Generalized Measure for Two Utility Distributions

Transcription

1 Proceedgs of the World Cogress o Egeerg 00 Vol III WCE 00, Jue 30 - July, 00, Lodo, U.K. Geeralzed Meure for Two Utlty Dstrbutos J. S. BHULLAR MEMBER IAENG, O. P. VINOCHA, MANISH GUPTA Abstract The meure I ( P : Q; U) ( ) log q ) addtve. I ths paper the meure s fd out, whch are o-addtve ad satsfy the o-addtve property by cosderg two probablty dstrbutos ad two utlty dstrbutos attached wth them. It s further show that the characterze a o-addtve geeralzed meure of relatve useful formato by usg weghted sum property ad two meures of J-dvergece for U ad V utlty dstrbutos. Idex Terms Dvergece meures; Relatve formato of type s; Relatve J-dvergece, Propertes Of J. I. INTRODUCTION The dvergece meure s appled o umerous are lke probablty meure, patter recogto, sgal processg, ecoomcs etc. These meures are ormally used to fd proper dstace or dfferece betwee two probablty dstrbuto. These meures ca be categorzed parametrc, o-parametrc ad etropy-type meures of formato. Parametrc meures: These type of formato, meure the amout of formato about a ukow parameter α spled by the data ad are fuctos of α. The best kow meure of ths type s meure of formato []. No-parametrc meures: These type of formato, gve the amout of formato spled by the data for dscrmatg favor of oe of a probablty dstrbuto agt the aother, or for meurg the dstace or affty betwee probablty dstrbuto. Kullback ad Lebler [3] showed that ths meure s the best kow ths cls. Etropy-type Meures: These type of formato express the amout of formato cotaed a dstrbuto, that s, the amout of ucertaty socated wth the outcome of a expermet. New dvergece meures ad ther relatoshps wth the well kow dvergece meures are also studed by Kumar, Chha [4], Kumar, Huter [3] ad Kumar, Johso [5]. J-dvergece equals the average of the two possble KL-dstaces betwee two probablty dstrbutos, although Mauscrpt receved November 0, 009. Dr. Jkara S. Bhullar s wth the Malout Isttute of Maagemet ad Iformato Techology(MIMIT), Malot, Puab, Ida. Ph (e-mal: bhullarkar@ yahoo.co.). Dr. O.P. Vocha s wth Ferozpur College of Egeerg ad Techology, Ferozpur, Puab, Ida. Mh Gta s wth the D.A.V. College, Bathda, Puab, Ida. Ph (e-mal: mh_gtabt@yahoo.com). ISBN: ISSN: (Prt); ISSN: (Ole) s Jeffreys [] dd ot develop t to symmetres the Kullback ad Lebler [3] dstace. Kumar ad Taea [6] dscussed o-symmetrc relatve J-dvergece meure ad ts propertes. Let P = (p, p,, p ), 0 < p, p be a fte probablty dstrbuto of set of evets E = (E, E,, E ) ad Q = (q, q,, q ), 0 < q, q be the revsed probablty dstrbuto of set of set evets F = (F, F,, F ). The Kullback ad Lebler [3] meures of drected dvergece defed I ( P/ Q) p log( p / q ).(.) Bels, Guu [6] have attached a utlty dstrbuto U = (U, U,, U ) to radom expermet E = (E, E,, E ) where U > 0 s the utlty of the th outcome E. Bhaker, Hooda [] characterzed the followg meure of useful drected dvergece or related formato log( p/ q) I( P: Q; U).(.) whe V = (v, v,, v ) s the utlty dstrbuto fucto of evets F = (f, f,, f ) of revsed expermets of E. The we defe aother ew useful formato meure log( pu/ qv) I( P: Q; U; V).(.3) whe u = v the (.3) (.) Bhaker, Hooda [] also studed geeralzed mea value characterzato of the followg meure of useful relatve formato of order. q ) I ( P: Q; U) ( ) log.(.4) The meure (.4) s also addtve. So t s terestg to fd out the meure, whch are o-addtve ad satsfy the o-addtvty property of the followg form. I(P* R : Q* S;U * V) I(P : Q;U) I(R;S;V) KI(P;Q;U) I(R;S;V).(.5) Where P, Q, R, S m, U ad V are utlty dstrbutos ad k ( 0) s ay real umber. Now we defe the useful relatve formato of order WCE 00

