Series of New Information Divergences, Properties and Corresponding Series of Metric Spaces

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1 Srs of Nw Iforao Dvrgcs, Proprs ad Corrspodg Srs of Mrc Spacs K.C.Ja, Praphull Chhabra Profssor, Dpar of Mahacs, Malavya Naoal Isu of Tchology, Japur (Rajasha), Ida Ph.d Scholar, Dpar of Mahacs, Malavya Naoal Isu of Tchology, Japur (Rajasha), Ida Absrac: Dvrgc asurs ar bascally asurs of dsac bw wo probably dsrbuos or hs ar usful for coparg wo probably dsrbuos. Dpdg o h aur of h probl, h dffr dvrgcs ar suabl. So s always dsrabl o cra a w dvrgc asur. Thr ar svral gralzd fucoal dvrgcs, such as: Csszar dvrgc, Ry- lk dvrgc, Brga dvrgc, Burba- Rao dvrgc c. all. I hs papr, w oba a srs of dvrgcs corrspodg o a srs of covx fucos by usg gralzd Csszar dvrgc. Furhr, w df h proprs of covx fucos ad dvrgcs, copar h dvrgcs ad lasly roduc h srs of rc spacs. Idx Trs: Srs of rc spacs, srs of w covx ad oralzd fucos, srs of dvrgc asurs, proprs of covx fucos ad dvrgcs. Mahacs Subjc Classfcao: 94A7, 6D5. L I. INTRODUCTION P p, p, p3..., p : p 0, p, p 0 for so,, 3,..., probably dsrbuos. If w ak b h s of all copl f dscr, h w hav o suppos ha 0 0 f 0 0 f 0. 0 Csszar [], gv h gralzd f- dvrgc asur, whch s gv by: p C f P, Q q f q (.) Whr f: (0,) R (s of ral o.) s ral, couous ad covx fuco ad P p, p, p3..., p, Q q, q, q3..., q Γ, whr p ad q ar probably ass fucos. May kow dvrgcs ca b obad fro hs gralzd asur by suably dfg h covx fuco f. So of hos ar as follows: If w ak f log, w g p K P, Q p log q = Kullback- Lblr dvrgc asur []. (.) Copyrgh o IJIRSET 4

2 f, w g If w ak PQ, p q = Ch- Squar dvrgc asur [3]. (.3) q If w ak f log, w g p F P, Q p log p q = Rlav JS Dvrgc [4]. (.4) If w ak f log, w g p q p q GP, Q log = Rlav AG Dvrgc [5]. (.5) p Slarly, w g ay ohrs dvrgcs as wll by dfg suabl covx fuco. May rsarch paprs hav b sudd of I.J. Taja, P. Kuar, S.S. Dragor, K.C. Ja ad ohrs, who gav h da of dvrgc asurs, hr proprs, hr bouds ad rlaos wh ohr asurs. Ths all ar vry usful bcaus dvrgc asurs ar appld vary of dscpls (od cocluso). W roducd a w for of hs asurs,.. rc spacs. W foud ha squar roo of all dvrgcs of Csszar s class, s a rc spac, whch s vry usful fucoal aalyss. W ca xd hs da fucoal aalyss. II. SERIES OF CONVEX FUNCTIONS AND THEIR PROPERTIES I hs sco, w shall dvlop so srs of covx fucos, ad wll sudy hr proprs. For hs, L f: (0, ) R (s of ral o.) b a appg, dfd as: Ad f,,,3,4... (.) / 6 f (.) / 4 3 / f ,,, Fro (.), w g h followg covx fucos a =,, 3, 4 rspcvly. 4 6 f, f /, f 3/ 3... (.4) 5/ Now by usg (.4), w g h followg srs of covx fucos as wll. 4 4 f, f f (.5) / 3/ 3/ (.3) Copyrgh o IJIRSET 5

