An Application of Generalized Entropy Optimization Methods in Survival Data Analysis

Transcription

1 Joural of Moder Physcs, 017, 8, hp://wwwscrporg/joural/jp ISSN Ole: X ISSN Pr: A Applcao of Geeralzed Eropy Opzao Mehods Survval Daa Aalyss Aladd Shalov 1, Cgde Kalahlparbl 1*, Sevda Ozder 1 Faculy of Scece, Depare of Sascs, Aadolu Uversy, Eskşehr, Turkey Ozalp Vocaoal School, Accouacy ad Tax Depare, Yuzucu Yl Uversy, Va, Turkey ow o ce hs paper: Shalov, A, Kalahlparbl, C ad Ozder, S (017) A Applcao of Geeralzed Eropy Opzao Mehods Survval Daa Aalyss Joural of Moder Physcs, 8, hps://doorg/10436/jp Receved: Augus 5, 016 Acceped: February 5, 017 Publshed: February 8, 017 Copyrgh 017 by auhors ad Scefc Research Publshg Ic Ths work s lcesed uder he Creave Coos Arbuo Ieraoal Lcese (CC BY 40) hp://creavecoosorg/lceses/by/40/ Ope Access Absrac I hs paper, survval daa aalyss s realzed by applyg Geeralzed Eropy Opzao Mehods (GEOM) I s kow ha all sascal dsrbuos ca be obaed as MaxE dsrbuo by choosg correspodg oe fucos owever, Geeralzed Eropy Opzao Dsrbuos (GEOD) he for of MMaxE,MaxMaxE dsrbuos whch are obaed o bass of Shao easure ad suppleeary opzao wh respec o characerzg oe fucos, ore exacly represe he gve sascal daa For hs reaso, survval daa aalyss by GEOD acqures a ew sgfcace I hs research, he daa of he lfe able for ege falure daa (1980) s exaed The perforaces of GEOD are esablshed by Ch-Square crera, Roo Mea Square Error (RMSE) crera ad Shao eropy easure, Kullback-Lebler easure Coparso of GEOD wh each oher he dffere seses shows ha alog of hese dsrbuos ( MMaxE ) 4 s beer he seses of Shao easure ad of Kullback- Lebler easure I s showed ha, ( MMaxE) (( MaxMaxE ) ) s ore 3 4 suable for sascal daa aog MMaxE, = 1,, 3, 4 MaxMaxE, = 1,, 3, 4 Moreover, ( MMaxE ) 3 s beer for sascal daa ha ( MaxMaxE ) 4 he sese of RMSE crera Accordg o obaed dsrbuo ( MMaxE ) 3 (( MaxMaxE ) 4 ) esaor of Probably Desy Fuco ˆf ( ), Cuulave Dsrbuo Fuco ˆF ( ), Survval Fuco Ŝ( ) ad azard Rae ĥ are evaluaed ad graphcally llusraed The resuls are acqured by usg sascal sofware MATLAB Keywords Survval Fuco, Cesored Observao, Geeralzed Eropy Opzao Mehods, MaxE,MMaxE,MaxMaxE Dsrbuos DOI: 10436/jp February 8, 017

