CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr theor uses algorthms and algebra to analse data It s used b socal scentsts to analse nteractons between actors and can be used to complement analses carred out usng game theor or other analtcal tools The concepts of fuzz matr have been defned n chapter A standard fuzz matr s the fuzz matr of the followng form where all the entres are less than or equal to : = () 3 Operatons on Two Fuzz Matrces Let us defne two fuzz matrces and Y of order as 3 7 8 9 3 () 6 8 9 6
9 3 6 Y (3) 8 9 6 3 7 Addton of Two Fuzz Matrces Two fuzz matrces and Y are compatble under matr addton f the are of same order For nstance for the fuzz matrces and Y gven b () and (3) we have, Clearl 3 7 7 Y () 9 9 6 8 9 Y s a matr, but not a fuzz matr ence we can conclude that addton of two fuzz matrces compatble under addton need not be a fuzz matr owever, addton of two standard fuzz matrces s a fuzz matr Mamum Operaton of Two Fuzz Matrces Two fuzz matres are conformable for mamum operaton f the are of the same order ence for two matrces and Y of order m n, mama of these two matrces s a matr c ma(, ) Ma (, Y) c of order m n, where ence for the matrces and Y gven b () and (3) we have, 7 8 9 6 Ma,Y () 8 9 8 9 6 7
Mnmum Operaton of Two Fuzz Matrces Two fuzz matres are conformable for mnmum operaton f the are of same order ence for two matrces these two matrces s a matr where mn(, ) and Y of order m n, fuzz mnma of Mn (, Y) c of order m n c ence for the matrces and Y gven b () and (3) we have, 3 9 3 3 Mn (, Y) (6) 6 6 3 In case of fuzz matrces, we have seen that the addton s not defned, where as the mama and mnma operatons are defned Clearl under the mamum and mnmum operatons the resultant matr s agan a fuzz matr of the same order and thus s n somewa analogous to our usual addton Product of Two Fuzz Matrces To fnd the product of two fuzz matrces wth Y gven b () and (3), where and Y are compatble under multplcaton; e the number of column of equal to the number of row of Y ; stll we ma not have the product Y to be a fuzz matr 38 6 Y 6 86 37 7 7 8 38 (7) 7 8 7 Clearl Y s not a fuzz matr Thus we need to defne a compatble operaton analogous to multplcaton of two fuzz matrces so that the product agan happens to be a fuzz matr owever, even for ths new operaton f the product Y s to be defned we need the number of columns
of s equal to the number of rows of Y e the fuzz matr should be compatble for multplcaton The two tpes of operatons whch we can have are ma-mn operaton and mn-ma operatonthese operatons are defned below: To fnd Y usng ma-mn operatons, we have c c 3 c c 3 c c 3 33 3 c c c c c3 c C (8) c c c c 3 where, c = ma {mn (3, ), mn (7, 3), mn (8, 8),mn (9, 3)} = ma {3, 3, 8, 3}= 8 c = ma {mn (3, ), mn (7, 6), mn (8, 9),mn (9, ) = ma {, 6, 8, }= 8 and so on Thus, we get 8 8 9 6 C (9) 6 7 9 8 6 7 9 Now suppose that the fuzz matrces and Y are gven b () and (3), on applng mn-ma operaton, we get where, Y 3 3 3 3 33 3 = mn {ma (3, ), ma (7, 3), ma (8, 8), ma (9, 3)} = mn {, 7, 8, 9} = 7 = mn {ma (3, ), ma (7, 6), ma (8, 9), ma (9, )} = mn {3, 7, 9, 9} = 3 and so on Thus, we have 3 () 3
7 3 7 3 3 () 3 6 3 From (9) and (), t s clear that C Some eperts ma lke to work wth mama-mnma value and some wth the mnma-mama value and accordngl the can adopt t ence we can have the product of two fuzz matrces Y = C or as per our requrement It s also observed that Y s defned but Y ma not be defned Conugate of Fuzz Matr Let = ( ) m n be a fuzz matr, the matr obtaned b replacng each element of b ts dual - ) ( s called conugate fuzz matr of and s denoted as Conugate of fuzz matrces and Y fuzz matr product of ma-mn and mn-ma operatons as gven below: () ma mn Y mn - ma Y () () mn ma Y ma - mn Y (3) () and (3) can be llustrated b the followng eample: Illustraton() Let and Y are two fuzz matrces gven b () and (3) respectvel Then, 3 ma mn Y () 6 3 mn - ma Y 6 3 3 ()
