Hidden Markov Models. A Specific Form of Process.. Doubly Stochastic Processes. What a sensible agent must do. A Common Trait

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1 -755 Mchne Lernng for Sgnl roceng Hdden Mrkov Model 04 Oc 0 redcon : holy grl hycl recore Auooble rocke hevenly bode Nurl phenoen Weher Fnncl d Sock rke World ffr Who gong o hve he ne prng? Sgnl Audo vdeo.. A Coon r Wh enble gen u do Lern bou he proce Sere d wh rend Sochc funcon of ochc funcon of ochc funcon of An underlyng proce h progree eengly rndoly E.g. Curren poon of vehcle E.g. curren enen n ock rke Curren e of ocl/econoc ndcor Rndo epreon of underlyng proce E.g wh you ee fro he vehcle E.g. curren ock prce of vrou ock E.g. do populce y que / proe on ree / opple dcor.. A B C 3 Fro whever hey know Bc requreen for oher procedure rck underlyng procee redc fuure vlue 4 A Specfc For of roce.. Doubly ochc procee One rndo proce genere n Rndo proce ; Q Second level proce genere obervon funcon of Rndo proce Y Y; f L Y Doubly Sochc rocee Doubly ochc procee re odel My no be rue repreenon of proce underlyng cul d Fr level vrble y be qunfble vrble oon/e of vehcle Second level vrble ochc funcon of poon Fr level vrble y no hve enng Senen of ock rke Confguron of vocl rc Y 5 6

2 Sochc Funcon of Mrkov Chn Mrkov Chn Fr level vrble uully brc S Y S S3 he fr level vrble ued o be he oupu of Mrkov Chn he econd level vrble funcon of he oupu of he Mrkov Chn Alo clled n HMM Anoher vrn ochc funcon of Mrkov proce Kln Flerng.. roce cn go hrough nuber of e Rndo wlk Brownn oon.. Fro ech e cn go o ny oher e wh probbly Whch only depend on he curren e Wlk goe on forever Or unl h n borbng wll Oupu of he proce equence of e he proce wen hrough 7 8 Sochc Funcon of Mrkov Chn Sochc funcon of Mrkov Chn HMMS S roble: S Oupu: Y Y ; f[ ] L Specfc o HMM: Y == Y Y Y Y ; f L S3 Lern he nure of he proce fro d rck he underlyng e Senc redc he fuure 9 0 Fun uff wh HMM.. he lle on beween he ll nd he cy A lle on beween he cy nd ll Inbound rn brng people bck fro he ll Mnly hopper Occonl ll eployee Who y hve hopped.. Oubound rn brng bck people fro he cy Mnly offce worker Bu lo he occonl hopper Who y be fro n offce..

3 he urnle One oble fernoon you ue yourelf by obervng he urnle he on Group of people e perodclly Soe people re werng cul oher re forlly dreed Soe re crryng hoppng bg oher hve brefce W he l rn n ncong rn or n ougong one he urnle One oble fernoon you ue yourelf by obervng he urnle he on. Wh you know: eople hop n cul re Unle hey hed o he hop fro work Shopper crry hoppng bg people fro offce crry brefce Uully here re ore hop hn offce he ll here re ore offce hn hop n he cy Oubound rn follow nbound rn Uully Modellng he proble Modellng he proble Inbound Oubound Inbound Oubound re luggge oubound =? Inbound rn fro he ll hve ore cully dreed people ore people crryng hoppng bg he nuber of people levng ny e y be ll Inuffcen o udge re luggge nbound =? oubound nbound =? nbound oubound =? If you know ll h how do you decde he drecon of he rn How do you ee hee er? 95 6 Wh n HMM A hough Eperen I u clled ou he 6 fro he blue guy.. go wch o pern.. robblc funcon of rkov chn Model dyncl ye Sye goe hrough nuber of e Followng Mrkov chn odel On rrvng ny e genere obervon ccordng o e pecfc probbly drbuon 7 wo hooer roll dce A cller cll ou he nuber rolled. We only ge o her wh he cll ou he cller behve rndoly If he h u clled nuber rolled by he blue hooer h ne cll h of he red hooer 70% of he e Bu f he h u clled he red hooer he h only 40% probbly of cllng he red hooer gn n he ne cll How do we chrcerze h? 8

