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A radial basis function (RBF) is a real-valued function ${\textstyle \varphi }$ whose value depends only on the distance between the input and some fixed point, either the origin, so that ${\textstyle \varphi (\mathbf {x} )=\varphi (\left\|\mathbf {x} \right\|)}$, or some other fixed point ${\textstyle \mathbf {c} }$, called a center, so that ${\textstyle \varphi (\mathbf {x} )=\varphi (\left\|\mathbf {x} -\mathbf {c} \right\|)}$. Any function ${\textstyle \varphi }$ that satisfies the property ${\textstyle \varphi (\mathbf {x} )=\varphi (\left\|\mathbf {x} \right\|)}$ is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection ${\displaystyle \{\varphi _{k}\}_{k}}$which forms a basis for some function space of interest, hence the name.

## Contents

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 were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, [1] [2] which stemmed from Michael J. D. Powell's seminal research from 1977. [3] [4] [5] RBFs are also used as a kernel in support vector classification. [6] The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications. [7] [8]

## Definition

A radial function is a function ${\textstyle \varphi$ :[0,\infty )\to \mathbb {R} }. When paired with a metric on a vector space ${\textstyle \|\cdot \|:V\to [0,\infty )}$ a function ${\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)}$ is said to be a radial kernel centered at ${\textstyle \mathbf {c} }$. A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes ${\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}}$

• The kernels ${\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}$ are linearly independent (for example ${\displaystyle \varphi (r)=r^{2}}$ in ${\displaystyle V=\mathbb {R} }$ is not a radial basis function)
• The kernels ${\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}$ form a basis for a Haar Space, meaning that the interpolation matrix
${\displaystyle {\begin{bmatrix}\varphi (\|\mathbf {x} _{1}-\mathbf {x} _{1}\|)&\varphi (\|\mathbf {x} _{2}-\mathbf {x} _{1}\|)&\dots &\varphi (\|\mathbf {x} _{n}-\mathbf {x} _{1}\|)\\\varphi (\|\mathbf {x} _{1}-\mathbf {x} _{2}\|)&\varphi (\|\mathbf {x} _{2}-\mathbf {x} _{2}\|)&\dots &\varphi (\|\mathbf {x} _{n}-\mathbf {x} _{2}\|)\\\vdots &\vdots &\ddots &\vdots \\\varphi (\|\mathbf {x} _{1}-\mathbf {x} _{n}\|)&\varphi (\|\mathbf {x} _{2}-\mathbf {x} _{n}\|)&\dots &\varphi (\|\mathbf {x} _{n}-\mathbf {x} _{n}\|)\\\end{bmatrix}}}$

is non-singular. [9] [10]

### Examples

Commonly used types of radial basis functions include (writing ${\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|}$ and using ${\textstyle \varepsilon }$ to indicate a shape parameter that can be used to scale the input of the radial kernel [11] ):

• Infinitely Smooth RBFs

These radial basis functions are from ${\displaystyle C^{\infty }(\mathbb {R} )}$ and are strictly positive definite functions [12] that require tuning a shape parameter ${\displaystyle \varepsilon }$

• Gaussian:
${\displaystyle \varphi (r)=e^{-(\varepsilon r)^{2}}}$
${\displaystyle \varphi (r)={\sqrt {1+(\varepsilon r)^{2}}}}$
${\displaystyle \varphi (r)={\dfrac {1}{1+(\varepsilon r)^{2}}}}$
${\displaystyle \varphi (r)={\dfrac {1}{\sqrt {1+(\varepsilon r)^{2}}}}}$
• Polyharmonic spline:
{\displaystyle {\begin{aligned}\varphi (r)&=r^{k},&k&=1,3,5,\dotsc \\\varphi (r)&=r^{k}\ln(r),&k&=2,4,6,\dotsc \end{aligned}}}
*For even-degree polyharmonic splines${\displaystyle (k=2,4,6,\dotsc )}$, to avoid numerical problems at ${\displaystyle r=0}$ where ${\displaystyle \ln(0)=-\infty }$, the computational implementation is often written as ${\displaystyle \varphi (r)=r^{k-1}\ln(r^{r})}$[ citation needed ].
• Thin plate spline (a special polyharmonic spline):
${\displaystyle \varphi (r)=r^{2}\ln(r)}$

These RBFs are compactly supported and thus are non-zero only within a radius of ${\displaystyle 1/\varepsilon }$, and thus have sparse differentiation matrices

${\displaystyle \varphi (r)={\begin{cases}\exp \left(-{\frac {1}{1-(\varepsilon r)^{2}}}\right)&{\mbox{ for }}r<{\frac {1}{\varepsilon }}\\0&{\mbox{ otherwise}}\end{cases}}}$

## Approximation

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

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

where the approximating function ${\textstyle y(\mathbf {x} )}$ is represented as a sum of ${\displaystyle N}$ 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[ citation needed ] in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation).

## RBF Network

The sum

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

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 ${\textstyle y(\mathbf {x} )}$ 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. [ citation needed ]

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