Quantifying Cognitive Processes in the Human Brain using Measures of Dependence

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Quantifying Cognitive Processes in the Human Brain using Measures of Dependence
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Fadlallah, Bilal H
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Principe, Jose C
Committee Members:
Wong, Tan Foon
Wu, Dapeng
Keil, Andreas

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Subjects / Keywords:
brain -- classification -- cognition -- dependence -- eeg -- graph -- itl
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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Abstract:
The question of localizing cognition in the human brain isan old and difficult one. Particularly challenging is to understand thefascinating human ability to recognize and identify faces. The exquisitecapacity to perceive facial features has been explained by the activity ofneurons particularly responsive to faces, found in the fusiform gyrus and theanterior part of the superior temporal sulcus. This dissertation hypothesizesand demonstrates that it is possible to detect automatically the recognition offaces solely from processed electroencephalograms (EEG) in an online fashionwith high temporal resolution using measures of statistical dependence appliedon steady-state visual evoked potentials (ssVEPs). EEG recordings are first modeled as an indexed family ofrandom variables belonging to a stochastic process. Measures of dependenceexploit bivariate distributions among pairwise channel recordings, and is amore realistic approach to quantify the joint spatio-temporal data distributionthan previous methods just working with the marginal distributions, since thelatter implicitly assume statistical independence between measurements.Standard and novel dependence measures were applied to estimate dependencewithin the filtered current source density (CSD) data. Based on previous andrecent literature, the analysis included measures of (i) linear and monotonecorrelation (Pearson's r, Spearman's rho and Kendall's tau), (ii) synchrony(using phase-locking statistics), (iii) mutual information (using k-nearestneighbors), and (iv) entropy (permutation and approximate entropy). Novelapproaches to quantify dependence are proposed using the concepts ofgeneralized association (GMA and TGMA) and weighted-permutation entropy (WPE). Dependencies between channel locations were assessed for twoseparate conditions elicited by distinct pictures flickering at a rate of 17.5Hz. Filter settings were chosen to minimize the distortion produced bybandpassing parameters on dependence estimation. A dynamic graph visualizingthe dependence evolution in time was generated for each condition anddependence measure. Several concepts from graph theory were adapted to analyzethe resulting graphs and identify the active recording sites. Measures ofcentrality were particularly useful in determining the main channels involvedin the cognitive response and a connected components analysis was employed tostudy in depth the network structure. A classification framework based on information theoreticalconcepts is further developed by computing a similarity measure between twomatrices storing the dependence information. This measure is then used todetermine whether or not two matrices share the same condition. Besides, statisticalanalysis was performed for automated stimuli classification on six participantsusing the Kolmogorov-Smirnov test. Results show active regions in theoccipito-parietal part of the brain for both conditions with a greaterdependency between occipital and inferotemporal sites for the face stimulus.This aligns with previous evidence suggesting re-entrant organization of theventral visual system, showing heightened re-entry when viewing meaningful orsalient stimuli. Further research should investigate whether the communicationpattern observed in this study is direct or enabled via one or moreintermediate sources.
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In the series University of Florida Digital Collections.
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by Bilal H Fadlallah.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Principe, Jose C.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-11-30

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QUANTIFYINGCOGNITIVEPROCESSESINTHEHUMANBRAINUSINGMEASURESOFDEPENDENCEByBILALH.FADLALLAHADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013BilalH.Fadlallah 2

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Formyparents,towhomioweeverything 3

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ACKNOWLEDGMENTS IshalltakethisopportunitytothankthepeoplewhohaveplayedasignicantroleinmyPh.D.journey.Iamindebtedtomyadvisor,Dr.JoseC.Prncipe,forhisguidance,insights,encouragementsandnancialsupportduringthepastyears.Heisatrueinspirationanditisdicultformetokeeptrackofthethingsilearnedfromhim.IhavealsobeenluckytoworkcloselywithDr.AndreasKeil,whointroducedmetocognitivepsychologyandresearchinthateld.Iwon'tforgethisenthusiasmorpatiencewithmystupidquestions.IwouldliketothankDr.TanWongandDr.DapengWuforbeingkindenoughtoserveonmycommittee,Dr.VladimirMiskovicforhelpingmewithcollectingEEGdata,Dr.FirasKobaissyforbeingthereallalong,andallmycolleaguesatCNELforbeingsuchgreatfriends,especiallythosewhohavebeenonthejourneysincethebeginningandtheFootballsuperstarsinF.C.Principe.Thefunmoments,researchandrandomdiscussions,andtheafter-workeventswillbehardtoforget.IamindebtedtotheLebaneseNationalCouncilforScienticResearch(CNRS),whichwasinstrumentalinprovidingnancialsupportduringmytimeatUF.IspecicallythankDr.CharlesTabetforhiscontinuousencouragementandsupportandthemanyinterestingandfruitfuldiscussions.SpecialthanksalsogotoErionHansanbelliu,RakeshChalasani,GoktugCinar,MiguelTeixeiraandGabrielNallathambifor\lendingmetheirheads"toperformtheEEGrecordingsusedinthisdissertation.Graduateyearswereveryimportanttomeinthattheyallowedmetobetterunderstandthenebalancethatexistsinresearchbetweenthesubjectivehumanapproachandthescienticreality.ItwasachallengingridewhereIhadtheopportunitytoattendmanyinspiringtalks,readseveralinterestingpapersandforemost,learntheimportanceofquestioningthebasicsandventuringintounexploredterritories.Finally,Iwouldliketothankmyfamily:myparents,twobrothersandtwosisters,forbeingtheultimatesourceofjoy,strength,support,andinspirationinmylife. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1Background ................................... 15 1.1.1BrainVisualizationTechniques ..................... 15 1.1.2TheElectroencephalogram(EEG) ................... 17 1.1.2.1History ............................ 17 1.1.2.2Neurophysiologicalconcepts ................. 17 1.1.2.3EEGversusMEG ....................... 18 1.1.3EEGforCognitiveStateEvaluation .................. 19 1.1.4TheSteady-StateVisualEvokedPotential(ssVEP) ......... 20 1.1.5FacePerceptionandAttentionSystems ................ 20 1.1.6ProblemStatementandResearchGoals ................ 23 1.2ProcedureOverview .............................. 23 1.2.1EEGSignalProcessingScales ..................... 23 1.2.1.1Timedomainmethods .................... 23 1.2.1.2Frequencydomainmethods ................. 24 1.2.1.3Time-frequencydomainmethods .............. 24 1.2.1.4Nonlineardynamics ...................... 25 1.2.1.5Timescalechoice ....................... 26 1.2.2SignalProcessingApproach ...................... 26 1.2.3MeasuresofDependence ........................ 30 1.2.3.1Brainnetworksandfunctionalconnectivity ......... 30 1.2.3.2Whymeasuresofdependence? ................ 31 1.2.3.3Previouswork ......................... 31 1.2.3.4Shortcomingsincurrentmethods .............. 33 1.2.3.5Proposedmethods ...................... 35 1.3DissertationOrganization ........................... 36 2EXPERIMENTALSETUPANDSIGNALPROCESSING ............ 37 2.1Methodology .................................. 37 2.1.1ExperimentalSetting .......................... 37 2.1.1.1Stimuliandrecordings .................... 37 2.1.1.2Data .............................. 39 5

