Radial basis function

A radial basis function (RBF) is a real-valued function ${\textstyle \phi }$ whose value depends only on the distance from the origin, so that ${\textstyle \phi \left(\mathbf {x} \right)=\phi \left(\left\|\mathbf {x} \right\|\right)}$; or alternatively on the distance from some other point ${\textstyle \mathbf {c} }$, called a center, so that ${\textstyle \phi \left(\mathbf {x} ,\mathbf {c} \right)=\phi \left(\left\|\mathbf {x} -\mathbf {c} \right\|\right)}$. Any function ${\textstyle \phi }$ that satisfies the property ${\textstyle \phi \left(\mathbf {x} \right)=\phi \left(\left\|\mathbf {x} \right\|\right)}$ is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible.

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; this was the context in which they originally surfaced, in work by David Broomhead and David Lowe in 1988, which stemmed from Michael J. D. Powell's seminal research from 1977. RBFs are also used as a kernel in support vector classification. The RBF implementation is ready enough to have the RBF exploited in different engineering applications.

Types

Commonly used types of radial basis functions include (writing ${\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|}$ and using ${\textstyle \varepsilon }$ to indicate the inverse of a critical radius):

• Gaussian:
  $$\phi \left(r\right)=e^{-\left(\varepsilon r\right)^{2}}$$

• Multiquadric:
  $$\phi \left(r\right)={\sqrt {1+\left(\varepsilon r\right)^{2}}}$$

• Inverse quadratic:
  $$\phi \left(r\right)={\dfrac {1}{1+\left(\varepsilon r\right)^{2}}}$$

• Inverse multiquadric:
  $$\phi \left(r\right)={\dfrac {1}{\sqrt {1+\left(\varepsilon r\right)^{2}}}}$$

• Polyharmonic spline:
  $${\begin{aligned}\phi \left(r\right)&=r^{k},&k&=1,3,5,\dotsc \\\phi \left(r\right)&=r^{k}\ln \left(r\right),&k&=2,4,6,\dotsc \end{aligned}}$$

• Thin plate spline (a special polyharmonic spline):
  $$\phi \left(r\right)=r^{2}\ln \left(r\right)$$

Approximation

Main article: Kernel smoothing

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

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

where the approximating function ${\textstyle y\left(\mathbf {x} \right)}$ is represented as a sum of $N$ radial basis functions, each associated with a different center ${\textstyle \mathbf {x} _{i}}$, and weighted by an appropriate coefficient ${\textstyle w_{i}}$. The weights ${\textstyle w_{i}}$ can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights ${\textstyle w_{i}}$.

Approximation schemes of this kind have been particularly used 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

Main article: radial basis function network

Two unnormalized Gaussian radial basis functions in one input dimension. The basis function centers are located at ${\textstyle x_{1}=0.75}$ and ${\textstyle x_{2}=3.25}$.

The sum

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

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 ${\textstyle N}$ of radial basis functions is used.

The approximant ${\textstyle y\left(\mathbf {x} \right)}$ is differentiable with respect to the weights ${\textstyle w_{i}}$. 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.

See also

• Matérn covariance function

Further reading

• Hardy, R.L. (1971). "Multiquadric equations of topography and other irregular surfaces". Journal of Geophysical Research. 76 (8): 1905–1915. Bibcode:1971JGR....76.1905H. doi:10.1029/jb076i008p01905.
• Hardy, R.L. (1990). "Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988". Comp. Math Applic. 19 (8/9): 163–208. doi:10.1016/0898-1221(90)90272-l.
• 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.; Hardy, R.L. (1995). "The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications". Journal of Applied Sciences and Computations. 1: 437–475.