Fast statistical model-based classification of epileptic EEG signals

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To cite this version:

[Quintero Rincon, Antonio and Pereyra, Marcelo Alejandro and D'Giano,

Carlos and Risk, Marcelo and Batatia, Hadj

Fast statistical

model-based classification of epileptic EEG signals. (2018) Biocybernetics and

Biomedical Engineering, 38 (4). 877-889. ISSN 0208-5216

Official URL:

https://doi.org/10.1016/j.bbe.2018.08.002

Fast

statistical

model-based

classification

of

epileptic

EEG

signals

Antonio

Quintero-Rincón

*

,

Marcelo

Pereyra

,

Carlos

D'Giano

,

Marcelo

Risk

,

Hadj

Batatia

a _{DepartmentofBioengineering,InstitutoTecnológicodeBuenosAires(ITBA),BuenosAires,Argentina} b _{SchoolofMathematicalandComputerSciences,Heriot-WattUniversity,Edinburgh,UK}

c _{CentroIntegraldeEpilepsiayTelemetría,FundaciónLuchacontralasEnfermedadesNeurológicasInfantiles(FLENI),}

BuenosAires,Argentina

d

ConsejoNacionaldeInvestigacionesCientíficasyTécnicas(CONICET),BuenosAires,Argentina

e _{UniversityofToulouse,IRIT–INPT,Toulouse,France}

a

r

t

i

c

l

e

i

n

f

o

Keywords:

GeneralizedGaussiandistribution Waveletfilterbanks

EEG Epilepsy

Leave-one-outcross-validation Linearclassifier

a

b

s

t

r

a

c

t

Thispaperpresentsasupervisedclassificationmethodtoaccuratelydetectepilepticbrain activityinreal-timefromelectroencephalography(EEG)data.Theproposedmethodhas three main strengths: it has low computational cost, making itsuitable for real-time implementationinEEGdevices;itperformsdetection separatelyforeachbrainrhythm orEEGspectralband,followingthecurrentmedicalpractices;anditcanbetrainedwith smalldatasets,whichiskeyinclinicalproblemswherethereislimitedannotateddata available.Thisisinsharpcontrastwithmodernapproachesbasedonmachinelearning techniques,whichachieveveryhighsensitivityandspecificitybutrequirelargetrainingsets withexpertannotationsthatmaynotbeavailable.Theproposedmethodproceedsbyfirst separatingEEGsignalsintotheirfivebrainrhythmsbyusingawaveletfilterbank.Eachbrain rhythm signalis then mapped to a low-dimensional manifold byusing a generalized Gaussianstatisticalmodel;thisdimensionalityreductionstepiscomputationally straight-forwardandgreatlyimprovessupervisedclassificationperformanceinproblemswithlittle training data available.Finally,this isfollowed byparallel linearclassificationson the statisticalmanifoldtodetectifthesignalsexhibithealthyorabnormalbrainactivityineach spectralband. Thegood performanceoftheproposedmethodisdemonstratedwithan applicationtopaediatricneurologyusing39EEGrecordingsfromtheChildren'sHospital Bostondatabase,whereitachievesanaveragesensitivityof98%,specificityof88%,and detectionlatencyof4s,performingsimilarlytothebestapproachesfromtheliterature.

*Corresponding author at: Department of Bioengineering, Instituto Tecnológicode Buenos Aires (ITBA),Av. Eduardo Madero 399, C1106ACDBuenosAires,Argentina.

E-mailaddress:aquinter@itba.edu.ar(A.Quintero-Rincón).

1.

Introduction

Epilepsyisadiseasethatproducesbrainactivitydisorders[1– 3].Itsdiagnosisreliesstronglyontheanalysisof electroen-cephalography(EEG)data,anon-invasiveandwidelyavailable biomedical modality that allows neurologists to monitor abnormalactivity and characterizeits nature.Inparticular, neurologistsanalyzecharacteristicwaveformstolocalizeand quantifytheepileptogeniczone.Thesebrainactivitydisorders canleadtoepilepticseizuresthathaveasuddenonset,spread quickly,andareverybrief.

EpilepticseizuredetectionmethodsbasedonEEGsignals stemfromtheobservationthatEEGsignaldescriptorsallow discriminatingbetweennormalandabnormalbrainactivity. This practice originated half a century ago with works by Viglioneetal.[4],Lissetal.[5],Ktonasetal.[6]andGotman etal.[7];andcontinuedwithIasemidisetal.[8,9]mainlyinthe medicalliteratureandbyusinganalogueEEGdevices.Later,as EEGsystemsadopteddigitalsignalprocessingcapacity,this stimulatedthedevelopmentofpatternrecognitionmethodsto detectandanalyseabnormalbrainactivityautomatically.A mainpracticaladvantageofEEGtechnologyisthatitisvery economicallyaccessible.Thishassignificantlycontributedto thewide adoptionof EEGindeveloping countries(whereas othermoreadvancedmodalities,suchas magnetoencepha-lography (MEG), are expensive and have not been widely adoptedasaresult).

TherearecurrentlyawiderangeofEEGsignalprocessing methodstodetectbrainseizuresaccurately.Mostmethodsuse classificationtechniquesfromthesupervisedmachine learn-ingliterature,suchassupport vectormachines[10–12]and discriminantanalysis[13],anddiffermainlyintermsoftheir feature extraction methods and the features classification approaches.Manymethodsusetime-frequencydescriptors, eitherexplicitly(e.g.,short-termFourierorwavelet represen-tations)[14–17,11,18,19],empiricalmodedecomposition[20– 22], orimplicitly bylearning neuralnetworks [23–25] or by usingcomponentanalysisorcommonspatialpatterns(seefor example[26–28]).Somealsousestatisticaldescriptorssuchas signalentropy[17,11,29–31]orfractaldimension[32,33].