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2.1.1.3Channelsubsets ........................ 40 2.1.2SignalProcessingOverview ....................... 41 2.2EEGSignalProcessing ............................. 43 2.2.1RawData ................................ 43 2.2.2CurrentSourceDensities ........................ 44 2.2.3SignalProcessing ............................ 45 2.2.3.1Notchlters .......................... 45 2.2.3.2Bandpassing .......................... 46 2.2.3.3Testing ............................ 47 2.3AnalysisintheFrequency-Domain ...................... 52 2.3.1Power .................................. 52 2.3.2Phase ................................... 53 3MEASURESOFDEPENDENCE .......................... 56 3.1MeasuresofDependenceintheLiterature .................. 56 3.1.1Dependence ............................... 56 3.1.1.1Correlation .......................... 56 3.1.1.2Mutualinformation ...................... 59 3.1.2MeasuresofCausality .......................... 60 3.1.2.1Whatiscausality? ...................... 60 3.1.2.2Cross-spectralanalysis .................... 61 3.1.2.3Grangercausality ....................... 63 3.1.3MeasuresofSynchrony ......................... 65 3.1.3.1Phase-lockingstatistics .................... 65 3.1.3.2Meanphasecoherence .................... 66 3.2NovelMeasuresofDependence ......................... 67 3.2.1PreliminaryConcepts .......................... 67 3.2.1.1Dependence .......................... 67 3.2.1.2Association .......................... 68 3.2.1.3Generalizedmeasureofassociation ............. 69 3.2.1.4Timeseriescomplexityandpermutationentropy ..... 71 3.2.2GeneralizedAssociationforTimeSeries ................ 74 3.2.2.1Motivation .......................... 74 3.2.2.2Proposedalgorithm ...................... 74 3.2.3Weighted-PermutationEntropyBasedDependence .......... 76 3.2.3.1Motivation .......................... 76 3.2.3.2Weighted-permutationentropy ............... 77 3.2.3.3WPE-baseddependence ................... 79 4THEDYNAMICDEPENDENCEGRAPH .................... 80 4.1GraphModel .................................. 80 4.2GraphTheoreticalConcepts .......................... 81 4.2.1BasicNotations ............................. 81 4.2.2NodeClusteringCoecient ....................... 81 6

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4.2.3MeasuresofCentrality ......................... 82 4.2.4LocalEciency ............................. 83 4.2.5ConnectedComponents ......................... 83 4.2.6GraphVisualization ........................... 84 4.3WeightsComputations ............................. 84 4.3.1Time-DelayEmbedding ......................... 85 4.3.2ImpactofFreeParameters ....................... 85 4.3.2.1Embeddingdimension .................... 85 4.3.2.2Timedelay .......................... 86 4.3.2.3Timewindow ......................... 86 4.3.2.4Timecomplexity ....................... 87 5INFORMATIONTHEORETICCLASSIFICATIONOFDEPENDENCEMATRICES ...................................... 88 5.1PreliminaryConcepts .............................. 88 5.2EstimatingEntropy-LikeQuantitieswithPositiveDeniteMatrices .... 89 5.2.1MatrixEntropyEstimator ....................... 89 5.2.2MatricesJointEntropyEstimatorusingHadamardProduct ..... 90 5.3TheDependenceMatrices ........................... 90 5.4ComputationalSteps .............................. 91 5.5Results ...................................... 94 5.6Classication .................................. 96 5.7Conclusions ................................... 97 6COMPUTATIONSANDRESULTS ......................... 98 6.1OverviewandComputationalSettings .................... 98 6.2TheKolmogorov-SmirnovTest ......................... 100 6.3Cronbach'salpha ................................ 100 6.4Results ...................................... 101 6.4.1SingleReferenceChannel ........................ 101 6.4.2AllChannels ............................... 109 6.4.2.1Displayingthedynamicdependencegraph ......... 109 6.4.2.2Localdescriptorsusinggraphtheoreticalmeasures .... 111 6.4.2.3Globaldescriptorsusinginformationtheoreticconcepts .. 114 6.4.3InternalConsistency .......................... 115 6.4.4PerformanceAnalysis .......................... 117 6.4.4.1TheKStest .......................... 117 6.4.4.2Classicationresults ..................... 120 6.5Discussion .................................... 123 7CONCLUSIONSANDFUTUREDIRECTIONS .................. 124 7.1SummaryandDiscussion ............................ 124 7.2FutureWork ................................... 125 7

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APPENDIX ......................................... 127 ACURRENTSOURCEDENSITY .......................... 127 A.1BasicConcepts ................................. 127 A.2ComputingtheCSD .............................. 128 A.2.1ForwardandInverseProblems ..................... 128 A.2.2CSDandCorticalMapping ....................... 129 BLINEAR-PHASEFIRFILTERDESIGN ...................... 132 CSECOND-ORDERBUTTERWORTHFILTERS .................. 133 DWEIGHTED-PERMUTATIONENTROPY .................... 134 D.1SyntheticData ................................. 134 D.2Single-ChannelEEGDataAnalysis ...................... 139 D.3EpilepsyDetection ............................... 139 EINFINITELYDIVISIBLEFUNCTIONS ...................... 143 E.1NegativeDeniteFunctionsandHilbertianMetrics ............. 143 E.2InnitelyDivisibleMatrices .......................... 143 REFERENCES ....................................... 145 BIOGRAPHICALSKETCH ................................ 162 8

