In mathematics 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 ${\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}$, or some other fixed point ${\textstyle \mathbf {c} }$, called a center, so that ${\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)}$. Any function ${\textstyle \varphi }$ that satisfies the property ${\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}$ 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.

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 ${\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}}$

Examplesedit

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]):