radial basis function について

is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible.
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 originally surfaced, in work by David Broomhead and David Lowe in 1988,〔(Radial Basis Function networks )〕 which stemmed from Michael J. D. Powell's seminal research from 1977.〔: "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."〕
RBFs are also used as a kernel in support vector classification.
== Types ==
Commonly used types of radial basis functions include (writing

r = \|\mathbf - \mathbf_i\|\;

):
* Gaussian:
The first term, that is used for normalisation of the Gaussian, is missing, because in our sum every Gaussian has a weight, so the normalisation is not necessary.
::

\phi(r) = e^\,

* Multiquadric:
::

\phi(r) = \sqrt

* Inverse quadratic:
::

\phi(r) = \frac

* Inverse multiquadric:
::

\phi(r) = \frac{\sqrt{1 + (\varepsilon r)^2}}

* Polyharmonic spline:
::

\phi(r) = r^k,\; k=1,3,5,\dots

\phi(r) = r^k \ln(r),\; k=2,4,6,\dots

* Thin plate spline (a special polyharmonic spline):
::

\phi(r) = r^2 \ln(r)\;

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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