hmm Viterbi+algorithm.docx

hmm Viterbi+algorithm.docx

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hmmViterbi+algorithm

Viterbialgorithm

Overview

Theassumptionslistedabovecanbeelaboratedasfollows.TheViterbialgorithmoperatesonastatemachineassumption.Thatis,atanytimethesystembeingmodeledisinsomestate.Thereareafinitenumberofstates.Whilemultiplesequencesofstates（paths）canleadtoagivenstate,atleastoneofthemisamostlikelypathtothatstate,calledthe"survivorpath".Thisisafundamentalassumptionofthealgorithmbecausethealgorithmwillexamineallpossiblepathsleadingtoastateandonlykeeptheonemostlikely.Thiswaythealgorithmdoesnothavetokeeptrackofallpossiblepaths,onlyoneperstate.

Asecondkeyassumptionisthatatransitionfromapreviousstatetoanewstateismarkedbyanincrementalmetric,usuallyanumber.Thistransitioniscomputedfromtheevent.Thethirdkeyassumptionisthattheeventsarecumulativeoverapathinsomesense,usuallyadditive.Sothecruxofthealgorithmistokeepanumberforeachstate.Whenaneventoccurs,thealgorithmexaminesmovingforwardtoanewsetofstatesbycombiningthemetricofapossiblepreviousstatewiththeincrementalmetricofthetransitionduetotheeventandchoosesthebest.Theincrementalmetricassociatedwithaneventdependsonthetransitionpossibilityfromtheoldstatetothenewstate.Forexampleindatacommunications,itmaybepossibletoonlytransmithalfthesymbolsfromanoddnumberedstateandtheotherhalffromanevennumberedstate.Additionally,inmanycasesthestatetransitiongraphisnotfullyconnected.Asimpleexampleisacarthathas3states—forward,stopandreverse—andatransitionfromforwardtoreverseisnotallowed.Itmustfirstenterthestopstate.Aftercomputingthecombinationsofincrementalmetricandstatemetric,onlythebestsurvivesandallotherpathsarediscarded.Therearemodificationstothebasicalgorithmwhichallowforaforwardsearchinadditiontothebackwardsonedescribedhere.

Pathhistorymustbestored.Insomecases,thesearchhistoryiscompletebecausethestatemachineattheencoderstartsinaknownstateandthereissufficientmemorytokeepallthepaths.Inothercases,aprogrammaticsolutionmustbefoundforlimitedresources:

oneexampleisconvolutionalencoding,wherethedecodermusttruncatethehistoryatadepthlargeenoughtokeepperformancetoanacceptablelevel.AlthoughtheViterbialgorithmisveryefficientandtherearemodificationsthatreducethecomputationalload,thememoryrequirementstendtoremainconstant.

Algorithm

SupposingwearegivenaHiddenMarkovModel（HMM）withstatesY,initialprobabilitiesπiofbeinginstateiandtransitionprobabilitiesai,joftransitioningfromstateitostatej.Sayweobserveoutputs

.Thestatesequence

mostlikelytohaveproducedtheobservationsisgivenbytherecurrencerelations:

[1]

HereVt,kistheprobabilityofthemostprobablestatesequenceresponsibleforthefirstt+1observations（weaddonebecauseindexingstartedat0）thathaskasitsfinalstate.TheViterbipathcanberetrievedbysavingbackpointerswhichrememberwhichstateywasusedinthesecondequation.LetPtr（k,t）bethefunctionthatreturnsthevalueofyusedtocomputeVt,kift>0,orkift=0.Then:

Thecomplexityofthisalgorithmis

.

Example

Considertwofriends,AliceandBob,wholivefarapartfromeachotherandwhotalktogetherdailyoverthetelephoneaboutwhattheydidthatday.Bobisonlyinterestedinthreeactivities:

walkinginthepark,shopping,andcleaninghisapartment.Thechoiceofwhattodoisdeterminedexclusivelybytheweatheronagivenday.AlicehasnodefiniteinformationabouttheweatherwhereBoblives,butsheknowsgeneraltrends.BasedonwhatBobtellsherhedideachday,Alicetriestoguesswhattheweathermusthavebeenlike.

AlicebelievesthattheweatheroperatesasadiscreteMarkovchain.Therearetwostates,"Rainy"and"Sunny",butshecannotobservethemdirectly,thatis,theyarehiddenfromher.Oneachday,thereisacertainchancethatBobwillperformoneofthefollowingactivities,dependingontheweather:

"walk","shop",or"clean".SinceBobtellsAliceabouthisactivities,thosearetheobservations.TheentiresystemisthatofahiddenMarkovmodel（HMM）.

