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|Table of Contents Multivariate Data Modeling Neural Networks Models of Neural Networks RBF Neural Networks RBF Network as Kernel Estimator|
|See also: RBF network|
RBF neural networks belong to the class of kernel estimation methods. These methods use a weighted sum of a finite set of nonlinear functions F(x-ci) to approximate an unknown function f(x). The approximation is constructed from the data samples presented to the network using the following equation:
where h is the number of kernel functions, F() is the kernel function, x is the input vector, c is a vector which represents the center of the kernel function in the n-dimensional space, and wi are the coefficients to adapt the approximating function f(x). If these kernel functions are mapped to a neural-network architecture, a three-layered network can be constructed where each-hidden node is represented by a single kernel function and the coefficients wi represent the weights of the output-layer.
When R equals 0, the kernel function is the classical Gaussian function (see figure below). A large R creates a flat top of the kernel which more and more approaches the form of a cylinder with increasing R.
The output layer of an RBF network combines the kernel function of all
hidden neurons with a linear-weighted sum of these functions. Depending
on various parameters, the response of the network can assume virtually
all thinkable shapes. Several possible response functions obtained from
a network with five hidden neurons by varying the S and the R parameters
are displayed below.
Last Update: 2006-Jšn-17