This document introduces a two parameter entropy measure for uncertain variables based on two parameter probabilistic entropy. It begins by reviewing concepts of uncertainty theory such as uncertainty space, uncertain variables, and independence of uncertain variables. It then defines entropy of an uncertain variable using Shannon entropy. Previous work on one parameter entropy of uncertain variables is summarized. The document proposes a new two parameter entropy for uncertain variables, providing its definition and examples of calculating it for specific uncertainty distributions. Properties of the two parameter entropy are discussed.

1. 1
Two Parameter Entropy of Uncertain
Variables
Surender Singh
Assistant Prof. , School of Mathematics
Shri Mata Vaishno Devi University
Katra –182320 (J & K)
Email:surender1976@gmail.com
International Conference on Mathematics & Engineering
Sciences (ICMES-14)
Chitkara University, Himachal Pradesh
20-22nd March, 2014

2. 1. INTRODUCTION
 Concept of entropy was founded by [Shannon, 1948]
Let S be the sample space belongs to random events. Compose this sample
space into a finite number of mutually exclusive events nn EEE ...,,,1 , whose
respective probabilities are nn ppp ...,,,1 , then the average amount of
information or Shannon s entropy is defined:
iiii ppEInfpPH log)()(
n
1i
n
1i
 

(1.1)
H(P) is always non negative. Its maximum value depends on n. It is equal to
nlog when all pi are equal.
 Maximum Entropy Principle
2

3. 3
Figure 1.1 General Communication System

4. 4
Cont…
A general communication system has a source which generates messages
symbol by symbol and chooses symbols with given probabilities. The
symbols are transmitted through a channel in which noise may disturb
them. The messages are received at destination.
)( ixp the probability that the source will generate and send symbol xi
)( jmp the probability that the destination will receive symbol mj
),( ji mxp the probability that symbol xi will be sent and symbol mj will
be received
)|( ij xmp the probability that symbol mj will be received when symbol xi
has been sent
)|( ji mxp the probability that symbol xi was sent if symbol mj has been
received.
Now using (1.1) we can define the entropies of the communication
system:

5. 5
Cont…

i i
i
xp
xpxH
)(
1
log)()(
(1.2)

i j
j
mp
mpmH
)(
1
log)()(
(1.3)

i j ji
ji
mxp
mxpmxH
),(
1
log),(),(
(1.4)

i j ij
iji
xmp
xmpxpxmH
)|(
1
log)|()()|(
(1.5)

i j ji
jij
mxp
mxpmpmxH
)|(
1
log)|()()|(
(1.6)
System in which
)|( ji mxp






jiwhen
jiwhen
xmp ij
0
1
)|(
(1.7)

6. 6
Cont...
is called noiseless systems. Consequently, in noiseless systems 0)|()|(  mxHxmH
and also, ).()( mHxH  In noisy communication systems the conditional entropy of the
symbols sent when the received symbols are known ),|( mxH has been taken to
measure the missing information due to noise in the channel. )|( mxH is called the
equivocation of the channel. The rate of actual transmission of the information can
be defined in three equivalent ways:
),()()(
)|()()|()(
mxHmHxH
xmHmHmxHxHR


(1.8)
R is always non negative. Its minimum is reached when ),()(),( jiji mpxpmxp  i.e when
the symbols sent and received are independent. This means that there is maximum
noise in the system. The maximum of R is )(xH and is reached when the system is
noiseless.

7. 7
Cont…
 [ De Luca and Termini, 1972] -- Fuzzy entropy by using Shannon function.
 [Liu, 2009] proposed the concept of entropy of uncertain variable
(Characterizes the uncertainty of uncertain variable resulting from
information deficiency.)
 [Liu et al., 2012] proposed one parametric measure of entropy of uncertain
variable
 [Wada and Suyari, 2007] proposed two parametric generalization of
Shannon-Khinchin axioms and proved that the entropy function satisfying
two parameter generalized Shannon Khinchin axioms is given by Eq.(1.1) ,



n
i
ii
n
C
pp
pppS
1 ,
21, ),,(


  (1.9)
where ,C is a function of  and  satisfying certain conditions.
 One parametric [Tsallis, 1988] entropy and [Shannon, 1948] entropy
recovered for specific values of  and  .

