Radial basis function

A radial basis function (RBF) is a real-valued function φ { extstyle varphi } whose value depends only on the distance from the origin, so that φ ( x ) = φ ( ‖ x ‖ ) { extstyle varphi (mathbf {x} )=varphi (left|mathbf {x} ight|)} ; or alternatively on the distance from some other point c { extstyle mathbf {c} } , called a center, so that φ ( x , c ) = φ ( ‖ x − c ‖ ) { extstyle varphi (mathbf {x} ,mathbf {c} )=varphi (left|mathbf {x} -mathbf {c} ight|)} . Any function φ { extstyle varphi } that satisfies the property φ ( x ) = φ ( ‖ x ‖ ) { extstyle varphi (mathbf {x} )=varphi (left|mathbf {x} ight|)} is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible.A radial function is a function φ : [ 0 , ∞ ) → R { extstyle varphi :[0,infty ) o mathbb {R} }  . When paired with a metric on a vector space ‖ ⋅ ‖ : V → [ 0 , ∞ ) { extstyle |cdot |:V o [0,infty )}   a function φ c = φ ( ‖ x − c ‖ ) { extstyle varphi _{mathbf {c} }=varphi (|mathbf {x} -mathbf {c} |)}   is said to be a radial kernel centered at c { extstyle mathbf {c} }  . A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes { x k } k = 1 n {displaystyle {mathbf {x} _{k}}_{k=1}^{n}}  Radial basis functions are typically used to build up function approximations of the formThe sum