Maximummean uncertain measure
 Xiang Li^{1}Email author
https://doi.org/10.1186/s4046701500277
© Li; licensee Springer. 2015
Received: 14 November 2014
Accepted: 1 January 2015
Published: 12 February 2015
Abstract
Uncertain measure, which is used to indicate the degree of belief that an uncertain event may occur, is a set function satisfying the normality, duality, and subadditivity axioms. Although the tools of uncertain measure, uncertainty space, and uncertain variable have been well developed and became the foundations of uncertainty theory, there is no study on how to construct the uncertainty space for an uncertain variable, which seriously hinders the applications of uncertainty theory in practice. In this paper, we propose an expert knowledgebased construction approach for uncertainty space with maximummean uncertain measure, which takes the maximum and mean operations on confidence function.
Keywords
Introduction
Uncertainty theory introduced by Liu [1] is a branch of nonadditive measure theory for describing human uncertainty. Compared with randomness, Liu’s uncertainty amounts to subject indeterminacy arising from information loss on some exact value of a certain quantity. The main difference between randomness and Liu’s uncertainty is that the former is used to model unfixed values, while the latter is used to model fixed but illknown values. For example, if we toss a coin in the air and choose between head and tail before it lands on the ground, the outcome is random since both head and tail are possible. However, if we make the choice after the coin has landed on the ground, the outcome is uncertain since it has been fixed although we do not know the result.
Nowadays, uncertainty theory has become a branch of axiomatic mathematics for modeling human uncertainty, and it has been well developed and applied to uncertain programming [2], uncertain statistics [3], uncertain risk analysis [4], uncertain set [5], uncertain logic [6], uncertain inference [7,8], uncertain process [9], uncertain renewal process [10], uncertain calculus [11], uncertain differential equation [12], uncertain finance [13], and so on. The latest results related to uncertainty theory may be found in [14].
Uncertain measure and uncertain variable are two basic concepts in uncertainty theory. Uncertain measure, which is used to indicate the degree of belief that an uncertain event may occur, was defined in [1] as a set function satisfying the normality, duality, and subadditivity axioms. The normality is used to ensure that degree of belief is consistent with human thinking, and the duality is an extension of the law of contradiction. For a sequence of countable events, subadditivity means that the entire degree of belief must be less than or equal to the sum of their individual degrees of belief. Liu [15] proved the monotonicity theorem, nulladditivity theorem, and asymptotic theorem, and Gao [16] studied some properties of continuous uncertain measures. The product uncertain measure of compound event was defined by Liu [17] as the minimum of uncertain measures of individual events, and the product uncertain measure theorem was proved by Peng and Iwamura [18]. In order to study the dependence of uncertain events, Liu [1] defined the conditional uncertain measure of an event given that some other event has occurred. Uncertain variable is used to represent quantity with uncertainty, which was defined in [1] as a measurable function from uncertainty space to the set of real numbers.
Possibility theory is another type of nonadditive measure theory for dealing with the subjective uncertainty. Possibility measure was defined by Zadeh [19] as a set function satisfying the normality, nonnegativity, and maximality. Necessity measure was also defined by Zadeh [20] as the dual part of possibility measure. It is proved that both possibility measure and necessity measure satisfy the properties of normality, nonnegativity, and monotonicity. However, neither of them are selfdual. Since the duality is intuitive and important in real problems, Liu and Liu [21] defined a credibility measure as the average of possibility measure and necessity measure, which was lately proved to be a set function satisfying normality, monotonicity, duality, and partial maximality [22]. As an extension of credibility measure, Liu [1] proposed the uncertain measure by slacking the partial maximality to subadditivity, which generalizes the uncertain model to describe illknown but fixed value.
Although the tools of uncertain measure, uncertainty space, and uncertain variable have been well developed and became the foundations for uncertainty theory, there is no study on how to construct the uncertainty space for an uncertain variable. This seriously hinders the applications of uncertainty theory in practice. The purpose of this paper is to provide a general construction approach for uncertainty space with maximummean uncertain measure based on the expert knowledge. The rest of this paper is organized as follows. The ‘Preliminaries’ section reviews some basic definitions and properties on uncertainty theory. The ‘Confidence function’ section introduces a concept of confidence function. In the ‘Maximummean uncertain measure’ section, we propose the concept of maximummean uncertain measure based on the maximum and mean operations on confidence function, and we provide a general procedure on generating an uncertainty space with the maximummean uncertain measure. The ‘An illustrative example’ section provides a practical example to illustrate how to construct an uncertainty space for an uncertain quantity. Finally, a brief summary is given.
Preliminaries