4 blue red blue red A hough Eperen A hough Eperen he do nd rrow repreen he e of he cller When he on he blue crcle he cll ou he blue dce When he on he red crcle he cll ou he red dce he hogr repreen he probbly drbuon of he nuber for he blue nd red dce 9 When he cller n ny e he cll nuber bed on he probbly drbuon of h e We cll hee e oupu drbuon A ech ep he ove fro h curren e o noher e followng probbly drbuon We cll hee rnon probble he cller n HMM!!! 0 Wh n HMM Hdden Mrkov Model HMM re cl odel for cul procee he odel ue h he proce cn be n one of nuber of e ny e nn he e of he proce ny e nn depend only on he e he prevou nn culy Mrkovn A ech nn he proce genere n obervon fro probbly drbuon h pecfc o he curren e he genered obervon re ll h we ge o ee he cul e of he proce no drecly obervble Hence he qulfer hdden A Hdden Mrkov Model con of wo coponen A e/rnon bckbone h pecfe how ny e here re nd how hey cn follow one noher A e of probbly drbuon one for ech e whch pecfe he drbuon of ll vecor n h e h cn be fcored no wo epre probblc ene A probblc Mrkov chn wh e nd rnon A e of d probbly drbuon oced wh he e 755/8797 Mrkov chn D drbuon HMM ued o be generng d e equence e drbuon obervon equence How n HMM odel proce 3 HMM reer he opology of he HMM Nuber of e nd llowed rnon E.g. here we hve 3 e nd cnno go fro he blue e o he red he rnon probble Ofen repreened r here he probbly h when n e he proce wll ove o he probbly p of begnnng ny e he coplee e repreened p he e oupu drbuon

5 HMM e oupu drbuon he e oupu drbuon he drbuon of d produced fro ny e ypclly odelled Gun he Dgonl Covrnce Mr Full covrnce: ll eleen re non-zero Dgonl covrnce: off-dgonl eleen re zero p Gun ; Q e d Q 0.5 Q Q - - S - / he preeer re nd Q More ypclly odelled Gun ure K w Gun ; Q 0 Oher drbuon y lo be ued E.g. hogr n he dce ce 5 For GMM frequenly ued h he feure vecor denon re ll ndependen of ech oher Reul: he covrnce r reduced o dgonl for he deernn of he dgonl Q r ey o copue 6 hree Bc HMM roble Wh he probbly h wll genere pecfc obervon equence Gven obervon equence how do we deerne whch obervon w genered fro whch e he e egenon proble Copung he robbly of n Obervon Sequence wo pec o producng he obervon: rogreng hrough equence of e roducng obervon fro hee e How do we lern he preer of he HMM fro obervon equence 7 8 HMM ued o be generng d rogreng hrough e robbly h he HMM wll follow prculr e equence e equence he proce begn oe e red here Fro h e ke n llowed rnon o rrve he e or ny oher e Fro h e ke noher llowed rnon And o on he probbly h he proce wll nlly be n e he rnon probbly of ovng o e he ne e nn when he ye currenly n Alo denoed by erler 9 30

6 Generng Obervon fro Se HMM ued o be generng d robbly h he HMM wll genere prculr obervon equence gven e equence e equence known e equence o o o o o o e drbuon obervon equence Copued fro he Gun or Gun ure for e o he probbly of generng obervon o when he ye n e A ech e genere n obervon fro he e n h e 3 3 HMM ued o be generng d roceedng hrough Se nd roducng Obervon robbly h he HMM wll genere prculr e equence nd fro prculr obervon equence e equence e drbuon o o o o o o o o o obervon equence A ech e produce n obervon nd ke rnon robbly of Generng n Obervon Sequence he prece e equence no known All poble e equence u be condered Copung Effcenly Eplc ung over ll e equence no rcble A very lrge nuber of poble e equence o o o... 3 ll. poble e. equence o o o Ined we ue he forwrd lgorh A dync progrng echnque. ll. poble e. equence o o o