2 A Shalov e al 1 Iroduco Eropy Opzao Mehods (EOM) have pora applcaos, especally sascs, ecooy, egeerg ad so o There are several exaples he leraure ha kow sascal dsrbuos do o cofor o sascal daa; however, he eropy opzao dsrbuos cofor well Geeralzed Eropy Opzao Mehods (GEOM) have suggesed dsrbuos he for of MMaxE whch s he closes o sascal daa, ad MaxMaxE whch s he furhes fro eoed daa he sese of forao heory [1] [], respecvely For hs reaso, GEOM ca be ore successfully appled Survval Daa Aalyss Dffere aspecs ad ehods of vesgaos of survval daa aalyss are cosdered [3]-[8] I parcular he paper [6], s vesgaed several probles of hazard rae fuco esao based o he axu eropy prcple The poeal applcaos clude developg several classes of he axu eropy dsrbuos whch ca be used o odel dffere daa-geerag dsrbuos ha sasfy cera forao cosras o he hazard rae fuco I order o represe he resuls of our vesgaos, we gve soe auxlary coceps ad facs frs Survval Aalyss Survval e ca be defed broadly as he e o he occurrece of a gve eve Ths eve ca be he develope of a dsease, respose o a reae, relapse or deah [9] Cesorg: The echques for reducg expereal e are kow as cesorg I survval aalyss, he observaos are lfees, whch ca be defely log So que ofe he expere s so desged ha he e requred for collecg he daa s reduced o aageable levels Le T be a couous, o-egave valued rado varable represeg he lfee of a u Ths s he e for whch a dvdual (or u) carres ou s appoed ask sasfacorly ad he passes o faled or dead sae hereafer [10] The probablsc properes of he rado varable are suded hrough s cuulave dsrbuo fuco F or oher equvale fucos defed below [9]: Cuulave Dsrbuo Fuco: F( ) = P{ T < } = f ( u) d u, 0< < 0 Survval Fuco: Ths fuco s deoed by S( ), s defed as he probably ha a dvdual survves loger ha : S( ) = 1 F( ) { } d, 0 S = P T > = f u u < < Probably Desy Fuco: Lke ay oher couous rado varable, he survval e T has a probably desy fuco defed as he l of he 350

3 A Shalov e al probably ha a dvdual fals he shor erval + per u wdh, or sply he probably of falure a sall erval per u e I ca be expressed as f ds( ) df = = d azard Rae: Ths fuco s defed as he probably of falure durg a very sall e erval, assug ha he dvdual has survved o he begg of he erval, or as he l of he probably ha a dvdual fals a very shor erval, +, gve ha he dvdual has survved o e : h f = S 3 Geeralzed Eropy Opzao Mehods (GEOM) Eropy Opzao Proble (EOP) [11] ad Geeralzed Eropy Opzao proble (GEOP) [10] ca be forulaed he followg for ( 0 EOP: Le f ) ( x ) be gve probably desy fuco (pdf) of rado varable X, L be a eropy opzao easure ad g be a gve oe vecor fuco geerag oe cosras I s requred o oba he dsrbuo correspodg o g, whch gves exree value o L ( 0 GEOP: Le f ) ( x ) be gve probably desy fuco of rado varable X, L be a eropy opzao easure ad K be a se of gve oe ( 1) vecor fucos I s requred o choose oe vecor fucos g, ( ) g K such ha g ( 1) defes eropy opzao dsrbuo ( 1 f ) ( x ) closes o ( 0 ) ( ) f ( x ), g defes eropy opzao dsrbuo ( f ) ( x ) ( 0 furhes fro f ) ( x ) wh respec o eropy opzao easure L If L s ( 1 ake as Shao eropy easure, he f ) ( x ) s called he MMaxE dsrbuo, ad ( f ) ( x ) s called he MaxMaxE dsrbuo [1] [] [1] [13] [14] The ehod of solvg GEOP s called as GEOM 31 MaxE Fucoal The proble of axzg eropy fuco ( p) = p l, ( 1,, ) 1 p p = p p = (1) subjec o cosras p 1, 0,1,,, 1 = pg 1 j x = µ j j= = () = where µ = 1, g ( x) = 1, p 0, = 1,,, ; + 1 <, has soluo 0 0 where j ( j 0,1,,, ) ( p p p ) j 0λ jgj( x) d = p = e, = 1,,, (3) λ = are Lagrage ulplers Fdg he dsrbuo 1,, whch axzes fuco (1) subjec o cosras geeraed by equaos () s a opzao proble I he leraure, here have bee uerous sudes ha have calculaed hese ulplers [1] I hs sudy, we use 351

