Online neuro-fuzzy CANFIS hidden-node teaching

On-line first-order backpropagation (BP) has been widely employed for optimizing a multi-layer neural network and a fuzzy neural network. When BP is applied to a TSK fuzzy system, the interpretability of fuzzy rules may be lost. We describe a simple and effective remedy for the loss by casting the posed optimization problem into a general Bolza-type optimal-control mold so as to include stage costs on top of the terminal cost; this is what we called the (on-line) hidden-node teaching. Our on-line learning scheme turns out to be useful not only in optimizing a large model, a so-called CANFIS neuro-fuzzy modular network, but also in enhancing its generalization capacity. The fundamental concept is that each (local) expert network is supervised individually by hidden-node teaching for minimizing a stage cost while all local-expert modules are encouraged to cooperate in reducing the terminal cost simultaneously. Furthermore, we show how to construct a CANFIS modular network in order to alleviate the so-called curse of dimensionality that frequently hampers the design of fuzzy systems. In simulation, our concepts have been demonstrated in the letter recognition benchmark problem as well as in small regression and XOR-classiflcation tasks.