1 Introduction
Uncertainty exists extensively in the real world, and many theories have been developed for representing and dealing with uncertainty, such as probability theory, imprecise probabilities, fuzzy logic, possibility theory, DempsterShafer theory, setmembership approach, and so on. For any theory of uncertainty reasoning, the quantification of uncertainty degree to a piece of information is a crucial issue
klir1991generalized ; klir2005uncertainty . Regarding this issue, different branches of theories have given different solutions. For example, the Shannon entropy is widely accepted to be the uncertainty measure of probabilities. However, in many other uncertainty reasoning theories, there are not well acknowledged and satisfactory measures that can effectively quantify the uncertainty of information.In this paper, we put our attention on a special theory for uncertainty reasoning which is called D numbers theory (DNT) deng2012DJICS99 ; deng2017d09186 . This theory is a recently proposed theoretical framework that has generalized DempsterShafer theory (DST) Dempster1967 ; Shafer1976 . In DST, every piece of knowledge or information is abstracted as a mass function or basic probability assignment (BPA), and each BPA is defined on a mutually exclusive and collectively exhaustive set which is called frame of discernment (FOD). In contrast, DNT generalizes the DST to a set with nonexclusive elements and does not strictly require the knowledge or information is complete. For more details about DNT and its differences with DST, please refer to references xydeng2017DNCR ; deng2014environmental412 ; deng2017d09186 and introductions given in the next section. As for this new theory DNT, how to measure the uncertainty of D numbers has not been studied yet.
Since DNT is a generalization of DST, previous studies about uncertainty measures in DST would be useful for the design of uncertainty measures in DNT. In DST, measuring the uncertainty degree of a BPA is still an open issue. Some representative total uncertainty measures are aggregated uncertainty (AU) harmanec1994measuring224 , ambiguity measure (AM) jousselme2006measuring , and other entropies discussed in abellan2008requirements ; jirouvsek2018new ; deng2016deng91 , and recently proposed belief interval based uncertainty measures yang2016new94 ; AnImprovedAPIN2017 ; wang2017uncertaintySY , to name but a few. These uncertainty measures have studied the uncertainty quantification of BPAs from different perspectives for example axiomatization, desirable properties and behaviour, consistency with probabilities or only taking into account the framework of DST. These perspectives and existing uncertainty measures for DST are also valuable for the design of an uncertainty measure for D numbers. However, different from DST that only contains two types of uncertainty factors which are discord and nonspecificity yager1983entropy94249 , in DNT there exists a new type of uncertainty caused by the nonexclusiveness among elements. In addition, since a D number can be informationincomplete which means that the knowledge or information is incomplete, a rational uncertainty measure for D numbers must be able to deal with the case of incomplete information or knowledge.
In this paper, inspired by distancebased uncertainty measures yang2016new94 ; AnImprovedAPIN2017 for BPAs in DST, a total uncertainty measure for D numbers, denoted as , is proposed based on the belief intervals of D numbers. The proposed can simultaneously capture the discord, and nonspecificity, and nonexclusiveness involved in a D number. Meanwhile, the possible incompleteness of information or knowledge is also considered in by introducing a new notation representing unknown event. Moreover, some basic properties of including range, monotonicity, generalized set consistency, are presented.
2 Preliminaries
2.1 Basics of DempsterShafer theory
DempsterShafer theory (DST) Dempster1967 ; Shafer1976 , also called belief function theory or evidence theory, is a popular tool for uncertainty reasoning because of its advantages in expressing uncertainty. As a theory of reasoning under the uncertain environment, DST has an advantage of directly expressing the “uncertainty” by assigning the basic probability to a set composed of multiple objects, rather than to each of the individual objects. For completeness of the explanation, a few basic concepts in DST are introduced as follows.
Let be a set of mutually exclusive and collectively exhaustive events, indicated by
(1) 
where set is called a frame of discernment (FOD). The power set of is indicated by , namely
(2) 
The elements of or subsets of are called propositions.
Let a FOD be , a mass function defined on is a mapping from to , formally defined by:
(3) 
which satisfies the following condition:
(4) 
In DST, a mass function is also called a basic probability assignment (BPA). The assigned basic probability measures the belief exactly assigned to and represents how strongly the evidence supports . If , is called a focal element, and the union of all focal elements is called the core of the mass function.
Given a BPA, its associated belief measure and plausibility measure express the lower bound and upper bound of the support degree to each proposition in that BPA, respectively. They are defined as
(5) 
(6) 
Obviously, for each , and is called the belief interval of in .
2.2 D numbers theory
D numbers theory (DNT) is a new theoretical framework for uncertainty reasoning which generalizes the DST from two aspects: on one hand, the elements within FOD are not required to be mutually exclusive in DNT; on the other hand, the providing information in DNT can be incomplete in contrast to in DST. For more theoretical details about DNT and its recent advances, please refer to literatures deng2012DJICS99 ; xydeng2017DNCR ; deng2017d09186 . Furthermore, some applications of DNT can be found in references fan2016hybrid44 ; liu2014failure4110 ; xiao2016intelligent3713518 ; wang2017modifiedIJFS ; deng2014environmental412 ; Deng2017Fuzzy2086 . A few of basic concepts in DNT are introduced as follows deng2017d09186 .
Let be a nonempty finite set , a D number is a mapping formulated by
(7) 
with
(8) 
where is the empty set and is a subset of .
It is worthy noting that a D number can be defined on a set with nonexclusive elements, which means that any pair of elements in , for example , are not required to be strictly exclusive, i.e. . Here, we still call as a FOD, but should note that a FOD in DNT is a set consisting of nonexclusive elements. Besides, according to Definition 2.2, in a D number the information is not required to be complete. If , we say that the D number is informationcomplete. By contrast, if the D number is informationincomplete. In order to transform a D number with incomplete information to the informationcomplete case, a new notation is imported to represent the unknown event, and there is not any restriction for the relationship between and . So, a new definition about D numbers is given as below. A D number defined on a nonempty finite set and unknown event is a mapping satisfying
(9) 
In the latter definition of D numbers, i.e. Definition 2.2, represents the unknown (or incomplete) part in a D number. Regarding the requirement of nonexclusiveness in DNT, a membership function is developed to measure the nonexclusive degrees in .
Given , the nonexclusive degree between and is characterized by a mapping :
(10) 
with
(11) 
and
(12) 
where . If letting the exclusive degree between and be denoted as , then .
According to Definition 2.2, the nonexclusive degree between and is 1 if and have intersections, otherwise is taking a value from . Obviously, if for any , the FOD in DNT is degenerated to classical FOD in DST.
In order to express the bound of uncertainty in a D number, in a very recent study deng2017d09186 we have developed a belief measure and a plausibility measure for D numbers.
Let represent a D number defined on where represents unknown event, for any proposition , its belief measure is defined as
(13) 
and its plausibility measure is defined as
(14) 
where .
For the above definition, because for , the plausibility measure can also be written as
(15) 
As same as DST, is called the belief interval of in DNT, which expresses the lower bound and upper bound of support degree to proposition . And it is easy to find that the and for D numbers will degenerate to classical belief measure and plausibility measure in DST if the associated D number is a BPA in fact.
3 Proposed total uncertainty measure for D numbers
How to measure the uncertainty of information is an important issue in the theories of uncertainty reasoning. Up to now, uncertainty measure for D numbers is an unsolved problem in DNT. With respect to this problem, there are two considerable aspects, as graphically shown in Figure 1. At first, since a D number consists of two parts, called known part and called unknown part, a rational uncertainty measure must be able to model the total uncertainty that contains known uncertainty caused by known part and unknown uncertainty from unknown part. This is the first difference between DNT and DST in the design of uncertainty measure. At second, in DST the uncertainty consists of discord and nonspecificity yager1983entropy94249 , in DNT, however, the nonexclusiveness among elements in becomes a new source of uncertainty. Therefore, a rational uncertainty measure for D numbers should simultaneously capture discord, nonspecificity, and nonexclusiveness.
In this paper, a belief interval based total uncertainty measure, called , is proposed for D numbers. This idea of is inspired by distancebased uncertainty measures yang2016new94 ; AnImprovedAPIN2017 for mass functions in DST.
Let represent a D number defined on where and expresses the unknown part in , the total uncertainty of is
(16) 
with
(17) 
(18) 
where
(19) 
and is a function associated with the cardinality of and expresses the overall uncertainty in , and are the belief measure and plausibility measure of D numbers, respectively.
The underlying assumption of this total uncertainty measure is that an element in has the largest uncertainty degree if its belief interval is . Let us analyze the :