4 A Shalov e al he MATLAB progra o calculae Lagrage ulplers If (3) s subsued o (1), he axu eropy value s obaed: j= 0λ jgj( x) ax = λ j 0 j j λµ = = j j = 1 j= 0 p e g x (4) If dsrbuo ( 0) ( 0) ( 0) p = p1,, p s calculaed fro he daa, he oe µ = 1, µ 1, µ ca be obaed for each oe vecor fuco g( x) = ( 1, g1 ( x),, g ( x) ) Thus, ax s cosdered as a fucoal of g( x ) U g o vecor value ad called he MaxE fucoal Therefore, we use he oao deoe he axu value of correspodg o ( 1, 1,, ) g x = g x g x 3 MMaxE ad MaxMaxE Dsrbuos Le K be he copac se of oe vecor fucos g( x) U g reaches s leas ad greaes values hs copac se, because of s couy propery For hs reaso, Cosequely, ( 1) ( ) g K U g = U g ; ax U g = U g g K U( g 1 ) U g Dsrbuos ( 1) ( 1) ( 1) p = p,, 1 p ad ( ) ( ) ( ) p = p1,, p correspodg 1 o he g x ad g x, respecvely, are called MMaxE ad MaxMaxE dsrbuos [1] MMaxE MaxMaxE ehod for a fe se of characerzg oe fucos ca be defed followg for Le K0 = { g,, 1 gr} be he se of characerzg oe vecor fucos ad all cobaos of r elees of K 0 ake elees a a e be K 0, We oe ha, each elee of K 0, s vecor g wh copoes Solvg he MMaxE ad he MaxMaxE probles requre o fd vecor fucos g, g ( 1 ) 0 ( x ),, ( ) 0 ( 1) ( ) g ( x) 1, g K, g K g g x, where 0 = 0, 0, zg ad axzg U( g ) accordgly wh respec o Shao eropy easure I should be oed ha U( g ) reaches s u value subjec o cosras geeraed by fuco g ( x ) g x g K, I oher words, - ad all -desoal vecor fucos 0, u value of U( g ) s leas value of values g( x), g K0, If ( 1) g g gves he u value o (, 0 ) (,, 1 ) U g he dsrbuo ( 1) ( 1) ( 1) p p p ax correspodg o = correspodg o ( 1) g, 0 g s called he MMaxE dsrbuo MMaxE ehod represes probably dsrbuo he for of MMaxE dsrbuo I a slar way, U( g ) reaches s axu value subjec o cosras geeraed by fuco g0 ( x ) ad all -desoal vecor fucos g( x), g K0, I oher words, axu value of U( g ) s greaes value of values ax correspodg o g x g K If ( ) U g he dsrbu- 0, ( 1,, ),, 0 o ( ) ( ) ( ) g g gves he axu value o (, 0 ) p = p p correspodg o ( ) g g s called he 0 35

5 A Shalov e al MaxMaxE dsrbuo MaxMaxE ehod represes probably dsrbuo he for of MaxMaxE dsrbuo I should be oed ha boh dsrbuos ca be appled solvg proper probles survval daa aalyss 4 Applcao of MMaxE ad MaxMaxE Mehods o Survval Daa 41 MMaxE ad MaxMaxE Dsrbuos for Fe Se of Characerzg Moe Fucos I he prese research, he daa of he lfe able for ege falure daa (1980) gve Table 1 s cosdered [10] I our vesgao, he expere s plaed for 00 ubers of paes survvg a begg of erval bu he presece of cesorg fro he plag paes 97 dvduals say ou he expere Ths suao s ake o accou Table I should be oed ha, he presece of cesorg he survval es leads o a suao where he su of observao probables sads less ha 1 for he Table 1 The daa of he lfe able for ege falure daa (1980) Survval Te (year) Workg a he begg of erval Faled durg he erval d Cesored durg he erval c Table Observed ad correced probables d c Observed probables p Correced probables p

