Introduction to Radial Basis Functions

The hidden units of a radial basis function network are not the same as used for a multilayer perception, and the weights between input and hidden layer have different meanings. Transfer functions typically used include the Gaussian function, spline functions and various quadratic functions they all are smooth functions, which taper off as distance from a center point increases. In two dimensions, the Gaussian is the well-known bell-shaped curve in three dimensions it forms a hill. 

For radial basis function networks, each hidden unit represents the center of a cluster in the data space. Input to a hidden unit in a radial basis function is not the weighted sum of its inputs but a distance measure a measure of how far the input vector is from the center of the basis function for that hidden unit. Various distance measures are used, but perhaps the most common is the well-known Euclidean distance measure. 

If x and p. are vectors, the Euclidean distance between them is given by 

Where D. is the Euclidean distance between an input vector and the location vector for 

Radial basis function networks with more than one input unit have more parameters for each hidden node e.g.,. if there are two input units, then the basis function for each hidden unit j needs two location parameters, pij and p2j, for the center, and, optionally, two parameters, Oij and a2j, for variability. The dimension of the centers for each of the hidden units matches the dimension of the input vector. 

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