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Gaussian Observation HMM for EEG

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Gaussian Observation Hidden Markov models for EEG analysis
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  GaussianObservationHiddenMarkovmodelsfor EEGanalysis  WilliamD.PennyandStephenJ.RobertsTechnicalReport,NeuralSystemsResearchGroup,DepartmentofElectricalandElectronicEngineering,ImperialCollegeofScience,TechnologyandMedicine,LondonSW72BT.,U.K. w.penny@ic.ac.uk,s.j.roberts@ic.ac.uk  October5,1998  Abstract Weshow,usinganumberofsyntheticdatasets,thatHMMscandetectthechangesinDClevels,correlation,frequencyandcoherencethataretypicalofthenonstationary changesinanEEGsignal.WealsoshowthatthenumberofhiddenstatesinanHMM canbechosenusingaclusteranalysisofderivedfeatures.Ourexperimentsarebasedon theuseofHMMswithGaussianobservationdensitiestrainedonautoregressive(AR)ormultivariateautoregressive(MAR)coecients.Theextractionofthesecoecientsrequiresthatthesignalbewindowedandthecoecientscalculatedforeachwindow.Thewindowingprocedureis,however,fundamentallyawedsomewindowscontain informationfrommorethanonestateandsocanleadtoalargernumberofinferred statesthanisnecessaryandtoincorrectestimatesofstateandstatetransitions. 1Introduction  InareviewofspatialpatternrecognitionmethodsforEEGanalysis,Nunez2]stressestheneedtodealwithnon-stationaryandnon-localizedsourcesofinformation.InthisbriefreportweshowthatHiddenMarkovModels(HMMs)texactlythisrequirementtheyaretheperfectmodelforEEGanalysis.IntheHMM,thereanumberofunderlyingstateseach ofwhichisassociatedwithstationarydynamics.Themodelsassociatedwitheachstatemaybemultivariateandrelyonsomelocalordistributedsubsetofelectrodes.Transitionsbetweenstatesallowfortrackingofnonstationarydynamics.Inthisreport,wefocusonHMMswithGaussianobservationdensities.TheHMMsaretrainedusingtheEMalgorithmandthemostlikelystatesequencesareidentiedusing single-stateorViterbidecoding.FulldetailsofthealgorithmsareavailableinthetutorialbyRabiner5].TheHMMcanbeappliedeithertotheoriginalelectroderecordingsortoatime-seriesofautoregressive(AR)ormultivariateautoregressive(MAR)coecientsderivedfromtheoriginaldata(seeAppendicesAandBfordetails).Inthisshorttechnicalnote,wedemon-strateafewpropertiesoftheHMM(onsyntheticdata)whichmakeitparticularlysuitableforEEGanalysis.1   0 10 20 30 40 50 600.20.40.60.811.21.41.61.8 Figure1: DCchanges .HMMwithtwostates,one-dimensionalobservations,dierentmeans,commonvariances.Thesolidlineshowstheoriginaltimeseries.ThedottedlineshowsthemeanoftheGaussiandistributionassociatedwiththeHMMstatechosenby Viterbidecoding. 2DCchanges  Figure1showsatimeseriesgeneratedfromanHMMwithtwohiddenstateswithmeans   =0  : 5   1  : 5].Thevarianceswere Cov  =0  : 02   0  : 02].Thehiddenstatesequenceisstate1 for20steps,state2for20stepsthenstate1againforthenal20steps.Giventhesestatedurationsitispossibletocalculatetheoptimaldiagonalelementsinthestatetransition matrix5]andaswehaveonlya2by2matrix,theodiagonalelementsarealsocalculablegiving  A  =0  : 95   0  : 050  : 05   0  : 95].AnHMMwastrainedonthetimeseries(theobservationmodelwasconstrainedsuch thatthe(co)varianceswerethesameforallstates).Thelearnedparameterswere^   = 1   0],^ A  =0  : 97   0  : 030  : 05   0  : 95],^   =0  : 52   1  : 52]and ^ Cov  =0  : 02   0  : 02].Viterbidecoding identiedthecorrecthiddenstatesequence. 