7 Se nde Se nde Se nde Se nde Illurve Eple Dveron: he rell Eple: generc HMM wh 5 e nd ernng e. Lef o rgh opology = for e nd 0 for oher he rrow repreen rnon for whch he probbly no 0 Noon: = We repreen o = b for brevy 37 - Feure vecor e he rell grphcl repreenon of ll poble ph hrough he HMM o produce gven obervon he Y repreen HMM e repreen obervon Every edge n he grph repreen vld rnon n he HMM over ngle e ep Every node repreen he even of prculr obervon beng genered fro prculr e 38 he Forwrd Algorh he Forwrd Algorh... e... e - Cn be recurvely eed rng fro he fr e nn forwrd recuron - e he ol probbly of ALL e equence h end e e nd ll obervon unl - - ' ' ' e cn be recurvely copued n er of he forwrd probble e he Forwrd Algorh olprob he borbng e e In he fnl obervon he lph ech e gve he probbly of ll e equence endng h e Generl odel: he ol probbly of he obervon he u of he lph vlue ll e Obervon equence re ued o end only when he proce rrve n borbng e No obervon re produced fro he borbng e 4 4

8 Se nde he Forwrd Algorh olprob borbng roble : Se egenon Gven only equence of obervon how do we deerne whch equence of e w followed n producng? borbng ' borbng ' e ' Aborbng e odel: he ol probbly he lph copued he borbng e fer he fnl obervon he HMM generor Se re hdden HMM ued o be generng d HMM ued o be generng d e equence e equence e drbuon e drbuon obervon equence he proce goe hrough ere of e nd produce obervon fro he 45 obervon equence he obervon do no revel he underlyng e 46 HMM ued o be generng d e equence e drbuon obervon equence he e egenon proble Eng he Se Sequence Mny dfferen e equence re cpble of producng he obervon Soluon: Idenfy he o probble e equence he e equence for whch he probbly of progreng hrough h equence nd generng he obervon equence u.e u o o o Se egenon: Ee e equence gven obervon 47 48

9 Eng he e equence Eng he e equence Once gn ehuve evluon pobly epenve Bu once gn ple dync progrng Once gn ehuve evluon pobly epenve Bu once gn ple dync progrng oluon vlble o o o oluon vlble o o o o o o o o o Needed: Needed: rg... o o o rg... o o o he HMM generor he e equence HMM ued o be generng d e equence he probbly of e equence????y endng e nd producng ll obervon unl o o..-???? oy = o..-???? oyy e drbuon obervon equence he be e equence h end wh y wll hve probbly equl o he probbly of he be e equence endng - e oyy Ech encloed er repreen one forwrd rnon nd ubequen eon 5 5 e equence Eendng he e equence y rell he grph below how he e of ll poble e equence hrough h HMM n fve e nn e drbuon obervon equence he probbly of e equence????y endng e nd producng obervon unl o o..-o???? y = o..-???? oyy 53 e 54

10 he co of eendng e equence he co of eendng e equence endng only dependen on he rnon fro o y nd he obervon probbly y he co of eendng e equence he be ph o y hrough ply n eenon of he be ph o Beo..-???? oyy oyy y y e 55 e 56 he Recuron he overll be ph o y n eenon of he be ph o one of he e he prevou e he Recuron rob. of be ph o y = M Beo..-???? oyy y y e 57 e 58 Fndng he be e equence Verb Serch cond. he ple lgorh u preened clled he VIERBI lgorh n he lerure Afer A.J.Verb who nvened h dync progrng lgorh for copleely dfferen purpoe: decodng error correcon code! 59 Inl e nlzed wh ph-core = b e 5 Oc 0 All oher e hve core 0 nce = 0 for he 755/

11 Verb Serch cond. Verb Serch cond. Se wh be ph-core Se wh ph-core < be Se whou vld ph-core = [ - b ] Se rnon probbly o = [ - b ] Se rnon probbly o Score for e gven he npu e ol ph-core endng up e e Score for e gven he npu e ol ph-core endng up e e e e 6 6 Verb Serch cond. Verb Serch cond. e 63 e 64 Verb Serch cond. Verb Serch cond. e 65 e 66

12 Verb Serch cond. Verb Serch cond. e 67 e 68 Verb Serch cond. Verb Serch cond. HE BES SAE SEQUENCE IS HE ESIMAE OF HE SAE SEQUENCE FOLLOWED IN GENERAING HE OBSERVAION e 69 e 70 roble3: rnng HMM preer We cn copue he probbly of n obervon nd he be e equence gven n obervon ung he HMM preer Bu where do he HMM preer coe fro? hey u be lerned fro collecon of obervon equence Lernng HMM preer: Sple procedure counng Gven e of rnng nnce Iervely:. Inlze HMM preer. Segen ll rnng nnce 3. Ee rnon probble nd e oupu probbly preer by counng 7 7