The main approaches from the state of the art are summarisedinTable1,togetherwiththeirdetection perfor-manceonatestdataset.Observethatmostmodernmethods performremarkablywellandachievetruepositiverates(TPR) orsensitivitiesoftheorderof95–99%,andtruenegativerates or specificities of the order of 85–95%, depending on the specificmethod and dataset considered. This good perfor-mance is achieved by using advanced signal processing techniquesthataregenerallyverycomputationallyintensive. As a result, state-of-the-art detection methods cannot be incorporated into EEG devices to perform detection inreal time. For example, the method [26] uses common spatial patterns that require estimating covariance matrices and performingsingularvaluedecompositionsateachdetection step. This limitation is motivating the development of detectionmethodsthatusecloud computingtechnology to performdetectiononahighperformancecomputingserver thatisaccessedremotely(seeforexample[28]).Thisstrategyis potentiallyveryinterestinginsomesettings,butitwouldbe

difficulttoimplement indevelopingcountrieswhere many hospitals still have limited Internet access and poor IT infrastructure.

Anotherlimitationofstate-of-the-artmethodsisthatthey pullinformationfromallspectralbandstoimprovedetection performance [26]. Whilebeneficialin termsof classification accuracy,thiscanbeproblematicinmanyclinicalapplications wherethecurrentpracticeistodetectseizuresindependentlyin eachphysiologicalspectralbandorbrainrhythm(thesebands arespecifiedinSection2).Finally,state-of-the-artmethodsalso relyincreasinglyonlargetrainingdatasets,whichisadrawback inclinicalapplicationswherethereislimitedannotateddata available. Also, many existing methods use feature-based classificationtechniques,withasignificantnumberoffeatures inordertohandletheinherentvariabilityofsuchfeatures.

Thispaperseekstoaddresstheselimitationsoftheexisting methodology by developing an automatic EEG detection technique that has low computational cost, that performs detection independently in each brain rhythm following currentclinicalpractice,andthatcanbetrainedwithsmall datasets,withadetectionperformancethatissimilartothatof state-of-the-artalgorithms.Incontrastwithexistingmethods, the proposed method adopts a model-based classification approach.Model-basedclassificationhasbeenusedinvarious applications [34–36]. The idea is to capture the statistical propertiesofthesignalusingtheparametersofaprobabilistic model. This approach is interesting compared to feature-basedclassification,especiallywhenfeaturesarenumerousor exhibit largevariability. Itcan beviewed asan interesting dimensionality reduction technique facing the curse of dimensionality and leading to low computational cost classification.Despiteitsinterest,thisapproachhasnotbeen widely investigated inEEG signal processing.Precisely,our classification method is driven by a parametric statistical modelthatcaptures thestatisticalpropertiesof thesignals andtheirevolutionintime,withthemodelparametersacting as classification features. This approach is an interesting alternativetothenon-parametricfeatures(e.g.,signalpower spectrum, variance, entropy, etc.) commonly used in the literaturebecausetheparametricstructureofthemodelacts asadimensionalityreductionmechanismthatregularizesthe classificationproblemandconsequentlyimprovesthe stabili-tyandrobustnessoftheclassification,andwhichatthesame timesignificantlyreducestheassociatedcomputationalcost. Despite its advantages, to the best of our knowledge this promisingapproachhasnotbeeninvestigatedforEEGsignal classification.Notehoweverthatstatisticalapproacheshave beensuccessfullyappliedtootherchallengingEEGprocessing problems(seeforexample[37,38]).

The remainder of the paper is structured as follows. Section 2 introduces notation and specifies the detection problemconsidered.Section3presentstheproposedmethod, withitsthreemainstepsdetailedinSections3.1–3.3.Section4

presentsarangeofexperimentalresultswithEEGrecordings from the Children's Hospital Boston database and reports detectionperformanceintermsofsensitivity,specificity,and latency. Advantages and limitations, Conclusions and per-spectivesforfutureworkarefinallyreportedinSections5and 6respectively.

Table1–State-of-the-artmethodstoperformseizuredetectionautomaticallyinEEGsignals,summarisedintermsofthe classificationtechniquesandfeaturesusedandtheirreportedperformanceonatestdataset.Theperformancemetricsare theTruePositivesRateorSensitivity(TPR),theTrueNegativeRateorSpecificity(TNR)andtheAccuracy(ACC).

Classificationmethod Features Testdata Performance Ref.

Learningvectorquantization Signalentropyfromwavelet coefficients

400epochsfrom5normalsubjects and5epilepticpatients

TPR:98% [17]

SupportVectorMachine Matchingpursuitalgorithm 133EEGfromRigshospitalet UniversityHospitaldatabase (Copenhagen,Denmark)

TPR:78%,TNR:84% [10]

SupportVectorMachine SpectralandEntropyAnalysis 3datasetsfromEEGUniversity HospitalBonndatabase

TPR:90% [11]

Fuzzyclassification Amplitude,frequencyandentropy descriptors

56iEEGfrom20patientsfrom UniversityofFreiburgdatabase

TPR:95.8%,TNR:74% [18]

HiddenMarkovModel Segmentationoftopographicmaps oftimevaryingspectral

10EEGpatientsfromEPILEPSIAE[39] TPR:94.59%,TNR:92.22% [19]

SupportVectorMachine Third-ordertensordiscriminant analysis:spectral,spatial,and temporaldomains

36EEGpatientsfromChildren's HospitalBostondatabase

TPR:98%,TNR:94% [13]

K-meansclustering SpatiotemporalAnalysisas morphologicalfilter

10EEGpatientsfromUniversityof FloridaHospitaldatabase

TPR:87.4% [40]

