Lower Bounds of Functional Mean Code Word Length in Fuzzy Set

9

I

https://doi.org/10.22214/ijraset.2021.32951

January 2021

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue I Jan 2021- Available at www.ijraset.com

Lower Bounds of Functional Mean Code Word Length in Fuzzy Set Sangeeta Pandey1, Dr. R. P. Dubey2, Dr. P. Jha3 1

2

Department of Mathematics, Rajiv Gandhi Govt. P.G.College,Ambikapur (C.G.)-India Department of Mathematics, Dr. C.V. Raman university Kargi Road, Kota, Bilaspur (C.G.)- India 3 Department of Mathematics, Govt. D. K. P. G. College, Balodabazar (C.G.)-India

Abstract: In the present paper, we find lower bound of functional mean code word length in fuzzy set with existing knowledge of different code word length. We know that lower bound of functional mean code word length is measure of fuzzy entropy which satisfy all different properties of itself. By finding lower bounds of different mean code word length, different measure of entropies in fuzzy set can be found which are very important in present .

Keywords: Mean code word length, Kraft’s inequality, measure of entropy and fuzzy entropy, measure of directed divergence and fuzzy directed divergence, I. INTRODUCTION Let (x1, x2,x3,..........xn) be n inputs which have to be encoded in terms of an alphabet of size D. Let l1,l2,l3,.......,ln be the n codeword lengths and let p1, p2,.........,pn be the probabilities ,the arithmetic mean L of codeword length is n

L   li pi

........................ (1)

i 1

n

Shannon showed that the minimum value of L subject to Kraft’s inequality

D

 li

 1 .....(2)

i 1

n

lies between S(P) and S(P)+1,where S(P) is given by S ( P )  

 p log i

D

pi

i 1

After this Campbell considered more general exponential mean codeword length (1 ) li    L  log   pi D   ,   0,   1 1   and showed that subject to (2) the minimum value of L lies between R ( P ) and

R ( P )  1 where R ( P) 

n 1 log  pi ,   0,   1 1 i 1

Extending the scope of study, Kapur has several mean codeword length .

One of Kapur’s mean codeword length is

 n f ( pi ) D ( 1)li   1 Lf  log  i 1 n  1  f ( pi )   i 1

    

Here we introduced lower bound of Kapur’s mean codeword length with the help of Kraft’s inequality in terms of fuzzy set.

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MAIN RESULT f

LOWER BOUND OF KAPUR ‘S MEAN CODEWORD LENGTH L SUBJECT TO KRAFT’S

INEQUALITY

We know that according to Kapur’s mean codeword length given in equation (1) ,the Fuzzy Mean Codeword length is

 n  f (  A ( xi ))  f (1   A ( xi )) D ( 1)li   1 Lf  log  i 1 n  1   f (  A ( xi ))  f (1   A ( xi ))   i 1

   ............... (3)   n

Now, we find the lower bound of this mean codeword length with respect to Kraft’s inequality

D

 li

 k 1

i 1

........................................ (4) Suppose

 f ( A ( xi ))  f (1   A ( xi ))  Ai

& D

 li

 yi

 n  Ai yi(1 )    1 f  ..........(5) Then eqn(5.2) & eqn(5.3) become L  log  i 1 n  1  Ai    i 1  n

&

y

i

 k 1

i 1

Applying Lagrange’s method, we get,

   n  Ai yi(1 )      n   1        yi  k    0 log  i 1 n yi    1   i 1  Ai      i 1    n

  Ai yi 

i 1 n

 0 1 i

Ay i

i 1

n

 Ai yi    Ai yi1 i 1 n

 Ai D li    Ai yi1 i 1

1

A  n   i l     Ai yi1  D i  i 1  ©IJRASET: All Rights are Reserved

1 

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1

1

1  1  l1

1  2  l2

1 

A A A  n   1l  2 l  ............  i  l     Ai yi1  1 2 i D D D  i 1  n



A D



A D

 ............ 

Ai  D  li

A

1 

1 

i

i 1 n

D

 li

i 1

n



1  1  l1

A D



1  2  l2

A D

1 

 ............ 

Ai  D  li

A

1 

i

i 1

k

1

Minimum value of D

 li

 yi 

kAi  n

A

1 

i

i 1

f

Substituting this value in equation(5.4), we get minimum value of L

   n  kA 1  Ai  n i  1 i 1   Ai    i 1 1 Min Lf  log  n  1  Ai   i  1    

1

    

1  n 1    Ai k Ai  1  i 1 f Min L  log   1 n  1    A 1    i 1 i  

 n 1   Ai  k 1 1 f Min L  log  i 1   1   n 1 1    Ai       i 1 f

Min L 

           

      

      

  n 1   1 log    Ai     log k   1   i 1  

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Min

Lf 

  n 1  log   Ai    log k   1  i 1 

1 n  f (  ( x ))  f (1   ( x ))    A i A i  Min L  log    1 n i 1   f 

f

f

    1 log n   1 k  1 

f

Thus Min L lies between M  (  A ( xi )) and M  (  A ( xi )) +1 1  n  f (  A ( xi ))  f (1   A ( xi ))     f Where M  (  A ( xi ))  log  i 1  1  n  

which is generalized measure of entropy of order

   ..........(6)   

 for any function in terms of fuzzy

set.

f

Generalized measure of entropy M  (  A ( xi )) has different value for different function of

 A ( xi ) .

Suppose f (  A ( xi )) 

 A ( xi ) ,then by eqn(6)

1  n     ( x )  (1   ( x ))   A i  A i  M f (  A ( xi ))  log  i 1  1  n  

   ........ (7)   

  1,   1 III. 1)

PROPERTIES OF

M f (  A ( xi ))

M f (  A ( xi )) is the minimum value of an exponentiated mean so it will always be non – negative.

