不确定性系统的统一性  

The United Theory of Uncertainty Systems

作　　者：李洪兴[1] 

机构地区：[1]北京师范大学数学科学学院,北京100875

出　　处：《工程数学学报》2007年第1期1-21,共21页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(60474023);教育部博士点基金(20020027013);973国家重大基础研究发展规划基金(2002CB312200)

摘　　要：本文综述不确定性系统的统一性。首先揭示Fuzzy系统的概率论意义,指出Fuzzy系统中常用的清晰化方法,即重心法是合理的且在平均平方意义下是最优的方法。基于不同的Fuzzy蕴涵算子,给出几种典型的概率分布,如Zadeh分布、Mamdani分布、Lukasiewicz分布等,它们充当Fuzzy系统的“系统内核”作用。此外,根据Fuzzy系统概率分布的一些性质,论证了由Zadeh提出的构造Fuzzy系统的CRI算法是基本合理的且有效的。此外还刻画了均匀概率分布在Fuzzy系统中的特殊作用。随后揭示了随机系统的Fuzzy推理意义。首先,相对于不确定性系统,给出了随机系统的定义,它视为对一个不确定系统的逼近。然后指出,对于任意给定的一个随机系统,总能将它转化为一组Fuzzy推理规则,由此可构造一个Fuzzy系统,并且证明了这样构造的Fuzzy系统能逼近给定的随机系统到指定的精度。还讨论了Fuzzy系统与随机系统转换中的还原性。最后概述了不确定性系统的统一性。A kind of united theory of uncertainty systems is introduced. Firstly, the probability significance of fuzzy systems is revealed. It is pointed out that the COG method, a defuzzification technique used commonly in fuzzy systems, is reasonable and is the optimal method in the sense of mean square. Based on different fuzzy implication operators, several typical probability distributions such as Zadeh distribution, Mamdani distribution, Lukasiewicz distribution, etc. are given. Those distributions act as ＂inner kernels＂ of uncertainty systems. Furthermore, by some properties of probability distributions of fuzzy systems, it is also demonstrated that CRI method, proposed by Zadeh, for constructing fuzzy systems is reasonable basically and effective. Besides, the special action of uniform probability distributions in fuzzy systems is characterized. Secondly, the fuzzy reasoning significance of stochastic systems is also revealed. What is a stochastic system is defined with respect to uncertainty systems, which is regarded as an approximation to an uncertainty system. Then it is pointed out that a arbitrarily given stochastic system Can be transformed into a group of fuzzy inference rules, and a fuzzy system is made by means of the group of fuzzy inference rules and the fuzzy system can approximate the given stochastic system for reaching arbitrarily given precision. Thereafter the reducibility in transformation between fuzzy systems and stochastic systems is discussed. At last, the unification of uncertainty systems is summarized.

关 键 词：不确定性系统 随机系统 Fuzzy系统 条件数学期望 Zadeh分布 Mamdani分布 Lukasie-wicz分布 还原性 不确定性系统的统一性 

分 类 号：O159[理学—数学]