Logisticclassifier Stackedautoencodersneural network

36EEGpatientsfromChildren's HospitalBostondatabase

TPR:100% [23]

SupportVectorMachine Fractionallinearprediction 100singlechannelEEGsegments fromTheBern-BarcelonaEEG database

TPR:96%,TNR:95% [41]

LeastSquaresSupportVector Machine

Phasespacerepresentation 100segmentsfromtheEEG UniversityHospitalBonn

TPR:100%,TNR:96% [42]

SupportVectorMachine Empiricalmodedecomposition 51EEGsegmentsfrom17patients fromUniversityofFreiburg (Germany)

TPR:98.6%,TNR:88.6% [43]

1-NearestNeighbor 1D-localbinarypatternsfrombank ofGaborfilters

100ECoGsegmentsfromUniversity HospitalBonndatabase

TPR:98.33% [44]

SupportVectorMachine CommonSpatialPattern 36EEGpatientsfromChildren's HospitalBostondatabase

TPR:100% [26]

RelevanceVectorMachine Multifractalformalism 21EEGpatientsfromtheEpilepsy CenteroftheUniversityHospitalof Freiburg

TPR:92.94%,TNR:97.47% [45]

Regressionneuralnetwork Statisticaldescriptorsofdual-tree complexwavelettransform coefficients

100segmentsfromUniversityof Bonndatabaseand21patientsfrom SirGangaRamHospital(NewDelhi)

TPR:92%,TNR:98% [24]

K-NearestNeighbor,linear discriminantanalysis,naive Bayesian,logisticregressionand SupportVectorMachine

Time,frequency,time-frequency andnonlinearfeatures

100segmentsfromUniversity HospitalBonndatabase

TPR:99.25% [46]

1-NearestNeighbor Convolutionalneuralnetwork 5patientsfromtheEEGUniversity HospitalBonn

TPR:95%,TNR:88.67% [27]

RandomForest,C4.5,Functional tree,Bayesian-network, Naive-BayesandK-nearestneighbours

Meanofjointinstantaneous amplitude,Meanmonotonic absolutechangeandVarianceof monotonicabsolutechangefrom Empiricalwavelettransform

36EEGpatientsfromChildren's HospitalBostondatabase

TPR:97.91%,TNR:99.57% [47]

Multilayerperceptronneural network

Time-frequencylocalized three-bandsynthesiswaveletfilterbank andsubbandnorm

100segmentsfromUniversity HospitalBonndatabase

ACC:99.66% [48]

SupportVectorMachine PyramidofdifferenceofGaussian filteredsignalsandLocalbinary patterns

100segmentsfromtheEEG UniversityHospitalBonn

TPR:100%,TNR:100% [49]

SupportVectorMachine Randomsubspaceensemblemethod andInfiniteIndependent

ComponentAnalysis

208ECoGfromUniversityof PennsylvaniaandtheMayoClinic

TPR:98%,TNR:96% [28]

Least-SquareSupportVector Machine

Time-frequencyrepresentation basedontheimprovedeigenvalue decompositionofHankelmatrixand Hilberttransform

100segmentsfromtheEEG UniversityHospitalBonn

2.

Problem

statement

LetX2RMN_{denoteatime-discretizedEEGsignalrecordedby} anarraycomposedofMchannelsoveraperiodofTseconds, andusingasamplingperiodofT/Nseconds.EachrowofXis associatedwithonechannelofthearrayandcontainsallthe samplingpointscorrespondingtotheEEGsignalrecordedby that channel, whereas each column is associated with a samplingpointandcontainsthevectorsignalacquiredbythe full array at that time instant. Moreover, to analyse the differentfrequencycomponentsofX,wedenotebyXd,Xu,Xa,

Xb,andXgthespectralcomponentsrelatedtothed(0–4Hz),u

(4–8Hz),a(8–16Hz),b(16–32Hz),andg(32–64Hz)frequency bands.Asmentionedpreviously,eachofthesebandsisrelated todifferentneurologicalfunctionsandisthereforeassociated withspecificneurologicaldisorders.

This paper considers the problem of detecting epileptic seizureactivityinEEGsignalsinreal-time,andidentifyingthe frequencybandswheretheseizureoccurs.Formally,forany timeinstantn2{1,N},defineband-specificbinarylabelszd(n),

zu(n), za(n), zb(n), and zg(n) that take value 1 toindicate the

presenceofanepilepticseizureattheirspectralband,and0to indicatenormalactivity.Givensomeexpertannotatedtraining datafXðkÞ0 g

K0

k¼1andfX ðkÞ

1 gKk¼11 correspondingtoshortEEG record-ingsofhealthyandepilepticseizureactivity,weconsiderthe supervisedclassificationproblemofestimatingthevaluesof zd(n),zu(n),za(n),zb(n),andzg(n)inreal-timeasXisacquiredbythe

EEGarray.Similarlyto[14],becauseweareinterestedinclinical applicationswherethisinformationisrequiredinreal-time,we focusonclassifiersthathavelowcomputationalcomplexity.

3.

Proposed

method

This section presents a new method to detect epileptic seizures in EEG signals and simultaneously identify the frequency bands where the seizure occurs. As mentioned previously,themainstrengthsofthemethodologyarethatit can betrained withsmall training datasets, and that it is computationally very efficient and suitable for performing detectioninreal-time.

Theproposedmethodhasapipelinestructurecomposedof thefollowingthreesteps:afilterbankthatseparatesXintoits Xd, Xu, Xa, Xb, and Xg spectral components, followed by a

statistical dimensionality reduction step that maps these components into a low-dimensional representation where pathological brain activity is easily detected, and finally a classification step based on a thresholding approach. This structureissummarisedinthediagraminFig.1.

