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IOSR Journal of Electronics and Communication Engineering (IOSR-JECE) e-ISSN: 2278-2834,p- ISSN: 2278-8735.Volume 8, Issue 2 (Nov. - Dec. 2013), PP 52-57 www.iosrjournals.org

‘Useful’ R-norm Information Measure and its Properties D.S. Hooda 1, Keerti Upadhyay 2 and D.K.Sharma3 Jaypee University of Engineering and Technology A.B. Road, Raghogarh – 473226 Distt. Guna – M.P. (India).

Abstract : In the present communication, a new ‘useful’ R-norm information measure has been defined and characterized axiomatically. Its particular cases have been discussed. Properties of the new measure have also been studied. Keywords: non-additivity, R-norm entropy, stochastic independence, utility distribution.

I.

Introduction

Let us consider the set of positive real numbers, not equal to 1 and denote this by    R : R  0, R 1 . Let 

defined as

 n with n  2 is the set of all probability distributions

n   P   p1 , p 2 ,, p n , pi  0 , for each i and  pi  1. i 1  

[1] studied R-norm information of the distribution P defined for R   by: 1 R   n R  R  H R P   1    pi   R  1   i 1   

(1) 

The R-norm information measure (1) is a real function  n   defined on  n ,where n  2 and R is the set of real positive numbers. The measure (1) is different from entropies of [2], [3], [4] and [5]. The main property of this measure is that when R  1 (1) approaches to Shannon’s entropy and when R   , H R P   1  max pi , where i  1,2,, n . The measure (1) can be generalized in so many ways. [6] Proposed and characterized the following parametric generalization of (1.1):  R   n  2  H R P   1  p  i R    2   i 1   R

   

2  R

   , 0    1,  

R 0  1

(2)

The above measure (2) was called generalized R-norm information measure of degree β and it reduces to (1) when β=1. Further when R=1 (2) reduces to: 1  1   H 1 P   1   pi2    2   

In case

 

   

2 

  , 0   1  

(3)

1 reduces to: 2

1   n  1    (4)    H P  1  p   ,    1   1   i 1 i   2   This is an information measure which has been given by [7]. It can be seen that (4) also reduces to Shannon’s entropy when   1 .

[8] Proposed and studied the following parametric generalization of (1):

 ,

HR

 R R   n 2   ( P)  1  p  i R    2   i 1  

   

2   R

   ,   1, 0    1, R 0  1, 0  R    2  

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(5) 52 | Page


‘Useful’ R-norm Information Measure and its Properties They called (5) as the generalized R-norm information measure of type  and degree  . (5) reduces to (2) when   1 and it further reduces to (1) when   1 .Recently, [9] have applied (5) in studying the bounds of generalized mean code length. In order to distinguish the events E1 , E2 ,, En with respect to a given qualitative characteristic of physical system taken into account, we ascribe to each event E i a non- negative number uEi   ui  0 directly proportional to its importance. We call u i , the utility or importance of event E i where probability of occurrence is

p i .In general u i is independent of p i (see [10]).

[11] characterized a quantitative-qualitative measure which was called ‘useful’ information by [10] of the experiment E and is given as: H P;U   H  p1 , p2 ,, pn ; u1 , u 2 ,, u n 

  u i pi log pi ,

u i  0,

0  pi  1,

p

i

(6)

1

Later on [12] characterized the following measure of ‘useful’ information:

H P;U  

  u i pi log pi

u

i

(7)

pi

Analogous to (1) we consider a measure of ‘useful’ R-norm information as given below:

 R R    u i pi H R P;U   1 R  1    u i pi 

   

1

R

 ,  

(8)

where U  u1 , u 2 ,u n  is the utility distribution and u i  0 is the utility of an event with probability p i . It may be noted that if

R  1, then (8) reduces to (7). Further let

 n be a set of utility distributions

s.t. U  n is utility distribution corresponding to P   n . In the present paper we characterize the ‘useful’ R-norm information measure (8) axiomatically in section 2. In section 3 we study the properties of the new measure of ‘useful’ R-norm information measure.

II.

Axiomatic Characterization

Let Sn  n    R , n  2,3, and G n be a sequence of functions of  n

'

u i s , i  1,2,, n , defined over

'

pi s and

S n satisfying the following axioms:

n

Axiom 2.1. Gn  P : U   a1  a2  h  pi , ui  , where

a1 and a 2 are non zero constants, and

i 1

p, u  J  0,1  0, 0, y ;0  y  1  , y  : 0  y   . This axiom is also called sum property. Axiom 2.2. For P   n ,U  n , P   m , and U   m , Gmn satisfies the following property: Gmn PP  : UU   Gn P : U   Gm P  : U  

1 Gn P : U Gm P  : U . . a1

h p, u  is a continuous function of its arguments p and u.   Axiom 2.4. Let all pi s and u i s are equiprobable and of equal utility of events respectively, then Axiom 2.3.

