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

Comments on recent publications about the use of orthogonal transforms to order and select rules in a fuzzy rule base. The techniques are well-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. They are essentially subset selection methods, and we show that such methods do not produce an "importance ordering", contrary to what has been stated in the literature. The orthogonal least-squares (OLS) method, which evaluates the contribution of the rules to the output, is more attractive for systems modeling. However, it has been shown to sometimes assign high importance to rules that are correlated in the premise. This hampers the generalization capabilities 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. Further, 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 researchers.