3Correlationchanges  Two-dimensionaldatawasgeneratedfroma2-componentGaussianMixtureModel(GMM)asshowninFigure2a.Therstcomponentwasspherical(ie.thevariableswereuncor-related)andthesecondcomponentwaselliptical.Thedatawasthenorderedsuchthatweget200pointsfromcomponent(i),200fromcomponent(ii)andthen200fromcom-ponent(i)again.ThisisshowninFigure2b.Thetimeseriesareuncorrelatedexceptin themiddleportion.Themodelparametersare   =1   0], A  =0  : 995   0  : 0050  : 005   0  : 995](idealtransitionmatrixcalculatedfromstatedurations),   1 =1   10]and    2 =1   10]and  Cov  1 =1   00   1]and  Cov  2 =1   0  : 90  : 9   1].AsecondGMMwasthentrainedonthedata.Thelearnedmeanandcovarianceparam-eterswere^   1 =0  : 95   9  : 80]and ^   2 =1  : 07   10  : 12]and ^ Cov  1 =0  : 794   ;  0  : 04 ;  0  : 04   0  : 741]and ^ Cov  2 =1  : 43   1  : 281  : 28   1  : 35].Theclassicationerrorratewas26.00percent.Thisis'ntgreatas2/3ofthedataarefromclass/state1thenalwaysguessingclass1wouldgiveanerrorof33.00percent.AnHMMwasthenseededwiththemeanandcovariancevaluesobtainedfromtheGMM andtrainedwiththeEMalgorithmtone-tunethemeansandcovariancesandtoestimatetheinitialstatedistributionandstatetransitionprobabilities.Thelearnedparameterswere^   =0  : 001   0  : 999],^ A  =0  : 995   0  : 0050  : 021   0  : 979],^   1 =0  : 98   9  : 84]and ^   2 =0  : 998   10  : 02]2   (a) −4 −2 0 2 4 67891011121314 (b) 0 50 100 150 200 250 300−4−202468101214 (c) 0 50 100 150 200 250 30000.511.522.53 (d) 0 50 100 150 200 250 30000.511.522.53 Figure2: Correlationchanges .HMMwithtwostates,two-dimensionalobservations,samemeans,dierentcovariances:(a)dataasgeneratedbyaGMMwithamixtureof(i)anuncorrelatedGaussianand(ii)acorrelatedGaussian(b)thedatainplot a  hasbeen orderedsuchthatweget100pointsfromcomponent(i),100fromcomponent(ii)andthen 100fromcomponent(i)again.Thetimeseriesareuncorrelatedatthebeginningandend butarecorrelatedinthemiddle,(c)classicationsfromaGaussianMixtureModel,(d)classicationsfromanHMM.and ^ Cov  1 =0  : 831   ;  0  : 013 ;  0  : 013   0  : 822]and ^ Cov  2 =0  : 831   ;  0  : 013 ;  0  : 013   0  : 822].Single-statedecodingthenidentiedthehiddenstatesequencewithanerrorof1.00percent(3 errorsoutof300).Viterbidecodingreducedthisto0.667percent(2outof300).TheGMMmakesaclassicationateachtimestepusingthecurrentobservationonly whereastheHMMuses all observationstomakeeachclassication.Eectively,theGMM triestodiscriminatebetweendistributionsbasedonasingledatapointwhereastheHMM isusing   d  j =100datapoints,where  d  j istheaveragestatedurationdensity.Thisiswhy theHMMdoessomuchbetteratrecognisingthehiddenstates. 