Single Parameter Entropy of Uncertain Variables *

Uncertainty theory is a new branch of axiomatic mathematics for studying the subjective uncertainty. In uncertain theory, uncertain variable is a fundamental concept, which is used to represent imprecise quantities (unknown constants and unsharp concepts). Entropy of uncertain variable is an important concept in calculating uncertainty associated with imprecise quantities. This paper introduces the single parameter entropy of uncertain variable, and proves its several important theorems. In the framework of the single parameter entropy of uncertain variable, we can obtain the supre-mum of uncertainty of uncertain variable by choosing a proper . The single parameter entropy of uncertain variable makes the computing of uncertainty of uncertain variable more general and flexible.


Introduction
The concept of entropy was founded by Shannon [1] in 1949, which is a measurement of the degree of uncertainty of random variables.In 1972, De Luca and Termini [2] introduced the definition of fuzzy entropy by using Shannon function.Inspired by the Shannon entropy and fuzzy entropy, Liu [3] in 2009 proposed the concept of entropy of uncertain variable, where the entropy characterizes the uncertainty of uncertain variable resulting from information deficiency.
Tsallis Entropy initiated by Tsallis [4][5][6] in 1988, this is based on the following single parameter generalization of the Shannon entropy: where is a conventional positive constant, which is usually set equal to 1, is the total number of microsopic configurations, and is the set of associated . For the equiprobability dis- , the value of Tsallis entropy Henceforth, many scholars conduct to research the tsallis entropy, such as S. Abe [7], S. Abe and Y. Okamoto [8], R. J. V. dos Santos [9] and so on.
Uncertainty theory was founded by Liu [10] in 2007 and refined by Liu [11] in 2010, which is a branch of mathematics based on normality, monotonicity, selfduality, countable subadditivity, and product measure axioms.It is a effectively mathematical tool disposing of imprecise quantities in human systems.In recent years, Uncertainty theory was widely developed in many disciplines, such as uncertain process [12], uncertain calculus [3], uncertain differential equation [3], uncertain logic [13], uncertain inference [14], uncertain risk analysis [15], and uncertain statistics [11].Meanwhile, Liu [16] proposed a spectrum of uncertain programming and applied it into system reliability design, facility location problems, vehicle routing problems, project scheduling problems and so on.
In order to provide a quantitative measurement of the degree of uncertainty in relation to an uncertain variable, Liu [3] proposed the definition of uncertain entropy resulting from information deficiency.Dai and Chen [17] investigated the properties of entropy of function of uncertain variables.The principle of maximum entropy for uncertain variables are introduced by Chen and Dai [18].Besides, there are many literature concerning the definition of entropy of uncertain variables, such as Chen [19], Dai [20], etc.
where q is a monotonic increasing function of n , is a real number.It is clearly that in the limit , recovers the Shannon entropy formula: Inspired by the tsallis entropy, this paper introduces a new type of entropy, single parameter entropy in the framework of uncertain theory and discusses its properties.Consequently, we generalize the entropy of uncertain variable.The rest of the paper is organized as follows.In Section 2, we recall some basic concepts and theorems of uncertain theory.In Section 3, the definition of single parameter entropy of uncertain variables is proposed.In addition, some examples of the single parameter entropy are illustrated.In Section 4, several properties of single parameter entropy are proved.In Section 5, gives some discussions of single parameter entropy.In Section 6, some examples of single parameter entropy are given.At last, a brief summary is drawn.

Preliminaries
In this section, we will recall several basic concepts and theorems in the uncertain theory.Let be a nonempty set, and a    -algebra over .Each element is called an event.Uncertain measure was introduced as a set function satisfying the following five axioms ( [10]): whenever .
1 2 Axiom 3. (Self-Duality Axiom) for any event . Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of events   i  , we have   Axiom 5. (Product Measure Axiom) Let k be nonempty sets on which k are uncertain measures , respectively.Then the product uncertain measure is an uncertain measure on the product where ., 1,2, , We will introduce the definitions of uncertain variable and uncertainty distribution as follows.
Definition 2.1 (Liu [10]) Let be a nonempty set, and be a where and e  are real num- bers with 0 Then we will recall the definition of inverse uncertainty distribution as follows.
Definition 2.4 (Liu [11]) An uncertainty distribution  is said to be regular if its inverse function 1   exists and is unique for each .

 
0,1   Definition 2.5 (Liu [11]) Let  be an uncertain variable with uncertainty distribution .Then inverse function Definition 2.6 (Independence of uncertain variable Liu [10]) The uncertain variables 1 2 , , , m for any Borel sets of real numbers., , , n     , respectively.If f be a strictly increasing with respect to 1 2 , , , m x x  x n and strictly decreasing with respect to x x

Single Parameter Entropy
In this section, we will introduce the definition and theorem of single parameter entropy of uncertain variable.
For the purpose, we recall the entropy of uncertain variable proposed by Liu [3].Definition 3.1 (Liu [3]) Suppose that  is an uncertain variable with uncertainty distribution .Then its entropy is defined by where We set throughout this paper.A , if its incredible degree is 0 or 1, then the incident is no uncertainty.Conversely, when this event confidence level is 0.5, the uncertainty of the event is maximums.Therefore, the function   must increases on   0, 0.5 and decreases on   0.5,1 .By the enlightenment of Tsallis entropy, we try to define the single parameter entropy of uncertain variable as follows.
Definition 3.2 Suppose that  is an uncertain vari- able with uncertainty distribution  .Then its single parameter entropy is defined by where q is a positive real number.For , it is immediately verified H  is the quadratic entropy of uncertain variable [20].Figure 2 illustrates Definition 3.2.
Remark 3.1 From the plot of and typical values of , we notice that q   x   q is a monotonic function of .From Definition 3.2 and the Figure 2, we can see the difference between entropy of uncertain variable and single parameter entropy, because the single parameter entropy introduces a adjustable parameter , which makes the computing of uncertainty of uncertain variable more general and flexible.

