Fuzzy basis functions for triangle-shaped membership functions: Universal approximation - MISO case
چکیده
In this paper, the universal approximation property of one of the most frequently used type of fuzzy systems is proved. The type of fuzzy system in the present work employs triangle-shaped fuzzy membership functions (TSMF) for its input variables. The proof of the universal approximation property does not use the Stone-Weierstrass theorem because the TSMFs are not closed for the product; instead, it is based on the " Density lemma for linear subspaces of (the space of all bounded continuous real-valued functions on )". This lemma is a powerful tool to prove the universal approximation property for other classes of fuzzy systems. IIntroduction functions (TSMF) in fuzzy systems, because they are normal The approximation problem of multi-input-single-output are easier to generate and probably they are the most frequently (MISO) fuzzy systems is discussed in this work. A multi-output used functions in applications. The proof of the universal system can always be separated into several systems of a single approximation property based on the Stone-Weierstrass output. From a mathematical point of view, fuzzy inference theorem, limits the type of membership functions to be used systems are functions which map inputs to outputs and can be merely to Gaussian function forms, because the latter ones are represented by a linear combination of fuzzy basis functions closed for the product. (FBFs), [6]. The function approximation problem using fuzzy The present paper analyzes the MISO case and considers systems has recently been addressed in [1], [5], [6], [7], [8] and the fuzzy systems class with singleton fuzzifier, product [9], and was motivated by publications on the approximation inference, centroid defuzzifier, and triangle-shaped capacity of neural networks and by the fact that problems membership functions. The proof of universal approximation found in designing fuzzy systems can be regarded as property for the fuzzy systems class considered here, does not approximation problems as well. The authors considered the use the Stone-Weierstrass theorem because the TSMFs are not universal approximation property for some types of fuzzy closed for the product. The proof is based on the " Density systems, that is, any continuous function on a compact set can lemma for linear subspaces of ", [3], where is be uniformly approximated by fuzzy systems with an arbitrary the space of all bounded continuous real-valued functions on degree of accuracy. . This lemma is a powerful tool to prove the universal Based on the Stone-Weierstrass theorem, Wang and approximation property for other classes of fuzzy systems. Mendel [7], proved the approximation property by considering the fuzzy systems class with singleton fuzzifier, product IIFuzzy system specifications inference, centroid defuzzifier, and scaled Gaussian membership function. They have defined FBFs and expressed In this paper a multi-input, single-output (MISO) fuzzy the above class fuzzy systems by means of a linear inference system (FIS) is considered. Multi-output systems combination of FBFs. It is clear that the approximation (MIMO) can always be separated into single-output systems. property of fuzzy systems is closely related to FBFs properties. A MISO fuzzy system performs a static mapping Nguyen and Kreinovich [6], proved via Stone-Weierstrass, , where is a compact set. The main theorem the universal approximation property of the linear elements of fuzzy systems in this work are: singleton fuzzifier, combination of FBFs corresponding to the fuzzy systems class, Mamdani's fuzzy rule base, fuzzy product inference engine and with singleton fuzzifier, min-inference, centroid defuzzifier, and centroid defuzzifier. The former performs a mapping from the Gaussian membership function. Zeng and Singh, [9], crisp input space to the fuzzy sets defined in the considered the SISO case for a broad class of membership compact set usually employed in the functions (pseudo trapezoid-shaped (PTS) membership approximation theory ( the spaces and are function). In their paper, they have carried out a detailed homeomorphic, where is a dense set ). The fuzzy set analysis of the FBFs properties, and they also have proven A defined in is characterized by a membership function important approximation properties of fuzzy systems. , with , and it is labelled by a linguistic Scaled Gaussian membership functions are not normal, term defined in the term set of the considered linguistic hence they do not have the consistency property, [9]. On the variable. Next, a summary of the characteristics of the fuzzy other hand, it is desirable to use triangle-shaped membership inference systems used in this paper is presented. and they also have the consistency property. Besides, TSMF n xi x ' ( x1 x2 ... xn ) T 0 [ 0 , 1 ]n y ' F( x ) 0 U p n m ' ' p n y aj 0 U