paper-1-dec-it-2013

IJITE Vol.01 Issue-07, (December, 2013) ISSN: 23211776

International Journal in IT and Engineering

http://www.ijmr.net

JOINT AND CONDITIONAL R-NORM INFORMATION

MEASURE

*Dr Meenakshi Darolia

The present paper depicts the joint and conditional probability distribution of two

random variables and having probability distributors P and Q over the Sets

{ }nxxxX ..,.,, 21= and { }myyyY ..,.,, 21= respectively. Then the R-norm information

of the random variables is denoted by ( ) ( )PHH RR = and ( ) ( )QHH RR = , where

( ) nixPp ii ,...,2,1, === , ( ) mjyPp ji ,...,2,1, ===

are the probabilities of the possible values of the random variables. Similarly, we

consider a two-dimensional discrete random variable (,) with joint probability

distribution ( )n1,12,11 ... = ,

where ( ) m2...., 1,jn,1,2.....,i ,, ===== jiij yxP is the joint probability for the

values ( )ji yx , of ( )., We shall denote conditional probabilities by pij and qij such

that ijijijij pqqp == And = =

==m

j

n

j

ijiiji qandp1 1

.

*Assistant Professor, Isharjyot Degree College Pehova



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DEFINITION: The joint R-norm information measure for + RR and is given by

( )

= ==

Rm

j

R

ij

n

i

RR

RH

1

11

11

, (1.1)

Proposition 1: ( ) ,RH is symmetric in and .

Proof: The joint R-norm information measure is defined by

( )

= ==

Rm

j

R

ij

n

i

RR

RH

1

11

11

,

( )

==

= ==

Rm

j

ji

Rn

i

yxPR

R

1

11

,11

( )

==

= ==

Rm

j

i

R

i

Rn

i

yPxPR

R

1

11

)(11

( )

==

= ==

Rm

j

i

R

i

Rn

i

xPyPR

R

1

11

)(11

( )

==

= ==

Rm

j

ji

Rn

i

xyPR

R

1

11

,11

( ) ,11

1

11

R

Rm

j

R

ji

n

i

HR

R=

= ==

This implies that ( ) ,RH is symmetric in , .



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Proposition 2: If and are stochastically independent. Then the following holds

( ) ( ) ( ) ( ) ( ) RRRRR HHR

RHHH

1,

+= (1.2)

Proof: Since the joint R-norm information measure for + RR and is given by

( )

= ==

Rm

j

R

ij

n

i

RR

RH

1

11

11

,

( )

==

= ==

Rm

j

ji

Rn

i

yxPR

R

1

11

,11

( )

==

= ==

Rm

j

i

R

i

Rn

i

yPxPR

R

1

11

)(11

( )

==

= ==

Rm

j

i

R

i

Rn

i

yPxPR

R

1

11

)(11

(1.3)

Since and are stochastically independent, thus (1.3) becomes

( )

=

=

=

==

Rm

j

j

RRn

i

i

R

R yPxPR

RH

1

1

1

1

)(.)(11

,

( ) ( )

= RR H

R

RH

R

R

R

R

R

R 11

11

11

( ) ( ) ( ) ( ) RRRR HHR

RHH

1+= Thus finally

( ) ( ) ( ) ( ) ( ) RRRRR HHR

RHHH

1,

+= (1.4)



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In the limiting case R1 we find the additive form of Shannons information

measure for independent random variables.i.e. when R1 in (1.4), then we get

( ) ( ) ( ) ( ) ( ) RRRRR HHHHH1

11,

+= , ( ) ( ) ( ) RRR HHH += ,

To construct a conditional R-norm information measure we can use a direct and an

indirect method. The indirect method leads to next definition

DEFINITION: The average subtractive conditional R-norm information of

given for + RR and is defined as

( ) ( ) RRR HHH = ,)/(

= ===

Rn

i

R

i

Rm

j

R

ij

n

i

pR

R

R

R1

1

1

11

11

11

(1.4)

=

= ==

Rn

i

m

j

R

ij

Rn

i

R

ipR

R

1

1 1

1

11 (1.5)

= == =

Rn

i

R

i

Rn

i

m

j

R

ij pR

R

R

R1

1

1

1 1

11

11

(1.6)

( ) RR HH += )/(

Thus ),( RH ( ) RR HH += )/(



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A direct way to construct a conditional R-norm information is the following.