From ()and () we conclude () holds Illustraton: () Let and Y are two fuzz matrces gven b () and (3) respectvel Then, 3 7 6 3 mn ma Y 7 7 7 6 9 6 7 8 6 6 3 7 6 3 ma - mn Y 7 7 7 6 9 6 7 8 6 6 From (6) and (7) we conclude () holds Fuzz matr Theor (FST) has been appled to man felds such as control, sgnal and mage processng, medcne, the econom, etc The results show that FMT elds effcent solutons to varous problems In crsp set theor, a member of a set s represented b or So, n a crsp matr, a member ether belongs or doesn't belong to a class owever, n FMT, a member of a set s represented b a degree between and The degree s called membershp degree whch shows belongng degree of the member to the class The membershp degree s computed usng the membershp functon obtaned b the eperts on the subect or a pror knowledge In the present chapter we prove that the set of fuzz matrces forms a lattce under matr mnma and matr mama bnar fuzz operatons n secton In secton 3 we ntroduce and characterze a new fuzz nformaton measure on fuzz matr and ts propertes have been studed n secton In secton we defne a new measure of nformaton on fuzz bnar relaton A Lattce of Fuzz Matrces A partal ordered set n whch ever par of elements has both least upper bound and greatest lower bound s called a lattce refer to Trembla and Manohar (997) or n
other words Algebra L,, s called a lattce f L s a non empt set, and are bnar operatons on L, satsfng (A-) Idempotent ( A A= A or A A = A), (A-) Commutatve ( A B = B A or A B = B A) (A-3) Assocatve ( A ( B C ) = ( A B ) C or A ( B C )=( A B ) C ) (A-) Absorpton law ( A ( A B ) = A or A ( A B ) = A) Theorem Let A s a set of m n fuzz matrces where J,,, then A J under matr mnma and matr mama operatons A forms a lattce Proof To prove that the gven set A of fuzz matrces forms a lattce, we shall show that A satsfes the four propertes (A-) to (A-) (A-) Idempotent Law For an fuzz matr A, the followng holds: mn( A, A ) A and ma( A, A ) A ence Idempotent Law s satsfed (A-) Commutatve Law It can be easl verfed that for all the fuzz matrces A anda A, the followng holds: mn( A, A ) mn( A, A ) and ma( A, A ) ma( A, A ) Ths proves that Commutatve Law s satsfed (A-3) Assocatve Law For an fuzz matrces A, A A A, t can be proved that, k mn ( A, A ), A mn( A, A, A ) and ma ( A, A ), A ma( A, A, A ) k k ence we can conclude that A s assocatve under matr mama and matr mnma operatons k k 6
(A-) Absorpton Law For an fuzz matrces A, A, we can prove that ma A,mn A, A A mn A,ma A, A A and ence Absorpton Law holds Snce t satsfes all the four propertes (A-) to (A-) of the lattces, therefore the set of the matrces s a lattce under matr mama and matr mnma Ma(A,A,A k ) Ma(A, A ) Ma(A, A k ) Ma(A, A k ) A A A k Mn(A, A ) Mn(A, A k Mn(A, A k ) Mn(A,A, A k ) Fgure A lattce of fuzz matrces under matr mama and matr mnma Fgure s the pctoral representaton of the lattce wth the fuzz matrces A, A and A k wth the least element mnma ( A, A, A k ) and the greatest element mama ( A, A, A k ) 3 Informaton Measure on Fuzz Matr Fuzz nformaton measures the degree of fuzzness of a fuzz set It s pecular to mathematcs, nformaton theor and computer scence It s an mportant concept n fuzz set theor and has been successfull appled to pattern recognton, mage processng, classfer desgn and neural network structure etc 7
