Radial basis function
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,[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 . 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
- The kernels are linearly independent (for example in is not a radial basis function)
- The kernels form a basis for a Haar Space, meaning that the interpolation matrix
Examples[]
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[11]):
- Infinitely Smooth RBFs
These radial basis functions are from and are strictly positive definite functions[12] that require tuning a shape parameter
- :
- :
- :
- :
- :
- :
- Polyharmonic spline:*For even-degree polyharmonic splines , to avoid numerical problems at where , the computational implementation is often written as [citation needed].
- Thin plate spline (a special polyharmonic spline):
- Compactly Supported RBFs
These RBFs are compactly supported and thus are non-zero only within a radius of , and thus have sparse differentiation matrices
Approximation[]
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).
RBF Network[]
The sum
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]
See also[]
References[]
- ^ Radial Basis Function networks Archived 2014-04-23 at the Wayback Machine
- ^ Broomhead, David H.; Lowe, David (1988). "Multivariable Functional Interpolation and Adaptive Networks" (PDF). Complex Systems. 2: 321–355. Archived from the original (PDF) on 2014-07-14.
- ^ Michael J. D. Powell (1977). "Restart procedures for the conjugate gradient method". Mathematical Programming. 12 (1): 241–254. doi:10.1007/bf01593790. S2CID 9500591.
- ^ Sahin, Ferat (1997). A Radial Basis Function Approach to a Color Image Classification Problem in a Real Time Industrial Application (M.Sc.). Virginia Tech. p. 26. hdl:10919/36847.
Radial basis functions were first introduced by Powell to solve the real multivariate interpolation problem.
- ^ Broomhead & Lowe 1988, p. 347: "We would like to thank Professor M.J.D. Powell at the Department of Applied Mathematics and Theoretical Physics at Cambridge University for providing the initial stimulus for this work."
- ^ VanderPlas, Jake (6 May 2015). "Introduction to Support Vector Machines". [O'Reilly]. Retrieved 14 May 2015.
- ^ Buhmann, Martin Dietrich (2003). Radial basis functions : theory and implementations. Cambridge University Press. ISBN 978-0511040207. OCLC 56352083.
- ^ Biancolini, Marco Evangelos (2018). Fast radial basis functions for engineering applications. Springer International Publishing. ISBN 9783319750118. OCLC 1030746230.
- ^ Fasshauer, Gregory E. (2007). Meshfree Approximation Methods with MATLAB. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 17–25. ISBN 9789812706331.
- ^ Wendland, Holger (2005). Scattered Data Approximation. Cambridge: Cambridge University Press. pp. 11, 18–23, 64–66. ISBN 0521843359.
- ^ Fasshauer, Gregory E. (2007). Meshfree Approximation Methods with MATLAB. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 37. ISBN 9789812706331.
- ^ Fasshauer, Gregory E. (2007). Meshfree Approximation Methods with MATLAB. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 37–45. ISBN 9789812706331.
Further reading[]
This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations. (June 2013) |
- Hardy, R.L. (1971). "Multiquadric equations of topography and other irregular surfaces". Journal of Geophysical Research. 76 (8): 1905–1915. Bibcode:1971JGR....76.1905H. doi:10.1029/jb076i008p01905.
- Hardy, R.L. (1990). "Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988". Comp. Math Applic. 19 (8/9): 163–208. doi:10.1016/0898-1221(90)90272-l.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 3.7.1. Radial Basis Function Interpolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
- Sirayanone, S., 1988, Comparative studies of kriging, multiquadric-biharmonic, and other methods for solving mineral resource problems, PhD. Dissertation, Dept. of Earth Sciences, Iowa State University, Ames, Iowa.
- Sirayanone, S.; Hardy, R.L. (1995). "The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications". Journal of Applied Sciences and Computations. 1: 437–475.
- Artificial neural networks
- Interpolation
- Numerical analysis