A ‘useful’ fuzzy measure of order α and type β is developed. Its validity established with a numerical example

Baig, Mirza Abdul Khalique

Peerzada, Safina

Sofi, Saima Manzoor

English

Digital Commons@Wayne State University

Journal of Modern Applied Statistical Methods Volume 18 Issue 2 Article 18 8-19-2020 A New Two-Parametric ‘Useful’ Fuzzy Information Measure and its Properties Saima Manzoor Sofi University of Kashmir, Srinagar, India, saimam.stsc@gmail.com Safina Peerzada University of Kashmir, Srinagar, India, sapezad@gmail.com Mirza Abdul Khalique Baig University of Kashmir, Srinagar, India, baigmak@gmail.com Follow this and additional works at: https://digitalcommons.wayne.edu/jmasm  Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Sofi, S. M., Peerzada, S., & Baig, M. A. K. (2019). A new two-parametric ‘useful’ fuzzy information measure and its properties. Journal of Modern Applied Statistical Methods, 18(2), eP2805. doi: 10.22237/jmasm/1604189580 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. A New Two-Parametric ‘Useful’ Fuzzy Information Measure and its Properties Cover Page Footnote The authors are thankful to the reviewers for their valuable suggestions and comments. This regular article is available in Journal of Modern Applied Statistical Methods: https://digitalcommons.wayne.edu/jmasm/vol18/iss2/18 Journal of Modern Applied Statistical Methods November 2019, Vol. 18, No. 2, eP2805. doi: 10.22237/jmasm/1604189580  Copyright © 2020 JMASM, Inc. ISSN 1538 − 9472    doi: 10.22237/jmasm/1604189580 | Accepted: March 18, 2018; Published: August 19, 2020. Correspondence: Saima Manzoor Sofi, saimam.stsc@gmail.com    2 A New Two-Parametric ‘Useful’ Fuzzy Information Measure and its Properties Saima Manzoor Sofi University of Kashmir Srinagar, India Safina Peerzada University of Kashmir Srinagar, India Mirza Abdul Khalique Baig University of Kashmir Srinagar, India   A ‘useful’ fuzzy measure of order α and type β is developed. Its validity established with a numerical example.  Keywords: Shannon's entropy, fuzzy set, fuzzy entropy, ‘useful’ information measure  Introduction Zadeh (1965) presented fuzzy set theory. The degree of fuzziness in a fuzzy set is measured by using the concept of entropy. Ebanks (1983) and Pal and Bezdek (1994) called it fuzzy entropy, which is an important concept for measuring fuzzy information. It has a vital role in fuzzy systems such as neural networks, pattern recognition, decision making, knowledge base, communication, etc. This led to further developments, such as Kaufmann (1975), Pal and Pal (1989), Parkash and Sharma (2002, 2004), Bhat and Baig (2016a, b), Bhat, Baig, and Salam (2016), and Bhat, Bhat, et al. (2017). Let X = {x1, x2,…, xn} be a universal set defined in the universe of discourse. A fuzzy subset ‘A’ in ‘X’ is defined as A = {(xi, μA(xi)): xi ∈ X, μA(xi) ∈ [0, 1]} where μA(xi) is a membership function which is defined as   ( )0 if  and there is no ambiguity,μ 0.5 if there is maximum ambiguity whether  or ,1 if  and there is no