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LISTOFTABLES Table page 6-1Two-sampleKolmogorov-SmirnovtestresultsforPEandWPE ......... 104 6-2Cronbach'salphapersubjectfordierentdependencemeasures&conditions .. 117 6-3Cronbach'salphaacrosssubjects,dependencemeasuresandconditions ..... 120 6-4KStestresultsonaveragedmatricesforallsubjects ................ 122 6-5ClassicationresultsforSubject6. ......................... 122 D-1Ratioofaverageentropiesbetweenepileptic&non-epilepticsegments ...... 141 9

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LISTOFFIGURES Figure page 1-1SpatialversusTemporalResolutionfordierentbrainanalysistools. ...... 16 1-2FirstrecordedhumanEEGfromtherstpublicationofHansBerger. ...... 18 1-3Diagramvisualizingfaceperceptionandattentionsystems. ............ 22 1-4FunctionalBrainNetworks. ............................. 32 2-1128-channelHCGSNmontageused. ......................... 38 2-2ThetwotypesofstimuliusedtoinstigatessVEPs. ................ 39 2-3SensormapcorrespondingtothesettinginFigure 2-1 .............. 40 2-4Sensormapcorrespondingtoa256-channelsGSNmontage. ........... 41 2-5The10-20systemtodescribethelocationofEEGelectrodes. ........... 42 2-6SensorsinFigure 2-3 groupedaccordingtoa10-20scheme. ............ 42 2-7Standardparametersofabandpasslter. ...................... 44 2-8OriginalCSDsignalatchannellocation72averagedoverthe15trials. ..... 46 2-9ChangeinnumberofchannelswhereKScorrectlydiscriminatesconditions. ... 50 2-10GMAvs.qualityfactorplotfor3dierentlterorders. .............. 51 2-11FilteredCSDsignalatchannellocation72averagedoverthe15trials ...... 51 2-12FFTpowerinelectrodespaceforlteredsignalsaveragedovertrials. ...... 52 2-13Assessingpowervariabilitypertrialattheickeringpeak. ............ 53 2-14PowerdistributionsfortheFaceandGaborpatchconditions. .......... 54 2-15FFTphaseinelectrodespaceforlteredsignalsaveragedovertrials. ...... 54 2-16Averagepowerat17:50:3Hzforeachchannelacrosssubjects. ......... 55 3-1SimpleillustrationofGMA. ............................. 72 3-2GMAvs.TGMAandeectoftemporalstructure. ................. 76 3-3Twoexamplesofpossiblem-dimensionalvectorssharingsamemotif. ...... 77 4-1Anundirectedgraphconsistingofthreeconnectedcomponents. ......... 83 4-2Distributingtheverticesofthedependencegraphonacircle. ........... 84 10

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4-3EectofFreeParametersSelection. ......................... 87 5-1ComputedvaluesofQFGavgforTGMA,Kendall'stauandSpearman'srho. .... 95 5-2DistributionsofQFGm;nandQFm;nforTGMA,Kendall'stauandSpearman'srho. 96 6-1Sensornetworkandchannel72highlightedinred. ................. 102 6-2Dependencemapsinsensorspace. .......................... 103 6-3Averagedphase-lockingvaluesforthefaceandGaborpatchconditions. ..... 105 6-4Distributionsoftheaveragedphase-lockingvaluesforthetwoconditions. .... 105 6-5Usingabsolutecorrelationtoweightgraphconnectionsforchannel72. ..... 106 6-6UsingGMAtoweightgraphconnectionsforchannelPOzor72. ......... 107 6-7UsingGMAtoweightgraphconnectionsforthesecondsubject. ......... 108 6-8UsingMItoweightgraphconnectionsforchannelPOzor72. .......... 108 6-9Electrodesubsetconsideredacrosstimewindows. ................. 109 6-10UsingSpearman'scorrelationtoweightgraphconnectionsforchannel72. .... 109 6-11EmpiricalCDFsperconditionforPEandWPE. ................. 110 6-12UsingTGMAandKendall'stautoweightgraphconnectionsforchannel72. .. 110 6-13GeneratedNullHypothesesandObtainedp-values. ................ 111 6-14ObtainedTGMAvaluesversusKendall'stau. ................... 112 6-15DynamicdependencegraphusingTGMA(Facecondition) ............ 113 6-16DynamicdependencegraphusingTGMA(Gaborpatchcondition) ........ 114 6-17Dynamicdependencegraphusingabsolutecorrelation(Facecondition) ..... 115 6-18Dynamicdependencegraphusingabsolutecorrelation(Gaborpatchcondition) 116 6-19GraphtheoreticalmeasuresextractedfromtwographsconstructedusingTGMA. 117 6-20GraphtheoreticalmeasuresfromagraphconstructedusingSpearman'srho. .. 118 6-21BetweennesscentralityforagraphconstructedusingTGMA. .......... 118 6-22Channelsandtheircorrespondingconnectedcomponentsizes. .......... 119 6-23Cardinalityofeachchannel'sconnectedcomponentmappedtosensorspace. .. 120 6-24Empiricalcumulativedistributionfunctionsforthethreedependencemeasures. 121 11

