The Theory,Arithmetic And Applications Of Two Dimensional Belief Functions

The research of reasoning in artificial intelligence indicates that the human brain is not entirely based on precise reasoning in many times.The real-word information is imperfect.On the one hand,the information is often characterized by uncertainty.This implies that we often impose soft constraints on values of variables of interest.On the other hand,it is worth noting that it is inadequate to only consider uncertainty when processing imperfect information.The other property of information is its partial reliability.Indeed,any estimation of values of interest,be it precise or soft,are subject to the confidence in sources of information.Thus,uncertainty from the one side and partial reliability from the other side are strongly associated to each other.In order to take into account this fact,Zadeh suggested the concept of Z-number as a more adequate formal construct for description of real-world information.In recent years,some researchers in the fuzzy areas have carried out the research of theory and applications of Z-numbers.The mathematical model of Z-numbers be implementing step by step,but it is seldom used in practice due to its high computational complexity.In 2018,the concept of Two Dimensional Belief Function(TDBF)was proposed based on evidence theory as another model similar to but simpler than Z-number to describe real-world information.Compared with classical belief function,TDBF can carry more information,is more flexible.With the proposed of Two Dimensional Belief Function,it is an obvious and urgent problem to establish a perfect theory and application basis based Two Dimensional Belief Function.We mainly focus and address the issues of TDBF as follows,(1)The theoretical system of TDBF.This paper constructed a complete theoretical sys-tem of TDBF,including the definition of TDBF,combination rule of TDBF and the method of converting TDBF to classical belief function.(2)The general framework of computations over TDBF.The computations over TDBF is the basis of dealing with the TDBF-based-information.In this paper,the general framework of computations over TDBF is suggested,including addition,subtrac-tion,multiplication,division,square and square root of TDBFs.It is the synergistic result of belief functions and fuzzy set operations.(3)An approach for determining TDBF based on data.The exiting generation of TDBF is given by experts directly,which is subjective.In this paper,an approach for determining TDBF is presented based on the improved similarity measure of fuzzy numbers.