13 Lernng by counng eple Eplnon by eple n ne few lde e HMM Gun DF e 3 obervon equence Eple how ONE eron How o coun fer e equence re obned Eple: Lernng HMM reer We hve n HMM wh wo e nd. Obervon re vecor h equence h vecor We re gven he followng hree obervon equence And hve lredy eed e equence Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c Eple: Lernng HMM reer Inl e probble uully denoed p: We hve 3 obervon of hee begn wh S nd one wh S ps = /3 ps = /3 Eple: Lernng HMM reer rnon probble: Se S occur e n non ernl locon Of hee followed by S e I followed by S Y e S S = / ; S S = y / Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c Eple: Lernng HMM reer rnon probble: Se S occur e n non ernl locon Of hee followed edely by S 6 e I followed by S Y e S S = / ; S S = y / Eple: Lernng HMM reer rnon probble: Se S occur e n non ernl locon Of hee followed edely by S 6 e I followed edely by S 5 e S S = / ; S S = y / Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c

14 Eple: Lernng HMM reer rnon probble: Se S occur e n non ernl locon Of hee followed edely by S 6 e I followed edely by S 5 e S S = 6/ ; S S = 5 / Eple: Lernng HMM reer rnon probble: Se S occur 3 e n non ernl locon Of hee followed edely by S 6 e I followed edely by S 5 e S S = 6/ ; S S = 5 / Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c Eple: Lernng HMM reer rnon probble: Se S occur 3 e n non ernl locon Of hee followed edely by S 5 e I followed edely by S 5 e S S = 6/ ; S S = 5 / Eple: Lernng HMM reer rnon probble: Se S occur 3 e n non ernl locon Of hee followed edely by S 5 e I followed edely by S 8 e S S = 6/ ; S S = 5 / Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S S Ob Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 8 8 Eple: Lernng HMM reer rnon probble: Obervon Se S occur 3 e n non ernl locon Of hee followed edely by S 5 e I followed edely by S 8 e S S = 5 / 3; S S = 8 / 3 e e S S S S S S S S S S Ob S S = 6/ = 0.545; S S = 5/ = S S = 5/3 = 0.385; S S = 8/3 = 0.65 reer lern o fr Se nl probble ofen denoed p ps = /3 = 0.66 ps = /3 = 0.33 Se rnon probble Obervon Obervon 3 e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 Repreened rnon r A S S S S S S Ech row of h r u u o.0 S S

15 Eple: Lernng HMM reer Se oupu probbly for S here re 3 obervon n S Se oupu probbly for S here re 3 obervon n S Segrege he ou nd coun Eple: Lernng HMM reer Copue preer en nd vrnce of Gun oupu deny for e S Obervon e e S S S S S S S S S S Ob e e S S S S S S Ob Q S ep d p Q Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 e e S S S Ob b3 b4 b9 3 b 4 b 9 6 c 7 c 9 c 4 0 c 5 b 3 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 e e S S S S Ob c c c4 c5 Q 3... b3 b3 b4 b c c c c Eple: Lernng HMM reer Se oupu probbly for S here re 4 obervon n S Se oupu probbly for S here re 4 obervon n S Segrege he ou nd coun Eple: Lernng HMM reer Copue preer en nd vrnce of Gun oupu deny for e S Obervon e e S S S S S S S S S S Ob e e S S S S Ob Q S ep d p Q Obervon e e S S S S S S S S S Ob b b b3 b4 b5 b6 b7 b8 b9 e e S S S S S S Ob b b b5 b6 b7 b8 Obervon 3 e e S S S S S S S S Ob c c c3 c4 c5 c6 c7 c8 87 e e S S S S Ob c c6 c7 c b6 4 b7 5 b8 8 c b Q 88 c6 b c7 b5 c 8 We hve lern ll he HMM preer Upde rule ech eron Se nl probble ofen denoed p ps = 0.66 ps = /3 = 0.33 p No.of obervon equence h r e olno.of obervon equence Se rnon probble A Se oupu probble Se oupu probbly for S Se oupu probbly for S 0.5 Q S ep d p Q 0.5 Q S ep d p Q ob : e.&. e Q ob : e ob : e. Aue e oupu DF = Gun ob : e ob. ob ob : e ob : e ob. For GMM ee GMM preer fro collecon 89 of obervon ny e 90