2 Proceedgs of the World Cogress o Egeerg 00 Vol III WCE 00, Jue 30 - July, 00, Lodo, U.K. I ( ) log ( u/ q ).(.6) The meure (.6) s also addtve. We defe aother o-addtve useful formato meure of order ( / ) I.(.7) The meure (.7) satsfy the o-addtvty property of the followg form I(P* P : Q* Q;V * V) I(P : Q;U :V) I(P : Q;U;V) KI(P;Q;U :V).(.8) Where P, Q, P, Q m, U, U ad V, V are utlty dstrbutos ad k ( 0) s ay real umber. II. CHARACTRIZATION OF USEFUL DIRECTED DIVERGENCE We cosder the o-addtvty of the type (.8) let h be real valued fucto defed or R R. We sume that h ( ) I.(.) Whe p, q? ad u v ; h ( u,? u) - ad h(u/, u/ ) = 0.(.) The property (.) s called weghted sum property. The fuctoal equato (.8) together wth (.) reduces to u' p' hu ( p) h ( ) u' p' h( u' p' v' q' ) h ( ) u' p' h( u' p' v' q' ) u' p'.(.3) It ca be ely proved Hooda [] that the cotuous fucto whch satsfes the fuctoal equato (.3) s also cotuous soluto of the followg fuctoal equato. hu ( ' p'; ' ') h (, ) hu ( ' p', ' ') kh( ; ) h( u ' p ', v ' q ').(.4) where pp, qq, uu, vv R ad k s a o-zero real umber. Multplyg both sdes of (.4) by k ad addg () we have +kh( ; )=+kh(, )+kh(, )+kh(, )h(, ) +kh( ; )=[+kh(, )][+kh(, )].(.5) Puttg +kh(, )=f(, ) (.5) F( ; ) = f(, )f(, ).(.6) Puttg = r, = r, = s, = s (.6) f(rr; ss) = f(r, s) f(r, s) s cotuous real valued fuctoal equato. Aczel [5] h gve oe of the cotuous real soluto s gve by f(r, s) = rs f(, ) = ()() Where ad are real or complex umbers. It mples + kh(, ) = ()() or h (, ) [( ) ( ) ].(.7) k Applyg sutable codtos (.) (.7), we get + = 0 ad k = -- = - h(, ) =.(.8) ( ) Wthout the loss of geeralty we ca replace by - (.8), we get h(, ) =.(.9) ( ) Substtutg (.9) (.) we get ( / Vq) I.(.0) III. MEASURE OF USEFUL J-DIVERGENCE The Useful J-dvergece meure correspodg to (.3) s gve by ( / Vq) log( / ) JPQUV ( : ; : ).(3.) I ce utltes are gored or u = v = for each () the (3.) reduces to J ( P: Q) p log( p / q ) q log( q / p ).(3.) The meure (3.) w studed by Kullback [4] ad h foud wde applcatos statstcs ad patter recogto [7]. The meure (3.) ca also be wrtte J I I( P: Q; V : U).(3.3) Thus correspodg to (.0) a meure of J-dvergece s defed J ( / ) log( / ) ( u p) /( ) ( ) ( ) u p.(3.4) ISBN: ISSN: (Prt); ISSN: (Ole) WCE 00

3 Proceedgs of the World Cogress o Egeerg 00 Vol III WCE 00, Jue 30 - July, 00, Lodo, U.K. It may be oted that (3.4) reduces to (3.) whe ad so t ca be called geeralzed useful J-dvergece of degree. I ce utltes are gored, the meure (3.4) reduces to J ( P: Q) pq q p, { p p }.(3.5) The meure (3.5) w studed by Sheg ad Rathee, [8] ad Burbea et al. [9] ad w called J-dvergece meure of degree. Hece to call meure (3.4) useful J-dvergece meure of degree ustfed. Aother geeralzato of (3.) ca be cosdered of the followg form J ( P: Q; U : V ) (α ) log (u p ) (v q ) (v q ) u p ) α α α α.(3.6) We see that (3.6) reduces to (3.) ce utltes are gored or u = p = for each, (3.6) becomes J ( P: Q) ( ) log p q q p,.(3.7) whch w characterzed by Taea [0]. Let W ( u p) ( ) ( ) ( u p).