ambiguityA ix Ax x A x Ax A=      Some important concepts related to fuzzy sets are given below:  SOFI ET AL 3 • Sum of A and B (A + B) is given as   ( ) ( ) ( ) ( ) ( )μ μ μ μ μ ,A B i A i B i A i B i ix x x x x x X+ = + −   ;   • Product of A and B (AB) is given as   ( ) ( ) ( )μ μ μ ,AB i A i B i ix x x x X=   ;   • Equality of A and B (A = B) is given as   ( ) ( )μ μ ,A i B i ix x x X=   ;   • Containment of A and B (A ⊂ B) is given as   ( ) ( )μ μ ,A i B i ix x x X   ;   • Complement of A (A′) is defined as   ( ) ( )μ 1 μ ,A i A i ix x x X = −   ;   • Union of A and B (A ∪ B) is defined as   ( ) ( ) ( ) μ Max μ ,μ ,A B i A i B i ix x x x X =   ;   • Intersection of A and B (A ∩ B) is defined as:   ( ) ( ) ( ) μ Min μ ,μ ,A B i A i B i ix x x x X =      where A and B are two fuzzy subsets of X with membership functions μA(xi) and μB(xi), respectively. TWO-PARAMETRIC NEW ‘USEFUL’ FUZZY INFORMATION 4 Shannon’s Entropy Let X = (x1, x2,…, xn) be a discrete random variable with probability distribution P = (p1, p2,…, pn) such that pi ≥ 0 ∀ i = 1, 2,…, n and 11niip== . Then the Shannon’s information measure, called entropy, is defined as (Shannon, 1948)   ( )1H logni D iiP p p== − .  (1)  Corresponding to Shannon’s measure of entropy, De Luca and Termini (1972) gave a measure of fuzzy entropy given as   ( ) ( ) ( ) ( )( ) ( )( )1H μ logμ 1 μ log 1 μnA i A i A i A iiA x x x x= = − + − −  .  (2)  The fuzzy entropy measure should satisfy the following four properties, given by De Luca and Termini (1972):  1. Sharpness: H(A) is minimum if and only if A is a crisp set. 2. Maximality: H(A) is maximum if and only if A is most fuzzy set. 3. Resolution: H(A) ≥ H(A*), where A* is sharpened version of A. 4. Symmetry: H(A) = H(A′), where A′ is the complement of A. ‘Useful’ Fuzzy Information Measure Let U = (u1, u2,…, un) be a set of non-negative numbers such that ui > 0 and ui represents the utility of the occurrence of element xi. In general, utility is independent of probability pi. The information scheme given by   1 21 211 2; 0, 0 & 1n nn i i iinx x xp p p u p pu u u=  =   =   U   (3)  is called as utility information scheme. Corresponding to the scheme (3), Belis and Guiasu (1968) gave the following measure of information:  SOFI ET AL 5  ( )1H ; logni i D iiP u p p== −U .  (4)  The measure defined in (4) is called ‘useful’ entropy. This measure can be taken as a satisfactory measure for the average quantity of ‘useful’ information provided by the information scheme (3). For any fuzzy set A, the ‘useful’ fuzzy entropy is defined as   ( ) ( ) ( ) ( )( ) ( )( ) 1H ; μ log μ 1 μ log 1 μni A i D A i A i D A iiA u x x x x== − + − −U .  (5) Proposed ‘Useful’ Fuzzy Information Measure and Its Properties The proposed ‘useful’ fuzzy information measure is   ( )( ) ( ) ( )( )( ) 1111μ 1 μH ; log ;10 1,0 1, , 0ni A i A iiD niiiu x xAuu     −−== + − =  −       U  (6)  For (6) to be a valid ‘useful’ fuzzy information measure, it should satisfy the four properties given by De Luca and Termini (1972).  