16 rnng by egenon: Verb rnng Inl odel Segenon Model Converged? ye Inlze ll HMM preer no Alernve o counng: SOF counng Epecon zon Every obervon conrbue o every e Segen ll rnng obervon equence no e ung he Verb lgorh wh he curren odel Ung eed e equence nd rnng obervon equence reee he HMM preer h ehod lo clled egenl k-en lernng procedure 9 Upde rule ech eron Upde rule ech eron p Ob e olno. of Ob obervon equence p Ob e olno. of Ob obervon equence Ob Ob e Ob e Ob e Ob Ob Ob e e Ob Ob Ob Ob e Ob e Ob e Ob Ob Ob e e Ob Ob Q Ob e Ob Ob Ob e Every obervon conrbue o every e Ob Ob Q Ob e Ob Ob Ob e Where dd hee er coe fro? Ob Ob e Ob he probbly h he proce w when genered gven he enre obervon Droppng he Ob ubcrp for brevy e... he probbly h he HMM w n prculr e when generng he obervon equence he probbly h followed e equence h ped hrough e e... e... We wll copue e... fr h he probbly h he proce ved e whle producng he enre obervon e 95 96

17 e... h cn be decopoed no wo ulplcve econ he econ of he lce ledng no e e nd he econ ledng ou of he Forwrd h he probbly of he red econ he ol probbly of ll e equence endng e e h ply Cn be copued ung he forwrd lgorh e e he Bckwrd h he blue poron repreen he probbly of ll e equence h begn e e Lke he red poron cn be copued ung bckwrd recuron b he Bckwrd Recuron... e bn b b Cn be recurvely eed rng fro he fnl e e nn bckwrd recuron e b + b ' ' ' e b he ol probbly of ALL e equence h depr fro e nd ll obervon fer b = he fnl e nn for ll vld fnl e ' he coplee probbly oeror probbly of e b... e bn he probbly h he proce w n e e gven h we hve oberved he d obned by ple norlzon - b e e Ob ' e e h er ofen referred o he g er nd denoed by g ' ' b b ' 0 0

18 Upde rule ech eron Upde rule ech eron p Ob e olno. of Ob obervon equence p Ob e olno. of Ob obervon equence Ob Ob e Ob e Ob e Ob Ob Ob e e Ob Ob Ob Ob e Ob e Ob e Ob Ob Ob e e Ob Ob Q Ob hee hve been found e Ob Ob Ob e Ob Ob Q Ob e Ob Ob Ob e Where dd hee er coe fro? Ob Ob e e '... e e '... + e 05 + e 06 e e '... e e '... ' ' ' ' b ' + e 07 + e 08

19 he poeror probbly of rnon Upde rule ech eron p Ob e olno. of Ob obervon equence e e ' Ob b ' ' b ' Ob Ob e Ob e Ob e Ob Ob Ob e e Ob Ob he poeror probbly of rnon gven n obervon Q Ob hee hve been found e Ob Ob Ob e Ob Ob 09 0 rnng whou eplc egenon: Bu Welch rnng Every feure vecor oced wh every e of every HMM wh probbly Inl odel Se ocon probble Model Converged? ye no HMM Iue How o fnd he be e equence: Covered How o lern HMM preer: Covered How o copue he probbly of n obervon equence: Covered robble copued ung he forwrd-bckwrd lgorh Sof decon ken he level of HMM e In prcce he egenon bed Verb rnng uch eer o pleen nd uch fer he dfference n perfornce beween he wo ll epeclly f we hve lo of rnng d Mgc nuber How ny e: No nce uoc echnque o lern h You chooe For peech HMM opology uully lef o rgh no bckwrd rnon For oher cyclc procee opology u reflec nure of proce No. of e 3 per phonee n peech For oher procee depend on eed no. of dnc e n proce Applcon of HMM Clfcon: Lern HMM for he vrou cle of e ere fro rnng d Copue probbly of e e ere ung he HMM for ech cl Ue n Byen clfer Speech recognon von gene equencng chrcer recognon e nng redcon rckng 3 4

20 Applcon of HMM Segenon: Gven HMM for vrou even fnd even boundre Sply fnd he be e equence nd he locon where e dene chnge Auoc peech egenon e egenon by opc geneoe egenon 5

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