A New Measure of Probabilistic Entropy. and its Properties

Transcription

1 Appled Mathematcal Sceces, Vol. 4, 200, o. 28, A New Measure of Probablstc Etropy ad ts Propertes Rajeesh Kumar Departmet of Mathematcs Kurukshetra Uversty Kurukshetra, Ida rajeesh_kuk@redffmal.com Subash Kumar Departmet of Mathematcs S.S.M.I.T, Daagar, Ida Al Kumar Departmet of Mathematcs S.S.C.E.T., Badha Pathakot, Ida INTRODUCTION The cocept of etropy commucato theory was frst troduced by Shao [3] ad t was the realzed that etropy s a property of ay stochastc system ad the cocept s ow used wdely dfferet dscples. The tedecy of the systems to become more dsordered over tme s descrbed by the secod law of thermodyamcs, whch states that the etropy of the system caot spotaeously decrease. Today, formato theory s stll prcpally cocered wth commucatos systems, but there are wdespread applcatos statstcs, formato processg ad computg. A great deal of sght s obtaed by cosderg etropy equvalet to ucertaty, the geeralzed theory of whch has well bee explaed by Zadeh [5].

2 388 R. Kumar, S. Kumar ad A. Kumar The ucertaty assocated wth probablty of outcomes, kow as probablstc ucertaty, s called etropy, sce ths s the termology that s well etreched the lterature. Shao [3] troduced the cocept of formato theoretc etropy by assocatg ucertaty wth every probablty dstrbuto P ( p, p2,..., p ) ad foud that there s a uque fucto that ca measure the ucertaty, s gve by (P) p I p (.) The probablstc measure of etropy (.) possesses a umber of terestg propertes. Immedately, after Shao gave hs measure, research workers may felds saw the potetal of the applcato of ths expresso ad a large umber of other measures of formato theoretc etropes were derved. Rey [] defed etropy of order α as: α α I p p α, α, α > 0 (.2) whch cludes Shao s [3] etropy as a lmtg case as α. Zyczkowsk [6] explored the relatoshps betwee the Shao s [3] etropy ad Rey s [] etropes of teger order. avrada ad Charvat [5] troduced frst o-addtve etropy, gve by: α p α, α, α > 0 (.3) α 2 Kapur [7] geeralzed Rey s [] measure further to gve a measure of etropy of order α ad type β, vz., α+ β β α, β I p p, α, α > 0, β > 0,α + β > 0 (.4) α The measure (.4) reduces to Rey s [] measure whe β, to Shao s [3] measure whe β, α. Whe β, α, t gves the measure l pmax May other probablstc measures of etropy have bee dscussed ad derved by Brssaud [], Chakrabart [2], Che [3], Garbaczewsk [4], erremoes [6], Laveda [8], Nada ad Paul [9], Rao, Yume ad Wag [0], Sergo [2], Sharma ad Taeja [4] etc. The applcatos of the results obtaed by varous authors have bee provded to varous felds of Mathematcal Sceces. I secto 2, we have troduced a ew geeralzed probablstc formato theoretc measure.

3 A ew measure of probablstc etropy A New Geeralzed Iformato Theoretc Measure based upo Probablty Dstrbutos I ths secto, we propose a ew geeralzed formato measure for a probablty dstrbuto P ( p, p2,..., p), p 0, p ad study ther essetal ad desrable propertes. Ths geeralzed etropy depedg upo real parameters α, α2,..., α s gve by the followg mathematcal expresso: α + α α p α, α,..., α α α... α 2 (2.) where α + α, α + α > ad α, α > 0, α 0 (2.2) If α 0, the α, α > 0.Thus, we see that the proposed measure (2.) becomes p α α 2 α (2.3) whch s avrada ad Charvat s [5] measure of etropy of order α. The measure (2.3) aga reduces to Shao s [3] measure of etropy as α. Thus, we see that the measure proposed equato (2.) s a geeralzed measure of etropy. Next, we study some mportat propertes of ths geeralzed measure. The measure (2.) satsfes the followg propertes: () It s cotuous fucto of p, p2,..., p, so that t chages by a small amout whe p, p2,..., p chage by small amouts. () It s permutatoally symmetrc fucto of p, p2,..., p, that s, t does ot chage whe p, p2,..., p are permuted amog themselves. () 0 αα,,..., α (v) + (,,...,,0),,..., p p 2 p αα α α + α α p α α... α 2