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6-25Teststatisticsandwindowsize. ........................... 121 A-1Potentialmapvs.CSDmapofavisualevokedpotential(VEP). ......... 131 D-1PEversusWPEinthecaseofanimpulse. ..................... 135 D-2DierententropymeasuresappliedonaGaussian-modulatedsinusoidaltrain. 137 D-3NormalizedPEandWPEvaluesfordierentSNRlevels. ............. 138 D-4WPE-basedanalysisperformedonlteredEEGdata. ............... 140 D-5WPE-basedanalysisperformedonasinglechannelprocessedEEGsignal. ... 141 D-6Dierententropy-basedmeasuresappliedonepilepticEEG. ........... 142 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyQUANTIFYINGCOGNITIVEPROCESSESINTHEHUMANBRAINUSINGMEASURESOFDEPENDENCEByBilalH.FadlallahMay2013Chair:JoseC.PrncipeMajor:ElectricalandComputerEngineering Thequestionoflocalizingcognitioninthehumanbrainisanoldanddicultone.Particularlychallengingistounderstandthefascinatinghumanabilitytorecognizeandidentifyfaces.Theexquisitecapacitytoperceivefacialfeatureshasbeenexplainedbytheactivityofneuronsparticularlyresponsivetofaces,foundinthefusiformgyrusandtheanteriorpartofthesuperiortemporalsulcus.Thisdissertationhypothesizesanddemonstratesthatitispossibletodetectautomaticallytherecognitionoffacessolelyfromprocessedelectroencephalograms(EEG)inanonlinefashionwithhightemporalresolutionusingmeasuresofstatisticaldependenceappliedonsteady-statevisualevokedpotentials(ssVEPs). EEGrecordingsarerstmodeledasanindexedfamilyofrandomvariablesbelongingtoastochasticprocess.Measuresofdependenceexploitbivariatedistributionsamongpairwisechannelrecordings,andisamorerealisticapproachtoquantifythejointspatio-temporaldatadistributionthanpreviousmethodsjustworkingwiththemarginaldistributions,sincethelatterimplicitlyassumestatisticalindependencebetweenmeasurements.Standardandnoveldependencemeasureswereappliedtoestimatedependencewithinthelteredcurrentsourcedensity(CSD)data.Basedonpreviousandrecentliterature,theanalysisincludedmeasuresof(i)linearandmonotonecorrelation(Pearson'sr,Spearman'sandKendall's),(ii)synchrony(usingphase-lockingstatistics),(iii)mutualinformation(usingk-nearestneighbors),and(iv)entropy 13

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(permutationandapproximateentropy).Novelapproachestoquantifydependenceareproposedusingtheconceptsofgeneralizedassociation(GMAandTGMA)andweighted-permutationentropy(WPE). Dependenciesbetweenchannellocationswereassessedfortwoseparateconditionselicitedbydistinctpicturesickeringatarateof17:5Hz.Filtersettingswerechosentominimizethedistortionproducedbybandpassingparametersondependenceestimation.Adynamicgraphvisualizingthedependenceevolutionintimewasgeneratedforeachconditionanddependencemeasure.Severalconceptsfromgraphtheorywereadaptedtoanalyzetheresultinggraphsandidentifytheactiverecordingsites.Measuresofcentralitywereparticularlyusefulindeterminingthemainchannelsinvolvedinthecognitiveresponseandaconnectedcomponentsanalysiswasemployedtostudyindepththenetworkstructure. Aclassicationframeworkbasedoninformationtheoreticalconceptsisfurtherdevelopedbycomputingasimilaritymeasurebetweentwomatricesstoringthedependenceinformation.Thismeasureisthenusedtodeterminewhetherornottwomatricessharethesamecondition.Besides,statisticalanalysiswasperformedforautomatedstimuliclassicationonsixparticipantsusingtheKolmogorov-Smirnovtest.Resultsshowactiveregionsintheoccipito-parietalpartofthebrainforbothconditionswithagreaterdependencybetweenoccipitalandinferotemporalsitesforthefacestimulus.Thisalignswithpreviousevidencesuggestingre-entrantorganizationoftheventralvisualsystem,showingheightenedre-entrywhenviewingmeaningfulorsalientstimuli.Furtherresearchshouldinvestigatewhetherthecommunicationpatternobservedinthisstudyisdirectorenabledviaoneormoreintermediatesources. 14

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CHAPTER1INTRODUCTION Thehumanbrainisoneofthemostcomplexself-organizingsystemsknown.Itroughlyconsistsof100billionneurons,eachhavingonanaverage7000synapticconnectionswithitssurroundingneurons.Theenormousstructuralconnectivityofthisintricatenetworkcomplicatesthecompleteunderstandingofitsvariousprocesses,andmakesmeasuringthefunctioninghumanbrainoneofthemostformidableendeavorseverundertakeninscienceorengineering[ Gevins 1984 ]. 1.1Background 1.1.1BrainVisualizationTechniques Thequestionoflocalizingcognitioninthehumanbrainisanoldanddicultone[ Posneretal. 1988 ].Currentanalysesoftheoperationsinvolvedincognitionarerelyingonnovelandpowerfulbrainimagingtechniquesduringcognitivetasks.Today,themostwidespreadusedneuroimagingtoolintheclinicalandresearchworldsisfunctionalmagneticresonanceimaging(fMRI).fMRIisbasedonnuclearmagneticresonanceandisahemodynamictechnique,i.e.itinvestigatesneuralactivitybytrackingchangesinbloodow.Anotherhemodynamictechnique,positronemissiontomography(PET),isgainingmoreacceptanceasanuclearmedicinetoolusefulinthecontextofevaluatingtumorsnoninvasively[ OtteandHalsband 2006 ].Themaindrawbackofusingthesetechniquesisthattheircorrespondingtemporalresolutionisboundtothehemodynamicresponseofneuralactivityandhencesuersgrievouslyfromtheslowresponseinducedbythelatter.Ontheotherhand,theyhaveanexcellentspatialresolution,notexceedingafewmillimeters. Theneedforotheralternativestoanalyzeneuralactivitywithinveryshorttimescalesmotivatestheusageofelectromagnetictechniques.Suchtechniqueshavebeenunderdevelopmentformorethanacenturynowandcircumventthelatencyandintrinsiclowtemporalresolutioninducedbythebloodstreaminthehumanbody,by 15