Aliceknowsthegeneralweathertrendsinthearea,andwhatBoblikestodoonaverage.Inotherwords,theparametersoftheHMMareknown.TheycanbewrittendowninthePythonprogramminglanguage:

states=（'Rainy','Sunny'）

observations=（'walk','shop','clean'）

start_probability={'Rainy':

0.6,'Sunny':

0.4}

transition_probability={

'Rainy':

{'Rainy':

0.7,'Sunny':

0.3},

'Sunny':

{'Rainy':

0.4,'Sunny':

0.6},

}

emission_probability={

'Rainy':

{'walk':

0.1,'shop':

0.4,'clean':

0.5},

'Sunny':

{'walk':

0.6,'shop':

0.3,'clean':

0.1},

}

Inthispieceofcode,start_probabilityrepresentsAlice'sbeliefaboutwhichstatetheHMMisinwhenBobfirstcallsher（allsheknowsisthatittendstoberainyonaverage）.Theparticularprobabilitydistributionusedhereisnottheequilibriumone,whichis（giventhetransitionprobabilities）approximately{'Rainy':

0.57,'Sunny':

0.43}.Thetransition_probabilityrepresentsthechangeoftheweatherintheunderlyingMarkovchain.Inthisexample,thereisonlya30%chancethattomorrowwillbesunnyiftodayisrainy.Theemission_probabilityrepresentshowlikelyBobistoperformacertainactivityoneachday.Ifitisrainy,thereisa50%chancethatheiscleaninghisapartment;ifitissunny,thereisa60%chancethatheisoutsideforawalk.

AlicetalkstoBobthreedaysinarowanddiscoversthatonthefirstdayhewentforawalk,ontheseconddayhewentshopping,andonthethirddayhecleanedhisapartment.Alicehasaquestion:

whatisthemostlikelysequenceofrainy/sunnydaysthatwouldexplaintheseobservations?

ThisisansweredbytheViterbialgorithm.

#HelpsvisualizethestepsofViterbi.

defprint_dptable（V）:

print"",

foriinrange（len（V））:

print"%7s"%（"%d"%i）,

print

foryinV[0].keys（）:

print"%.5s:

"%y,

fortinrange（len（V））:

print"%.7s"%（"%f"%V[t][y]）,

print

defviterbi（obs,states,start_p,trans_p,emit_p）:

V=[{}]

path={}

#Initializebasecases（t==0）

foryinstates:

V[0][y]=start_p[y]*emit_p[y][obs[0]]

path[y]=[y]

#RunViterbifort>0

fortinrange（1,len（obs））:

V.append（{}）

newpath={}

foryinstates:

（prob,state）=max（[（V[t-1][y0]*trans_p[y0][y]*emit_p[y][obs[t]],y0）fory0instates]）

V[t][y]=prob

newpath[y]=path[state]+[y]

#Don'tneedtoremembertheoldpaths

path=newpath

print_dptable（V）

（prob,state）=max（[（V[len（obs）-1][y],y）foryinstates]）

return（prob,path[state]）

Thefunctionviterbitakesthefollowingarguments:

obsisthesequenceofobservations,e.g.['walk','shop','clean'];statesisthesetofhiddenstates;start_pisthestartprobability;trans_parethetransitionprobabilities;andemit_paretheemissionprobabilities.Forsimplicityofcode,weassumethattheobservationsequenceobsisnon-emptyandthattrans_p[i][j]andemit_p[i][j]isdefinedforallstatesi,j.

Intherunningexample,theforward/Viterbialgorithmisusedasfollows:

defexample（）:

returnviterbi（observations,

states,

start_probability,

transition_probability,

emission_probability）

printexample（）

Thisrevealsthattheobservations['walk','shop','clean']weremostlikelygeneratedbystates['Sunny','Rainy','Rainy'],withprobability0.01344.Inotherwords,giventheobservedactivities,itwasmostlikelysunnywhenBobwentforawalkandthenitstartedtorainthenextdayandkeptonraining.

TheoperationofViterbi'salgorithmcanbevisualizedbymeansofatrellisdiagram.TheViterbipathisessentiallytheshortestpaththroughthistrellis.Thetrellisfortheweatherexampleisshownbelow;thecorrespondingViterbipathisinbold:

WhenimplementingViterbi'salgorithm,itshouldbenotedthatmanylanguagesuseFloatingPointarithmetic-aspissmall,thismayleadtounderflowintheresults.Acommontechniquetoavoidthisistotakethelogarithmoftheprobabilitiesanduseitthroughoutthecomputation,thesametechniqueusedintheLogarithmicNumberSystem.Oncethealgorithmhasterminated,anaccuratevaluecanbeobtainedbyperformingtheappropriateexponentiation.

Ifyouwouldliketoimplementthisalgorithmsuchthatitcanacceptanynumberofstates,observations,andprobabilitiesfromthekeyboard,substitutetheparametersdeclarationpartwiththefollowingcode:

#TakesterminalinputtocreateHMM.

states=[]

number_of_states=input（'Howmanystates?

Pleaseenteranintegerdifferentfrom0or1'）

index=0

whileindex

states.append（str（raw_input（'Giveanametothestatenumber'+str（index））））

index=index+1

observations=[]

number_of_observations=input（'Howmanyobservations?

Pleaseenteranintegerdifferentfrom0or1'）

index=0

whileindex

observations.append（str（raw_input（'Giveanametotheobservationnumber'+str（index））））

index=index+1

start_probability={}

forstateinstates:

start_probability[state]=input（'Giveavalueforthestartingprobabilityofthestate'+state）

transition_probability={}

forinitial_stateinstates:

transition_probability[initial_state]={}

forfinal_stateinstates:

transition_probability[initial_state][final_state]=input（'Giveavalueforthetransitionprobabilityfromthestate'+initial_state+'tothestate'+final_state）

emission_probability={}

forstateinstates:

emission_probability[state]={}

for