3.1. Spectraldecompositionbywaveletfilterbank WeuseaDauchebies(Db4)waveletfilterbanktoseparateX intothefivespectralcomponentsXd,Xu,Xa,Xb,andXg[51](we

useDb4becauseitoffersthenumberofvanishingmoments thatallowrepresentingthesignalwithsufficientsmoothness). Performingwaveletdecompositionfits naturally thedyadic structureoftheneurologicalspectralbands, andprovidesa computationally efficient filtering algorithm that can be

implementedstraightforwardlyonreal-timesignalprocessing hardware.Becauseourdataisacquiredata256Hzsampling rate,inourexperimentsweuseawaveletfilterthrough tree-basedtopology,withsixscales.Theupperfivescalesmatch withthespectralbandsofinterest(theremainingscalerelated tothe64-128Hzbandhasverypoorsignal-to-noiseratioandis discarded). The output of this stage are 5 sets of wavelet coefficientsVd,Vu,Va,Vb,Vg(observethatthisapproachcan

be straightforwardly generalized to higher sampling rates byusingordiscardinganyadditionalbands).

3.2. Statisticalmodelofthespectralcomponents

Designingaclassifiertodetectpathologicalbrainactivitydirectly fromtheEEGsignals (ortheirwaveletrepresentation)is very challenging due tothe high-dimensionality of thedata, and because it would requirea largetrainingset anda complex classification methodology. Todetectabnormal brain activity withlimitedannotatedtrainingdata,particularlyinthecontext ofclassifierswithlowcomputationalcomplexitysuitablefor real-timeimplementations,itisnecessarytomaptheEEGdatatoa meaningfulcompactrepresentationthathighlightsthe informa-tion able to discriminate normal and abnormalactivities. A successfulrepresentationshouldalsoprovidethe low-dimen-sionalstructureandfavorableregularitypropertiesthatenablea simpleclassificationscheme,suchasthreshold-basedmethods. Hereweconstructthisrepresentationbyusingaparametric statisticalmodeltosummarizetheempiricaldistributionofthe wavelet coefficients associated with each spectral band. Precisely, we adopt a sliding window approach and fit a parametricstatisticalmodeltothewaveletcoefficients associ-atedwiththelast2sofX.Becausethesignalsweconsiderinour experiments areacquiredwithM=23channelarray,the2-s windowcorrespondsrespectivelytothelast8192,4096,1024, 512and256coefficientsofVd,Vu,Va,Vb,andVg.Wemodeleach

setofwaveletcoefficientswithzero-meangeneralizedGaussian distribution(GGD)withdensitygivenby

fðs;s;tÞ¼ t 2sGðt1_Þexp j s sj t   (1) wheres2Rþisascaleparameter,t2Rþisashapeparameter thatcontrolsthedensitytail,andGðÞistheGammafunction (weestimatethevaluesofsandtforeachspectralbandby maximumlikelihoodestimation,whichwesolve straightfor-wardlybyusingaNewton–Raphsonalgorithm[52]).Therefore, atagiventimepointn,theP13_j¼92j_{¼15872waveletcoefficients} correspondingtothe2-swindowaremappedtoa 10-dimen-sionalrepresentations(n)=[sd(n),su(n),sa(n),sb(n),sg(n)],t(n)=

[td(n),tu(n),ta(n),tb(n),tg(n)].Inadditiontobringingsignificant

dimensionality reduction, theexperiments reportedin Sec-tion4showthatthisapproachmapseachneurological spec-tral band onto a two-dimensional representation where seizuresare easilydiscriminatedandcanbedetected accu-ratelywithalinearclassifier.FromaEEGphysicsviewpoint, theparameterss(n)andt(n)capturethestatisticaldistribution ofthepoweroftheEEGsignalarrayattimenineachspectral band;s(n)measurestheaveragepowerineachband,andt(n) thedeviationsofpowerfromtheseaveragevalues[53].

As mentioned previously, parametric approaches are regularizedbytheirparametricstructureandasaresultthey

achieve good performance results with smaller datasets compared to non-parametric approaches. However, non-parametricstrategiestendtobesuperiorwhenalargeamount oftrainingdataisavailablebecausetheyaremoreflexibleand consequentlycanbetterexploittheinformationprovidedby thedata(whereasparametricmodelsareconstrainedbytheir structureandhencesufferfromintrinsicestimationbiasdue to model misspecification). Similarly, over-parameterised strategies,e.g.,machinelearningtechniquesbasedonneural networks,alsoperformverywellindatarichsettings.

Finally, it is mentioning that we also considered other statistical models for the wavelet coefficients, namely the logistic,t-locationscale,andsymmetrica-stabledistributions. WefoundthatthegeneralizedGaussianmodelprovidedthe bestmodel-fit-to-data(weconductedthesecomparisonsusing realdatafromtheChildren'sHospitalBostondatabase[54,14]– seeresultsinAppendixA).

