A radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a center, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection 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,which stemmed from Michael J. D. Powell's seminal research from 1977. RBFs are also used as a kernel in support vector classification. The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications.
A radial function is a function . When paired with a metric on a vector space a function is said to be a radial kernel centered at . A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes
Commonly used types of radial basis functions include (writing and using to indicate a shape parameter that can be used to scale the input of the radial kernel ):
These radial basis functions are from and are strictly positive definite functions that require tuning a shape parameter
These RBFs are compactly supported and thus are non-zero only within a radius of , and thus have sparse differentiation matrices
Radial basis functions are typically used to build up function approximations of the form
where the approximating function is represented as a sum of radial basis functions, each associated with a different center , and weighted by an appropriate coefficient The weights can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights .
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).
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 of radial basis functions is used.
The approximant is differentiable with respect to the weights . 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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In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions. RBF interpolation is a mesh-free method, meaning the nodes need not lie on a structured grid, and does not require the formation of a mesh. It is often spectrally accurate and stable for large numbers of nodes even in high dimensions.
Radial basis functions were first introduced by Powell to solve the real multivariate interpolation problem.
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