Radial basis function

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]