Incremental Neuro-fuzzy Systems

The poor scaling behavior of grid-partitioning fuzzy systems in case of increasing data dimensionality suggests using fuzzy systems with a scatter-partition of the input space. Jang has shown that zero-order Sugeno fuzzy systems are equivalent to radial basis function networks (RBFNs). Methods for nding scatter partitions for RBFNs are available, and it is possible to use them for creating scatter-partitioning fuzzy systems. A fundamental problem, however, is the structure identiication problem, i.e., the determination of the number of fuzzy rules and their positions in the input space. The supervised growing neural gas method uses classiication or regression error to guide insertions of new RBF units. This leads to a more eeective positioning of RBF units (fuzzy rule IF-parts, resp.) than achievable with the commonly used unsupervised clustering methods. Example simulations of the new approach are shown demonstrating superior behavior compared with grid-partitioning fuzzy systems and the standard RBF approach of Moody and Darken. 1. GRID-PARTITIONING FUZZY SYSTEMS In this introductory section we remind of some fundamental properties of fuzzy systems. Moreover, we describe the standard approach of partitioning the input space of a fuzzy system using a rectangular grid. Fuzzy systems can model continuous input/output relationships. The purpose can be, for example, function approximation (regression), non-linear control, or pattern classiication. In the latter case the output of a fuzzy system can sometimes be interpreted as posterior class probabilities which can be used to achieve a classiication by means of the standard Bayes rule. A basic component of a fuzzy system is a fuzzy rule. Sometimes these rules are expressed using linguistic labels such as the rule IF (pressure is low) AND (temperature is medium) THEN (valve opening is 0.7): (1) Fuzzy membership functions (MFs) associate linguistic labels (e.g. low) with a particular area of one of the input or output variables (e.g. \pressure"). In the example shown above the THEN-part of the rule does not consist of a membership variable but of the \crisp" value 0.7. Fuzzy systems consisting of this kind of rules are called zero-order Sugeno fuzzy systems. In an n-th order Sugeno fuzzy system the THEN-part of each rule consists of a polynomial of degree n in the input variables. In this article we will concentrate on zero-order Sugeno fuzzy systems since they have the interesting property of being equivalent to radial basis function networks (RBFNs) as is discussed in section 4. Diierent shapes of the MFs …