Neural Networks-Radial Basis Functions

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•Using nonlinear functions, we can convert a nonlinearly separable problem into a linearly separable one.

•From a function approximation perspective, this is equivalent to implementing a complex function (corresponding to the nonlinearly separable decision boundary) using simple functions (corresponding to the linearly separable decision boundary)


•Implementing this procedure using a network architecture, yields the RBF networks, if the nonlinear mapping functions are radial basis functions.


•Radial Basis Functions:
–Radial: Symmetric around its center
–Basis Functions: A set of functions whose linear combination can generate an arbitrary function in a given function space.

Principle of Operation



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•What do these parameters represent?
–Physical meanings:

• phi: The radial basis function for the hidden layer. This is a simple nonlinear mapping function (typically Gaussian) that transforms the d- dimensional input patterns to a (typically higher) H-dimensional space. The complex decision boundary will be constructed from linear combinations (weighted sums) of these simple building blocks.

•uji: The weights joining the first to hidden layer. These weights constitute the center points of the radial basis functions.

•sigma: The spread constant(s). These values determine the spread (extend) of each radial basis function.

•Wjk: The weights joining hidden and output layers. These are the weights which are used in obtaining the linear combination of the radial basis functions. They determine the relative amplitudes of the RBFs when they are combined to form the complex function.