S q q
Example 3.1 Let  be an uncertain variable with uncertain distribution Essentially,  is constant.It follows from the defini- tion of single parameter entropy that The different entropy value of uncertain variable with parameter q 1 = 0.5, q 2 = 2, and q 3 = 4.
This means that a constant has no uncertainty.
Example 3.2 Suppose  be a linear uncertain variable with uncertain distribution Then its single parameter entropy is Then its single parameter entropy is

Properties of Single Parameter Entropy
Assuming the uncertain variable with regular distribution, we obtain some theorems of single parameter entropy as follows.
Theorem 4.1 Let  is an uncertain variable.Then the single parameter entropy where the equality holds if  is a constant.
Proof: From Figure 2, the theorem is clear.As an uncertain variable tends to a constant, the single parameter entropy tends to the minimum 0. Theorem 4.2 Let  be an uncertain variable, and $c$ a real number.Then that is, the single parameter entropy is invariant under arbitrary translations.
Proof: Write the uncertainty distribution of  as  , then


From this equation, we get the uncertainty distribution of uncertain variable as follow: Using the definition of the single parameter entropy, we find The theorem is proved.Theorem 4.3 Let  be an uncertain variable, and let be a real number, then k Theorem 4.4 Let  be an uncertain variable with uncertain distribution  , then where and noting that the uncertain variable  has a regular uncertain distribution , we have The theorem is proved.
Theorem 4.5 Let  and  be independent uncertain variables, then for any real numbers and b , we have Proof: Suppose that  and  have uncertainty distribution and , respectively, and inverse uncertainty distribution and , respectively.Note that the inverse distribution of From Theorem 4.4, we have The theorem is proved., , , n , , , m be independent uncertain variables with uncertainty distribution 1 2 , respectively.If the function $f$ is a strictly increasing with respect to , , , n     , , m m and strictly decreasing with respect to , n x x where The theorem is proved. , respectively.Since the function is strictly increasing with respect to 1 x and strictly decreasing with respect to 2 x .From the Theorem 2.2, the inverse uncertainty distribution of the function 1 2 therefore, its single parameter entropy is

Discussions of Single Parameter Entropy
Theorem 5.1 Let  be a uncertain variable with un- where the equality holds if uncertain distribution where the equality holds if We complete the proof.In according to Theorem 5.1, we obtain three situa-tions as follows.
from Theorem 4.1, we get since the constant is no uncertainty.

 
According to the fact, we can find the appropriate to describe the uncertainty of uncertain variable.Especially, when . That is, the single parameter entropy measures the uncertainty of uncertain variable more flexible than the entropy of uncertain variable.
Situation 5.3 Suppose uncertain variable  is an impossible event.If we choose , we have from Theorem 4.1, we get   0 q H   .It is consistent with the reality, which the impossible event can be interpreted that it has no uncertainty.

of Single Parameter Entropy
By the expert's experimental data or people's subjective judgment, we can choose a appropriate to judge the relation of For some case, the entropy of uncertain variable is invalid.However, the single parameter entropy of uncertain variable works well.The follow example shows the point.
Example 6.2 Assume that the uncertain variable  has uncertain distribution as follow   we get the entropy of uncertain variable as follow: The example illustrate that we can obtain the supremum of uncertainty of uncertain variable by choosing a proper .So the application of single parameter entropy is more extensive.

Conclusion
In this paper, we recalled the entropy of uncertain variable and its properties.On the basis of the entropy of uncertain variable, and inspired by the tsallis entropy, we introduce the single parameter entropy of uncertain variable and explored its several important properties.We have generalized entropy of uncertain variable because of the singe parameter entropy of uncertain variable, which makes the calculating of uncertainty of uncertain variable more general and flexible by choosing an appropriate .q Finally we will recall their theorems about the operational law of independent uncertain variables.Theorem 2.2(Liu [11]) Let 1 2 , , , n     be independent uncertain variables with uncertainty distribution 1 2

Example 2 . 4
Let 1  and 2  be independent and positive uncertain variables with uncertainty distribution 1  and 2  , respectively.Then the inverse uncertainty Copyright © 2012 SciRes.AM distribution of the quotient 1 2

Figure 1
illustrates Definition 3.1.0 ln 0 0  Through observing Definition 3.1 and Figure 1, we find that the selection of function is very important.For an uncertain event

)Proof:
Denote the uncertain distribution function of  by  .If 0 k  , then the uncertain variable k has an uncertain distribution function   x k  .It follows from the definition of single parameter entropy that

Example 4 . 1
Let 1  and 2  be independent uncertain variables with regular uncertainty distribution 1  and 2

2 qH
 .Furthermore, we can obtain the relation of  and  .For instance, if two persons' age years old than  . q It is clear that entropy of uncertain variable is infinite.So we consider the single parameter entropy of uncertain variable.