基于等价关系的信息熵及概率分配函数

The Information Entropy and Probability Assignment Based on Equivalent Relations

Pawlak在1982年提出的粗糙(Rough)集是基于等价关系的理论,粗糙集的发展推动了人们对等价关系的研究。等价关系上的信息熵具有最为简单、规范的性质。本文研究基于等价关系上的信息熵及概率分配函数,讨论基于等价关系上的信息熵的基本性质,为等价关系的信息熵的各种应用提供理论基础,比如等价关系的信息熵在信息系统的约简方面可能发挥重要作用。文章主要从两方面进行论证:①等价关系的粗细对信息熵的影响,这点通过8个命题来说明;②等价关系与证据理论之间的联系。证据理论主要是通过概率分配函数、信任函数及似然函数来表述,从某种意义上说粗集理论继承和发展了证据理论。另外,本文的讨论均在有限论域U={u1,u2,…,uU}上进行,用具体的例子来说明抽象的数学命题,使之更容易理解。

The rough set theory, proposed by Pawlak in 1982, is based on equivalence relation. Recently, the equivalence relation becomes more significant because of rough sets. Information entropy of an equivalence relation has many regular and interesting properties. In this paper we study the basic properties of information entropy and probability of assignment based on equivalent relation. This study gives a theoretical basis for the applications of information entropy based on equivalent relation. For example, on a reduction in information systems, the information entropy based on equivalent relation has an important effect. This paper demonstrates two main lemmas. Firstly, we focus on the influence of the inclusion of equivalent relation on the information entropy, which is explained through 8 propositions. Secondly, we also investigate the relation between the Shafer＇s evidence theory and equivalent relation. The evidence theory is mainly stated through the probability distribution function, and the function of trust and likelihood. In a sense, the rough set theory is inherited and has developed the evidence theory . Moreover, the discussion in this article is limited in the domain U = { u1, u2,..., u｜v｜ } , by using specific examples to illustrate abstract mathematical proposition, in order to make it more easier to understand.