Influences Of Uncertainties Of Training Pattern Pairs On Fuzzy Systems

The uncertainty of the world is one of the important factors, which leads to its complexity. Constructing a fuzzy system, the training patterns are necessary to describe the basic performances of the system. Generally, there are some small variances (perturbations) between collected patterns and real objective patterns, which result in adverse influence for the performance of the system in many aspects. Therefore we evaluate the influence for kinds of fuzzy neural networks and typical fuzzy reasonings. The work in the paper is useful for analysis of fuzzy systems, the choice of learning algorithms and fuzzy implication operators, and the guidance to training pattern pair acquisition for fuzzy systems.The problems have been researched as follow:(1) Two kinds of morphological associative memory networks were analyzed by many scholars both have very good fault-tolerant abilities of erosive/dilative noise. But in the paper, our investigation reveals that two kinds of networks have different robustness to perturbation of training patterns. The robustness is good to the one kind of fuzzy morphological associative memory networks, while this property of the other one is relatively bad.(2) A learning algorithm is proposed for a class of fuzzy morphological bi-directional associative memories (FMBAMs). It is proved theoretically that, for any given set of pattern pairs, if there exist pairs of connection weight matrices which make the set to become a set of the equilibrium states of FMBAMs, the presented learning algorithm can give the maximum of all such pairs of weight matrices. And the learning algorithm can ensure that, the FMBAMs with this maximal pair of connection weight matrices can be convergent to an equilibrium state in one iterative process for any input. Any equilibrium state of FMBAM is Lyapunov stable. The robustness of FMBAM is good, when training pattern pairs have perturbations.(3) Based on the fuzzy composition of Max operation and any triangular norm T, a type of fuzzy bi-directional auto-associative memory (Max-T FBAM) is proposed. By means of concomitant implication operator of a triangular norm, a general learning algorithm is proposed for a class of such Max-T FBAMs. It is proved theoretically that, the learning algorithm can ensure that the Max-T FBAMs have maximal pair of connection weight matrices, which can be convergent to an equilibrium state in one iterative process for any input, also have reliable store capabilities. When triangular norms satisfy Lipschitz condition, Max-T FBAMs will have good robustness of small perturbations of training pattern pairs by the learning algorithm.(4) In this paper, an efficient learning algorithm is proposed for a class of fuzzy Hopfield networks (Max-T FHNNs) based on T-norms. For any given set of patterns, the learning algorithm can find the maximum of all connection weight matrices that can make the set become a set of the equilibrium points of the Max-T FHNN when T is a left-continuous T-norm. This maximal matrix is idempotent matrix in sense of Max-T composition, with which the Max-T FHNN can be convergent to a stable state in one iterative process for any input vector. It is proved theoretically that arbitrary set of patterns can become a set of the equilibrium points of every Max-T FHNN, if only the T is left-continuous T-norm. Max-TL FHNN has universally good robustness to perturbation of training patterns.(5) For the first time, we research carefully on a kind of inference algorithm, which is reverse triple I method of fuzzy reasoning proposed lately, and hold the continuity properties. Moreover, investigate also how the approximation errors are propagated by the inference algorithm.Therefore, a fuzzy inference algorithm is viewed as a mapping from one fuzzy set to the other fuzzy set; Hamming distance formula is used as computing distance between two fuzzy sets. We prove that the algorithm holds continuity properties in the case of fuzzy modus ponens and fuzzy modus tollens. We also point out that reverse triple I method of fuzzy reasoning make approximation errors magnified; the biggest enlargement range is 2. Multiple fuzzy reasonings and multi-dimensional fuzzy reasonings hold the continuity properties and approximation properties.We prove that kinds of fuzzy reasonings hold continuity properties in the case of fuzzy modus ponens and fuzzy modus tollens. We point out algorithms hold approximation hold approximation property only if the algorithms hold consistency property; the classes of fuzzy reasonings do not make approximation errors magnified when they hold approximation property.(6) When constructing the neural networks, training pattern pairs obtained by a certain acquisition way always have perturbations. In the paper, a new measure method of perturbations of fuzzy sets is first proposed to measure the perturbations. Constructing Max-T FAM where T is a triangular norm, Max-T FAM can be actually a mapping from a vector space to another vector space, so the storage ability and pairs of connection weight matrices and robustness of small perturbations of training pattern pairs of the Max-T FAM, where T is a triangular norm, is analyzed with the aid of analyses of its value domain. Based on the new measure method, two kinds of morphological associative memory networks, one is good to the perturbation of training pattern pairs, the other is relatively bad.