6 A Shalov e al survval daa For hs reaso, solvg ay probles, s requred o supplee he su of observao probables up o 1 Sce he su of observed probables p Table s 08155, accordg o he uber of cesorg, suppleeary probably = s uforly dsrbued o each cesorg daa ad correced probables p are obaed As we oed ha above, MMaxE ad MaxMaxE dsrbuos ca be appled solvg proper probles survval daa aalyss I our vesgao as copoes of K 0 characerzg oe vecor fuco g1 x = x, g x = x, g3 x = l x, g4 x = lx, g5 ( x) = l ( 1+ x ) are chose The se of oe fucos s chose fro he characersc oes whch are osly used Sascs K = g,, g For exaple, f = 3 he Cosequely, { } gves he leas value o U( g ) ad ( 1) 1 ( g0, g ) = 1, xx,, ( l x), g K0,3 ( ) ( ) ( g0, g ) = ( 1, x, l x, l ( 1 + x )), g K0,3 gves he greaes value o U( g ) The MaxE dsrbuos correspodg o ( g, g), g ( x) 1, g K, 1,,,4 0 0 = 0, = ad ax values are show Tables 3-6 I hese ables, MMaxE ad MaxMaxE dsrbuos correspodg o MMax ad MaxMax are represeed wh bold fo By vrue MaxMaxE, of hese ables are also obaed ( MMaxE), = 1,,3,4 dsrbuos whch are show Table 7 ad Table 8 I order o oba he perforace of he eoed dsrbuos, we use varous crera as Roo Mea Square Error (RMSE), Ch-Square, eropy values All of dsrbuos The acqured resuls are deosraed Table 9 ad Table 10 MMaxE, MaxMaxE, = 1,,3, 4 dsrbuos are accepable o survval daa he sese of Ch Square crera I he sese of RMSE crera each ( MMaxE) ( = 1,,3, 4) dsrbuo s beer ha correspodg ( MaxMaxE) ( = 1,,3, 4) dsrbuo More- MaxMaxE ad over, ( MMaxE ) 1 s earer o sascal daa ha 1 Table 3 The predced probables for he MaxE dsrbuo correspodg o ( g, g), g ( x) = 1, g K ad ax values 0 0 0,1 ( g, g 0 ) ( g, g ) 0 1 ( g, g 0 ) ( g, g 0 3) ( g, g 0 4) ( g, g 0 5) ax ( 0) p for MaxE Dsrbuo

7 A Shalov e al Table 4 The predced probables for he MaxE dsrbuo correspodg o ( g, g), g ( x) = 1, g K ad ax values 0 0 0, ( g, g 0 ) ( g, g, g ) 0 1 ( g, g, g 0 1 3) ( g, g, g 0 1 4) ( g, g, g 0 1 5) ( g, g, g 0 3) ax ( 0) p for MaxE Dsrbuo ,, ( g, g 0 ) ,, ,, ,, g, g, g g g g ( g g g ) ( g g g ) ( g g g ) ax ( 0) p for MaxE Dsrbuo Table 5 The predced probables for he MaxE dsrbuo correspodg o ( g, g), g ( x) = 1, g K ad ax values 0 0 0,3 ( g, g 0 ) ( g, g, g, g ) ( g, g, g, g ) ( g, g, g, g ) ( g, g, g, g ) ( g, g, g, g ) ax ( 0) p for MaxE Dsrbuo ,,, ( g, g 0 ) ,,, ,,, ,,, g, g, g, g g g g g ( g g g g ) ( g g g g ) ( g g g g ) ax ( 0) p for MaxE Dsrbuo

8 A Shalov e al Table 6 The predced probables for he MaxE dsrbuo correspodg o ( g, g), g ( x) = 1, g K ad ax values 0 0 0,4 ( g, g 0 ) ( g, g, g g g ) 0 1, 3, 4 ( g, g, g g g 0 1, 3, 5) ( g, g, g g g 0 1, 4, 5 ) ( g, g, g g, g 0 1 3, 4 5) ( g, g, g g, g 0 3, 4 5) ax ( 0) p for MaxE Dsrbuo Table 7 Dsrbuos of ( MMaxE ), = 1,,3, 4 d c p ( MMaxE ) 1 ( MMaxE ) ( MMaxE ) 3 ( MMaxE ) Table 8 Dsrbuos of ( MaxMaxE ), = 1,,3, 4 d c p ( MaxMaxE ) 1 ( MaxMaxE ) ( MaxMaxE ) 3 ( MaxMaxE ) ( MaxMaxE ) dsrbuos; each MMaxE, 3, 4 all of ( MaxMaxE) ( 1,,3, 4) ha aog of dsrbuos ( MMaxE), ( 1,,3,4) = s beer ha = dsrbuos Fro hese resuls follows = he dsrbuo 356

