Rule base reduction: Some comments on the use of orthogonal transforms

This paper comments on recent publications about the use of orthogonal tran sforms to order and select rules in a fuzzy rule base. The techniques are w ell known from linear algebra, and we comment on their usefulness in fuzzy modeling. The application of rank-revealing methods based on singular value decomposition (SVD) to rule reduction gives rather conservative results. T hey are essentially subset selection methods, and we show that such methods do not produce an "importance ordering" contrary to what has been stated i n literature. The orthogonal least-squares (OLS) method, which evaluates th e contribution of the rules to the output, is more attractive for systems m odeling. However, it has been shown to sometimes assign high importance to rules that are correlated in the premise. This hampers the generalization c apabilities of the resulting model. We discuss the performance of rank-revealing reduction methods and advocate the use of a less complex method based on the pivoted QR decomposition. Fu rther, we show how detection of redundant rules can be introduced in OLS by a simple extension of the algorithm. The methods are applied to a problem known from the literature and compared to results reported by other researc hers.