(3.8) () The (3.4) ad (3.6) ca be wrtte J (P : Q;U : V) ( ) W (P : Q;U : V).(3.9) J(P:Q;U:V) (α ) log W (P : Q;U : V)/ α.(3.0) () The meure W ( P: Q; U : V ) gve by (3.8) plays a mportat role characterzato of the above two geeralzed meures of J- dvergece. So we characterzed W ( P: Q; U : V ). Whch ca be wrtte W ( P: QU ; : V) W ( P: QU ; : V) W ( P: QU ; : V) we see that W ad W satsfy the followg relato WP ( * P': Q* QU '; * U': V* V') WPQUVWP ( : ; : ) ( ': QU '; ': V').(3.) Let h be a real valued cotuous fucto o R R such that g (, ) W.(3.) g(, u p) ad W ( Q: P; V : U).(3.3) The fuctoal eq (3.) together wth (3.) ad (3.3) gves ' ' ' ' ' ' uu p p g( p p, qq ; uu, vv ) ' ' uu p p ' ' ' ' ' ' g (, ) g (, ). ' '.(3.4) Theorem : Oe of the cotuous soluto of the fuctoal equato (3.4) uder boudary codtos p =q = ½ ad u, v.e. g(u/, u/) = s gve by g(, ) = ()- ()- 0, > 0. Proof: It ca be ely verfed that cotuous fucto whch satsfy the fuctoal equato (3.4) s the cotuous soluto of the followg fuctoal equato g(pp, qq ; uu, vv) = g(p, q; u, v) g(p, q; u, v).(3.5) Where p, p, q, q u, u, v, v R The most geeral cotuous soluto of the fuctoal equato (3.5) s gve by g(, ) = ()().(3.6) where, are real or complex umbers. Now (3.6) together wth codto g(u/, u/) = ad thus we get g(, ) = ()(), > 0 Wthout loss of geeralty, we ca replace by - ad thus we get g(, ) = ()-()-, 0 > 0.(3.7) Ths proves the theorem. Puttg (3.7) ad (3.) we have ( ) ( ) ( ).(3.8) Smlarly, we ca prove ( ) ( u p) ( Q: P; V : U) ( ).(3.9) ad we have ( u p) ( ) ( ) ( ) u p ( ) ( ).(3.0) Substtutg (3.0) (3.9) we get requred geeralzed meures J (P:Q;U:V) ad J(P:Q;U:V) respectvely. ISBN: ISSN: (Prt); ISSN: (Ole) WCE 00

4 Proceedgs of the World Cogress o Egeerg 00 Vol III WCE 00, Jue 30 - July, 00, Lodo, U.K. IV. PROPERTIES OF J(P:Q;U:V) () No-Negatvty Theorem : J (P:Q;U:V) s o-egatve ad vhes ff p = q & u = v for all =,,. Proof: J (P:Q;U:V) ( u p) ( ) ( ) ( u p) = ( ) ( ), 0.(4.) for > by Holder s equalty ( ) ( ) ( ) ( ) ad ( ) ( ) ad also Therefore J (P:Q;U:V) s o-egatve for >. for < by Holder s equalty ( ) ( ) ( ) ( u p) ad ( ) ( ) ad also Therefore J (P:Q;U:V) s o-egatve for >. It s trval that f p = q, u = v the J (P:Q;U:V) vhes coversely. let J (P:Q;U:V) = 0 ( ) ( ) ( ) ( ) The ( ) ( ) ( ) ( u p) ad ( ) ( ) ( ) ( ) for p = q ad u = v for =,, sce. Hece J (P:Q;U:V) vhes ff p = q ad u = v for =,,. () Symmetry J ((p, p, p ) : (q, q, q ); (u, u, u ): (v, v, v )) = J ((pa, pa,,pa ):(qa, qa,,qa );(ua, ua,,ua ):(va, va,,va )) Where (a, a,,, a ) s a arbtrary permutato of (,, ). Ths me that the permutato of par-wse labelg of evets does ot chage the value of useful J-dvergece of degree. () Expblty J {(p, p,,p, 0):(q, q,,q, 0);(u, u,,u, u +):(v, v,,v, v -)) = J ((p, p,,p : (q, q,,q ; u, u,,u : (v, v,,v ) Thus corporatg a evet of probablty zero does ot chage the value of meure. (v) Recursvty Let A, A be two evets havg probabltes p, p ad utltes u, u respectvely ad, be aother two evets havg probabltes q, q ad utltes v, v. The we defed utlty U of the compoud evet A A ad utlty V of the compoud evet u p U( A A ) ad p p v q V( B B ) q q Theorem 3: Uder the composto law (4.) the followg holds J ( p, p,..., p, p', p'') : ( q, q,..., q, q', q''); ( u, u,..., u, u', u'') : ( v, v,..., v, v', v'')} J p' p'' q' q'' ( p' p'') ( q', q'') I,,,, u', u'', v', v'' p' p'' p' p'' q' q'' q' q'' q' q'' p' p'' ( qq ', '') ( p' p'') I,,,, uuvv ', '', ', '' q' q'' q' q'' p' p'' p' p'' Where p = p+p, q = q+q pu ' ' p'' u'' qv ' ' q'' v'' ad u, v p' p'' q' q'' Proof : ( ) ( ) ( ) ( ) u p u p c c L.H.S.