Radial basis function

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A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that $\phi ({\mathbf {x}})=\phi (\|{\mathbf {x}}\|)$ ; or alternatively on the distance from some other point c, called a center, so that $\phi ({\mathbf {x}},{\mathbf {c}})=\phi (\|{\mathbf {x}}-{\mathbf {c}}\|)$ . Any function $\phi$ that satisfies the property $\phi ({\mathbf {x}})=\phi (\|{\mathbf {x}}\|)$ is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible. For example, using Lukaszyk-Karmowski metric, it is possible for some radial functions to avoid problems with ill conditioning of the matrix solved to determine coefficients w_i (see below), since the $\|{\mathbf {x}}\|$ is always greater than zero.^[1]

Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network. RBFs are also used as a kernel in support vector classification.

RBF types

Commonly used types of radial basis functions include (writing $r=\|{\mathbf {x}}-{\mathbf {x}}_{i}\|\;$ ):

Gaussian:

The first term, that is used for normalisation of the Gaussian, is missing, because in our sum every Gaussian has a weight, so the normalisation is not necessary.

$\phi (r)=e^{{-(\varepsilon r)^{2}}}\,$

Multiquadric:

$\phi (r)={\sqrt {1+(\varepsilon r)^{2}}}$

Inverse quadratic:

$\phi (r)={\frac {1}{1+(\varepsilon r)^{2}}}$

Inverse multiquadric:

$\phi (r)={\frac {1}{{\sqrt {1+(\varepsilon r)^{2}}}}}$

Polyharmonic spline:

$\phi (r)=r^{k},\;k=1,3,5,\dots$

$\phi (r)=r^{k}\ln(r),\;k=2,4,6,\dots$

Thin plate spline (a special polyharmonic spline):

$\phi (r)=r^{2}\ln(r)\;$

Approximation

Radial basis functions are typically used to build up function approximations of the form

$y({\mathbf {x}})=\sum _{{i=1}}^{N}w_{i}\,\phi (\|{\mathbf {x}}-{\mathbf {x}}_{i}\|),$

where the approximating function y(x) is represented as a sum of N radial basis functions, each associated with a different center x_i, and weighted by an appropriate coefficient w_i. The weights w_i can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.

Approximation schemes of this kind have been particularly used^{[citation needed]} in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour, 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation).

RBF network

See also: radial basis function network

Two unnormalized Gaussian radial basis functions in one input dimension. The basis function centers are located at x₁=0.75 and x₂=3.25.

The sum

$y({\mathbf {x}})=\sum _{{i=1}}^{N}w_{i}\,\phi (\|{\mathbf {x}}-{\mathbf {x}}_{i}\|),$

can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number N of radial basis functions is used.

The approximant y(x) is differentiable with respect to the weights w_i. The weights could thus be learned using any of the standard iterative methods for neural networks.

Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly.

References

↑ Lukaszyk, S. (2004) A new concept of probability metric and its applications in approximation of scattered data sets. Computational Mechanics, 33, 299-3004. limited access

Buhmann, Martin D. (2003), Radial Basis Functions: Theory and Implementations, Cambridge University Press, ISBN 978-0-521-63338-3 .
Hardy, R.L., Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 76(8):1905–1915, 1971.
Hardy, R.L., 1990, Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988, Comp. math Applic. Vol 19, no. 8/9, pp. 163 208
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 3.7.1. Radial Basis Function Interpolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
Sirayanone, S., 1988, Comparative studies of kriging, multiquadric-biharmonic, and other methods for solving mineral resource problems, PhD. Dissertation, Dept. of Earth Sciences,Iowa State University, Ames, Iowa.
Sirayanone S. and Hardy, R.L., "The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications," Journal of Applied Sciences and Computations Vol. 1, pp. 437-475, 1995.

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