1 R  where 1 R  1  R Gn  , , , u, , u   1  n , n n R  1    

n  2,3,, and R 0  1

First of all we prove the following three lemmas to facilitate to prove the main theorem: www.iosrjournals.org

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‘Useful’ R-norm Information Measure and its Properties Lemma 2.1. From axiom 2.1 and 2.2, it is very easy to arrive at the following functional equation:

  h p n

m

i 1

j 1

i

  a2 p j , u i u j     a  1

m  n ,   h pi , u i  h p j , u j  j 1  i 1

where  pi , ui ,  p j , u j   J for i  1,2,, n and

(9)

j  1,2,, m.

Lemma 2.2. The continuous solution that satisfies (9) is the continuous solution of the functional equation:   a2  , (10)       h pp , uu



 a1

h p, u h p , u 

,

Proof: Let a, b, c, d and a, b, c, d  be positive integers such that 1  a  a,1  b  b,1  c  c , and 1  d  d  . Setting n  a  a  1  c  c  1 and m  b  b  1  d   d  1, pi 

ui 

1 i  1,2,, a  a , a

p a  a1 

a , a

1 c  i  1, 2,, c  c  , ucc 1  , c c

pj 

1 b  j  1, 2,, b  b  , pb b1  , b b

 1  j  1,2,, d   d , u d d 1  d , d d From equation (9) we have:         a  a b  bh 1 , 1   b  bh a , c   a  a h b , d   h a b , c d   ab cd   ab cd   ab cd   ab cd    a2   1 1  a  c    1 1   b d    a  a h ,   h ,  b  b  ' h ,   h ,    a a c a c b d        b d    1  u j 

Taking a  b  c  d   1in (11), we get:  1 1    a2   1 1   1 1  h , h ,  . h ,     ab cd   a1   a c   b d 

(11)

(12)

Taking a  c  1 in (11) and using (12), we have:

 b d     a 2 h ,     ab cd   a1

  1 1   b d   h , h ,  .  a c  b d 

Again taking b  d   1 in (11) and using (12), we get:  a c    a2   1 1   a c  h , h ,  h ,     ab cd   a1   b d   a c  Now (11) together with (12), (13) and (13) reduces to:  a b c d     a 2   a  c    b d   h , h , h ,      ab cd   a1   a c   b d  a c b d  p,  u,  p ,  u  in (15), we get the required results (10). Putting a c b d

(13)

(14)

(15)

Next we obtain the general solution of (10). Lemma2.3. One of the general continuous solution of equation (10) is given by:   a1  p  u    h p, u     a 2  pu  and h p, u   0

1

Proof: Taking g  p, u     a 2  a  1

, where   0,   0

(16) (17)

 in (10), we have:   h p , u  

g  pp , uu   g  p, u g  p , u 

(18)

The most general continuous solution of (18) (refer to [13]) is given by: www.iosrjournals.org

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‘Useful’ R-norm Information Measure and its Properties  p  u  g  p, u     pu    

1

,   0 and   0

(19)

and g  p, u   0

(20)

On substituting g  p, u     a 2 h p, u  in (19) and (20) we get (16) and (17) respectively. This proves the    a1 

lemma 2.3 for all rationals p  0,1 and u  0 , However, by continuity, it holds for all reals p  0,1 and u  0 . Theorem2.1. The measure (8) can be determined by the axiom 2.1 to 2.4. Proof: Substituting the solution (16) in axiom 2.1 we have: 1  n  p  u    (21) Gn P;U   a1 1    i i  ,   0   p u i 1  i i    1 Taking pi  and u i  u for each i in (21) we have: n 1   1   1 1     , Gn  , , , u, , u   a1  1  n u   n n   

n  2,3,  ,

(22)

Axiom (2.4) together with (22) gives:  a1 1  n  

1 

u

 1 

1 R      R 1  n R   R  1   

It implies a1 

R , R 1

  R ,  1

Putting these values in (21) we have   n R    u i pi R  G n P;U   1   i n1  R 1     u i pi  i 1  

 H R P;U 

     

1

R

      

Hence this completes the proof of theorem 2.1. Particular cases: (a) When utilities are ignored i.e. u i  1 for each i, (8) reduces to (1). (b) Further R  1 , (1) reduces to Shannon’s entropy [13].

III.

