Radial basis function network: Difference between revisions

Revision as of 22:52, 16 April 2008

Revision as of 10:32, 4 September 2008

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:<math> \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) = \exp \left[ -\beta \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert ^2 \right] = \exp \left[ -\beta \left ( x(t) - c_i \right ) ^2 \right] </math>.

Since the input is a [[Scalar (mathematics)|scalar]] rather than a [[Vector (spatialgeometric)|vector]], the input dimension is one. We choose the number of basis functions as N=5 and the size of the training set to be 100 exemplars generated by the chaotic time series. The weight <math> \beta </math> is taken to be a constant equal to 5. The weights <math> c_i </math> are five exemplars from the time series. The weights <math> a_i </math> are trained with projection operator training:

:<math> a_i (t+1) = a_i(t) + \nu \big [ x(t+1) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \frac {\rho \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} {\sum_{i=1}^N \rho^2 \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} </math>