Equivalence of fuzzy systems and radial basis function networks

 Jang[2] showed that under certain mild restrictions a fuzzy inference is equivalent to an RBFN. The following must be fulfilled for the equivalence to be valid:

• The number of RBF units is equal to the number of fuzzy IF-THEN-rules.
• The output of each fuzzy rule is a constant (the fuzzy system is a zero-order Sugeno fuzzy system).
• The MFs within each rule are chosen as Gaussian functions with the same variance.
• The T-norm operator used to compute the activation of each rule is multiplication.
• Both the RBFN and the fuzzy inference system under consideration use the same method to derive their overall output, i.e. weighted average (with normalization) or weighted sum (without normalization).

The generalization from strictly radial basis functions to ones with a diagonal covariance matrix with possibly different elements in the diagonal is straightforward. In this case the squared Euclidean distance $\Vert\mbox{\bf x}-\mbox{\bf c}\Vert^2$ used for computing the Gaussian activations must be replaced by the Mahalanobis distance $(\mbox{\bf x}-\mbox{\bf c})^T C^{-1}(\mbox{\bf x}-\mbox{\bf c})$using the inverse of the covariance matrix C of the respective Gaussian unit.

Keeping the equivalence between Sugeno fuzzy systems and RBFNs in mind we now discuss a standard RBFN training method. Thereafter, we describe an incremental RBFN which, according to Jang's results, is also an incremental neuro-fuzzy system.