Properties of ‘useful’ R-norm Information Measure

The ‘useful’ R-norm information measure H R P;U  satisfies the following properties:

Property 3.1. H R P;U  is symmetric function of their arguments provided that the permutation of p i  s and

 u i s are taken together. H R  p1 , p2 ,, pn1 , pn ; u1 , u 2 ,u n1 , u n   H R  pn , p1 , p2 ,, pn1 ; u n , u1 , u 2 ,, u n1 

Property 3.2. H  1 , 1 ;1,1  1 R 8 8

    u i pi R Proof: R  H R P;U   1    u i pi R 1    

for

   

1

R

    

i  1,2

 R  u1 p1 R u 2 p2 R  H R P;U   1    u p R 1  u 2 p2  1 1 

   

1

R

   

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‘Useful’ R-norm Information Measure and its Properties Taking p1  1 , 8

p2 

1 , 8

u 2  1 and

u1  1,

  1  2     8 1 1  2 H R  , ;1,1  1      8 8 1    1     8 

R2

   1     Property 3.3. Addition of two events whose probability of occurrence is zero or utility is zero has no effect on useful information, i.e. 1   8 1 8

2

      

1

2

H R  p1 , p2 ,, pn ,0; u1u 2 ,u n1   H R  p1 , p2 ,, pn ; u1 , u 2 ,, u n   H n  p1 , p2 ,, pn1 ; u1 , u 2 ,, u n ,0.

Proof: Let us consider 1  R R R R  u n pn 0 R .u n 1  u 2 p2 R   u1 p1   H R  p1 , p 2 , p n ,0; u1 , u 2 , , u n 1   1      R 1  u 2 p2 u n pn 0.u n 1    u1 p1   

 H R P;U 

Similarly we can prove that H n  p1 , p2 ,, pn , pn1 ; u1 , u 2 ,, u n , u n1   H R P;U  Property 3.4. H R P;U  satisfies the non-additivity of the following form: R 1 H R P;U H R Q;V  R where P  Q   p1q1 ,, p1qm , p2 q1 ,, p2 qm , pn q1 ,, pn qm  , and H R P  Q;U  V   H R P;U   H R Q;V  

U  V  u1v1 ,, u1vm , u 2 v1 ,, u 2 vm , u n v1 ,, u n vm 

R H R P;U H R Q;V  R 1 1 1    R R R R R R    u i pi   R    v j q j   R    u i pi  1  1  1 R  1    u i pi   R  1    v j q j   R  1    u i pi     

Proof: R.H.S= H R P;U   H R Q;V  

 R R    u i pi  1 R  1    u pi 

   

1

R

 v jq j R 1   v jq j 

   u i pi R R   1    u I pi R 1   

   

1

R

   

 q jv j   q jv j 

1

R

R

   

  u i pi R 1    u I pi  1

R

 H R P  Q;U  V  = L.H.S.

   

1

R

 q jv j R   q jv j 

   

1

R

   

1

R

  R  1    v j q j    v j q j   

  u i pi R    u I pi 

 m R  n   ui v j  pi q j      R   i 1 j 1   n m 1   R  1      ui v j pi q j   i 1 j 1 

     

1

   

R

1

R

 q jv j R   q jv j 

   

    1

1

R

R

   

   

      

Property 3.5. Let Ai , A j be two events having probabilities p i , p j and utilities u i , , u j respectively, then we define the utility u of the compound event Ai  A j as:

u Ai  A j  

u i pi  u j p j

pi  p j Theorem 3.1 Under the composition law (23), the following holds:

(23)

 p  p H R  p1 , p2 ,, pn 1 , p, p; u1u2 , , un 1 , u, u   n H R  P,U    p  p  H R  , ; u , u   ,  p  p p  p      p u  p u where pn  p  p and un  n 1

p  p

Proof:

    n 1 H R  p1 , p 2 , , p n 1 , p , p ; u1 , u 2 , , u n 1 , u , u 

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‘Useful’ R-norm Information Measure and its Properties  n H R  p1 , p 2 , , p n 1 ; u1, u 2 , u n 1  

n H R

1  u p  R R  u p  R  R  1    R 1  u p     u p   

 R R    p  p      u    u  p   p   R    p   p     p   p     R 1   p  p    u    u      p   p   p   p      

      

1

R

    p   p    

  p p   n H R   p   p H R  , ; u , u         p  p p  p   This completes the proof of theorem 3.1.

IV.

Conclusion

R- norm information measure is defined and characterized when the probability distribution P belong to R- norm vector space. This is a new addition to the family of generalized information measures. In present paper we have considered that physical system has qualitative characterization in addition to quantitative and have defined and characterized a new measure R-norm information measure . This measure can further be generalized in many ways and can be applied in source coding when source symbols have utility also in addition to probability of occurrence.

References [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [11]. [12]. [13].

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