Axiom 1. (Normality) M{Γ}=1;

Axiom 2. (Duality) M{A}+M{A ^{ c }} = 1 for any A∈Λ;

Axiom 3. (Subadditivity) For every countable sequence of events {A _{ i }}, we have:$$\mathrm{M}\left\{\bigcup_{i=1}^{\infty}A_{i}\right \}\leq\sum_{i=1}^{\infty}\mathrm{M}\{A_{i}\}.$$
The normality is used to ensure that degree of belief is consistent with human thinking. The duality is an extension of the law of contradiction, which ensures that an event and its complement cannot occur simultaneously. For a sequence of countable events, subadditivity means that the entire degree of belief must be less than or equal to the sum of their individual degrees of belief.
Definition 2.1.
Let Γ be a nonempty set, and let Λ be a σalgebra (Liu [1]). The set function M is called an uncertain measure if it satisfies the normality, duality, and subadditivity axioms.
Liu [15] proved the monotonicity theorem for uncertain measure. That is, for any events A _{1}⊆A _{2}, we have M{A _{1}}≤M{A _{2}}. In addition, the nulladditivity theorem and asymptotic theorem were also proved. Liu [17] defined the product uncertain measure for compound event, and Liu [1] defined a conditional uncertain measure of an event after it has been learned that some other event has occurred.
Definition 2.2.
Suppose that Γ is a nonempty set, Λ is a σalgebra, and M is an uncertain measure (Liu [1]). Then, the triplet (Γ,Λ,M) is called an uncertainty space.
Definition 2.3.
Example 2.1.
Suppose that Γ is a nonempty set containing three points γ _{1}, γ _{2}, and γ _{3}. Let Λ be the power set of Γ. Define M{∅}=0, M{γ _{1}}=0.2, M{γ _{2}}=0.4, M{γ _{3}}=0.6, M{γ _{1},γ _{2}}=0.4, M{γ _{2},γ _{3}}=0.8, M{γ _{1},γ _{3}}=0.6 and M{Γ}=1. It is easy to prove that M satisfies the normality, duality, and subadditivity axioms. Then, M is an uncertain measure, and the triplet (Γ,Λ,M) is an uncertainty space. Define a real function ξ(γ _{1})=x _{1}, ξ(γ _{2})=x _{2} and ξ(γ _{3})=x _{3}. Since Λ is the largest σalgebra over Γ, for any Borel set B of ℜ, it is clear that {ξ∈B}∈Λ. Thus, ξ is a measurable function, that is, an uncertain variable, on uncertainty space (Γ,Λ,M).
Confidence function
Let ξ be an uncertain variable. For theoretical studies, we always assume that we can define an uncertainty space (Γ,Λ,M) with known uncertain measure such that ξ is a measurable function from this space to the set of real numbers. However, it is difficult to construct an uncertainty space in practice since there is no efficient construction approach for uncertain measure.
If Λ is finite and has a small number of elements, we may define an uncertain measure by enumerating the degree of beliefs for all events according to the expert knowledge. However, if the cardinality of Λ is infinite, it is impossible to do in this way since the number of events is unmanageable. Therefore, an efficient construction approach for uncertain measure should be necessary. This paper defines a maximummean uncertain measure and proposes a construction approach based on the maximum and mean operations on confidence function. First, we introduce the concept of confidence function.
Let Γ={x _{1},x _{2},⋯ } be the universal set, i.e., the collection of candidates. Suppose that there are m experts with confidences {ν _{1},ν _{2},⋯,ν _{ m }}, which are given by an expert of experts. Without loss of generality, we assume that each candidate associated with one and only one confidence under the following rules: (i) if some candidate receives no support from any expert, we assign it with confidence 0; (ii) if some candidate receives support from one and only one expert, for example, expert i, we assign ν _{ i } to it; (iii) if some candidate receives supports from more experts, for example, experts \(\phantom {\dot {i}\!}i_{1},i_{2},\cdots,i_{m_{i}}\), we assign it with the confidence \(\phantom {\dot {i}\!}\max \left \{\nu _{i_{1}},\nu _{i_{2}},\cdots,\nu _{i_{m_{i}}}\right \}\).
The confidence function indicates a relative preference among different candidates for the uncertain quantity. Generally speaking, we prefer to take the candidate that has the largest confidence value.
Maximummean uncertain measure
This section defines a maximummean uncertain measure based on the confidence function, which expresses the degree of belief that an uncertain event will occur.
First, let us consider the confidence on a set of candidates. As we know, each uncertain quantity has one and only one correct value. In other words, there is only one expert providing the correct estimation, and the others all provide the wrong estimation. The uncertainty arises from the difficulty that we do not know who is the correct expert. Therefore, if we believe that the uncertain quantity takes a value belonging to set A, the confidence should be the maximum value among all confidences on candidates in set A.
Example 4.1.
Since candidate 30 has the maximum confidence, we prefer to believe that ξ will take value 30. On the other hand, for the uncertain event {ξ≠30}, the confidence should be the maximum confidence on candidates 10, 15, and 20, that is, 0.3∨0.5∨0.7=0.7.
If μ _{ A }>0, we believe that it is possible that ξ takes a value in set A, and the degree of belief should have a strong positive correlation with respect to the confidence such that a larger confidence leads to a larger degree of belief.
Theorem 4.1.
Then, the set function M is an uncertain measure (called the maximummean uncertain measure), and the triplet (Γ,Λ,M) is an uncertainty space.
Remark 4.1.
As a special uncertain measure, the maximummean uncertain measure also satisfies the monotonicity theorem, nulladditivity theorem, and asymptotic theorem.

