1. 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 [4-6] 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 [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.
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.
2. 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]):
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 [10]) Let 
be a nonempty set, and 
be a 
-algebra over
, and 
an uncertain measure. Then the triplet 
is called an uncertainty space.
Definition 2.2 (Liu [10]) An uncertain variable is a measurable function from an uncertainty space 
to the set of real numbers.
Definition 2.3 (Liu [10]) The uncertainty distribution 
of an uncertain variable 
is defined by
.
Theorem 2.1 (Sufficient and Necessary Condition for Uncertainty distribution [21]) A function 
is an uncertainty distribution if and only if it is an increasing function except 
and
.
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 [11]) An uncertainty distribution 
is said to be regular if its inverse function 
exists and is unique for each
.
Definition 2.5 (Liu [11]) Let 
be an uncertain variable with uncertainty distribution
. Then inverse function 
is called the inverse uncertainty distribution of
.
Example 2.2 The inverse uncertainty distribution of normal uncertain variable 
is
.
Definition 2.6 (Independence of uncertain variable Liu [10]) The uncertain variables 
are said to be independent if
.
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 [11]) Let 
be independent uncertain variables with uncertainty distribution
, respectively. If 
be a strictly increasing with respect to 
and strictly decreasing with respect to
. Then
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
.
3. 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
(1)
where
.
We set 
throughout this paper. Figure 1 illustrates Definition 3.1.
Through observing Definition 3.1 and Figure 1, we find that the selection of function 
is very important. For an uncertain event
, 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 
and decreases on
. 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 variable with uncertainty distribution
. Then its single parameter entropy is defined by
(2)
Figure 1. The entropy value of uncertain variable if and only if q = 1.
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 [20]. Figure 2 illustrates Definition 3.2.
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 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.
Example 3.1 Let 
be an uncertain variable with uncertain distribution
Essentially, 
is constant. It follows from the definition of single parameter entropy that
Figure 2. The different entropy value of uncertain variable with parameter q1 = 0.5, q2 = 2, and q3 = 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
especially,
.
Example 3.3 Suppose 
be a zigzag uncertain variable 
with uncertain distribution
Then its single parameter entropy is
especially,
.
4. 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
(3)
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
(4)
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
(5)
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
(6)
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
(7)
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
(8)
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
5. Discussions of Single Parameter Entropy
Theorem 5.1 Let 
be a uncertain variable with uncertain distribution
, then
(9)
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
(10)
from Theorem 4.1, we get 
since the constant is no uncertainty.
Situation 5.2 Let uncertain variable
, then
(11)
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
(12)
from Theorem 4.1, we get
.
It is consistent with the reality, which the impossible event can be interpreted that it has no uncertainty.
6. Example of Single Parameter Entropy
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.
7. 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
.
NOTES