3.3. Seizuredetectionbylineardiscriminantanalysis classification

The final stage of the proposed seizure detection pipeline isaclassifierthatlabelsthestatisticalparametersassociated

witheachspectralbandasseizureornon-seizure.Precisely,five independenttwo-parameterclassifiersareusedinparallelto classifythepairs[sd(n),td(n)], [su(n),tu(n)], [sa(n),ta(n)],[sb(n),

tb(n)],and[sg(n),tg(n)]generatedbythestatistical

dimension-ality reductionstep. Thisallowstosimultaneously identify seizureactivity andthespectral bandswhere itoccurs.For simplicity we use linear classifiers derived from a linear discriminant analysis. Precisely, we adopt a supervised approachwhereeachclassifierisband-specificandhasbeen trainedbyperformingalineardiscriminantanalysisonexpert annotateddata.Weperformthediscriminantanalysisonan augmented vector ½s;t;n2R3, where n=s2G(3/t)/G(1/t) is a varianceparameter.Includingninthediscriminantanalysis embeds(s,t)inanon-linearmanifoldinR3 _whereabetter linearclassificationispossible(notethatnavailableforfreeas aby-productoftheNewton-Raphsonmethodthatestimatess and t, hence this augmentation does not introduce any additionalcomputationalcost).Theresultinglinearclassifiers arespecifiedbythreeparameters(a,b,c)definingaplanethat splits R3 _{in two regions related to seizure and non-seizure} events,andwhichessentiallyoperateasathree-dimensional thresholdforthetripletss,t,n.Lastly,similarlytothechoiceof thestatisticalmodel,itispossibletoconsidermoreadvanced Fig.1–Block-diagramrepresentationoftheproposedmethodtodetectepilepticseizuresinEEGsignals.Themethodconsists ofthreemainsteps:separationofbrainrhythmsusingwavelet-bankfiltering,statisticalmodel-baseddimensionality reductionusingageneralizedGaussiandistribution(GGD),andseizuredetectionusingclassificationthroughlinear discriminantanalysis.

Fig.2–Scatterplotsforthestatisticalparameterssandtforseizuresignals(redcross)andnon-seizuresignals(bluecircles) foreachspectralband,showingthegooddiscriminationpropertiesoftheproposedrepresentation.

classificationsschemes.However,suchclassifiersalsoinvolve moreparametersandhencearemorepronetoover-fittingand morecomputationallyexpensive.

4.

Experimental

results

We now demonstrate the proposed methodology with

experiments with real data and comparisons with other approachesfromtheliterature.Fortheexperimentsweused data from the Children's Hospital Boston database [54], previously considered in [14], which consists of 36 EEG recordingsfrompediatricsubjectswithchallengingseizures. (Thesignalswereacquiredwitha23-channelarrayoperating ata256Hzsamplingrate). Fromthis database,weused13 seizuresignalsorepochsselectedbyanexperienced neurolo-gist,seeTable4.Thesecorrespondto13seizureeventsfrom9 differentsubjects,andarebetween1and5minlong(theother dataexhibitedstrongartifactsrelatedtomuscleactivityand werediscardedasaconsequence).Theneurologistannotated eachsignaltoindicatethebeginningandendingoftheseizure epochs, which weuse as groundtruth. Moreover, for each seizureepoch,theneurologistalsoselectedtwoadjacent non-seizure signal segments of the same length to represent challenging non-pathological brain activity. The resulting datasetconsistedthereforeof 39signalsegmentsrelatedto 13seizuresand26non-seizuresignals,andofvariablelengthin therangeof1–5min.

Toillustratethecapacityofthestatisticalparameterssand ttodiscriminateseizureeventsandnon-seizuresignals,Fig.2

showsscatterplotsforeachspectralbandconstructedusing thesignalsinthedatabaseandtheexpertannotations (non-seizuresignalsarerepresentedusingbluecirclesandseizure signalsusingred crosses).Observethat this representation providesaverygoodlineardiscriminationoftheseizureand non-seizure groups. Inparticular, one notices thatthe scale parametersisparticularlyusefulfordiscrimination(seealso discriminationtestsinAppendixB).

Moreover, to assess the performance of the proposed methodology, we adopted a supervised testing approach

and used the 39 signal segments described above to train andtestthemethod.Becausethedatasetisrelativelysmallwe used an exhaustive cross-validation technique based on a leave-one-out approach. Precisely, at each iteration of the cross-validation process we trained the 5 classifiers (each definedby3parameters)withdatafrom13seizuresignalsand 26non-seizuresignals,andthenassessedclassification perfor-manceontheremaining3signals(theseare1seizureand2 non-seizure signals). In each iteration of the cross-validation process the classification performance was assessed by splittingthetestsignalsintosequencesof2sandclassifying eachsequenceindividually;theseresultswerethenusedto assessclassificationperformance.Precisely,wemeasurethe method'struepositiverate(TPR)orsensitivity,falsepositive rate(FPR),truenegativerate(TNR)orspecificity,andoverall accuracy(ACC),expressedasthepercentageofgoodclassi fi-cation.Foreachfigureofmeritwereportthemeanvalueand thestandarddeviation.TheseresultsarereportedinTable2. We alsoreportlatency(timedelay) betweentheannotated seizureonsetandthedetectionbythemethodinTable3.We compareclassificationaccuracyandlatencywiththe state-of-the-art methods [13,26,23], which also report classification performanceandlatencyforthe Children'sHospitalBoston database. We emphasise again that these state-of-the-art methods are significantly more computationally expensive than the proposed method.Forexample, [13]uses a third-order tensor discriminant analysis, [23] a stack of neural networks combined withalogisticclassifier, and [26] com-putessingularvaluedecompositionsofcovariancematricesat each detection step. Neither of these methods can be implemented in real-time in a standard EEG system as a consequence.

Observe from Table 2 that, despite the computational simplicity, the proposed method achieves an excellent sensitivityof the order of 97–99% for allspectral bands on the testdataset. Thisisclose tothe state-of-the-art perfor-mances of 98–100% reported in [13,26,23] for this dataset. Moreover,thespecificityoftheproposedmethodis approxi-mately90%.Thisisslightlybellowthe94%specificityof[13]

(theworks[26,23]donotreportspecificity).However,notice

Table2–Seizuredetectionperformanceforeachbrainrhythmandfor39events(13seizureand26non-seizure) oftheChildren'sHospitalBostondatabase,intermsof:TPR=TruePositivesRateorSensitivity;TNR=TrueNegative RateorSpecificity;FPR=FalsepositiveRate;andACC=Accuracy[averagevalueWstandarddeviation].