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recordingelectricalormagneticinformationsimultaneouslyfrommultiplelocationsonthehumanscalp.Electromagnetictoolscanbeelectroencephalographic(EEG)ormagnetoencephalographic(MEG)andhavetheadvantageofprovidinganexcellenttemporalresolution(afewmilliseconds)butusuallypoorspatialresolution(afewcentimeters).Giventhatcommunicationbetweenneuronsandneuralensemblestypicallyrangesintimescalesbetween1msand100msdependingontheneuronscharacteristics[ Anderson 2004 ],fMRIorPETarenotwellsuitedtoassesscognitivetaskswheretimescalesofinterestdonotexceed100ms.Inthiscase,EEGandMEGrepresentaprecioustooltotrackthedynamicsofneuralactivitywithhightemporalresolutionandhencegatherabetterunderstandingofthetemporalneuralcorrelatesofcognitiveprocesses[ CabezaandNyberg 2000 ].Figure 1-1 1showsthespatialandtemporalresolutionsofdierenttoolsusedforneuralinformationextraction. Figure1-1. SpatialversusTemporalResolutionfordierentbrainanalysistools. 1Thisgurehasbeenreprintedfrom ParasuramanandRizzo [ 2008 ]withslightmodications. 16

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1.1.2TheElectroencephalogram(EEG) 1.1.2.1History Theelectroencephalogram(EEG)isanoninvasiverecordofbrainelectricpotentialfromthescalpusingone,oranarrayof,electrodes.ThebasicsofelectrophysiologycanbetracedtothecontributionsGalvani,Volta,OhmandFaradaymadetounderstandingelectricalpotentialandelectricalcurrent.Theseconceptswerewellestablishedbythemiddleofthenineteenthcentury[ Collura 1993 ],whichallowedRichardCaton,asearlyas1875[ Caton 1875 ],torecordelectricalactivityfromthebrainsofrabbitsandmonkeysusingamirrorgalvanometer.Itwasnotuntil1929,though,thatHansBerger[ Berger 1929 ]performedtherstrecordingofahumanEEGusingaSiemensdouble-coilgalvanometer(Figure 1-2 ).Eversince,theEEGdisciplinehasknownacontinuousevolutionandhasbeencombinedwithothercomponentsinmulti-modalapproachestoimprovethelevelofknowledgeaboutbrainactivity.Forexample,combiningEEGrecordingswithfMRIimaginghasbeensuggestedby Huang-Hellingeretal. [ 1995 ], Babilonietal. [ 2005 ], Gotmanetal. [ 2006 ],andothers.SuchmethodhoweverintroducesresidualartifactsintheEEG,thatismainlycausedbythecardioballistogram(BCG)andthechangingeldsappliedduringthefMRIimageacquisition[ Allenetal. 2000 ],thusmakingtheprocessofanalyzingtheresultingsignalsevenahardertask. 1.1.2.2Neurophysiologicalconcepts Thehumancerebralcortexisafoldedstructure,2to5mmthickandcontainingaround1010neurons[ NunezandSrinivasan 2006 ],whoseactivationcausestheowoflocalcurrents.EEGmeasuresmostlythecurrentsowingasaresultofdendriticexcitationsofpyramidalcells.Thesecurrentsessentiallyoriginatefromowsofions(Na+,K+,Ca++,andCl-)inneuronmembranes[ AtwoodandMacKay 1989 ].Electricpotentialdierencesarethencausedbysummedpostsynapticpotentialscreatingelectricdipolesbetweensoma(neuroncellbodies)andapicaldendrites(neuralcellbranches). 17

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Figure1-2. FirstrecordedhumanEEGfromtherstpublicationof Berger [ 1929 ].Thetopsubgureshowswhathewouldlatercallthe\betarhythm",withtheECGsignalanda10Hzsinewavetorepresenttime.Thebottomsubgureshowswhathewouldlatercallthe\alpharhythm".BothsampleswerecollectedasrecordingsfromhissonKlaus. Thenumeroustypesofsynapsesandthevarietyofneurotransmittersmakethegeneralpicturemoresophisticated[ Teplan 2002 ].Largepopulationsofneuronsareneededtogeneratedetectableelectricsignalsontheheadsurface.Moreover,giventhemultiplelayersbetweenthescalpandneuronsgeneratingcurrentow(liketheskinandtheskull),therecordedsignaldetectedbyscalpelectrodesisweakandhastobemassivelyamplied[ Teplan 2002 ].TheelaborategenerationofEEGfromtheneuronsisbeyondthescopeofthisdissertation.Furtherdetailscanbefoundin DaSilva [ 1991 ]and Buzsakietal. [ 2003 ].InAppendix A ,wealsooutlinethebasicunderlyingprinciplesthatgoverntheEEGneurophysicsbecauseoftheirrelevancetosomestagesofthiswork. 1.1.2.3EEGversusMEG AsabyproductofEEGresearch,MEGtechnologystarteddevelopinginthelate1960s[ Cohen 1968 ],andmaderapidadvances.WhileMEGandEEGshareimportantcharacteristics,theydierdrasticallyinotheraspects.MEGismoreexpensivetosetupandrequiresadedicatedlaboratorywithmagneticshielding.InanEEGsetting, 18

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recordingsensorsareattachedtotheparticipant'sscalpandthelatterisallowedrelativemobility,whereasinanMEGsetting,participantsmustremainstillandsensorsaresetinhelmet-shapeddewars.Besides,recordedEEGsignalshavehigherordersofmagnitudethantheirMEGcounterparts(mVversusfT)[ EconomidesandGetov 2012 ].Inthisdissertation,andsincetemporalresolutioniscriticalforanaccurateassessmentofthehumanbrainactivity,werelyonEEGtoanalyzethebrainresponsetospeciccognitivetasks.Inparticular,weresorttodensearrayEEGforthatpurpose.ThelatterisamethodemployedtorecordEEGwithanumberofelectrodesfarexceedingtheusualnumberofelectrodesutilizedwithstandardtechniques(approximately20electrodes).ThemotivationforthisapproachistoincreasethespatialresolutionofscalpEEG[ Holmes 2008 ]. 1.1.3EEGforCognitiveStateEvaluation Scalpelectroencephalographyhasbeenwidelyusedforassessingcognitivefunctioningandevaluatingdierentcognitivestates.Examplesincludestudyingcognitiveimpairmentinpreseniledementia[ Johannessonetal. 1979 ],genderdierencesincognitiveabilities[ Corsi-Cabreraetal. 1989 ],andtheperformanceofcognitiveandmemoryability[ Klimesch 1999 ].EEGhasalsobeenusedtoderivemodelsforthesimulationofcorticalactivityduringcognitivetasks[ Zavagliaetal. 2006 ]andanalyzecognitiveinformationprocessinginschizophrenicpatients[ Kirschetal. 2000 ]. TheimprovedunderstandingofthebrainresponsetooscillatorystimulienabledthedesignofaclassofexperimentswheretheimpactofnoiseandartifactsislessimportantintherecordedEEG.Theseexperimentsexploitthefactthatstimulioscillatingatagivenrateinduceanoscillatoryresponsewiththesamefrequency(ormultiplesofit).Theseinducedsignalsarereferredtoassteady-statevisualevokedpotentials(ssVEPs)anddiscussedinmoredetailsinthenextsection. 19

