A Research On Combination Of Belief Functions With Applications In Evidence Theory

In practice, all information, no matter data collected by sensors or knowledge of experts, is uncertain or incomplete to some extent. It is of great significance to research the modelling and reasoning of uncertain information. Evidence theory is a well known uncertainty reasoning theory. It uses belief function assigned on the power set of the discernment frame to model information, which can distinguish equal probability and ignorance, thus it is more flexible and accurate than the traditional probability theory. Besides, its reasoning process is similar to the experts’ mode of thinking. As a conse-quence, evidence theory has become an important information fusion method and has been applied in many areas. Thanks to its capability of modelling uncertain knowledge, it is also combined with other methods in various studies, among which the belief rule based (BRB) reasoning approach is a typical example.In the reasoning process of classical evidence theory, multiple bodies of evidence are aggregated by Dempster’s rule. However, some assumptions, like evidence distinct-ness and evidence reliability, which are not always satisfied in practical problems, are made when implementing this rule. Additionally, a counterintuitive result may be gen-erated when aggregating two bodies of conflicting evidence. To eliminate counterintu-itive results and to meet the practical conditions, this dissertation researches the combi-nation of belief functions, including analysis and combination of conflicting evidence, modelling of evidence importance and reliability and combination of non-distinct bod-ies of evidence. Considering that the training process of BRB model is slow and its rule reduction approaches are complicated, a new BRB structure is put forward.Firstly, typical counterexamples are analyzed and a new discounting approach is put forward along with a new evidence conflict measure. By use of the canonical de-composition, typical counterexamples are analyzed and the common point among them is summarized, which is proved to can be eliminated by evidence discounting. Con-sequently, a fuzzy discounting approach is put forward. Both the conflict between bodies of evidence and the uncertainty of evidence itself are taken into account. Thus it achieves fast belief convergence and is robust to high conflicting evidence. In terms of the evidence conflict and uncertainty, the conflict measure based on singular value and the discriminability measure based on pignistic probability are defined, respectively.Secondly, some properties of the evidential reasoning (ER) rule are researched as well as its flaw, and it is applied to investigate the fault diagnosis problem of the vi-bration system. The relationship between ER rule and evidence discounting approach is explored, and the effect of weights normalization on the combination result is ana-lyzed. Then a modification is made after revealing the defect of the original ER rule. For the fault diagnosis problem of the vibration system, methods for evaluating evi-dence importance and reliability are introduced, respectively. The appropriate feature is selected to generate bodies of evidence, which are then combined by the modified ER rule to obtain the final diagnosis result. The proposed diagnosis method is justified by experiments of multiple types of faults.Thirdly, combination of non-distinct bodies of evidence is researched when the dependent source is known or unknown, respectively. When the dependent source is unknown, the cautious conjunctive rule is extended to the weight cautious conjunctive rule by use of the evidential reasoning rule. Besides, two other combination methods are defined based on the least committed principle and the order of plausibility func-tion and commonality function, respectively. Optimal methods are designed to find a corresponding function which meets it definition. As to the case when the dependent source is given, the commonality function form of Dempster’s rule is used to propose a direct method for aggregating non-distinct bodies of evidence. The dependent source is not necessary to be non-dogmatic.Finally, the BRB method using weighted averaging operator to inference output is researched. Calculation results of some belief rule bases in literature show that be-lief distributions of the activated rules are often in high conflict, which is suitable for weighted averaging operator. Consequently, weighted averaging operator is adopted in the inference process and the corresponding BRB structure is derived. Its universal approximation capability is also proved. To avoid overfirting and to reduce model com-plexity, an effective attribute reduction approach is put forward by use of the special structure of the new belief rule base. At last, an online parameters updating algorithm is derived for time variant systems under the assumption that the outputs obey nor-mal distribution. Simulations are conducted to justify the new BRB structure with the proposed attribute reduction approach and the parameters updating algorithm.