DEFINITION: The average conditional R-norm information of given

is for + RR defined as

( )

= ==

RRm

j

iji

n

i

R qpR

RH

1

11

* 11

/ (1.7)

Or alternatively

( )

= ==

RRm

j

iji

n

i

R qpR

RH

1

11

** 11

/ (1.8)

The two conditional measure given in (1.7) and (1.8) differ by the way the

probabilities ip are incorporated. The expression (1.7) is a true mathematical

expression over, whereas the expression (1.8) is not.

Theorem: If and are statistically independent random variables then for

RR+

(1)

=

===

Rm

j

R

j

Rn

i

R

i

Rn

i

R

iR qppR

RH

11

11

1

1

.1

)/(

(2) ( ) ( ) ( ) ( ) ( ) RRRRRR HHR

RHHHH

1,)/(

==

(3) ( ) )(/* RR HH =



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(4) ( ) )(/** RR HH =

Proof: (I) Since the average subtractive conditional R-norm information of

given is for + RR defined as

)/( RH

=

= ==

Rn

i

m

j

R

ij

Rn

i

R

ipR

R

1

1 1

1

11 (1.9)

Substitute jijij qp= in (1.9), we get

)/( RH

=

= ==

Rn

i

m

j

R

jij

Rn

i

R

i qppR

R

1

1 1

1

1

)(1

(1.10)

Since and are stochastically independent. Thus (1.10) becomes

)/( RH

=

===

Rm

j

R

j

Rn

i

R

i

Rn

i

R

i qppR

R

1

1

1

1

1

1

.1

(II) Since we know that if and are stochastically independent. Then the

following holds

( ) ( ) ( ) ( ) ( ) RRRRR HHR

RHHH

1,

+=

( ) ( ) ( ) ( ) ( ) RRRRR HHR

RHHH

1,

= (1.11)



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And we know

( ) ( ) RRR HHH = ,)/( (1.12)

Using (1.12) in (1.11), we get

( ) ( ) ( ) ( ) ( ) RRRRRR HHR

RHHHH

1,)/(

==

(III) Since the average conditional R-norm information of given for + RR

and is defined as

( )

= ==

RRm

j

iji

n

i

R qpR

RH

1

11

* 11

/ (1.13)

Substitute jji qq = in (1.13), we get

( )

= ==

RRm

j

ji

n

i

R qpR

RH

1

11

* 11

/

)(11

1

1

RRRm

j

j HqR

R=

= =

=

=

11

n

i

ip

Hence ( ) )(/* RR HH = (1.14)

(IV) Since the average conditional R-norm information of given for + RR



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and is defined as

( )

= ==

RRm

j

iji

n

i

R qpR

RH

1

11

** 11

/ (1.15)

Substitute jji qq = in (1.15), we get

( )

= ==

RRm

j

ji

n

i

R qpR

RH

1

11

** 11

/

)(11

1

1

RRRm

j

j HqR

R=

= =

=

=

11

n

i

ip

Hence ( ) )(/** RR HH =

From this theorem we may conclude that the measure )/( RH , which is obtained

by the formal difference between the joint and the marginal information measure,

does not satisfy requirement (I). Therefore it is less attractive than the two other

measure. In the next theorem we consider requirement (II), for the conditional

information measures ( ) /* RH and ( ) /**

RH .

Theorem: If and are discrete random variables then for + RR then the

following results hold.

(I) ( ) ( ) RR HH /* (II) ( ) ( ) RR HH /

**



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(III) ( ) ( ) // *** RR HH (IV) ( ) ( ) ( ) RRR HHH //***

The equality signs holds if and are independent. Proof:

(I) Here we consider two cases: Cases I: when R < 1

We know by [4] that for 1>R .