Metric DeltaBand(d) ThetaBand(Q) AlphaBand(a) BetaBand(b) GammaBand(g)

TPR 0.970.06 0.990.01 0.990.02 0.970.05 0.990.01

TNR 0.920.07 0.790.23 0.910.08 0.900.10 0.910.08

FPR 0.080.07 0.210.23 0.090.08 0.100.10 0.090.08

ACC 95.134.64 92.697.61 96.592.99 94.485.77 97.002.88

Table3–Averagelatencybetweenseizureonsetanddetection(inseconds),fortheproposedmethodoneachspectral band,andforthestate-of-the-artmethods[13,26,23].

Proposed State-of-the-art

Deltaband(d) Thetaband(u) Alphaband(a) Betaband(b) Gammaband(g) [13] [26] [23]

thattoachievethishigherspecificity,themethod[13]pulls together all spectral bands, and as a result it does not discriminatebetweenseizuresindifferentbands.Ourmethod performsclassificationindependentlyoneachbandbecause thisisusefulinclinicalpractice,attheexpenseofaslightlya lowerspecificity.

Furthermore, observe from Table 3 that our method achieves an average latency of approximately 4s for all spectral bands, outperforming the state-of-the-art meth-ods[13,23] and closeto the fastest available method [26]. We emphasise at this point that all the latency values

reported in the literature measure the delay of the

detection algorithm offline, without taking into account any overhead related to the methods' computing times. Therefore,the fact that different methods achieve similar latencydoesnotindicatethat theyhavesimilar computa-tionalcomplexity.

Note that we do not report computing times for these experiments for two reasons. First, because we have con-ductedtheseproof-of-concepttestsinMATLAB,and proces-sing each 2-sEEG signalwindow requiredlessthan 50ms. Second,becausewedonothaveaccesstothe implementa-tionsof[13,26,23],andthereforethecomparisonswouldnotbe fair.However,asexplainedpreviously,thesemethodsclearly haveasignificantlyhighercomputationalcomplexitybecause ofthesophisticatedmathematicaloperationsinvolved(e.g., third-order tensor discriminant analysis, singular value decompositionsofcovariance matrices,stackedneural net-works, etc.). A real-time implementation of the proposed methodiscurrentlyunderdevelopment.

Finally,weemphaseatthispointthatthesensitivityand specificvaluesreportedaboveareforthespecifictestdataset

[54],whichislimitedinmanyways.Toreliablydeterminethe sensitivityandspecificityofthemethodinaclinicalsettingit isnecessarytoconductathoroughevaluationbyusing long-term and continuous EEG signals (see for example the

protocols adopted in [55]). A thorough evaluation of the performanceoftheproposedmethodsisamainperspective forfuturework.

5.

Advantages

and

limitations

of

the

proposed

method

Through the use of a statistical model-based classification technique,theproposedmethodhasthreemainadvantages. First, it requires onlyestimatingand classifyingtwo scalar parametersallowingittobeimplementedindedicatedreal timehardware.Second,itcanbetrainedusingareasonably smalldatasetduepreciselytothefactthatitusedonlytwo classificationparameters.Thiscontrastswithmethodsusinga numberoffeaturesthatwouldrequirelargetrainingdatasets. Third, it allows seizure detection simultaneously in the different brain rhythms, complying with current medical practices.

Nevertheless, the proposed method has three main

limitations.First,duetotheveryhighdynamicsofepileptic signals,definingtheslidingtime-windowandtheoverlapof epochs isdifficult. Second, it needs definingregularization parameters for the training stage in order to take into considerationrandompeaks,noiseandartefactsthatmight lead to false positives. Third, seizures have variable and dynamic offsets corresponding to the complex nature of differentepilepsy types.Asanexample, whenbrain waves slow down,change fromseizure tonon-seizure isdifficultto trackandcangenerateclassificationerrors.

6.

Conclusions

Thispaperpresented anewclassificationmethodtodetect epileptic brain activity in EEG signals, with a focus on

Table4–Lengthofthe18Seizuresusedinthisstudyandthecorrespondingnumberof2-ssegments.Anoffsethasbeen usedforeachepochtoavoidleadingandtrailingsignalsthatwerenoisy.Consequently,thenumberofwindows isirregularbetweenepochs.

Processingdurationinms

Epoch Seizure Duration Segments Delta Theta Alpha Beta Gamma

01 04 1m30sec 181 7 6 7 7 8 02 05 1m41sec 203 7 7 7 8 9 03 10 1m04sec 129 7 7 7 7 9 04 11 1m07sec 137 7 6 7 7 8 05 12 2m00sec 241 7 7 7 8 8 06 13 1m57sec 235 7 7 7 8 9 07 17 1m26sec 173 7 7 7 8 8 08 18 2m23sec 287 7 7 7 8 9 09 19 3m09sec 321 7 6 6 7 8 10 20 3m46sec 343 7 6 6 7 8 11 21 5m38sec 529 7 6 7 7 8 12 22 1m04sec 129 7 6 7 7 8 13 23 1m03sec 125 7 9 9 7 13 14 26 1m05sec 131 7 9 10 11 35 15 27 1m02sec 117 7 7 7 7 9 16 28 1m16sec 153 7 7 7 8 9 17 29 1m29sec 179 7 6 7 7 8 18 30 0m32sec 65 7 7 7 7 8

applicationsinvolvingreal-timeconstraintsandsmall train-ingdatasets.Anadditionaladvantageofthemethodisthat detectionisperformedindependentlyforeachbrainrhythm, followingthecurrentmedicalpractices.Detectionisachieved byfirstseparatingtheEEGsignalsintothefivebrainrhythms byusingawaveletdecomposition,andthenusinga general-izedGaussianstatistical modeltomap signalsontoa low-dimensional representation where classification can be performedefficientlybylineardiscriminantanalysis. Experi-ments with 39 signals from 9 patients of the Children's

Hospital Boston database and comparisons with other

approaches from the literature indicate that the method achievesaverygoodsensitivityandagoodspecificity.Future work will focus on a thorough evaluation of the proposed method in a pre-clinical setting, by using long-term and continuousEEGsignalsandtheprotocols presentedin[55]. Another important perspective consists in studyingspatial EEGsource location informationtocharacterizethe spatio-temporalpatternsofepilepticactivity.Areal-time implemen-tationoftheproposedmethodonanEEGmonitoringsystemis currentlyunderdevelopment.

