【作者】 高欣；

【导师】 刘宝碇；

【作者基本信息】 清华大学 ， 数学， 2009， 博士

【摘要】 随机性是现实世界中最基本的客观不确定性,而概率论是用来处理随机现象的数学工具.随着人们对不确定现象了解的日益深入,一些学者开始挑战概率测度的可列可加性,先后提出了容度、模糊测度、可能性测度等完全非可加测度.完全非可加测度在否定可加性的同时,也否定了自对偶性,从而违背了数学科学中最基本的法则矛盾律和排中律.为了解决这一问题,Liu提出一类具有规范性、单调性、自对偶性、可列次可加性的部分可加的不确定测度,并发展出了一套与概率论平行的不确定理论.本文首先证明了不确定测度连续性的判定条件,随后研究了不确定变量在测度连续情况下的性质,包括分布函数的连续性、关键值不等式以及期望值的单调收敛定理、Fatou引理、有界收敛定理等.其次,本文提出了不确定环境下的极大熵原理,并证明了一系列矩约束情况下不确定变量的极大熵定理.另外,为了度量一个不确定变量相对于另一个不确定变量的偏离程度,我们定义了不确定变量的互熵,并且把互熵的定义扩展到不确定向量上.进一步地,定义了不确定变量的广义互熵和广义熵,并分别研究了熵与互熵、以及广义熵与广义互熵之间的关系.在文章的最后,以条件不确定测度和辨识函数为基础,本文还建立了各种基于刘氏推理规则的不确定推理模型.本文的创新点主要有:1.研究了连续不确定测度及其数学性质;2.研究了不确定变量的极大熵原理,得到矩约束情况下的极大熵定理;3.研究了多个前件、多条准则的不确定推理规则,建立了相应的不确定推理模型.更多还原

【Abstract】 Randomness is a fundamental type of uncertainty present in the real world, andprobability theory is a branch of mathematics for studying random phenomena. Asan improvement of our understanding about uncertain phenomena, researchers havechallenged the“countable additivity”condition, for example, in capacity theory, fuzzymeasure theory, possibility theory, and so on. However, since this class of non-additivemeasures does not consider the self-duality property, all of those measures are incon-sistent with the law of contradiction and law of excluded middle, which are basic rulesin mathematics. In order to solve this important problem, Liu (2007) founded an un-certainty theory that is a branch of mathematics based on normality, monotonicity,self-duality, countable subadditivity, and product measure axioms.This dissertation first proposes the judgement conditions of continuous uncertainmeasures, then studies some important concepts about uncertain variables when the un-certain measure is continuous; more specifically, uncertainty distribution, critical value,expected value are studied, among others. Furthermore, this dissertation discusses themaximum entropy principle for uncertain variables and proves some maximum entropytheorems with moment constraints. Also defined and studied are cross-entropy of un-certain variables, generalized cross-entropy and generalized entropy. Finally, based ontools such as conditional uncertainty and identification functions, some expressions ofLiu’s inference rule for uncertain systems are derived.In conclusion, this dissertation contributes to the research area of uncertainty the-ory in the following aspects: 1. continuous uncertain measure is proposed and somebasic properties for uncertain variables in uncertainty space with continuous uncertainmeasures are studied; 2. maximum entropy principle for uncertain variables is investi-gated; 3. Liu’s inference rule with multiple antecedents and with multiple if-then rulesis considered.更多还原