3 f,3 f f3 (.6) 3/ 5/ 5/ I hs way, w ca wr: 3 4 f, f f (.7) / / / Whr, =,, 3, 4 Sc, w kow ha h lar cobao of covx fucos s also a covx fuco. s a covx fuco as wll, whr a, a, a 3... ar posv cosas... a f a f a f So, w g aohr srs of covx fucos by usg (.4), dfd as follows: Cas-I: f w ak log b log b 3 a, a log b, a3, a4..., b, h w hav! 3! 3 logb logb g f log b f f3 f4...! 3! 3 log b log b g log... / b 3/ 5/ 7/! 3! 4 3 log log 6 b b log... / b 3! 3! Cas-II: f w ak Copyrgh o IJIRSET 6 b, b (.8) / log b log b 3 a 0, a, a3 log b, a4, a5..., b, h w hav! 3! 3 log b log b g log... 3/ b 5/ 7/ 9/! 3! log log 6 b b log... 3/ b 3! 3! 4 b, b (.9) 3/ I hs way, w ca wr: g b, b ad,,3,4... (.0) / Spcal Cas: If w ak b.788, h fro (.0), w oba h followg srs:

4 g xp,,,3,4... / / Proprs of fucos dfd by (.), (.7) ad (.), ar as follows: a. Sc f f g 0 f, f ad g ach.,, (.) ar oralzd fucos for b. Sc f 0 0,,,3,4... f ar covx fucos ad so f g ar as wll.,, c. Sc f 0 a 0, ad f 0 a f ar ooocally dcrasg d. ooocally crasg,,, for ach valu of ad f 0. 0, ad Fgur : Graph of fucos f. ` Fgur : Graph of fucos f, Copyrgh o IJIRSET 7

5 f. Fgur 3: Graph of fucos g Fgur, ad 3 shows ha f, f ad g rspcvly., hav a sppr slop for crasg valus of III. CORRESPONDING SERIES OF DIVERGENCES AND PROPERTIES: I hs sco, w shall oba srs of dvrgc asurs corrspodg o covx fucos dfd sco, ad wll sudy h proprs. Th followg hor s wll kow lraur []. Thor : If h fuco f s covx ad oralzd,.., boh o-gav ad covx h par of probably dsrbuo Now, pu (.) (.), w g h followg dvrgcs:.. p q q f / q f 0, h, p C P, Q P, Q,,,3,4... C P Q ad s adjo C, PQ,. Copyrgh o IJIRSET 8 f f Q P ar (3.) 4 6 p q p q p q / 3/ (3.a) 5/ P, Q, P, Q, P, Q pq q pq q pq q Now, pu (.7) (.), w g h followg dvrgcs: p q p p q pq q f / pq q (3.) C P, Q P, Q,,, PQ, p q p p q pq q (3.a) 3/ 4 pq q 4 p q p p q pq q PQ,... (3.b) 5/ 6 pq q Now, pu (.) (.), w g h followg dvrgcs:

6 p q p q / 3 p q q pq C f P, Q P, Q xp,,, PQ p q p q, xp / 3 pq q pq (3.3) (3.3a) 4 p q p q PQ 3/ 4 3 pq q pq, xp... (3.3b) Proprs of dvrgcs dfd by (3.), (3.) ad (3.3), ar as follows: a. I vw of hor, w ca say ha P Q P Q P Q probably dsrbuo PQ,.,,,,, 0 ad ar covx h par of b. c. Sc P, Q Q, P, P, Q Q, P, P, Q Q, P P, Q, P, Q, P, Q ar o- syrc dvrgc asurs. d. P, Q P, Q P, Q 0 f P Q or p q (Aas s u valu) a Fgur 4: Coparso of dvrgcs Fgur 4 shows h bhavor of P, Q, P, Q, K P, Q ad P, Q p a, a, q a, a, whr a 0, PQ, has sppr slop for crasg valus of ad has a spr slop ha K P, Q ad P, Q Kullback- Llbr dv. (.) Ch-squar dv (.3) Nw dv. a = (3.a) Nw dv. a = (3.a). W hav cosdrd. I s clar fro fgur 4 ha h w pararc dvrgc. Copyrgh o IJIRSET 9