9 A Shalov e al ( MMaxE ) 3 s ore suable ad aog of dsrbuos ( = 1,,3,4) he dsrbuo 4 MaxMaxE, MaxMaxE s ore covee for sascal daa These resuls are also corroboraed by graphcal represeao (see Fgures 1-4) Cosequely, we shall cosder Probably Desy Fuco ˆf ( ), Cuulave Dsrbuo Fuco ˆF ( ), Survval Fuco Ŝ( ) ad azard Rae ĥ( ) for oly ( MMaxE ) 3 ad ( MaxMaxE ) 4 dsrbuos Alhough he dsrbuo wh he larges uber of oe fucos eds o f beer, should be oed ha soe cases, he se of oe fucos wh fewer elees s ore forave he a dffere se of oe fucos wh ore uber of elees Table 9 The obaed resuls for ( MMaxE), = 1,,3,4 Dsrbuo of MMaxE Calculaed value of Ch-Square Probably of Ch-Square value ( MMaxE ) ( MMaxE ) ( MMaxE ) ( MMaxE ) Table value of RMSE Ch-Square χ = , α χ = , α χ = 159 6, α χ = , α Table 10 The obaed resuls for ( MaxMaxE), = 1,,3,4 Dsrbuo of MMaxE Calculaed value of Ch-Square Probably of Ch-Square value ( MaxMaxE ) ( MaxMaxE ) ( MaxMaxE ) ( MaxMaxE ) Table value of RMSE Ch-Square χ = , 0349 α χ = , 0349 α χ = 159 6, 0104 α χ = , α (a) (b) Fgure 1 Graphc of ( MMaxE ) 1 ad ( MaxMaxE ) 1 dsrbuos 357

10 A Shalov e al (a) (b) Fgure Graphc of ( MMaxE ) ad ( MaxMaxE ) dsrbuos (a) (b) Fgure 3 Graphc of ( MMaxE ) 3 ad ( MaxMaxE ) 3 dsrbuos (a) (b) Fgure 4 Graph of ( MMaxE ) 4 ad ( MaxMaxE ) 4 dsrbuos 358

11 A Shalov e al 4 Avalably of GEOD o Survval Daa he Sese of Shao Measure I order o esablsh avalably of GEOD o survval daa he sese of Shao easure s requred o cosder eropy values of GEOD Fro Table 3 s see ha he MMaxE (he MaxMaxE ) dsrbuo s realzed by vecor fuco ( g0, g) = ( 1, x )(( g0, g3) = ( 1, l x) ) ad ( ( MMaxE) ) = 3854( ( ( MaxMaxE) 1 1) = 3319) Fro Table 4 s see ha he MMaxE (he MaxMaxE ) dsrbuo s realzed by vecor fuco ( g ) ( ( )) 0, g, g3 = 1, x, l x g0, g1, g4 = 1, x, l x ad ( ( MMaxE) ) = 3041( ( ( MaxMaxE) ) = 391) Fro Table 5 s see ha he MMaxE (he MaxMaxE ) dsrbuo s realzed by vecor fuco ( g ) ( )( ( )) 0, g1, g, g4 = 1, x, x, l x g0, g, g3, g5 = 1, x, l x, l 1+ x ad ( ( MMaxE) ) = 3000( ( ( MaxMaxE) 3 3) = 3155) Fro Table 6 s see ha he MMaxE (he MaxMaxE ) dsrbuo s realzed by vecor fuco ( ( )) g 0, g1, g3, g4, g5 = 1, x,l x, l x,l 1 + x g0, g1, g, g3, g4 = 1, x, x,l x, l x ad ( ( MMaxE) ) = 3193( ( ( MaxMaxE) 4 4) = 31937) Coparso of GEOD wh each oher he sese of Shao easure shows ha alog of hese dsrbuos ( MMaxE ) 4 s beer The resuls of our vesgao accordg o usg kow characerzg oe vecor fucos fro K 0, are suarsed he for of followg Corollary Corollary 1 If by ( MaxMaxE) (( MMaxE) ) deoe he MaxMaxE (he MMaxE ) dsrbuo correspodg o oe g x g K, he equaly, codos geeraed by oe fucos 0, ( MaxMaxE) > MaxMaxE ( ) 1 ( ( ( MMaxE) ) > ( ( MMaxE) )) 1 s fulflled, whe 1 < I oher words, eropy value of he MaxMaxE (he MMaxE ) dsrbuo depedg o he uber of oe codos decreases Moreover for ay he equaly akes place (( MaxMaxE) ) > ( MMaxE) 43 Avalably of GEOD o Survval Daa he Sese of Kullback-Lebler Measure Now, we calculae he dsace bewee observed dsrbuo p = ( p1, p,, p10 ) gve Table ad dsrbuos p ( ) ( ) = ( MMaxE ), p ax = ( MaxMaxE ), = 1,, 3, 4 gve Table 7 ad Table 8 respecvely 359