It separately calculates the known uncertainty and unknown uncertainty involved in a D number.

For the known uncertainty, employs the Euclidean distance function to calculate the distance between belief interval of each singleton (i.e., ) and the most uncertain interval , then expresses the contribution of to the total uncertainty of by using .

For the unknown uncertainty, a fictitious function is assumed to represent the overall uncertainty of , and is only associated the the size of . Since is an unknown set, we suppose and for any and . Then, the belief interval of is calculated to be , so in the contribution of to the total uncertainty is and the overall unknown uncertainty is positively correlated with , namely . The principle behind is logically consistent with that of .

Discord, and nonspecificity, and nonexclusiveness, are all considered in constructing belief intervals of singletons and simultaneously captured in .
Based on the above analysis, for the sake of simplicity the total uncertainty of a D number, , can be represented by a tuple , as graphically shown in Figure 2, where is a coefficient with respect to . Some basic properties of the proposed are given as follows.
Theorem 3.1.
Range. Given a D number defined over , the total uncertainty of , denoted as , is limited by ranges and .
Theorem 3.2.
Monotonicity. Let and be two arbitrary D numbers defined on , if
then
Theorem 3.3.
Generalized set consistency. When a set , exists such that then
namely
where is the difference between and .
4 Conclusion
This paper has studied the issue of uncertainty quantification of knowledge or information in DNT. At first, three types of uncertainty factors, namely discord, and nonspecificity, and nonexclusiveness, are identified for D numbers. Then, on the basis of our previous defined belief intervals for D numbers and distance based uncertainty measures for DST, a total uncertainty measure is proposed in the framework of D numbers theory. At last, some basic properties of are presented. In the future study, on one hand other properties of will be investigated, on the other hand the applications of the proposed are given more attention.
Acknowledgments
The work is partially supported by National Natural Science Foundation of China (Program Nos. 61703338, 61671384), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2016JM6018), Project of Science and Technology Foundation, Fundamental Research Funds for the Central Universities (Program No. 3102017OQD020).
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