is referenced in an algorithm that I'm looking at.

The basis function for the multiquadratic RBF is given as $\phi(x)=\sqrt{||x-x_i||^2+R^2}$, where $R$ is a shape parameter. While it's certainly possible to solve the set of linear equations in a basis function and get a valid result for any equation, this choice of basis functions seems odd. Compared to a Gaussian basis function, where the effect of a basis function with a large value of $||x-x_i||$ approaches zero, multiquadratics grow increasingly rapidly as the radius increases.

Intuitively, this seems like a strange choice for a basis function, since distant points in a training data set would seemingly influence the results of a local solution more than nearby points. Why is this function used as a basis function, and what advantages does it have over something like a Gaussian basis function?

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...a strange choice for a basis function, since distant points in a training data set would seemingly influence the results of a local solution more than nearby points. Why is this function used as a basis function?

The sum of their influences quickly fades away. It's well illustrated in 1d, where we can represent a spline either piece-wise, locally controlled, or as one, 'global' RBF sum with $|x|$ kernel (see here for example). This might seem strange as each $c_i|x-x_i|$ term is unbounded. Yet, together they cancel out automagically at a distance. You can see it by following the derivation at the link. Another example is a system of electric charges overall neutral.

what advantages does it have over something like a Gaussian basis function?

Multiquadric and other globally supported RBF have a good hole-filling property: if you have holes that are irregular with unknown scales, they're going to be smoothly interpolated over according to surrounding trends from neighborhoods of corresponding scales. If your dataset has a regular sparsity of one radius you can estimate, you can use instead compactly supported RBF, or Gaussian RBF which is practically local.

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