8. Cont…
 Uncertainty theory was founded by [Liu, 2007] and refined by
[Liu, 2010] .
 Uncertainty theory was widely developed in many disciplines,
such as uncertain process [Liu, 2008], uncertain calculus,
uncertain differential equation [Liu, 2009], uncertain risk analysis
[Liu, 2010a], uncertain inference [Liu, 2010b], uncertain statistics
[Liu, 2010c] and uncertain logic [Li and Liu, 2009].
 [Liu, 2009] proposed the definition of uncertain entropy resulting
from information deficiency. [Dai and Chen, 2009] investigated
the properties of entropy of function of uncertain variables and
introduced maximum entropy principle for uncertain variables.
 Inspired by two parametric probabilistic entropy, this paper
introduces a new two parameter entropy in the framework of
uncertain theory and discusses its properties.
8

9. 2. Preliminaries
Let  be non-empty set and  is a  algebra over . Each element  is called
an event. Uncertain measure M was introduced as a set function satisfying the
following five axioms [Liu, 2007]:
Axiom 1. (Normality Axiom) M{ }=1 for universal set  .
Axiom 2. (Monotonicity Axiom) M{ 1 } M{ 2 } whenever 21  .
Axiom 3. (Self-Duality Axiom) M{  }+ M{ c
 } = 1 for any event  .
Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of
events  i , we have
M{



1i
i } 

1i
M i
Axiom 5. (Product Measure Axiom) Let k be non empty sets on which Mk are
uncertain measures nk ,...,2,1 , respectively. Then product uncertain measure on
the product  algebra n 21 satisfying
M{

n
k
k
1
} nk1
min 

1i
Mk k .
Where kk  , nk ,...,2,1 . 9

10. Cont…
Definition 2.1 [Liu, 2007] Let  be non-empty set and  is a  algebra over  ,
and M an uncertain measure. Then the triplet ( ,, M) is called an uncertainty
space.
Definition 2.2 [Liu, 2007] An uncertain variable is a measurable function from an
uncertainty space ( ,, M) to the set of real numbers.
Definition 2.3 [Liu, 2007] The uncertainty distribution  of an uncertain variable
 is defined by
 )(x M{ x }.
Theorem 2.1 (Sufficient and Necessary Condition for Uncertainty distribution
[Peng and Iwamura, 2010]) A function  is an uncertainty distribution if and only
if it is an increasing function except 0)(  x and 1)(  x .
Example 2.1 An uncertain variable is called normal if it has a normal uncertainty
distribution
1
3
)(
exp1)(
















 


 xe
x
denoted by N ( ,e ) where e and σ are real numbers with σ > 0. Then we recall the
definition of inverse uncertainty distribution as follows.
10

11. Cont…
Definition 2.6 (Independence of uncertain variable [Liu, 2007] )The uncertain
variables n ,, 21 are said to be independent if
M









m
i
iB
1
 = ni1
min M  iB
for Borel sets nBBB ,,, 21  of real numbers.
Theorem 2.2 [Liu, 2007] Assume ξ1, ξ2, . . . , ξn are independent uncertain
variables with regular uncertainty distribution Φ1,Φ2, . . . ,Φn, respectively. If
n
f : is a strictly increasing function, then uncertain variable ξ = f (ξ1, ξ2, . . .
, ξn) has inverse uncertainty distribution
10)),(,),(),(()( 11
2
1
1
1
 
ttttft n
Theorem 2.3 [Liu, 2007] Assume ξ1, ξ2, . . . , ξn are independent uncertain
variables with regular uncertainty distribution Φ1,Φ2, . . . ,Φn, respectively. If
n
f : is a strictly decreasing function, then uncertain variable
ξ = f (ξ1, ξ2, . . . , ξn) has inverse uncertainty distribution
10)),1(,),1(),1(()( 11
2
1
1
1
 
ttttft n
11

12. Cont…
Theorem 2.4 ([Liu, 2007]) Let n ,, 21 be independent uncertain variables with
regular uncertainty distribution ,,,, 21 n  respectively. If n
f : be strictly
decreasing function with respect to mxxx ,, 21 and strictly increasing function with
,,, 21 nmm xxx  then ),,( 21 nf   is uncertain variable with uncertainty
distribution with inverse uncertainty distribution
.10)),1(,),1(),(,),(),(()( 11
1
11
2
1
1
1
 


ttttttft nmm 
12

13. 3. Two Parameter Entropy
In this section, we will introduce the definition and results of two parameter
entropy of uncertain variable. For the purpose, we recall the entropy of uncertain
variable proposed by [Liu, 2009].
Definition 3.1 ([Liu, 2009]) Suppose that ξ is an uncertain variable with
uncertainty distribution . Then its entropy is defined by
H(ξ)= dxxS


