Erratum to “Automatic tuning of complex fuzzy systems with Xfuzzy”: [Fuzzy Sets and System 158 (2007) 2026–2038

Abstract

This work was supported in part by the Spanish CICYT Projects DPI2005-02293 and TEC2005-04359, and by the Projects TIC2006-635 and TEP2006-375 from the Andalusian Regional Government. Furthermore, due to an unfortunate error in the final stages of the publication process three paragraphs (the first of Section 2, the first of Section 2.1 and the last of Section 4) were reproduced entirely in italic font. The correct presentation should, respectively, have been as follows. Xfsl is the tool of Xfuzzy 3 that allows the user to apply supervised learning algorithms to complex fuzzy systems. A formal specification language, named XFL3, has been developed to support this complexity [13]. An important characteristic of XFL3 is that the functional and logical structures of a fuzzy system are described independently. Xfsl admits many of the supervised learning algorithms reported in the literature. Regarding gradient-descent algorithms, it admits Steepest-Descent, Backpropagation, Backpropagation with Momentum, Adaptive Learning Rate, Adaptive Step Size,Manhattan,QuickProp andRProp. Since the convergence speed ofBackPropagation is slow, several modifications were proposed such as using a different learning rate for each parameter, adapting the control variables of the algorithm heuristically [8], or taking into account the gradient value of two successive iterations [6,19]. Xfsl performs different methods to compute or estimate the gradient as well as to maintain the constraints of the parameters of the system. In addition to the supervised algorithms, the simplification processes offered by xfsl are very helpful for optimizing the structure of the fuzzy system under design since they can reduce the number of rules and membership functions. Moreover, the capability of xfsl to tune hierarchical fuzzy systems makes it also possible to simplify the description of a system because complex behaviors can be usually generated by composing simple rule bases. Besides, xfsl can adjust any system described by the language XFL3, which allows for simple knowledge base specifications thanks to the use of expressive rules. In particular, xfsl can adjust systems that employ linguistic hedges, which is very interesting for maintaining the linguistic meaning of a given rule base or for extracting linguistic knowledge from a set of numerical data.