The concept of nformaton measures was developed b Shannon (98) to measure the uncertant of a probablt dstrbuton The concept of fuzz set was ntroduced b Zadeh (966) who also developed hs own theor to measure the ambgut of a fuzz set Let,,, n be the unverse set of dscourse and ( ) A be membershp functon defned on A Then A ( ), A ( ), A ( n ) le between (, ) and these are not probabltes because ther sum s not unt owever, A( ) A ( ) n,,,, n, (3) A( ) s a probablt dstrbuton Thus Kaufman (98) defned entrop of a fuzz set A havng n support ponts as n ( A) A( ) log A( ) (3) log n Correspondng to entrop due to Shannon (98), eluca and Termn (97) suggested the followng measure of fuzz entrop: n ( A) ( )log ( ) ( ) log ( ) (33) A A Correspondng to (33) we propose the followng nformaton measure defned on fuzz matrces: A A m n ( ) log log, (3) where s the th (, ) element of the standard fuzz matr Theorem The fuzz nformaton measure gven b (3) s a vald measure Proof To prove that the gven measure s a vald measure of fuzz nformaton, we shall show that (3) satsfes the followng four propertes (P-) to (P-): 8
(P-) ( ) f onl f s non fuzz matr or crsp matr We know that log and log f onl f ether or, =,, It mples ( ) f onl f s non fuzz or crsp matr (P-) ( ) s mamum f and onl f s the most fuzz matr, e for all =,, fferentatng ( ) wth respect to, we have d d m n whch vanshes at log log (3) = Agan dfferentatng ( ) wth respect to, we have d d m n (36) Puttng n (36), we have d ( ) d ence ( ) s mamum f and onl f s most fuzz matr e : =,, (P-3) Sharpenng reduces the value of nformaton measure Let us consder, then d d m n log log (37) It mples ( ) s an ncreasng functon of n the regon Smlarl, we can prove that ( ) s a decreasng functon of n the regon ence we can conclude that ( ) s a concave functon 9
Let = ( such that ) m n then, * = (* ) m n : [,] such that be * for all =,, * Then * s called the sharpened verson of fuzz matr and snce ( ) s ncreasng functon of n the regon, therefore () f * ( *) ( ) n [,) (38) Smlarl f = ( such that then, * = n ) m n ( * ) m : [,] such that be * for all =,, * Then * s called the sharpened verson of fuzz matr and snce ( ) s decreasng functon of n the regon, therefore () f * ( *) ( ) (39) (38) and (39) together gves ( *) ( ) Thus (P-3) s proved (P-) ual propert e ( ) ( ) It s evdent from the defnton that ( ) ( ) ence ( ) satsfes all the essental four propertes of fuzz nformaton measure Thus t s a vald measure of fuzz nformaton We can call ths measure as fuzz matr nformaton measure Propertes of Fuzz Matr Informaton Measure Propert For an standard fuzz matrces and Y compatble under mama and mnma operatons, the followng holds: mn {, Y} ma {, Y} and mn {, Y} ma {, Y} Proof Let and Y are the two standard fuzz matrces Snce fuzz matr nformaton measure ( ) s an ncreasng functon of n the regon Therefore
mn {, Y} ma {, Y} and mn {, Y} ma {, Y} or mn {, Y} Y ma {, Y} Propert For an two fuzz matrces and Y, such that Y and Y ests, we have Y Y and Y Y Y Proof of ths s evdent, whch can be llustrated n table, consderng dfferent pars of fuzz matrces and Y Table Y Y Y Y Y Y 86 736 7 78 66 33 3 36 3 8 3 Propert 3 For an fuzz matr we have, T, where T s the transpose of The followng table llustrates the proof for dfferent fuzz matr : Table T 9 86 736 7 3 39 T 9 86 736 7 3 39 Propert For an fuzz matr of order n n, let n be the set of square sub matrces, such that, e the nested form of the sub 3