====

RR

ij

m

j

n

i

RRn

i

ji

m

j

xx

1

11

1

11 (1.16)

Setting 0= jijix in (1.16), we have

====

RR

ij

m

j

n

i

RRn

i

ji

m

j

1

11

1

11

(1.17)

Since =

=m

j

ijiq1

and jiiji qp= (1.18)

Using (1.18) in (1.17), we get

===

Rm

j

R

iji

n

i

RR

j

m

j

pqq

1

11

1

1

)( (1.19)

It can be written as



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===

Rm

j

R

ji

n

i

i

RR

j

m

j

qpq

1

11

1

1

===

Rm

j

R

ji

n

i

i

RR

j

m

j

qpq

1

11

1

1

===

Rm

j

R

ji

n

i

i

RR

j

m

j

qpq

1

11

1

1

11 (1.20)

We know 01

>R

R if R > 1

Multiplying both sides of (1.20) by 1R

R, we get

===

Rm

j

R

ji

n

i

i

Rn

i

R

j qpR

Rq

R

R

1

11

1

1

11

11

(1.21)

But )/(11

1

11

RRm

j

R

ji

n

i

i HqpR

R +

==

=

and

)(11

1

1

RRm

j

R

j HqR

R=

=

Thus (1.21) becomes

( ) ( ) RR HH /* for R > 1 (1.22)



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Cases II: when 0



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( )

===

RR

iij

m

j

n

i

RR

j

m

j

pqq

1

11

1

1

11 (1.25)

We know 01

1

From Jensens inequality for R > 1,we find

Rj

R

iji

n

i

R

iji

n

i

qqpqp =

== 11

(1.28)

After summation over j and raising both sides of (1.28) by powerR

1, we have

R

R

j

m

j

RR

ij

m

j

i

n

i

qqp

1

1

1

11

===

R

R

j

m

j

RR

ij

m

j

i

n

i

qqp

1

1

1

11

===

R

R

j

m

j

RR

ij

m

j

i

n

i

qqp

1

1

1

11

11

===

(1.29)

Using 01

>R

Ras R > 1, Thus (1.29) becomes

== =

Rm

jj

R

j

Rn

i

m

j

R

jii qR

Rqp

R

R

1

1

1

1 1

11

11

(1.30)



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But )/(11

1

1 1

RRn

i

m

j

R

jii HqpR

R ++

= =

=

And )(11

1

1

RRm

jj

R

j HqR

R=

=

Thus (1.30) becomes

( ) ( ) RR HH /** for R > 1 (1.31)

Case II: when 0 < R



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Using 01

1 (1.36)



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R

R

ij

m

j

i

n

i

RR

ji

m

j

i

n

i

qpqp

1

11

1

11

====

R

R

ij

m

j

i

n

i

RR

ji

m

j

i

n

i

qpqp

1

11

1

11

11

====

(1.37)

Using 01

>R

R if R >1, then (1.37) becomes

= ===

Rn

i

m

j

R

jii

Rm

j

R

ji

n

i

i qpR

Rqp

R

R

1

1 1

1

11

11

11

(1.38)

But )/(11

1

11

RRm

j

R

ji

n

i

i HqpR

R +

==

=

And )/(11

1

1 1

RRn

i

m

j

R

jii HqpR

R ++

= =

=

Thus (1.38) becomes ( ) ( ) // *** RR HH for R > 1

(1.39) Case II: when 0 < R < 1 We know from Jensens inequality

R

R

ij

m

j

i

n

i

RR

ji

m

j

i

n

i

qpqp

1

11

1

11

====

for 0 < R < 1 (1.40)



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R

R

ij

m

j

i

n

i

RR

ji

m

j

i

n

i

qpqp

1

11

1

11

====

R

R

ij

m

j

i

n

i

RR

ji

m

j

i

n

i

qpqp

1

11

1

11

11

====

(1.41)

Using 01

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