Acknowledgements

Thiswork was supported by the SuSTaIN program, EPSRC grant EP/D063485/1 at the Department of Mathematics, Universityof Bristol;ITBACyTgrantDP.557No.34/2015, Insti-tutoTecnológicodeBuenosAires;07/15ProtocolatFLENI;and theDynBrainprojectsupportedbytheSTICAmSUD interna-tionalprogram.PartoftheworkwasconductedwhenMPheld

aMarieCurieIntra-EuropeanFellowshipforCareer Develop-mentattheSchoolofMathematicsoftheUniversityofBristol. Also,partoftheworkhasbeenpresentedattheIEEEIVMSP, IMFBE,andJournalofPhysicsconferences[56–58].

Appendix

A.

Goodness-of-fit

measures

for

the

statistical

models

of

the

EEG

wavelet

coefficients

V

,

V

,

V

,

V

,

V

.

Weassessedthegoodness-of-fitofthegeneralized Gauss-ianmodelfortheEEGwaveletcoefficientsVd,Vu,Va,Vb,Vgby

computingthe Kolmogorov–Smirnov (KS)score[59]andthe Cramer-von-Mises(CvM)score[60].Forcomparison,wealso computed the goodness-of-fit scorefor the following other two-parameterstatisticalmodelsthatarealsocommonlyused to modelwaveletcoefficients:logistic, t-locationscale, and alpha-stabledistributions.Moreover,wecomputedthescores foreachspectralbandandbyseparatingthedataintoseizure andnon-seizuregroups.Theresultingscoresaresummarizedin

Tables5and6,whichreportrespectivelythemeanandthe standarddeviationoftheKSandtheCvMscores.Observethat thegeneralizedGaussiandistributionclearlyprovidesthebest model-fit-to-data.

Appendix

B.

Model

based

characterization

Inordertousetheparameterssandtasfeaturestoclassify seizureandnon-seizureEEGsegments,wefirstproposetoassess

Table5–MeansoftheKSandCvMscoresobtainedforGGDpdfsestimatedwithallEEGsegmentsofthe39events.TheGGD showsthelowestscoreswithrespecttotheotherdistributions.

KS Means GGD Logistic t-Location alpha-Stable

delta Non-Seizure 0.002 0.007 0.007 0.007 Seizure 0.002 0.004 0.004 0.004 theta Non-Seizure 0.008 0.037 0.042 0.042 Seizure 0.005 0.018 0.021 0.021 alpha Non-Seizure 0.005 0.045 0.051 0.051 Seizure 0.003 0.021 0.024 0.024 beta Non-Seizure 0.002 0.024 0.027 0.027 Seizure 0.001 0.011 0.012 0.012 gamma Non-Seizure 0.003 0.022 0.027 0.027 Seizure 0.001 0.010 0.012 0.012

CvM Means GGD Logistic t-Location alpha-Stable

delta Non-Seizure <0.001 <0.001 <0.001 <0.001 Seizure 0.004 0.007 0.006 0.006 theta Non-Seizure <0.001 0.013 0.016 0.016 Seizure 0.001 0.006 0.008 0.008 alpha Non-Seizure <0.001 0.021 0.027 0.027 Seizure <0.001 0.005 0.006 0.006 beta Non-Seizure <0.001 0.009 0.012 0.012 Seizure 0.001 0.005 0.006 0.006 gamma Non-Seizure <0.001 0.01 0.016 0.016 Seizure <0.001 0.002 0.003 0.003

theirabilitytoseparatesuchsignals,ineachbrainrhythm.We consideradatasetcomposedofn1non-seizureEEGsegments

andn2seizuresegments.

Let sðNÞ_j ;tðNÞ_j   ¼ sðNÞ_1;j;...;sðNÞ_n 1;j n o ; tðNÞ_1;j;...;tðNÞ_n 1;j n o   denote

the set of parameters estimated from non-seizure

segments for a given rhythm j. Similarly, sðSÞ_j ;tðSÞ_j

  ¼ sðSÞ_1;j;...;sðSÞ_n 2;j n o ; tðSÞ_1;j;...;tðSÞ_n 2;j n o  

are the scale and shape pa-rametersoftheGGDdistributionsassociatedwiththewavelet coefficientsfrom seizuresegments.Itisassumed thatthese fourparametersareindependentandfollownormal distribu-tions sðNÞ_j NðmðNÞ_s j ;s ðNÞ sj Þ (2) sðSÞ_j NðmðSÞ_s j;s ðSÞ sjÞ (3) tðNÞ_j NðmðNÞ_t j ;s ðNÞ tj Þ (4) tðSÞ_j NðmðSÞtj;s ðSÞ tjÞ: (5)

AunivariateT-testwasdesignedtocomparethemeansmðNÞs andmðSÞs HðsjÞ 0 :mðNÞsj ¼m ðSÞ sj (6) HðsjÞ 1 :mðNÞsj 6¼m ðSÞ sj: (7)

