Radial basis function network: Difference between revisions

Revision as of 15:46, 30 March 2009

Revision as of 15:50, 30 March 2009

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===Interpolation===

RBF networks can be used to interpolate a function <math>y: \mathbb{R}^n \to \mathbb{R}</math> when the values of that function are known on finite number of points: <math>y(\mathbf x_i) = b_i, i=1, \ldots, N</math>. Taking the known points <math>\mathbf x_i</math> to be the centers of the radial basis functions and evaluating the values of the basis functions at the same points <math>g_{ij} = \rho(|| \mathbf x_j - \mathbf x_i ||)</math> the weights can be solved from the equation

:<math>\left[ \begin{matrix}

g_{11} & g_{12} & \cdots & g_{1N} \\

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\end{matrix} \right]</math>

It can be shown that the interpolation matrix in the above equation is non-singular, if the points ''<math>\mathbf x_i''</math> are distinct, and thus the weights ''<math>w''</math> can be solved by simple linear algebra:

:<math>\mathbf{w} = \mathbf{G}^{-1} \mathbf{b}</math>