12 A Shalov e al I s kow ha he Kullback Lebler dsace bewee dsrbuos p = ( p1, p,, p ) ad q = ( q1, q,, q ) s obaed by forula p D( pq ; ) = pl = 1 q By sarg hese forula Kullback-Lebler easures for he dsace bewee p = p, p,, p ad dsrbuos observed dsrbuo ( 1 10 ) ( ) ( ) p = MMaxE, p = MaxMaxE, = 1,, 3, 4 are gve Table 11 ax ad Table 1 respecvely Fro Table 11 ad Table 1 follows ha alog of GEOD 4 beer he sese of Kullback-Lebler easure MMaxE s The resuls of our vesgao accordg o usg kow characerzg oe vecor fucos fro K 0, are suarsed he for of followg Corollary Corollary If p = ( p1, p,, p10 ) observed dsrbuo ad p ( ) ( ) = ( MMaxE ), = 1,, 3, 4( p ax = ( MaxMaxE ), = 1,, 3, 4 ) deoe he MMaxE (he MaxMaxE ) dsrbuo correspodg o oe g x g K, he equaly, codos geeraed by oe fucos 0, ( 1) ( ) D( p ; p) > D( p ; p) ( 1) ( ) ( D( pax ; p) > D( pax ; p) ) s fulflled, whe 1 < I oher words, Kullback-Lebler value of he MMaxE (he MaxMaxE ) dsrbuo depedg o he uber of oe codos decreases Moreover for ay he equaly akes place ( ) ( ) D( pax; p) > D p; p Table 11 Kullback-Lebler easure of ( MMaxE ), = 1,,3, 4 dsrbuos ( MMaxE ), 1,,3, 4 = dsrbuos ( ) D p ; p, = 1,,3,4 ( MMaxE ) ( MMaxE) ( MMaxE) ( MMaxE ) Table 1 Kullback-Lebler easure of ( MaxMaxE ), = 1,,3, 4 dsrbuos ( MaxMaxE ), 1,,3, 4 = dsrbuos ( ) D p ; p, = 1,,3,4 ax ( MaxMaxE ) ( MaxMaxE) ( MaxMaxE ) ( MaxMaxE )

13 A Shalov e al 44 Survval Expresso of Dsrbuos ( MMaxE ) 3, ( MaxMaxE ) 4 I hs seco survval daa aalyss s coduced by ( MMaxE) (( MaxMaxE ) 3 4) dsrbuo sce he above acqured vesgaos ( MMaxE) (( MaxMaxE ) 3 4) s ore preseable for survval daa aog ( MMaxE) (( MaxMaxE) ), = 1,,,4 dsrbuos ( MMaxE ) 3 ad ( MaxMaxE ) 4 esaos of Probably Desy Fuco ˆf ( ), Cuulave Dsrbuo Fuco ˆF ( ), Survval Fuco Ŝ( ) ad azard Rae ĥ( ) are gve Table 13 & Table 14, respecvely O bass of he resuls gve Table 13 & Table 14, graphs of fˆ ( ), Fˆ ( ), ĥ are deosraed Fgures 5(a)-(c) & Fgures 6(a)-(c) Ŝ( ) ad Table 13 Survval aalyss by ( MMaxE ) 3 d fˆ = ˆF ( ) Ŝ( ) h c ( MMaxE) 3 ˆ fˆ = Sˆ Table 14 Survval aalyss by ( MaxMaxE ) 4 d ˆ f = ˆF ( ) Ŝ( ) h ˆ c ( MaxMaxE) 4 f ˆ ( ) = S ˆ

