key: cord-0227789-uviucg92 authors: Deng, Xinyang; Deng, Yong title: D numbers theory: a generalization of Dempster-Shafer theory date: 2014-02-14 journal: nan DOI: nan sha: 6a574024cf85a02a8a9eb9a0029cc9851ab0c612 doc_id: 227789 cord_uid: uviucg92 Dempster-Shafer theory is widely applied to uncertainty modelling and knowledge reasoning due to its ability of expressing uncertain information. However, some conditions, such as exclusiveness hypothesis and completeness constraint, limit its development and application to a large extend. To overcome these shortcomings in Dempster-Shafer theory and enhance its capability of representing uncertain information, a novel theory called D numbers theory is systematically proposed in this paper. Within the proposed theory, uncertain information is expressed by D numbers, reasoning and synthesization of information are implemented by D numbers combination rule. The proposed D numbers theory is an generalization of Dempster-Shafer theory, which inherits the advantage of Dempster-Shafer theory and strengthens its capability of uncertainty modelling. Since first proposed by Dempster [1] and then developed by Shafer [2] , Dempster-Shafer theory of evidence , also called Dempster-Shafer theory or evidence theory, has been paid much attentions for a long time and continually attracted growing interests. This theory needs weaker conditions than the Bayesian theory of probability, so it is often regarded as an extension of the Bayesian theory [3, 4, 5, 6] . Many studies have been devoted to further improve and perfect this theory in many aspects, for instance combination of evidences [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] , conflict management [20, 21, 22, 23, 24, 25, 26, 27] , independence of evidence [28, 29, 30, 31] , generation of mass function [32, 33, 34, 35] , similarity measure between evidences [36, 37, 38] , uncertainty measure of evidences [39, 40, 41, 42, 43, 44, 45, 46, 47] , and so on [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73] . Due to its ability to handle uncertain information, Dempster-Shafer theory has been extensively used in many fields, such as statistical learning [74, 75, 76, 77, 78, 79, 80, 81] , classification and clustering [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98] , Granular computing [99, 100, 101, 102] , uncertainty and knowledge reasoning [103, 104, 105, 106, 107, 108, 109, 110, 111] , decision making [112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126] , risk assessment and evaluation [127, 128, 129, 130] , knowledge-based systems and expert systems [131, 132, 133, 134, 135] , and so forth [136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146] . Even as a theory of reasoning under the uncertain environment, Dempster-Shafer theory has an advantage of directly expressing the "uncertainty" by assigning the probability to the subsets of the set composed of multiple objects, rather than to each of the individual objects. However, it is also constrained by many strong hypotheses and hard constraints which limit its development and application to a large extend. For one hand, the elements in a frame of discernment (FOD) are required to be mutually exclusive. It is called exclusiveness hypothesis. For another, the sum of basic probabilities of a mass function must be equal to 1, which is called completeness constraint. In the following of this paper, we will show how these conditions limit the application of Dempster-Shafer theory. To overcome these shortcomings in Dempster-Shafer theory and strengthen its capability of representing uncertain information, a novel theory called D numbers theory is systematically proposed in this paper. A novel data representation called D numbers [147, 148, 149 ] is used to model uncertain information. What's more, a D numbers combination rule is proposed to synthesize all the information expressed by D numbers and implement the uncertainty and knowledge reasoning. Actually, D numbers and D numbers combination rule is an extension of mass function and Dempster's rule of combination, respectively. If meeting certain conditions, they will degenerate to classical mass function and Dempster's rule of combination. Consequently, the proposed D numbers theory is an generalization of Dempster-Shafer theory. The rest of this paper is organized as follows. Section 2 gives a brief introduction about the Dempster-Shafer theory. In Section 3, the proposed D numbers theory is presented, mainly including D numbers and D numbers combination rule. Some numerical examples are given to show the application of D numbers theory in Section 4. Finally, conclusions are given in Section 5. For completeness of the explanation, a few basic concepts about Dempster-Shafer theory are introduced as follows. For a finite nonempty set Ω = {H 1 , H 2 , · · · , H N }, Ω is called a frame of discernment (FOD) when satisfying Let 2 Ω be the set of all subsets of Ω, namely 2 Ω is called the power set of Ω. For a FOD Ω, a mass function is a mapping m from 2 Ω to [0, 1], formally defined by: which satisfies the following condition: In Dempster-Shafer theory, a mass function is also called a basic probability assignment (BPA). Given a BPA, the belief function Bel : 2 Ω → [0, 1] is defined as The plausibility function P l : 2 Ω → [0, 1] is defined as whereĀ = Ω − A. These functions Bel and P l express the lower bound and upper bound in which subset A has been supported, respectively. Given two independent BPAs m 1 and m 2 , Dempster's rule of combination, denoted by m = m 1 ⊕m 2 , is used to combine them and it is defined as follows with Note that the Dempster's rule of combination is only applicable to such two BPAs which satisfy the condition K < 1. In the mathematical framework of Dempster-Shafer theory, there are several strong hypotheses and constraints on the FOD and BPA. However, these hypotheses and constraints limit the ability of Dempster-Shafer theory to represent uncertain information. First, a FOD must be a mutually exclusive and collectively exhaustive set, the elements of FOD are required to be mutually exclusive, as shown in Eq.(1). In many situations, however, it is very difficult to be satisfied. Take assessment as an example. In evaluating one object, it often uses linguistic variables to express the assessment result, such as "Very Good", "Good", "Fair", "Bad" and "Very Bad". Due to given by human, it inevitably exists intersections among these linguistic variables. Therefore, the exclusiveness hypothesis cannot be guaranteed precisely so that the application of Dempster-Shafer theory is questionable for such situations. There are already some studies about FOD with non-exclusive hypotheses [54, 150] . Second, the sum of basic probabilities of a normal BPA must be equal to 1, as shown in Eq.(4). We call it as completeness constraint. But in some cases, due to lack of knowledge and information, it is possible to obtain an incomplete BPA whose sum of basic probabilities is less than 1. For example, if an assessment is based on little partial information, the lack of information may result in a complete BPA cannot be obtained. Furthermore, in an open world [52] , the incompleteness of FOD may also lead to the incompleteness of BPA. Hence the completeness constraint is hard to completely meet in some cases and it restricts the application of Dempster-Shafer theory. To overcome these existing shortcomings in Dempster-Shafer theory and enhance its capability of expressing uncertain information, a novel theory, named as D numbers theory, is systematically proposed. D numbers theory looses FOD's exclusiveness hypothesis and BPA's completeness constraint, which is a generalization of Dempster-Shafer theory. with The proposed D numbers combination rule is a generalization of Dempster's rule of combination. If D 1 , D 2 are defined on a FOD and Q 1 = 1, to well handle the situation of information incompleteness. In the following, we will investigate this example using D numbers theory. Now let us consider this problem by using D numbers theory. According to the two pieces of information given by the diagnostic reports, two D numbers can be derived that It is noted that the unknown information is not assigned to any set. The constructed two D numbers are in the forms of information incompleteness. The intersection table of D 1 ⊙D 2 is shown in Table 2 . According to Table 2 , we can calculate that The result also shows the flu is the most probable disease the patient got. By comparison with Dempster-Shafer theory, however, in the proposed D numbers theory the unknown is inherited during the reasoning. D numbers theory has inherent advantage to handle the situation of information incompleteness, which is more natural and reasonable. The intersection table of m w ⊕ m s is shown in Table 3 . Then, we can obtain that Table 4 . So, we can calculate that In this paper, a novel theory called D numbers theory is systematically will be further studied. On the other hand, this theory will be applied to many real applications. 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