= + ( ) ( ) c c ( ' ') ( u' p') ( v'' q'') ( u'' p'') ( ' ') ( v'' q'') J ( p, p,..., p : q, q,..., q ; u, u,..., u, v, v,..., v ) = J ( u' p') ( v' q') ( u'' p'') ( v'' q'') u' p' u'' v'' ( ' ') ( u' p') ( v'' q'') ( u'' p'') u' p' u'' v'' ( ) ( ) ( ) ( ) ( ' ') ( ' ') ( u'' p'') ( '' '') u p ' ' '' '' ( ' ') ( u' p') ( v'' q'') ( u'' p'') ' ' v'' q'' J ( ) ( ) ( ) ( u p) ( ) ( ) ( ' '/ ) ( ' '/ ) ( u' p'/ u p ) ( v' q'/ v q ) u' p' u'' p''/ u p ( ) ( u p) ISBN: ISSN: (Prt); ISSN: (Ole) WCE 00

5 Proceedgs of the World Cogress o Egeerg 00 Vol III WCE 00, Jue 30 - July, 00, Lodo, U.K. ( ' '/ ) ( ' '/ ) ( '' ''/ ) ( u'' p''/ ) ' ' v'' q''/ J p' p'' q' q'' ( p' p'') ( q' q'') I,,,, u', u'', v', v'' p' p'' p' p'' q' q'' q' q'' q' q'' p' p'' ( q' q'') ( p' p'') I,,,, u', u'', v', v'' q' q'' q' q'' p' p'' p' p'' (v) Permutatoal symmetry of dstrbutos For P, Q, we have J J ( Q: P; U : V) (v) Cotuty J s a cotuous fucto of U varables V z, p, p,., p ; q, q,, q, u, u,., u, v, v,, v. V. CONCLUSION I ths paper we had defed log( pu/ qv) I( P: Q; U; V), the o addtve meure whch also satsfes the o addtve property. Further we defed the useful relatve formato of order. Aother o-addtve useful formato meure of order s also defed whch also satsfes the o-addtve property. REFERENCES [] R. A. Fsher, Theory of statstcal estmato. Proc. Cambrdge Phlos. Soc., 95. [] H. Jeffreys, A varat form for the pror probablty estmato problems, Proc. Roy. Soc. A, 86, 946, [3] S. Kullback ad R.A. Lebler, O Iformato ad Suffcecy, A. Math. Statst., vol., 95, [4] S. Kullback, Iformato theory ad statstcs. New York : Wlley, 959. [5] J. Aczel, Lectures o fuctoal equato ad ther applcatos, New York : Academc Press, 966. [6] M. Bels, ad S. Guu, A quattatve-qualtatve meures of formato cyberetc systems, IEEE Tra. Iformato Theory, vol. 4, 968, [7] G. I. Toursat, O the dvergece betwee two dstrbutos ad probablty of msclarfcato of several decso rules, Proc. d It. Jour. Cof. O Patter recogto, Copehego(Demark), 974, pp. -8. [8] L. T. Sheg ad P.N. Rathee, The J-dvergece of order, Joural of combatores, Iformato system sceces, vol. 6, 98, -9. [9] J. Burbea ad C.R. Rao, O the covexty of some dvergece meures bed o etropy fuctos, IEEE Tra. Iformato Theory, IT-8(3), 98, [0] T. J. Taea, O characterzato of J-dvegece ad ts geeralzato, Joural of combatores, Iformato system sceces, vol. 8, 98, 06-. [] D. S. Hooda, A o-addtve geeralazato relatve useful formato, Joural of PAMSXX, 984, [] U. S. Bhaker ad D. S. Hooda, Mea value characterzato of useful formato meures, Tamkag Joural of Mathematcs, vol. 4, 993. [3] P. Kumar ad L. Huter, O a formato dvergece meure ad formato equaltes, Carpatha Joural of Mathematcs, vol. 0(), 004, [4] P. Kumar ad S. Chha, A symmetrc formato dvergece meure of the Cssz ar s f-dvergece cls ad ts bouds, Computers ad Mathematcs wth Applcatos, vol. 49, 005, [5] P. Kumar ad A. Johso, O a symmetrc dvergece meure ad formato equaltes, Joural of Iequaltes Pure ad Appled Mathematcs, vol. 6(3), Artcle 65, 005, pp. -3. [6] P. Kumar ad I. J. Taea, Geeralzed Relatve J-Dvergece Meure ad Propertes, It. J. Cotemp. Math. Sc., vol., o. 3, 006, ISBN: ISSN: (Prt); ISSN: (Ole) WCE 00