Thevariancesofthedistributions(2)–(5)arenotequalandnot unknown.Consequently,wedesignedthetestasfollows.Let sðNÞ_j and sðSÞ_j denote theempiricalconditionalmeans ofsðNÞ_j

andsðSÞ_j ,andDsj¼s

ðNÞ j s

ðSÞ

j theirdifference.Denotingass2_sðNÞ j

and s2

sðSÞ_j the unbiased estimators of the variances in each groupofsegments(i.e.seizureandnon-seizure),thestandard deviationofDsjcanbeestimatedas:

s^sj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 sðNÞ_j n1 þ s2 sðSÞ_j n2 v u u t : (8)

ThestatisticsoftheT-testassociatedwith(6)and(7)isthen

TðsjÞ n ¼ sðNÞ_j sðSÞ_j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 sðNÞ j n1 þ s2 sðSÞ j n2 s ; (9)

which is distributed accordingto a Student's t-distribution withndegreesoffreedom,

n¼ s2 sðNÞ j n1 þ s2 sðSÞ j n2 0 @ 1 A 2 s4 sðNÞ j n2 1ðn11Þþ s4 sðSÞ j n2 2ðn21Þ : (10)

ThehypothesisHðs₀jÞisrejectedifjTðsjÞ

n j>Tt.Inthisstudy,we choseaprobabilityoffalsealarmt=0.05.Toassessthe

statis-Table6–StandarddeviationsoftheKSandCvMscoresobtainedforGGDpdfsestimatedwithallEEGsegments ofthe39events.TheGGDshowsthelowestscoreswithrespecttotheotherdistributions.

KS St.deviations GGD Logistic t-Location alpha-Stable

delta Non-Seizure 0.001 <0.001 <0.001 <0.001 Seizure 0.024 0.032 0.031 0.031 theta Non-Seizure <0.001 0.002 0.002 0.002 Seizure 0.014 0.022 0.023 0.023 alpha Non-Seizure <0.001 0.005 0.005 0.005 Seizure 0.008 0.007 0.007 0.007 beta Non-Seizure <0.001 0.004 0.004 0.004 Seizure 0.008 0.014 0.016 0.016 gamma Non-Seizure 0.001 0.004 0.005 0.005 Seizure 0.002 0.003 0.003 0.003

CvM St.deviations GGD Logistic t-Location alpha-Stable

delta Non-Seizure 0.001 <0.001 <0.001 <<0.001 Seizure 0.178 0.29 0.258 0.258 theta Non-Seizure 0.001 0.004 0.003 0.003 Seizure 0.013 0.064 0.078 0.078 alpha Non-Seizure <0.001 0.007 0.008 0.008 Seizure 0.005 0.016 0.015 0.015 beta Non-Seizure <0.001 0.006 0.007 0.007 Seizure 0.028 0.145 0.174 0.174 gamma Non-Seizure <0.001 0.007 0.011 0.011 Seizure 0.001 0.004 0.005 0.005

Table7–Decisionrulestoassesthestatistical signifi-canceofthedifferenceofmeansoftheGGDparameters forseizureandnon-seizuresignals.

p-Value Observeddifference

>0.10 notsignificant

0.10 marginallysignificant

0.05 significant

ticalsignificance,thep-valueofeachtesthasbeencalculated.

Table7showsthedecisionrulesthatwereapplied.

AsimilartesthasalsobeendesignedtocomparemðNÞtj and

mðSÞtj for each brain rhythm j. A bi-variate T-test has also

been designed for the pair (sj, tj). Its results were not

significant,thereforeitisnotreportedhere.Inaddition,to furthersupport the statistical significance givenby the p-value,wecalculatedtheBayesfactorindicatorfollowingthe methodproposedby[61].Thismethod establishesa corre-spondencebetweenfrequentistsignificancetests,suchasthe onesdesignedhere,withBayesiantests.Asaresultisallows onetoequatethesizeoftheclassicalhypothesistestswith evidencethresholdsinBayesiantests.Followingthiswork (and assuming equal variances), we calculated the Bayes factor(BF)thatprovidesthesameevidenceasthep-values givenbyourtests

BF¼ nþT ðsjÞ n nþ TðsjÞ n  ffiffiffiffiffiffiffiffiffiffiffi ðng _Þ p  2 0 B @ 1 C A ðn1þn2Þ=2 (11)

where the hypothesis H0 is rejected when t> ffiffiffiffiffiffiffing

p

with g*=g2/(n1+n21)1andg¼ððTðsjÞt Þ

2

=n1Þðn1þn2Þ=2.

ThetotalamountofEEGweren1=26non-seizuresegments

andn2=13seizuresegments.Table8showstheT-scores,and

theirassociatedp-valuesandBayesfactors.The correspond-ingthresholdsareshown.Weobservethatthet-scores(Ts

n) areallgreaterthatthethreshold(Tt).Thecorresponding

p-values(p)arealllowerthan0.01.TheequivalentBayesfactors (BF) are also all greater than the threshold (BFt). The Hs0 hypothesisisthereforerejectedforallbrainrhythms,with highlystatisticalsignificanceaccordingtothedecisionrules presentedinTable7.Thescaleparametersisagoodmarkerto distinguishseizureandnon-seizureEEGsegments.Contrarily, t-scoresfortarelowerthatthethreshold,exceptfortheDelta rhythm. Theassociated p-valuesare higher than 0.1. The Bayesfactorsarelowerthanthethresholds.Consequently,Ht 0 hypothesisisacceptedimplyingthattcannotdiscriminate seizureandnon-seizureEEGsegments.Basedontheseresults, itbecomescredibletoclassifyEEGsegmentsintotwoclasses seizureandnon-seizurebasedonthescaleparameterofthe GGD associated totheir wavelet coefficients ineachbrain rhythm.

r

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