4 390 R. Kumar, S. Kumar ad A. Kumar αα,,..., α Ths property says that etropy does ot chage by the cluso of a mpossble evet wth probablty zero. (v) Sce s a etropy measure, ts maxmum value must αα,,..., α occur. To fd the maxmum value, we proceed as follows: Let α + α α p ( ) f p λ p α α... α 2 The, we have α+ α α (... ) f α+ α + + α p λ p αα... α 2 α+ α α (... ) f α+ α + + α p 2 λ p αα... α α + α α (... ) p f α + α + + α λ p α α... α 2 For maxmum value, we take f f f p... 0 p 2 p whch gves α+ α α α+ α α α+ α α ( α+ α α ) p ( α+ α α ) p ( α+ α α ) p 2... αα... α αα... α αα... α whch s possble oly f p p... p Thus p gves p p... p

5 A ew measure of probablstc etropy 39 ece, we see that the geeralzed etropy measure (2.) possesses maxmum value ad ths value subject to atural costrat p arses whe p p 2... p. Ths result s most desrable. (v) The maxmum value s a creasg fucto of. To prove ths result, we have α α... α f ( p ) α α... α 2 Thus α α... α ( α α... α ) f ' ( p) α α... α 2 α + α α ( α + α α ).2 > 0 α + α α α + α α (2 2). sce α + α, α + α > ece maxmum value s a creasg fucto of. (v) Recursvty property : To prove that the measure (2.) s recursve ature, we cosder ( ) α + α α α + α α p + p + p 2 ( p + p, p p,..., p ) 3 αα,,..., α 2 3, 4 α α... α 2 ( ) α + α α α + α α α + α α p + p p p 2 2 α α... α 2 α + α α p + α α... α 2

6 392 R. Kumar, S. Kumar ad A. Kumar α + α α α + α α p p p p p p α + α + + α ( p p ) + 2 α α... α 2 α + α α p + α α... α 2 α+ α α p p 2 ( p + p ) + + ( p, p,..., p ) 2 2 p + p p + p αα,,..., α Thus, we have proved that (,,..., ) α (,..., ), α p p p p p p p p,..., 2,,..., 2 3, 4 α α α α +... p p 2 ( p p ) α + α + + α p + p p + p 2 2 Ths shows that the measure (2.) possesses recursvty property. (v) Addtve property: To show that the measure (2.) s o-addtve, we cosder m α + α α p q j, m j ( P Q ) α, α,..., α α α... α 2 α+ α α p m α+ α α q αα... α j 2 j α+ α α m α+ α α p q j j + αα... α + αα... α 2 2

7 A ew measure of probablstc etropy 393 αα α m αα α p q αα... α j j 2 αα... α αα... α 2 2 α+ α α m α+ α α p q j j + αα... α + αα... α 2 2 α α... α 2.. m ( Q) m + + ( Q) whch shows that the geeralzed etropy (2.) s o-addtve. Refereces [] Brssaud, J.B. (2005): "The meag of etropy", Etropy, 7 (), [2] Chakrabart, C.G. (2005):"Shao etropy: axomatc characterzato ad applcato, It. Jr. Math. Math.Sc., 7, [3] Che, Y. (2006): Propertes of quas-etropy ad ther applcatos, J.Southeast Uv. Nat.Sc., 36 (2), [4] Garbaczewsk, P. (2006): Dfferetal etropy ad dyamcs of ucertaty", J.stat.Phys.,23, [5] avrada, J.. ad Charvat, F. (967): "Quatfcato methods of classfcato process:cocept of structural α-etropy", Kyberetka, 3, [6] erremoes, P.(2006):"Iterpretatos of Rey etropes ad dvergeces", Phys.A., 365 (),

8 394 R. Kumar, S. Kumar ad A. Kumar [7] Kapur, J.N. (967): "Geeralzed etropy of order α ad type β", Maths. Sem., 4, [8] Laveda, B.. (2005) : "Mea Etropes", Ope System Ifor. Dy., 2, [9] Nada, A. K. ad Paul, P. (2006): "Some results o geeralzed resdual etropy", Iformato Sceces, 76 (), [0] Rao, M.C., Yume, V.B.C. ad Wag, F. (2004): "Commulatve resdual etropy : a ew measure of Iformato, IEEE Tras. Iform. Theory, 50 (6), [] Rey, A. (96): "O measures of etropy ad formato", Proc. 4th Ber. Symp. Math. Stat. ad Prob.,, [2] Sergo, Verdu (998): Ffty years of Shao theory, IEEE Tras. If. Theory, 44(6), [3] Shao, C. E.(948):"A mathematcal theory of commucato, Bell. Sys. Tech. Jr., 27, , [4] Sharma, B.D. ad Taeja, I.J. (975): "Etropes of type (α, β) ad other geeralzed measures of formato theory", Met., 22, [5] Zadeh, L.A. (2005) : "Towards a geeralzed theory of ucertaty (GTU)- a outle", Iformato Sceces, 72, -40. [6] Zyczkowsk, K. (2003) : Rey extrapolato of Shao etropy, Ope Syst. If. Dy., 0 (3), Receved: October, 2009