7 IV. SERIES OF METRIC SPACES (DISTANCE MEASURES): Sc, PQ, =,, 3, Bu h su of PQ, s a o- syrc ad pararc dvrgc wh parar or srs of dvrgcs wh ad s adjo s syrc, so p q q p P, Q Q, P P, Q / / PQ p, = q p q,,,3,4... Or W ca s ha, PQ, p q q q p p 6 (4.) / pq s syrc pararc dvrgc asur. Now, wh h hlp of hor ad quao (4.), w ca say h followgs: P, Q 0 P, Q. P, Q 0 ff P Q or p q,,3..., whr P, Q. P, Q Q, P PQ, Now, f h ragl qualy s sasfd by PQ, hav o prov h followg hor, whch s sad as: Thor : L s syrc, for ach N x p, q : R R R,,,3... b dfd as, p q p q pq 6 /., h hs wll bco rc spacs ovr R. For hs, w x p, q,,,3..., (4.) I vw of (4.), w ca wr Th P, Q x p, q (4.3) x p, q x p, r x r, q ragl qualy p, q, r R. (4.4) Proof: To prov h rsul (4.4), frs l us cosdr,, X r x p r x r q (4.5) pq Th, Copyrgh o IJIRSET 30 d x p, r x r, q X pq r X pq r (4.6) dr x p, r x r, q Now fro (4.) (afr pug q x p, r r ), w g p r p r pr 6 / (4.7)

8 Ad afr dffrg (4.7) w.r. r, w g h followg: (4.8) 6 6 r p r p r p x p, r / / p r p r p Pu p r,.. R (4.8), w g r 6 x p, r k / 6 pr Fro (4.7), w ca wr x, Fro (4.7) ad (4.0), w hav h followg rlao (4.0) 6 /,, x p r r x r l (4.) Whr, w ar assug x, l Now, dffra (4.9) w.r., w g Copyrgh o IJIRSET 3 (4.9) (4.) ,,,3, (4.3) k s, 0, l (4.4) 4 k 4 / Now, l w df a fuco Fro (4.0) ad (4.3), w coclud ha l x, 0 ad k 0 0, ad N... k s ooocally dcrasg fuco ad k 0, so s wll b dcrasg as wll s 0 or h aur of s dpds o h aur of k. Thrfor, w coclud ha chags h sg a =, ad s Now, suppos s 0, 0, q q q p u R u R, so (4.6) ca b wr as: p r p r pq 0, wh (4.5) r X r s s u (4.6) Now, w hav wo cass o u, as follows: Cas I: f w ar akg u or q p, h (by cosdrg ha s s dcrasg fuco):

9 For s ad s u s s u For s 0 ad s u 0 s s u s s u 0 u. For s 0 ad su 0 u s su.. X pq r chags h sg a or r p, so Xpq r aas s u valu a r or r p. Cas II: hs cas for u or q p, ca b do a slar ar. Slarly, rpag h abov procdur by cosdrg q p p p q s su R ad u R u R, h w g ha X pq r chags h r q r q r r sg a or r q X r aas s u valu a or r q., so pq Thrfor, rgh had sd of (4.4) has s u valu a p q r, p, q, r R. Hc proof h rsul (4.4) or hor. I vw of hs proof, w coclud ha h w pararc syrc dvrgc asur PQ, asur. Or, w g h srs of dsac asurs, as follows: P Q P Q P Q P Q P Q P Q Copyrgh o IJIRSET 3 s a dsac,,,,,,,,,,,... (4.7) Or, w g h srs of rc spacs ovr R, as follows: R R R R R,,,,,,,,,... V. CONCLUDING REMARKS (4.8) Dvrgc asurs hav b appld a vary of dscpls such as ahropology, gcs, fac, coocs ad polcal scc, bology, aalyss of cogcy abls, approxao of probably dsrbuos, sgal procssg ad par rcogo. I hs papr, w roducd a w srs of covx fucos, w srs of dvrgc asurs ad w srs of rc spacs. Th bouds of pararc dvrgc asur PQ, dscussd x papr. ad rlao wh ohr sadard dvrgc asurs wll b REFERENCES [] Csszar, I., Iforao yp asurs of dffrcs of probably dsrbuo ad drc obsrvaos, Suda Mah. Hugarca, Vol., pp , 967. [] Kullback S. ad Lblr R.A., O Iforao ad Suffccy, A. Mah. Sas., Vol., pp , 95. [3] Parso K., O h Crro ha a gv sys of dvaos fro h probabl h cas of corrlad sys of varabls s such ha ca b rasoabl supposd o hav ars fro rado saplg, Phl. Mag., Vol. 50, pp. 57-7, 900. [4] Sbso R., Iforao radus, Z. Wahrs. Udvrw. Gb., Vol. 4, pp , 969. [5] Taja I.J., Nw dvlops gralzd forao asurs, Chapr : Advacs Iagg ad Elcro Physcs, Ed. P.W. Hawks, Vol. 9, pp , 995.