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1.1.4TheSteady-StateVisualEvokedPotential(ssVEP) Scalp-recordedEEGsignalsareinherentlynoisy,andtherefore,werelyonthesteady-statevisualevokedpotential(ssVEP)fortwomainreasons:rst,itsusefulnessinelectrocorticalinvestigationsofattentionalprocessesovertime[ Morattietal. 2004 ; MullerandHillyard 2000 ];second,theeasewithwhichtheycanbeelicited.ssVEPsarecontinuousbrainresponsescausedbyashingvisualstimuli,generallymodulatedinintensitywithaxedrateusuallylessthan30Hzandnotsmallerthan3Hz.Thesescalppotentialscanbecapturedassignalsoscillatingwithafundamentalfrequencyequaltothestimuliashingrate.ssVEPshaverecentlybecomeapopulartoolincognitiveandclinicalneuroscience[ Keiletal. 2003 ; KeilandHeim 2009 ; Keiletal. 2005 ; 2009 ; Morattietal. 2004 ; Morganetal. 1996 ; Mulleretal. 2006 ]andbrain-machineinterfaces(BMIs)orbrain-computerinterfaces(BCIs)[ Chengetal. 2002 ; Materkaetal. 2007 ; Regan 1979 ; Wangetal. 2006 ; Zhangetal. 2010 ].ParadigmsusingssVEPshavebeendiscussedbyVialatteetal.inarecentcomprehensivereviewpaper[ Vialatteetal. 2010 ].ssVEPsareeasytoinduceandhaveseveralotheradvantageslikegoodsignal-to-noise(SNR)ratio,relativeimmunitytoartifactsandcosteectiveness.Inaddition,theyrequireminimalsubjecttraining,andsimplifyfeatureextractionandanalysisinfrequency-space.BeforeoutlininghowssVEPsareusedinourexperimentalparadigm,werstreviewhowattentionisinvolvedindetectingfacesandrecognizingfacialidentities. 1.1.5FacePerceptionandAttentionSystems Nowthatweprovidedbriefoverviewsofthemaincomponentsofourframework,wemovetodescribeitscognitiveandneuroanatomicaspects.Ofmostconcernhereishowhumansperceivedierentvisualstimuli.Facesforexamplecanbeconsideredtobethemostbiologicallyandsociallysignicantvisualstimuliforahumanbeing[ PalermoandRhodes 2007 ].Mostpeopletendtorecognizebirdsasbirds,tigersastigersandpenguinsaspenguins,i.e.automaticallyassignthevisualstimulitoaclassofobjects.Thisremainstrueforotherthingswemightperceivesuchastables,chairs,carsandbuildings,but 20

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notforhumanfaces.Wealwaystendtoperceiveahumanfaceassomeone'sface.Thebrainprocessedinvolvedinrecognizingfacesarealreadyinplaceatagetwoandgetfullydevelopedbyage10.Still,childrenmakemoremistakesthanadultsinfacerecognition,whichsuggeststhattimeisanimportantfactorinlearninghowtorecognizefaceswell[ SocietyForNeuroscience 2010 ].Thecoresystemforfaceperceptionconsistsofthreemaincomponents:thefusiformgyrus,thesuperiortemporalsulcusandtheinferioroccipitalgyri[ Haxbyetal. 2000 ]. Ahumanfacecanbeseenasasalientemotionalstimulus,regardlessofitsexpression,thusallowingdiscriminatingfamilymembersfromtotalstrangerswhileatthesametimeconveyingimportantinformationsuchasgender,race,directionofeyegazeetc.Thatbeingsaid,allfaces,eventhosethatcanbelabeledasneutralorbearingnoparticularexpressioncanbethoughtofhavingemotionalsignicanceandprobablyprivilegedaccesstovisualattentionresources[ PalermoandRhodes 2007 ].Figure 1-3 2illustratesthemainconstituentsofthefaceperceptionandattentionsystems. Event-relatedpotential(ERP)resultssuggestthatfacesarecategorizedaround100msafterstimulus,muchearlierthanthe200msrequiredtocategorizeobjectsandwords[ PalermoandRhodes 2007 ; Pegnaetal. 2004 ],whichsuggeststhatdetectingfacesisfastandecient.Moreover,weknowthattheamygdalaisinvolvedinrespondingtoallfacialstimuli.Howevertheresponseismoreaccentuatedforemotionalratherthanneutralfaces[ Streitetal. 2003 ].Extensiveresearchhasbeenperformedtodeterminewhetherfaceprocessingismandatory[ Lavieetal. 2003 ; O'Cravenetal. 1999 ],understandiffacesareregisteredwithoutconsciousawareness[ StoneandValentine 2003 ; 2004 ; Stoneetal. 2001 ],studyselectiveattentiontofaces[ CorbettaandShulman 2002 ; KastnerandUngerleider 2001 ],evaluatetheattentionalresourceallocationforfaceprocessing[ Huang 2Thegurehasbeenreprintedfrom PalermoandRhodes [ 2007 ]withminormodications. 21

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Figure1-3. Diagramvisualizingfaceperceptionandattentionsystems.Coloredrectanglesrepresentthecorecomponentsforfaceperception.Yellowreferstoidentityandsemanticinformationprocessing.Redreferstoemotionanalysis.Bluereferstothefronto-parietalcorticalnetworkinvolvedinspatialattention[ Hopngeretal. 2000 ].Solid(dashed)linesindicatecortical(subcortical)pathwaysforrapidemotionalexpressionprocessing.Multiplefeedbackconnectionsarethoughttoexistbetweenthedierentblocks. andPashler 2005 ; Maureretal. 2002 ],andanalyzeifabiasexistsforattendingfacesratherthanothertypesofobjects[ Johnsonetal. 1991 ; Roetal. 2001 ]. Haxbyetal. [ 1999 ]furthershowthatventralextrastriateregionsgetactivatedbesidesthesuperiortemporalsulcusandinferiorandmidoccipitalgyriregionsuponshowinganinvertedfaceforasubject.Nodierencehoweverwaswitnessedwheninvertingtheimageofahouse.Adetailedandcomprehensivelistofstudiescanbefoundinthepaperby PalermoandRhodes [ 2007 ]. 22