Step 1. Define the set of candidates Γ={x _{1},x _{2},⋯ }.

Step 2. Define a confidence function μ:Γ→[0,∞).

Step 3. Take a σalgebra Λ over Γ.

Step 4. Define a maximummean uncertain measure M according to formula (6).

Step 5. Define ξ(x _{ i })=x _{ i } for all x _{ i }∈Γ.
Remark 4.2.
If there is a candidate with confidence zero, for example, μ _{1}=0, then, we have M{ξ=x _{1}}=0. If all candidates have the same confidence, then, the degree of belief that ξ takes value x _{ i } is M{ξ=x _{ i }}=0.5 for each 1≤i≤n.
An illustrative example
Answers on Prof. Ning’s age
Expert  Candidate  Confidence ( ν _{ i } )  ≤45( α _{ i } )  >45 ( β _{ i } )  

1  Dr. Chen  47  0.7  0.4  0.6 
2  Dr. Gao  41  0.4  0.7  0.2 
3  Prof. Ke  45  0.6  0.8  0.6 
4  Dr. Li  45  0.8  0.5  0.4 
5  Prof. Qin  47  0.9  0.5  0.5 
6  Dr. Wen  45  0.4  0.3  0.7 
7  Prof. Zhou  45  0.3  0.9  0.1 
8  Prof. Zhang  46  0.9  0.0  1.0 
9  Prof. Li  44  0.8  0.6  0.4 
10  Prof. Zhu  45  0.7  0.8  0.8 
It follows from Table 1 that the confidence function is μ(41)=0.4, μ(44)=0.8, μ(45)=0.8, μ(46)=0.9, and μ(47)=0.9. Take Λ to be the power set of Γ. According to formula (6), we can define a maximummean uncertain measure M. Then, Ning’s age may be treated as the identity uncertain variable ξ on uncertainty space (Γ,Λ,M).

Prof. Ning is less than or equal to 45 with:$$\displaystyle\frac{\sum\limits_{i=1}^{10}\nu_{i}\alpha_{i}}{\sum\limits_{i=1}^{10}\nu_{i}\alpha_{i}+\sum\limits_{i=1}^{10}\nu_{i}\beta_{i}}=0.472. $$

Prof. Ning is larger than 45 with:$$\displaystyle\frac{\sum\limits_{i=1}^{10}\nu_{i}\beta_{i}}{\sum\limits_{i=1}^{10}\nu_{i}\alpha_{i}+\sum\limits_{i=1}^{10}\nu_{i}\beta_{i}}=0.528. $$
Therefore, the maximummean uncertain measure satisfies the experts’ knowledge.
Conclusion
The main contribution of this paper is the constructive approach of an uncertainty space with maximummean uncertain measure for an uncertain variable. It is the first such study on how to characterize a practical uncertain quantity mathematically based on expert knowledge, which makes it possible to apply uncertainty theory to reallife problems.
Declarations
Acknowledgements
This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20110009120036).
Authors’ Affiliations
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