Science & Technologies GENERAL BIRTH-DEATH PROCESS AND SOME OF THEIR EM (EXPETATION- MAXIMATION) ALGORITHM

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1 GEERAL BIRH-EAH ROCESS A SOME OF HEIR EM EXEAIO- MAXIMAIO) ALGORIHM Il Hl, Lz Ker, Ylldr Seer Se ery o eoo,, eoo Mcedo l.hl@e.ed.; lz.er@e.ed.; ylldr_@hol.co ABSRAC Brh d deh roce coo-e Mrco ch, h odel oege eger o rcle ye. he o h er o ele e-e ro role or eo o rrry rh d deh re geerl B. We r o ereo or Llce ror o hee ro role. I oe ecery o ele e-e ro role. For eo we e EM eecozo) lgorh cldg E-e y gg or ele le: le d geerlzed B, ler B gro, Mor odel d Regreo o co d. I he ed we ge corol o ercl error o ro role. Key word: Geerl Brh-deh roce, ro role, EM lgorh, E-Se, ercl error. Irodco he coo e rh d deh Mrco ch X h eher e or e e ce {,,,..., } or {,, }. Ae he el ro role o geerl rh d deh roce y., { X X }) ) ) or cely ll where, or,... d d ). I oe he ce h lo, ece, or ele whe here gro.,, I he e ce e, he he l ro r ) )) I dey r). I ll erl o e, o oe chge e c occr, eher rh or deh. I he olo ze, d rh occr, he, deh occr he. We coder he ro roly, [ ], [ ] [ ) ],,,,, [ ) ]...) I he ce o e e where he :, d he Vole V, er, 5 rl & Mhecl cece

2 ddg y [ d g he l, ) Kooro derel eo re oed or he geerl rh d deh roce. d o d d,, ), 4) I geerl hoe eo y d R where geeror r whe he e e : d whe he e e: For rh d deh ch, here ere rocedre or cog he ory roly dro. Krl d McGregor 957) how h or rrry rg e, ro role c e rereeed he or: ) ) d) 5) d Llce ror o 4) : ) L[ ] ) e d 6) Alyg he Llce ror o ), wh : Vole V, er, 5 rl & Mhecl cece

3 ,, oo d ) ) ) o, ) ),, ) ) ), ) ) ) oo ),,, ), ) ) ) ), ) ) )... Le, d, or, he :, )... or, )... d h coerge o ) ) A ) oo )... 7) B ). Eo EM lgorh We ow roceed wh eo o he reer o geerl B g dcree oero. Le Y X, X he or o oe or ore deede oero ro geerl B. For he gle relzo o he roce rg X d edg X, le e he ol e e e, e er o e rh) ro e d le e er o dow e deh) ro he e. Le he er o ol d dow e relzo o he roce e deoed y, d ol rcle e: I X d d Wol 965) how he -lelhood or cooly oered roce: 8) Whe B led dcreely ch h oly X d X re oered, he,, re ow or eery d we co ze he 8) who he. Vole V, er, 5 rl & Mhecl cece

4 I he EM lgorh we dee rroge oece co y g he eeco o 8), codol o he oered d Y d he reer le o Y d we or :. g he eeco o 8) codol, 9) Where or clry we he oed he deedece o he eeco o reer le h ere. ro he -. Cog eeco o he E-e I ddo he e-ce o B geerlly e, eg o roe e B lr roce o he e e-ce,,..., where choe o h he roly o he roce g e greer he ll. X X, X,, g ro role we he:, d, d d ) g Llce coolo roery, we rre he rereeo: L, L, L ) Where L deoe ere Llce roro. Vole V, er, 5 rl & Mhecl cece

5 4. Sle rh d deh roce For cely ll, he ro role or le rh d deh roce y:, X X,, ) Kooro derel eo: d d or,,...,wh ) I h roce rh d deh he co er-cl re, o, d he rroge co ecoe: d he E E E 4) Wh dero o 4) eg he rel o d olg or d we he: E E E E 5) 5. Ler B gro d egro Soee olo re o cloed d ew ddl c eer, we cll h co gro, where d Soe gro clde he le rh d deh roce co re. For cely ll, he ro role or le rh d deh roce wh gro y:, X X 6),, Vole V, er, 5 rl & Mhecl cece 4

6 Kooro derel eo: d d 7) he -lelhood ecoe : 8) 9) Mzg wh reec o dhe yeld he de: d ) Lewe, coder olo odel or he er o org re d oe ew egr lee re. I h ce, we dee he ler re. he roce coo le he gro ce. 6. Mor odel Mor 958) clde odel or he e-eolo o llelc loc whe he olo ze co hrogh e. h roce ee rc o he er o llele o cer ye llelc loc hlod olo o co ze. Cll he wo llele A d B, wh e d reecely. I he Mor odel wh o, A e B wh roly, d B e A wh roly. I we le X e he er o A llele he olo e, he re o ddo o ew A : d he re o reol o A : where,,,..., d wh whe Vole V, er, 5 rl & Mhecl cece 5

7 Vole V, er, 5 rl & Mhecl cece 6 We c ze he rh re : where: ) Lewe we ze he deh re, where: ) d we or he zg co H B H ) he de o 4) he olo o B ge he de o 7. Regreo o co d Coder colleco o deede B, X wh Z e d or,...,, ; where Z d ecor o core d core ecor o correodg deo d or ll. Leg X d X, he: Z Z e -lelhood or clcl oo regreo. he ewo-rho de or d 5)

8 where Z Z e Z d d Z Z e Z ' 8. ro roly error o d he ro roly rco he M -h coerge we e G. We c ele he e coed M M oe eger choe dyclly o h he error de rco M or eery ~ A e Re A e Re he dcree error A ~ A e e 6) A e Reerece:. Ae J. Wh W.,Cog Llce ror o ercl ero coed rco, 999, IFORMS J Co, 4): Wol R.W., role o Scl Ierece or Brh d eh eg Model, 965, Oero Reerch,4-57. Lee E.. d Wg J. W., Scl Mehod or Srl Aly, h ed.,, Wley, ew Yor 4. J. R. orr, Mro Ch, 998, Crdge Sere Scl d rolc Mhec 5. r Shoo, roly d Mhecl Sc,, ery o Lolle 6. Aho ol, roly Rdo Vrle d Sochc rocee h ed. 99, Mc Grw-Hll, c, ISB Vole V, er, 5 rl & Mhecl cece 7