Sharpness. ( )H ;A U  is minimum if and only if A is a crisp set i.e., ( )H ; 0A =U  iff μA(xi) = 0 or 1 ∀ i = 1, 2,…, n.  Proof. Suppose ( )H ; 0A =U , i.e.,   ( ) ( ) ( )( )( ) 1111μ 1 μlog 01ni A i A iiD niiu x xu  −−== + − = −     TWO-PARAMETRIC NEW ‘USEFUL’ FUZZY INFORMATION 6  ( ) ( ) ( )( )( ) 1111μ 1 μlog 0ni A i A iiD niiu x xu   −−== + −  =       ( ) ( ) ( )( )( ) 111 1μ 1 μn ni A i A i ii iu x x u   −−= = + − =    (7)  Because 0 < α < 1, 0 < β ≤ 1, and ui > 0, (7) will hold when either μA(xi) = 1 or μA(xi) = 0 ∀ i = 1, 2,…, n. Conversely, suppose   ( ) ( ) ( )( )( ) 1111μ 1 μlog 0ni A i A iiD niiu x xu   −−== + − =   .  (8)  Multiplying both sides of equation (8) by β / (1 – α),   ( ) ( ) ( )( )( ) ( )1111μ 1 μlog 01H ; 0ni A i A iiD niiu x xuA  −−== + − = −   =U   Hence, ( )H ; 0A =U  if and only if A is a crisp set.  Maximality. ( )H ;A U  is maximum if and only if A is most fuzzy set.  Proof. We have   ( )( ) ( ) ( )( )( ) 1111μ 1 μH ; log ;10 1,0 1, , 0ni A i A iiD niiiu x xAuu     −−== + − =  −       U  (9)  SOFI ET AL 7 Now, differentiating equation (9) with respect to μA(xi),   ( )( )( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) 1 11 12111μ 1 μH ;μ μ 1 μi A i A inA ii A i A iiu x xAx u x x    − −− −−−= − −  =   + −  U.   Let 0 ≤ μA(xi) < 0.5; then   ( )( )H ;0; 0 1,0 1, , 0μiA iAux          U.   Hence, ( )H ;A U  is an increasing function of μA(xi) whenever 0 ≤ μA(xi) < 0.5. Similarly, for 0.5 < μA(xi) ≤ 1,   ( )( )H ;0; 0 1,0 1, , 0μiA iAux          U.   Hence, ( )H ;A U  is a decreasing function of μA(xi) whenever 0.5 < μA(xi) ≤ 1, and for μA(xi) = 0.5,   ( )( )H ;0; 0 1,0 1, , 0μiA iAux    =      U.   Thus, ( )H ;A U  is a concave function which has a global maximum at μA(xi) = 0.5. This implies ( )H ;A U  is maximum iff A is most fuzzy set, that is, μA(xi) = 0.5 ∀ i = 1, 2,…, n.  Resolution. ( ) ( )H ; H ;A A   U U , where A* is sharpened version of A.  Proof. : Because ( )H ;A U  is an increasing function of μA(xi) whenever 0 ≤ μA(xi) < 0.5 and is a decreasing function of μA(xi) whenever 0.5 < μA(xi) ≤ 1,  TWO-PARAMETRIC NEW ‘USEFUL’ FUZZY INFORMATION 8  ( ) ( )( ) ( )  )μ μH ; H ;  in 0,0.5i A iAx xA A   U U  (10)  Also,   ( ) ( )( ) ( ) ( μ μH ; H ;  in 0.5,1i A iAx xA A   U U  (11)  Taking equation (10) and (11) together, ( ) ( )H ; H ;A A   U U .  Symmetry. ( ) ( )H ; H ;A A   =U U , where A′ is the compliment of A.  Proof. From the definition of ( )H ;A U  and μA′(xi) = 1 – μA(xi) ∀ xi ∈ X, we conclude that ( ) ( )H ; H ;A A   =U U . Because the proposed measure ( )H ;A U  satisfies all the four properties of fuzzy information measure, thus it is a valid measure of ‘useful’ fuzzy information. Illustration Sharpness From Table 1, conclude A is minimum (i.e., ( )H ; 0A =U ) iff A is a crisp set (i.e., when μA(xi) = 0 or μA(xi) = 1).   