2)

f 

M (  A ( xi )) is minimum iff A is a non fuzzy set. f

For  A ( xi )  0 , M  (  A ( xi ))  0 and when

 A ( xi )  1 we get M f (  A ( xi ))  0 f

3) When A is most fuzzy set, and then M  (  A ( xi )) is maximum.

For maximum,

M f ( A ( xi )) 0  A ( xi )

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue I Jan 2021- Available at www.ijraset.com We can write eqn (5.6) as

M f (  A ( xi )) 

1     n log    A ( xi ))  (1   A ( xi ))     log n   1  i 1   1

 1  1  1     (  ( x )  (1   ( x )) )  ( x ))  (1   ( x ))   M (  A ( xi ))  A i A i A i A i   n 1  A ( xi )  1      ( x ))  (1   ( x ))    A i A i  i 1 f 

    

................. (5.7)

M f ( A ( xi ))  0 gives (  A 1 ( xi )  (1   A ( xi ))  1  0  A ( xi )   A 1 ( xi )  (1   A ( xi ))  1

  A ( xi )  1   A ( xi )

  A ( xi )  So at

1 2

 A ( xi ) 

1 f , M  (  A ( xi )) has maximum or minimum value.

2

For this, we differentiate eqn(5.7) again

 1     ( x ))  (1   ( x ))  A i   M ( A ( xi ))  (  1)   2 A i  2    ( x )  (1   ( x ))   A i A i n 1  ( A ( xi ))2  1      ( x ))  (1   ( x ))    A i A i  i 1 2

f 

 1     ( x ))  (1   ( x ))   d  A i A i +  A 1 ( xi )  (1   A ( xi ))  1   n  1  1 dx      ( x ))  (1   ( x ))   A i A i   i 1

    

    

Which is less than zero for   1,   1

So ,at

 A ( xi ) 

1 f , M  (  A ( xi )) has maximum value.

2

 1  M f (  A ( xi ))   log 2 Max 1 

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4) Since

M

f 

M f ( A ( xi )) is an increasing function of  A ( xi ) for 0   A ( xi ) 

1 .i.e 2

(  A ( xi ))  A ( xi )  0  0

1   1  A ( xi )    log 2  0 for  1,   1 2 1  1 M f ( A ( xi ))  (x ) 5) Since  is decreasing function of A i for   A ( xi )  1 2 1  1  f i.e.  M  (  A ( xi ))  A ( xi )    log 2 and 2 1   

f

and  M  (  A ( xi ))

M So,

6)

f 

(  A ( xi ))  A ( xi )  1  0

M f ( A ( xi )) is a concave function.

M f (  A ( xi ))

 A ( xi ) is replaced by 1   A ( xi ) M f (  A ( xi )) = M f (1   A ( xi )) does not change when

f

f

Now we study the monotonic behavior of M  (  A ( xi )) . For this, different values of M  (  A ( xi )) by parameters has been calculated and further the generalized measure has been presented graphically.

We have from eqn (7) 1  n   A ( x i )  (1   A ( x i ))          where   1,   1 &   0 M f (  A ( x i ))  lo g  i 1  1 n       a) Suppose

1   ,  2 2  A ( xi ) M f (  A ( xi ))

0.7

0

0

0.1

0.172

0.2

0.334

0.5

0.3

0.473

0.4

0.568

0.5 0.6 0.7

0.602 0.568 0.473

0.4 M(µA(x)) 0.3

0.8

0.334

0.1

0.9

0.172

0

1

0 Table 1.1

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0.6

0.2

0

0.5

µ(x)

1

1.5

Figure 1.1

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b)

1    ,  2 2

 A ( xi )

M f (  A ( xi ))

0

0

0.1

0.057

0.2

0.111

0.3

0.158

0.4

0.189

0.5

0.201

0.6

0.189

0.7

0.158

0.8

0.111

0.9

0.057

1

0

0.25 0.2 0.15 M(µA(x)) 0.1 0.05 0 0

0.5

1.5

µ(x)

Table 1.2

c)

1

Figure 1.2

1    ,  3 2

 A ( xi )

M f (  A ( xi ))

0

0

0.1

0.091

0.35

0.2

0.189

0.3

0.3

0.288

0.4

0.369

0.25 M(µA(x)) 0.2

0.5

0.401

0.6

0.369

0.7

0.288

0.05

0.8

0.189

0

0.9

0.091

1

0

Table1.3

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0.45 0.4

0.15 0.1

0

0.5

µA(x)

1

1.5

Figure1.3

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  1,   3

 A ( xi )

M f (  A ( xi ))

0

0

0.1

0.069

0.2

0.142

0.3

0.216

0.4

0.276

0.2

0.5

0.301

0.15

0.6

0.276

0.1

0.7

0.216

0.8

0.142

0.9

0.069

1

0

0.35 0.3 0.25

0.05

Table 1.4

0 0

0.2

0.4

0.6

0.8

1

1.2

Figure 1.4

f

Thus, we found that value of M  (  A ( xi )) given by eqn(7) satisfy all properties of fuzzy entropy. Also by assuming different values of

f (  A ( xi )) , we get different fuzzy mean codeword length whose lower bounds are expressed as fuzzy entropies.

IV. CONCLUSION In this paper , by developing new fuzzy code word length ,we found lower bound of this fuzzy code word length which are expressed as measure of fuzzy entropies . We get this by existing knowledge. This generalisation are very important in present .Also we have proved properties of above fuzzy entropy and with the help of data, we have represented it graphically.

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