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1.1.6ProblemStatementandResearchGoals Wehenceproposetodesignanend-to-endframeworkthatprocessesEEGfromthestageofsignalextractiontothatoffeatureanalysisorclassicationinorderto:(1)discriminateacrossdierenttimescalestwocognitivetasksperformedbydierentsubjectsand(2)analyzeforeachconditionthedierentfunctionalconnectivitypatternsinvolvedbetweensensors.Todoso,weadoptacomparativeapproachbasedontheuseoftraditionalandnoveltime-basedmeasuresofdependence,usingatthesametimeadynamicgraphformulationoftheexaminedfunctionalnetworks. 1.2ProcedureOverview 1.2.1EEGSignalProcessingScales MostEEGanalysismeasurescanbeclassiedaccordingtotheirscalesoftimeandfrequency.Inthefollowing,wereviewthesescalesanddescribecommonwaysofextractingmeaningfulinformationthatcanlinkdierentEEGrecordings.Suchrecordingsmaybetakenfromthesameparticipantatdierenttrials,pointsintimeorspaceorfromdierentparticipants. Giventhecomplexityofcorticalactivityacrossrecordingsites,oneimportantstepistoquantifythefunctionalconnectivityamongscalplocationsand,ultimately,brainareas.Thiscanbedoneinseveralways,whichareusuallycategorizedastimedomain[ Heetal. 2007 ; Hjorth 1970 ; Keiletal. 2009 ; KleinandDavis 1981 ; Maynard 1979 ; SaltzbergandBurch 1971 ; Vidaurreetal. 2009 ],frequencydomain[ BaccalaandSameshima 2001 ; Makeig 1996 ; NunezandSrinivasan 2006 ; RappelsbergerandPetsche 1988 ],time-frequencydomain[ Blancoetal. 1995 ; Koenigetal. 2004 ; 2005 ]andnonlineardynamics[ Mayer-Kress 1986 ; Stam 2005 ]methods.InSections 1.2.1.1 1.2.1.2 1.2.1.3 and 1.2.1.4 ,wegiveabriefoverviewofeachscale. 1.2.1.1Timedomainmethods Time-domainmethodshaveseveraladvantagesthatmakethemappealingintime-varyingornonlinearsystems.Theyallowusingshortdatawindows,truncating 23

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theinputwaveformstocomeupwithsolutionscorrespondingtospecictimedurations,besidesmoreaccuratelystudyingtheeectofnon-stationaryartifacts.Atimedomainmethodpreservesthetimeresolutionperfectlyascomparedwithfrequencyortime-frequencymethods.Timedomainmethodshavebeenappliedby Sutter [ 1992 ]and FarwellandDonchin [ 1988 ]inthecontextofdesigningbrainmachineinterfaces(BMIs),by SaltzbergandBurch [ 1971 ]toestimateperiodanalyticdescriptorsand Keiletal. [ 2009 ]toexaminefunctionalrelationshipsamongstructuresinvolvedinemotionalperception. 1.2.1.2Frequencydomainmethods TheFastFourierTransform[ CooleyandTukey 1965 ]isaclevercomputationaltechniquebasedontheimportantconceptsintroducedby Fourier [ 1807 ]statingthat\anycontinuousperiodicsignalcouldberepresentedbythesumofproperlychosensinusoidalwaves".Togetherwithspectralanalysis,itcanbeseenasoneofthemainfrequency-basedmethodstoanalyzeEEG.TheresultsofEEGspectralanalysiscanbeoftengroupedintothetraditionalfrequencybands: whenf4Hz when4f7Hz when8f13Hz when14f30Hz whenf30Hz Coherence-basedmeasuresofdependenceareintroducedandfurtherdiscussedinChapter 3 .Furtheranalysiscanbefoundin GevinsandRemond [ 1987 ]and DaSilvaetal. [ 1986 ]. 1.2.1.3Time-frequencydomainmethods \EEGtime-frequencyanalysis"referstoallapproachesthatdecomposeEEGsignalsintomagnitudeandphaseinformationateachfrequency,andcharacterizetheirchangesovertime[ RoachandMathalon 2008 ].Theseapproachesaredierentastheycapture 24

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dierentaspectsofEEGmagnitudeandphaserelationships.AnoverviewanddiscussionofEEGtime-frequencyanalysisisgivenin RoachandMathalon [ 2008 ]. Ansari-Asletal. [ 2005 ]adoptedatime-frequencyapproachforthecharacterizationoftheinterdependencybetweenepilepticEEGsignals,basedonestimatingtheirlinearrelationshipusingthecross-correlationofnarrow-bandsignals. Mehrkanoonetal. [ 2011 ]usedtime-frequencycoherencyforassessingneuralinteractionsinEEGrecordings. Blancoetal. [ 1995 ]usedGaborandwavelettransformstoquantifyandvisualizethetimeevolutionoffrequencycontentsofEEGtimeseries. Durka [ 2003 ]alsoappliedamodiedwaveletapproachtoseveralEEGproblems. 1.2.1.4Nonlineardynamics TheEEGisanonlinearsignal[ Elbertetal. 1994 ].Theprogressinthetheoryofnonlineardynamicalsystemsor\chaostheory"hasreachedastagethatallowsitsapplicationtoneuraldata[ BasarandBullock 1990 ; Mayer-Kress 1986 ; West 1993 ]anditsusagetostudyself-organizationandpatternformationinbrainnetworks[ Stam 2005 ]. NonlinearEEGAnalysis:NonlinearEEGanalysiscanbetracedbacktoa1985paperby Rappetal. [ 1985 ]reportingcorrelationonhumansleepEEG.SinceEEGisatime-varyingsignal,itmightoccurthatnonlineardynamics,aeldthatwasrstintroducedbyHuygensin1669[ Huygens 1669 ]andfurtherestablishedbyPoincarein1889[ Poincare 1889 ],isanaturalpathtofollow.Briey,adynamicsystemisasystemthatchangesitsstateovertime.Here,astateisdeterminedbythevaluesofthevariablesinconsideration.Ifwehavenvariablesforexample,thestatecanberepresentedasapointinanndimensionalspace,commonlyreferredtoasstate-spaceorphase-space.Adynamicsystemisgovernedbyasetofequations(usuallycoupleddierentialequations)describinghowthestateofthesystemchangesovertime.However,thestartingpointofanyinvestigationinclinicalneurophysiologyisusuallynotasetofdierentialequations,butratherasetofobservationsintheformofanEEGorMEGrecord[ Stam 2005 ]. 25