Table 1. Behavior of ( )H ;βαA U  when μA(xi) = 1 and μA(xi) = 0 with respect to α and β  α β ui μA(xi) ( )H ;UβαA  μA(xi) ( )H ;UβαA  0.1 0.2 4 1 0 0 0   3 1 0 0 0   2 1 0 0 0   1 1 0 0 0 SOFI ET AL 9 Maximality From Table 2, conclude ( )H ;A U  is an increasing function of μA(xi) (i.e. ( ) ( )( )H ; μ 0A iA x  U ) whenever 0 ≤ μA(xi) < 0.5. From Table 3, conclude ( )H ;A U  is a decreasing function of μA(xi) (i.e. ( ) ( )( )H ; μ 0A iA x  U ) whenever 0.5 < μA(xi) ≤ 1. For μA(xi) = 0.5, α = 0.1, and β = 0.2,   ( )( )H ;0μA iAx=U  (12)  Thus, from Tables 2 and 3 and equation (12), conclude ( )H ;A U  is a concave function with global maximum at μA(xi) = 0.5.   Table 2. At 0 ≤ μA(xi) < 0.5 and with respect to α and β  α β ui μA(xi) ( )( )H ;UμβαA iAx 0.1 0.2 1 0.00 ∞   2 0.13 0.0202   3 0.27 0.0118   4 0.43 0.0039   Table 3. 0.5 < μA(xi) ≤ 1 and with respect to α and β  α β ui μA(xi) ( )( )H ;UμβαA iAx 0.1 0.2 1 0.55 –0.000810   2 0.70 –0.007520   3 0.85 –0.030181   4 1.00 –∞   TWO-PARAMETRIC NEW ‘USEFUL’ FUZZY INFORMATION 10 Resolution From Table 4, conclude ( ) ( )H ; H ;A A   U U  whenever ( ) ( )μ μA i iAx x  in [0, 0.5). From Table 5, conclude ( ) ( )H ; H ;A A   U U  whenever ( ) ( )μ μA i iAx x  in (0.5, 1]. Thus, from Tables 4 and 5, conclude ( ) ( )H ; H ;A A   U U , where A* is sharpened version of A.   Table 4. At [0, 0.5) and with ( ) ( ) *μ μA i iAx x   α β ui μA(xi) ( )H ;UβαA  ( )*μA ix  ( )H ;UβαA*  0.2 0.6 1 0.12 0.2539 0.00 0.2306   2 0.23  0.15    3 0.36  0.31    4 0.49  0.44    Table 5. At (0.5, 1] and with ( ) ( ) *μ μA i iAx x   α β ui μA(xi) ( )H ;UβαA  ( )*μA ix  ( )H ;UβαA*  0.2 0.6 1 0.65 0.1823 0.70 0.1226   2 0.83  0.83    3 0.89  0.94    4 0.96  1.00  Symmetry From Table 6, conclude that ( ) ( )H ; H ;A A   =U U , where A′ is the compliment of A.   Table 6. Verification of symmetry property  α β ui μA(xi) ( )H ;UβαA  1 – μA(xi) ( )H ;UβαA  0.2 0.6 1 0.65 0.1858 0.35 0.1858   2 0.78  0.22    3 0.89  0.11    4 0.96  0.04  SOFI ET AL 11 Behavior of Proposed ‘Useful’ Fuzzy Information Measure of Order α and Type β In order to study the behavior of the proposed ‘useful’ fuzzy information measure, fix β and observe the behavior of ( )H ;A U  at different values of α and vice-versa. Consider the membership function μA(xi) = {0.11, 0.45, 0.23, 0.65, 0.82, 0.31, 0.72, 0.56, 0.92} with the utilities ui = {1, 2, 3, 4, 5, 6, 7, 8, 9}.   Table 7. Behavior ( )H ;βαA U  of at different values of α and β = 1  α 0.15 0.29 0.36 0.40 0.53 0.61 0.70 0.85 0.90 ( )1H ;αA U  0.0978 0.2340 0.3278 0.3923 0.6858 0.9714 1.4846 3.7597 6.0577     Figure 1. Behavior ( )H ;βαA U  of at different values of α and β = 1    Table 8. Behavior ( )H ;βαA U  of at different values of β and α = 0.2  β 0.29 0.36 0.40 0.53 0.61 0.70 0.85 0.90 0.92 ( )0.2H ;βA U  0.1804 0.2045 0.2151 0.2348 0.2363 0.2288 0.1962 0.1800 0.1729  TWO-PARAMETRIC NEW ‘USEFUL’ FUZZY INFORMATION 12   Figure 2. Behavior ( )H ;βαA U  of at different values of β and α = 0.2   On observing the behavior of ( )H ;A U  at different values of β and fixed α, ( )H ;A U  increases up to α = 0.59 and after this value ( )H ;A U  starts decreasing. Conclusion The present communication introduces a new ‘useful’ fuzzy information measure i.e., ( )H ;A U , of order α and type β. The