 ))((
Where )).(1ln())(1()(ln)())(( xxxxxS 
We set 00ln0  throughout this paper. By the enlightenment of [Tsallis, 1988]
entropy, [Liu et al. , 2012] defined the single parameter entropy.
Definition 3.2 ([Liu et al. , 2012]) Suppose that ξ is an uncertain variable with
uncertainty distribution . Then its entropy is defined by
)(qH = dxxSq


 ))((
Where
)1(
]))(1())((1[
))((



qq
xx
xS
qq
q
q is a positive real number . Further, it can be seen that as 1q we have
)(qH  H(ξ).
13

14. Cont…
Now, in the light of two parametric probabilistic entropy function for
case  ,C given in Eq.(1.1) a new two parameter entropy of
uncertain variable can be proposed.
Definition 3.3 Suppose that ξ is an uncertain variable with uncertainty
distribution . Then its entropy is defined by
)(, H = dxxS


 ))((,
Where
)(
))((1()())((1()(
))((,






xxxx
xS
, are positive real number . Further, it can be seen that as 1  we
have
)(, H  H(ξ).
In subsequent examples two parameter entropy formulae for uncertain
variable ξ with different uncertainty distributions are obtained.
14

15. Cont…
Example 3.1 Let ξ be uncertain variable with uncertain distribution






ax
ax
x
,1
,0
)(
we have 0)(, H
Example 3.2 Let ξ be uncertain variable with uncertain distribution











ax
bxa
ab
ax
ax
x
,1
,
,0
)(
then ξ is called linear uncertain variable and is denoted by L (a, b) where
a and b are real numbers a< b.
Then two parameter entropy of linear uncertain variable is
)1)(1(
)(2
)(,





ab
H
15

16. Cont…
Example 3.3 Let ξ be uncertain variable with uncertain distribution


















cx
cxb
bc
bcx
bxa
ab
ax
ax
x
,1
)(2
2
)(2
,0
)(
then ξ is called Zig-Zag uncertain variable and is denoted by Z (a, b, c)
where a, b and c are real numbers a< b<c. Then two parameter entropy
of Zig-Zag uncertain variable is
)1)(1(
)(2
)(,





ac
H
16

17. 4. Properties of Two Parameter Entropy
Theorem 4.1 Let ξ be uncertain variable and ‘c’ be a real number. Then
)()( ,,   HcH 
that is two parameter entropy is invariant under arbitrary translations.
Theorem 4.2 Assume ξ is an uncertain variables with regular
uncertainty distribution Φ. If entropy )(, H exists, then
dttStH )()()( '
,
1
0
1
,   


Where 




 








1111
,
)1()1(
)(
1
)(
tttt
tS
17

18. 18
Cont…
Theorem 4.3 Let ξ1, ξ2, . . . , ξn are independent uncertain variables with
regular uncertainty distribution Φ1,Φ2, . . . ,Φn, respectively. If n
f :
is a strictly monotone function, then the uncertain variable ξ = f (ξ1, ξ2, . .
. , ξn) has an entropy
.)())(,),(),(()( ,
1
0
11
2
1
1, dttStttfH n    


Theorem 4.4 Let ξ and η be independent uncertain variables. Then for
any real numbers a and b,
.][||][||][ ,,,   HbHabaH 

19. 19
Conclusions
In this paper, the entropy of uncertain variable and its properties are recalled. On
the basis of the entropy of uncertain variable and inspired by two parameter
probabilistic entropy (1.9), two parameter entropy of uncertain variable is
introduced and explored its several important properties. Two parameter entropy of
uncertain variable Proposed in this paper, makes the calculation of uncertainty of
uncertain variable more general and flexible by choosing an appropriate values of
parameters α and β. Further, the results obtained by [Liu et al., 2012] for single
parameter entropy of uncertain variable and [Dai and Chen, 2009] for entropy of
uncertain variable are the special cases of results obtained in this paper.

20. 20
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21. 21
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22. 22