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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 supremum of uncertainty of uncertain variable by choosing a proper q. The single parameter entropy of uncertain variable makes the computing of uncertainty of uncertain variable more general and flexible.

The concept of entropy was founded by Shannon [

Tsallis Entropy initiated by Tsallis [4-6] 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 probabilities. For the equiprobability distribution, the value of Tsallis entropy, where is a monotonic increasing function of, is a real number. It is clearly that in the limit, recovers the Shannon entropy formula:

Henceforth, many scholars conduct to research the tsallis entropy, such as S. Abe [

Uncertainty theory was founded by Liu [

In order to provide a quantitative measurement of the degree of uncertainty in relation to an uncertain variable, Liu [

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.

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 ([

Axiom 1. (Normality Axiom) for the universal set.

Axiom 2. (Monotonicity Axiom) whenever.

Axiom 3. (Self-Duality Axiom) for any event.

Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of events, we have

.

Axiom 5. (Product Measure Axiom) Let be nonempty sets on which are uncertain measures, respectively. Then the product uncertain measure is an uncertain measure on the product -algebra satisfying

.

where.

We will introduce the definitions of uncertain variable and uncertainty distribution as follows.

Definition 2.1 (Liu [

Definition 2.2 (Liu [

Definition 2.3 (Liu [

.

Theorem 2.1 (Sufficient and Necessary Condition for Uncertainty distribution [

Example 2.1 An uncertain variable is called normal if it has a normal uncertainty distribution

denoted by where and are real numbers with.

Then we will recall the definition of inverse uncertainty distribution as follows.

Definition 2.4 (Liu [

Definition 2.5 (Liu [

Example 2.2 The inverse uncertainty distribution of normal uncertain variable is

.

Definition 2.6 (Independence of uncertain variable Liu [

.

for any Borel sets of real numbers.

Example 2.3 Let and be independent normal uncertain variables and, respectively. Then the sum is also normal uncertain variable for any real number and.

Finally we will recall their theorems about the operational law of independent uncertain variables.

Theorem 2.2 (Liu [

is an uncertain variable with inverse uncertain distribution

.

Example 2.4 Let and be independent and positive uncertain variables with uncertainty distribution and, respectively. Then the inverse uncertainty distribution of the quotient is

.

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 [

Definition 3.1 (Liu [

where

.

We set throughout this paper.

Through observing Definition 3.1 and

Definition 3.2 Suppose that is an uncertain variable with uncertainty distribution. Then its single parameter entropy is defined by

where

.

is a positive real number. For, it is immediately verified

This means that is entropy of uncertain variable. For, we have

It’s clear that is the quadratic entropy of uncertain variable [

Remark 3.1 From the plot of for and typical values of, we notice that is a monotonic function of. From Definition 3.2 and the

Example 3.1 Let be an uncertain variable with uncertain distribution

Essentially, is constant. It follows from the definition of single parameter entropy that

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

especially,.

Example 3.3 Suppose be a zigzag uncertain variable with uncertain distribution

Then its single parameter entropy is

especially,.

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

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

Proof: Denote the uncertain distribution function of by. If, then the uncertain variable has an uncertain distribution function. It follows from the definition of single parameter entropy that

when, we have.

Theorem 4.4 Let be an uncertain variable with uncertain distribution, then

where

especially,

Proof: It is obvious that is a derivable function with

Since

and noting that the uncertain variable has a regular uncertain distribution, we have

By Fubini theorem, we have

The theorem is proved.

Theorem 4.5 Let and be independent uncertain variables, then for any real numbers and, we have

Proof: Suppose that and have uncertainty distribution and, respectively, and inverse uncertainty distribution and, respectively. Note that the inverse uncertainty distribution of is

From Theorem 4.4, we have

Since, Theorem 4.3, we obtain

The theorem is proved.

Theorem 4.6 (Alternating Monotone function) Let be independent uncertain variables with uncertainty distribution, respectively. If the function $f$ is a strictly increasing with respect to and strictly decreasing with respect to, then has a single parameter entropy

where

Proof: Let be the uncertainty distribution function of, then it follows from Theorem 2.2 that

Since, Theorem 4.4, we have

The theorem is proved.

Example 4.1 Let and be independent uncertain variables with regular uncertainty distribution and, respectively. Since the function

is strictly increasing with respect to and strictly decreasing with respect to. From the Theorem 2.2, the inverse uncertainty distribution of the function is as follow

therefore, its single parameter entropy is

Theorem 5.1 Let be a uncertain variable with uncertain distribution, then

where the equality holds if uncertain distribution.

Proof: Let be a uncertain variable with uncertain distribution, then

where the equality holds if, that is. Then

We complete the proof.

In according to Theorem 5.1, we obtain three situations as follows.

Situation 5.1 If uncertain variable is a constant, that is, then

from Theorem 4.1, we get since the constant is no uncertainty.

Situation 5.2 Let uncertain variable, then

According to the fact, we can find the appropriate to describe the uncertainty of uncertain variable. Especially, when, as

. 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.

It is consistent with the reality, which the impossible event can be interpreted that it has no uncertainty.

Example 6.1 Let uncertain variable, then

By the expert’s experimental data or people’s subjective judgment, we can choose a appropriate to judge the relation of and. Furthermore, we can obtain the relation of and. For instance, if two persons’ age and they are about 25 years old, Suppose we obtain, then,. It is clear that is more close to 25 years old than.

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:

It is clear that entropy of uncertain variable is infinite.

So we consider the single parameter entropy of uncertain variable.

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.

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.