properties of ( )H ;A U  were considered via hypothetical data. Further, the behavior of ( )H ;A U  at different values of α and β were studied. References Belis, M., & Guiasu, S. (1968). A quantitative-qualitative measure of information in cybernetic system. IEEE Transactions on Information Theory, 14(4), 593-594. doi: 10.1109/TIT.1968.1054185 SOFI ET AL 13 Bhat, A. H., & Baig, M. A. K. (2016a). Coding theorems on generalized useful fuzzy inaccuracy measure. International Journal of Modern Mathematical Science, 14(1), 54-62. Bhat, A. H., & Baig, M. A. K. (2016b). Generalized useful fuzzy inaccuracy measures and their bounds. International Journal of Advanced Research in Engineering Technology and Sciences, 3(6), 28-33. Retrieved from http://ijarets.org/publication/24/IJARETS%20V-3-6-6.pdf Bhat, A. H., Baig, M. A. K., & Salam, A. (2016). Bounds on two parametric new generalized fuzzy entropy. Mathematical Theory and Modeling, 6(7), 7-17. Retrieved from https://iiste.org/Journals/index.php/MTM/article/view/31542 Bhat, A. H., Bhat, M. A., Baig, M. A. K., & Sofi, S. M. (2017). Noiseless coding theorems of generalized useful fuzzy inaccuracy measure of order α and type β. International Journal of Fuzzy Mathematical Archive, 13(2), 135-143. Retrieved from http://www.researchmathsci.org/IJFMAart/IJFMA-v13n2-4.pdf De Luca, A., & Termini, S. (1972). A definition of non-probabilistic entropy in the setting of fuzzy set theory. Information and Control, 20(4), 301-312. doi: 10.1016/S0019-9958(72)90199-4 Ebanks, B. R. (1983). On measures of fuzziness and their representations. Journal of Mathematical Analysis and Applications, 94(1), 24-37. doi: 10.1016/0022-247X(83)90003-3 Kaufmann, A. (1975). Introduction to theory of fuzzy subsets: Fundamental theoretical elements (Vol. 1). New York: Academic Press. Pal, N. R., & Bezdek, J. C. (1994). Measuring fuzzy uncertainty. IEEE Transaction on Fuzzy Systems, 2(2), 107-118. doi: 10.1109/91.277960 Pal, N. R., & Pal, S. K. (1989). Object-background segmentation using new definition of entropy. IEE Proceedings E - Computers and Digital Techniques, 136(4), 136-284. doi: 10.1049/ip-e.1989.0039 Parkash, O., & Sharma, P. K. (2002). A new class of fuzzy coding theorems. Caribbean Journal of Mathematical and Computing Sciences, 12, 1-10. Parkash, O., & Sharma, P. K. (2004). Noiseless coding theorems corresponding to fuzzy entropies. Southeast Asian Bulletin of Mathematics, 27(6), 1073-1080. Retrieved from http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200406&filename=14.pdf Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379-423. doi: 10.1002/j.1538-7305.1948.tb01338.x TWO-PARAMETRIC NEW ‘USEFUL’ FUZZY INFORMATION 14 Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353. doi: 10.1016/S0019-9958(65)90241-X 

A New Two-Parametric ‘Useful’ Fuzzy Information Measure and its Properties

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A New Two-Parametric ‘Useful’ Fuzzy Information Measure and its Properties

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