Research On The Properties And Applications Of Finite Precision Entropy

Information theory was a subject developed from communication engineering , which combined with communication technology, probability theory, stochastic processes and mathematical statistics. Dating from 1948, when C.E.Shannon published his famous paper, "A mathematical theory of communication", which made the theoretical foundation of information theory, then communication theory and technology had been under the guidance of it. Classical information theory, which made communication theory be its core, involved much more than the narrow sense of communication project domain. And it entered a burgeoning information scientific domain.Information entropy was one of the most important basic definitions of information theory, which could be divided into two kinds, continuous and discrete information entropies. Discrete information entropy had good properties, for example nonnegative, dimension-free, dimensional consistency principle, etc. But continuous entropy was divergent. After the infinite part was removed, differential entropy was got, which didn't had these good properties. Many scholars had been developing this definition. Finite precision entropy based on a set of real number equivalence classes, which was brought forward by a domestic scholar. This theory gave the hierarchized and discretized expressions of real numbers, and brought forward the definition of sets of real number equivalence classes, and then developed the discrete function system. Afterwards it was applied to the discussion of atomy's material structure. And then, based on the finite precision theory, a complete basis of signal theory and probability theory were brought forward, and the finite precision entropy came forth.So this thesis mainly researched the following aspects. Firstly, some representative improved Shannon entropies were researched on their definitions, properties, the relations between them and Shannon entropy, and their applications. For example, K-L entropy,continuous entropy based on variance,Renyi entropy,Kolmogorov complexity and weighted entropy. But so far, for the engineering applications, the best and the most sententious one still was Shannon entropy. Secondly, the physical meaning of probability relation and non-probability relation between the different values of a continuous random variable was analyzed, and properties of the finite precision entropy was also analyzed, for example nonnegative, dimension-free, dimensional consistency principle, etc. And it had all of the fine properties, an entropy should have. Thirdly, finite precision entropy was applied to mathematical model of the measuring information system, and the essence of information transmission was revealed. Although the mutual information had the same expression with the traditional one, the essence was totally different. Finally, the relations and differences between finite precision entropy and uncertainty theories were discussed, for instance DST,fuzzy measure,possibility theory. The differences and relations between probability relations and conflicts, non-probability relations and non-specificity , were illustrated by examples.In conclusion , finite precision entropy was a reasonable development of Shannon information entropy in mathematical sense, as it hurdled many disadvantages of differential entropy, while maintained the fine properties of discrete entropy. And it was applied to the measuring information system to illustrate the essence of information transmission. With comparing it to the uncertainty theory, the finite precision entropy had many excellent properties. This thesis still was in the initial stage of finite precision entropy. New directions in finite precision entropy were presented in the end of the thesis.