IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 51, NO. 2, APRIL 2002 217 SVD-Based Complexity Reduction of Rule-Bases With Nonlinear Antecedent Fuzzy Sets Orsolya Takács and Annamária R. Várkonyi-Kóczy, Senior Member, IEEE Abstract—With the help of the singular value decomposition (SVD) based complexity reduction method, not only can the redun- dancy of fuzzy rule-bases be eliminated, but further reduction can also be made, considering the allowable error. Namely, in the case of higher allowable error, the result may be a less complex fuzzy inference system, with a smaller rule-base. This property of the SVD-based reduction method makes possible the usage of fuzzy systems, even in cases when the available time and resources are limited. The original SVD-based reduction method was proposed for rule-bases with linear antecedent fuzzy sets. This limitation re- mained valid in the later extensions, as well. The purpose of this paper is to give a formal mathematical proof for the original for- mulas with nonlinear antecedent fuzzy sets and thus to end this limitation. Index Terms—Anytime systems, complexity reduction, fuzzy rule bases with nonlinear antecedent fuzzy sets, nonexact com- plexity reduction, singular value decomposition. I. INTRODUCTION T HE USE OF fuzzy inference systems in system modeling and control has several advantages. First, for the construc- tion of the inference system, both higher-level, symbolic knowl- edge and samples can be used as an information source. Namely, a priori information originated from experts can be used to form the basic outlines of the system, which usually means the construction of the rule base and the determination of the approximate fuzzy sets. Furthermore, for further improvement of the model—which usually means the adjusting of the fuzzy sets—even learning methods can be used. Second, fuzzy systems and fuzzy rules are close to human thinking, which means that a human operator usually finds it easier to survey a fuzzy system than other—classical or neural—systems. In contrast with other systems, in fuzzy systems, the effects of parameter changes are usually easier to estimate, and the cause of a difference between the outputs of the model and the system can be more easily determined. On the other hand, to achieve the required accuracy, nu- merous antecedent (input) membership functions must be defined, which results in a high number of rules and high complexity. There is no universal method for the determination of the necessary number of antecedent fuzzy sets, and, in practice, it is usually overestimated, which results in huge and redundant rule-bases. Manuscript received December 19, 2001; revised January 10, 2002. This work was supported by the Hungarian Fund for Scientific Research OTKA T 035190. O. Takács and A. R. Várkonyi-Kóczy are with the Department of Measure- ment and Information Systems, Budapest University of Technology and Eco- nomics, Budapest, Hungary (e-mail: takacs@mit.bme.hu; koczy@mit.bme.hu). Publisher Item Identifier S 0018-9456(02)02918-2. Since, today, there are more and more applications, where the computations must be carried out within a limited time, the complexity reduction of fuzzy inference systems is an important question. The complexity reduction based on singular value decom- position (SVD) was first proposed by Yam [1] as a method for creating fuzzy inference systems for the approximation of functions, based on grid-point sampling. Later, the method was proposed as a method for the rule-base reduction for certain types of fuzzy inference systems. The SVD-based complexity reduction method was extended to PSGS [2] (product-sum-gravity with Singleton consequences), PSGN [3] (product-sum-gravity with non-Singleton consequences), and Takagi-Sugeno [4] fuzzy systems, and recently even to generalized neural networks [5], [6]. The SVD-based reduction offers a way for the automatic elimination of the redundancy of the rule-base, so it can be used for exact complexity-reduction. Although the complexity-re- duction obtained in this way is not always enough, usually further, nonexact reduction is needed, as well. In the case of nonexact complexity-reduction, the error-bound (accuracy) of the reduced system must be known for the qualification of the results. This further, nonexact complexity reduction usually does not result in significant degradation of the quality of the results. Generally, the original fuzzy/neural model is not perfectly exact, and this further inaccuracy can be tolerated. On the other hand, as it will be shown, the amount of the error depends on the singular values of the matrix, describing the system. If there is a large difference between the singular values, then a good complexity-reduction can be made with a low error, but if the singular values are nearly equal, then the nonexact complexity-reduction cannot be carried out efficiently in this way. The SVD-based reduction makes possible this further, nonexact reduction, considering the allowable error. Namely, in the case of higher allowable error, the result will be a less complex fuzzy inference system, with a smaller rule-base. The complexity and inaccuracy of the resulting rule-bases can always be easily estimated. This property of SVD-based reduction makes possible the use of fuzzy systems or neural networks in anytime systems with a modular architecture [7]. The first SVD-based complexity reduction was proposed for PSGS fuzzy systems with linear (triangular) antecedent fuzzy sets, and the proof of the error-bound of the nonexact reduc- tion was only given for this case, as well. All of the later exten- sions were based on this first proof, which limits the use of the nonexact SVD-based reduction. 0018-9456/02$17.00 © 2002 IEEE