The structure and realization of a polygonal fuzzy neural network
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ORIGINAL ARTICLE
The structure and realization of a polygonal fuzzy neural network Xiaoping Li1 • Dan Li2
Received: 6 April 2015 / Accepted: 15 June 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract As a generalization of ladder fuzzy numbers, the polygonal fuzzy numbers can be approximately represented by a group of ordered real numbers, it not only keeps the closeness of arithmetic operations, and also inherits some excellent properties of ladder fuzzy numbers. To modify defects of the definition of the original polygonal fuzzy numbers, the method of an equidistant subdivision is firstly introduced, and the specific algorithm of general fuzzy numbers into the polygonal fuzzy numbers by two examples is explained in detail. Secondly, using the extension operations of polygonal fuzzy numbers, a polygonal fuzzy neural network is constructed by mathematical methods, and the coefficients transform formulas of the new network are obtained. In addition, it is also proved that the new network still possesses approximation with respect to a continuous polygonal fuzzy valued function. Finally, the approximation algorithm and realization process of the polygonal fuzzy neural networks to a twopolygonal fuzzy valued function are given by means of an example. Keywords Polygonal fuzzy numbers Polygonal fuzzy valued functions Activation functions Polygonal fuzzy neural networks Universal approximation
This work has been supported by National Natural Science Foundation China (Grant No. 61374009). & Xiaoping Li [email protected] 1
School of Management, Tianjin Normal University, Tianjin 300387, China
2
Foundation Department, Dalian University of Finance and Economics, Dalian 116622, China
1 Introduction As a particular type of pure fuzzy system, fuzzy neural networks can effectively handle natural language messages. In the real world, there are more data messages of digital type than language messages. Thus, we may obtain data messages with corresponding input–output relationship of a fuzzy system by measurement date and transmission. In recent years, a number of system theories established in the light of fuzzy numbers have been successfully applied to the fields of fuzzy control, fuzzy reliability, fuzzy system analysis and fuzzy neural networks etc. Unfortunately, the operations of fuzzy arithmetic are extremely complex, even then the operations for the simple trigonometric fuzzy numbers and ladder fuzzy numbers are very difficult. The reason is that the four arithmetic operations in Zadeh’s extension principles do not satisfy closeness. Thus, how to approximately finish nonlinear operations in view of general fuzzy numbers is an important question for research or discussion. In 1994, Buckley [1] conjectured that a regular fuzzy neural network is an universal approximator of a continuously-increasing fuzzy function class. Later, from the point of view of system approximations and learning algorithms, this class of networks was thoroughly and systematically studied by many scholars both domestically and interna
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