14 A Shalov e al (a) (b) Fgure 5 Survval expresso of dsrbuo ( MMaxE ) 3 5 Cocluso (c) I hs sudy, s esablshed ha survval daa aalyss s realzed by applyg Geeralzed Eropy Opzao Mehods (GEOM) Geeralzed Eropy Opzao Dsrbuos (GEOD) he for of MMaxE, MaxMaxE dsrbuos whch are obaed o bass of Shao easure ad suppleeary opzao wh respec o characerzg oe fucos, ore exacly represe he gve sascal daa For hs reaso, survval daa aalyss by GEOD acqures a ew sgfcace The perforaces of GEOD are esablshed by Ch-Square crera, Roo Mea Square Error (RMSE) crera ad Shao eropy easure, Kullback-Lebler easure Coparso of GEOD wh each oher he dffere seses shows ha alog of hese dsrbuos ( MMaxE ) 4 s beer he seses of Shao easure ad of Kullback-Lebler easure I 36

15 A Shalov e al (a) (b) Fgure 6 Survval expresso of dsrbuo ( MaxMaxE ) 4 (c) s showed ha, ( MMaxE) ( MaxMaxE ) s ore suable for sascal 3 4 daa aog ( MMaxE ), = 1,, 3, 4( ( MaxMaxE ), = 1,, 3, 4 ) Moreover, ( MMaxE ) 3 s beer for sascal daa ha ( MaxMaxE ) 4 he sese of RMSE crera Accordg o obaed dsrbuo ( MMaxE ) 3 (( MaxMaxE ) 4 ) esaor of Probably Desy Fuco ˆf ( ), Cuulave Dsrbuo Fuco ˆF ( ), Survval Fuco Ŝ( ) ad azard Rae ĥ( ) are evaluaed ad graphcally llusraed These resuls are also corroboraed by graphcal represeao Our vesgao dcaes ha GEOM survval daa aalyss yelds reasoable resuls 363

16 A Shalov e al Refereces [1] Shalov, A (006) A Develope of Eropy Opzao Mehods Wseas Trasacos o Maheacs, 5, [] Shalov, A (007) Geeralzed Eropy Opzao Probles ad he Exsece of Ther Soluos Physca A: Sascal Mechacs ad Is Applcaos, 38, hps://doorg/101016/jphysa [3] Kask, D ad Gesler, C (01) Survval Aalyss of Faculy Reeo Scece ad Egeerg by Geder Scece, 335, hps://doorg/10116/scece [4] Regold, EM, Rechle, ED ad Glahol, MG (01) eaher Sherda, Drec Lexcal Corol of Eye Movees Readg: Evdece fro a Survval Aalyss of Fxao Duraos Cogve Psychology, 65, [5] Wag, ad Da, S (01) Acceleraed Falure Te Models for Cesored Survval Daa uder Referral Bas Bosascs, 14, [6] Ebrah, N (000) The Maxu Eropy Mehod for Lfee Dsrbuos Sakhyā: The Ida Joural of Sascs, Seres A, 6, [7] Guyo, P, Ades, A, Ouwes, MJ ad Welo, NJ (01) Ehaced Secodary Aalyss of Survval Daa: Recosrucg he Daa fro Publshed Kapla-Meer Survval Curves BMC Medcal Research Mehodology, 1, 9 hps://doorg/101186/ [8] Joly, P, Gerds, TA, Qvs, V, Coeges, D ad Kedg, N (01) Esag Survval of Deal Fllgs o he Bass of Ierval-Cesored Daa ad Mul-Sae Models Sascs Medce, 31, 11-1 [9] Lee, ET ad Wag, JW (003) Sascal Mehods for Survval Daa Aalyss Wley-Ierscece, Oklahoa [10] Deshpade, JV ad Puroh, SG (005) Lfe Te Daa: Sascal Models ad Mehods, Seres o Qualy Vol 11, Relably ad Egeerg Sascs, Ida [11] Kapur, JN (199) Kesava, Eropy Opzao Prcples wh Applcaos [1] Shalov, A (009) Eropy, Iforao ad Eropy Opzao TC Aadolu Uversy Publcao, Esksehr [13] Shalov, A (010) Geeralzed Eropy Opzao Probles wh Fe Moe Fucos Ses Joural of Sascs ad Maagee Syses, 13, hps://doorg/101080/ [14] Shalov, A, Grfoglu, C, Usa, I ad Mer Kaar, Y (008) A New Cocep of Relave Suably of Moe Fuco Ses Appled Maheacs ad Copuao, 06, hps://doorg/101016/jac

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