4Frequencychanges  Figure3showsninesecondsofatimeseriesgeneratedfromanHMMwithtwohiddenstateswheretheobservationsweregeneratedusingautoregressive(AR)models(seeAppendixA)oforder p  =4.Thereare128samplespersecond.Thecoecientsofthetwomodelswere a  1 = ;  0  : 3704   ;  0  : 1740   0  : 2220   0  : 3189]and  a  2 = ;  0  : 0275   0  : 0097   0  : 0377   ;  0  : 0443].Thesecorrespondto(i)a10-HzsinusoidhiddeninGaussiannoise,and(ii)justGaussiannoise.Thehiddenstatesequenceisstate2for3seconds,state1for3secondsthenstate2again forthenal3seconds.3   (a) 0 1 2 3 4 5 6 7 8 9−4−3−2−101234 (b) 0 1 2 3 4 5 6 7 8 900.511.522.53 Figure3: Frequencychanges .HMMwithtwostates,one-dimensionalobservations.ThetwostatesusedierentARmodelscorrespondingto(i)a10-HzsinusoidhiddeninGaussian noise,and(ii)justGaussiannoise:(a)originaltimeseries,(b)classicationbyHMM.Theobservedtimeserieswasthensplitupintooverlappingtwo-secondblockswith anosetof0.2secondsbetweenblocks.WithineachoftheseblocksanAR(4)modelwastrained.Thisgeneratedaseriesof464-dimensionalfeaturesonwhichanHMMwastrained.Thelearnedparameterswere^   =0   1],^ A  =0  : 941   0  : 0590  : 036   0  : 964], a  1 = ;  0  : 356   ;  0  : 035   0  : 287   0  : 309]and  a  2 = ;  0  : 054   0  : 013   0  : 062   ;  0  : 056].Viterbidecodingidentiedthecorrecthiddenstatesequencetowithinthequantisationerrorintroducedbysplittingup thedataintoblocks.Anundesirableaspectofthisexampleisthatthetimeserieshadtobechoppedupinto discrete,overlappingblocks.Thisinvolvedtheadhocchoiceofblocksizeandblockosetparameters.Adesirableaspectofchoppingupthesignalintodiscreteblocksandderiving featuresfromthoseblocksisthatwehavedrasticallyreducedthedimensionalityofthespacethattheHMMoperatesin.Inthisexamplewehavereducedatimeseriescontaining (9secondstimes128samplespersecond)1152samplesintoafeaturevectorcontaining46 items.Thisreducesthecompututionbyafactorof25. 5Coherencechanges  Figure4showsafoursecondexcerptfroma15-secondtimeserieswhichissampledata rateof128samplespersecond.Therstandlastvesecondsweregeneratedfromtwo coherent10-Hzsinusoidsandthemiddlevesecondsweregeneratedbytwo  incoherent 10-Hzsinusoids.AlldatawascorruptedbyadditiveGaussiannoise.Thecohererencetimefortheincoherentsectionwas1second,thatis,after1secondthephaseofoneofthesignalschangedrandomlywithrespecttotheothersignal.Thislongcoherencetimewaschosen suchthattherewasnosignicantchangeinthespectraldensityofeachsignal-thustheonlywaytotellthesignalsapartisbylookingatthecoherency.Theobservedtimeserieswasthensplitupintooverlappingtwo-secondblockswithan osetof0.2secondsbetweenblocks.Withineachoftheseblocksamultivariateautoregres-siveMARmodel(seeAppendixB)wastrained,withamodelorderof p  =4.Thisgenerated aseriesof4616-dimensional(thedimensionoftheMARmodelis p  x  d  x  d  where d  =2isthedimensionofthetimeseries,and  p  =4-seeAppendixB)featuresonwhichaGaussian observationHMMwastrained.TheestimatedmeansoftheGaussiansineachofthestatescorrespondtoanMARmodelforthatstate.FromtheMARcoecientsitispossibletoreconstructthespectraldensitiesandthecross-spectraldensity(seeAppendixB).Figure5showstheestimatedcoherency 4 

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