matrces, where A B means order of B s less than order of A We have element 3 n and the greatest element The followng table3 verfes the result emprcall:, whch forms a lne lattce wth the least Table 3 3 9836 3 6 7 8 9 97 9833 973 99 9977 863 877 89333 88 73699 7 733 76 737 76 633 6 96 8 3976 36 7 68 368 333 333 373 369 3933 63 6 83 67 8977 73 787 93 9 686 3 3 673 38838 3 37878 86 86 979 33 33 6 6
Fgure s the pctoral representaton of the lne lattce / column lattce of the fuzz nformaton measures on fuzz matr of the nested matrces n Fgure The pctoral representaton of the lne lattce Propert For an fuzz matr of order m n and an arbtrar element a les between and, we have a a Fuzz Bnar Relaton Informaton Measure The word relaton suggests some famlar eamples of relatons such as the relaton of father to son, mother to son, etc Famlar eamples n arthmetc are the relatons such as greater than or less than and so on These eamples suggest that relatonshps est among two obects Such tpe of relatons s known as bnar relatons, between a par of obects e an set of order pars defnes a bnar relatons The relatonshp between three concdent lnes or a pont between two gven ponts s eamples of relatons among three obects Smlarl relatons can est among four or more obects 3
Fuzz Bnar Relaton Let,, 3,, and Y,, 3,, be two fnte sets, fuzz bnar relaton R whch ma assgn two or more elements of Y correspondng to each element of Let R be the fuzz bnar membershp relaton such as, 9,,,,,, 3, 3, 3,, 3,,,,,, () Operatons on Fuzz Bnar Relatons Fuzz bnar relaton s ver mportant because the can descrbe nteractons between varables () Intersecton of fuzz bnar relatons The ntersecton of two fuzz bnar relatons R and S s defned b, mn R S for each and () R S, () Unon of fuzz bnar relatons The unon of two fuzz bnar relatons R and S s defned b, ma R S for each and (3) R S, Fuzz bnar relatons pla an mportant role n fuzz modellng, fuzz dagnoss, and fuzz control These also have wde applcatons n felds such as pscholog, medcne, economcs, and socolog Fuzz Bnar Relaton Matr A Fuzz bnar relaton R from a fnte set to a fnte set Y can also be represented b a fuzz matr called the fuzz bnar relaton matr of R Fuzz bnar relaton matr s obtaned usng the membershp functon and the membershp degree Let,, 3,, and Y,, 3,, be two sets and the fuzz bnar
relaton defned on (,Y ) s gven b (), then the fuzz bnar relaton matr s as follows: R, Y 3 9 3 () The nverse fuzz bnar relaton of R s the relaton R from Y to, and the correspondng matr s equal to the transpose of the matr R, Y, e R Y, 3 6 9 3 R Y, T () Analogous to (3) we can defne the followng fuzz nformaton measure on fuzz bnar relaton matr (): m n R,Y ( R )ln ( R ) ( R ) ln ( R ), (6) th where, (R) s the, element of the bnar fuzz relaton matr R, Y On the lnes of proof of theorem, t can easl be verfed that (6) s a vald measure of fuzz nformaton owever, ths measure wll be called as fuzz bnar relaton nformaton measure
6 Concluson In the present chapter we have defned two bnar operatons on fuzz matrces We have also proved that the set of fuzz matrces forms a lattce under these bnar fuzz operatons Further we have ntroduced and characterzed a new nformaton measure on fuzz matr Propertes of ths proposed measure have also been studed Fuzz relaton plas an mportant role n fuzz modellng, fuzz dagnoss and fuzz control The also have applcatons n felds such as pscholog, medcne, economcs and socolog A fuzz bnar relaton R from a fnte set to a fnte set Y can also be represented b a fuzz matr called the fuzz bnar relaton matr of R The fuzz nformaton measure thus defned on fuzz bnar relaton matr s a vald measure of fuzz nformaton and can further be generalzed We can also stud ts applcaton; however, t s an open problem 6