13.3.1 The Basis Function in an RBF Network

The Gaussian function is the most commonly used basis function in RBF networks. There is, however, no a priori reason for this choice, and you may use the Neuron option in the initialization to define a different basis function.

Recall that a general RBF network with nb basis functions is described by the following equation.

In addition to this expression, you also may have a parallel linear part as described in Section 2.5.2, Radial Basis Function Networks. The default value of G(x) is Exp so that the Gaussian function is obtained.

Load the Neural Networks package.

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Look at the default basis function.

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Another possible choice for the basis function is one that decays linearly with the distance from the basis center. Such a choice can be obtained with the help of the SaturatedLinear function. Notice that G is a function of the distance squared in Eq. (13.4). To make the basis function linear with respect to the distance, you must compensate for this square by introducing a square root. This is done in the following example.

Plot a linear saturated basis function.

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Since a layer of linear parameters is included in the network, in , it does not matter that the basis function is inverted.

Now use this basis function in a small example.

Generate data and plot the function.

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Initialize an RBF network with the proposed basis function.

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Notice that a negative sign was used inside the square root to compensate for the negative sign in the function defined by Eq. (13.4).

Train the RBF network using the data.

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Plot the result.

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You can vary the number of basis functions and repeat the example. You can also try different types of basis functions.