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Time-DelayEmbedding:Themostimportantstepinnonlinearanalysisconsistsinreconstructing,startingfromacertainnumberoftimeseries,anattractorinthestatespacecorrespondingtotheunderlyingsystem.Embeddingatimeseriesisequivalenttoconvertingitintoasequenceofvectorsinanmdimensionalembeddingspace.Forasucientlyhighembeddingdimensionm, Takens [ 1981 ]hasshownthatthereconstructedattractorsharethesamedynamicalpropertiesasthetrueattractor.Time-delayembeddingisthemostwidespreadappliedembeddingtechniqueinthecontextoftimeseries.Itconsistsofreconstructing,foreachtimeseries,mdimensionalvectorsbyconsideringmconsecutivevaluesofthetimeseriesasvaluesforthemcoordinatesofthevector.Specically,ifweconsiderthetimeseriestxtuTt1,thecorrespondingtime-delayembeddingrepresentationwouldbe:xm;jtxj;xj)]TJ /F5 7.97 Tf 6.59 0 Td[(;:::;xjpm1quforj1;2;:::;Tpm1q,wheremanddenoterespectivelytheembeddingdimensionandtimedelay.Settingthevaluesofthemandhasbeenthesubjectofmuchdebate[ Celluccietal. 2003 ; Kenneletal. 1992 ; Rosensteinetal. 1994 ]andisbeyondthescopeofthisdissertation. 1.2.1.5Timescalechoice Inthisstudy,weconsidertherstapproachsinceitisfairlypossiblethatacognitivestateishighlyvolatile,andtherefore,itshouldbequantiedonlyoverashortperiodoftimei.e.withhightimeresolution.Thiscanbedonepreferablyinthetimedomain,sincefrequencydomainapproachessacricethetimeresolutioninordertoextractprecisefrequencyinformation.Anothermotivationforselectingatimedomainmethodistheexibilitythelatterprovideswiththehandlingoftimewindows. 1.2.2SignalProcessingApproach Asseenintheprevioussection,alotofknowledgehasbeenaccumulatedoverthepastdecadesonunderstandingthebrainprocessesandanatomicalcomponentsinvolvedintheperceptionandprocessingoffacialstimuli.Itisclearthatitistheevolutionintimeoftheinteractionamongstneuralmassesthatproducesthecognitivestate.Since 26

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thebrainisahierarchicalsystemwithphysicalextent,thetypeofsensingwillrestrictthetypeofactivitythatwillbecollected,anditsresolutionintimeandspace(inthehead).Butnomatterwhatthesensingmodalityis,oneshouldunderstandthattheinformationtoquantifycognitivebrainstatesiscontainedinthejointspacetimedistributionofthestochasticsignalsbeingcollected.RecordingscalpEEGcanbethoughtofastakingasnapshotofthefunctioningoftheseprocessesthatisblurredbythevolumeconductionpropertiesofthebrainsampledintimeandspace.ThersteectcanbemitigatedbytransformingEEGintocurrentsourcedensity(CSD)domain(RefertoAppendix A ).Usingdense-arrayEEGalsohelpsaddressthespaceresolutionconcern.ThereforeitcanbeclaimedthattheempiricaljointdistributionprovidedbythemultivariateEEGtimeseriescontainsmaximalinformationtoquantifytheunderlyingcognitivestates. Unfortunatelythisjointtimevaryingdistributionishardlyeverutilizedtoquantifybrainactivitybecauseofitshugedimensionality.Infact,astochasticprocessisanindexfamilyofrandomvariables,oneforeachtimesample.Moreover,thespatialdistributionoftheEEGisamultivariaterandomvariableofsizegivenbytheelectrodegrid(orthesources).Thisjointdistributiontobequantiedproperlywouldrequireanenormousamountofdatathatisimpossibletocollect,thereforeseveralassumptionsarenormallydone. Perhapsthemostimportantisstationarity,whichmeansthatthejointmomentsovertimearejustafunctionofthelags.Ergodicityallowsustoestimatestatisticalquantitiesbytemporalaverageswhichisalsouniversallyutilized.IfwefurtherassumeGaussianityinthestatistics,thenjustrstandsecondordermomentsarenecessarytoquantifythestatisticalbehaviorovertime.Asiswellknown,EEGisnotstationaryandprobablynotaGaussianprocesseither.However,experiencehasshownthatlocally(atapproximately1secondwindows),EEGstatisticsdonotchangeappreciably[ BullmoreandSporns 2009 ].Startingfromthispoint,wemodeleachrecordedEEGsignalasarealizationofastochasticprocess,whereeachEEGepochcorrespondstooneobservationofthestochastic 27

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process.Ifthenumberofchannelsisn,weendupwithnobservationsofthestochasticprocesswhereeachobservationcanbemodeledasasinglerandomvariable,hencetheresultingmodelconsistsofanindexedfamilyorvectorofreal-valuedrandomvariablesunderassumptionsofstationarityandergodicity.UnstationarityinEEGoriginatesbothfromexternalfactors(unstableelectrodecontacts,eyeblinks,muscleartefact,skinartefactandothertypesofartefact),aswellasfromwithinbrainsourcesbecauseoftheon-otypeofcommunicationbetweenneuralmasses.DenotingSourstochasticprocess,wecanexpressitas: StUt;tPIuwithIt1;:::;Tu(1{1) whereeachUtisarandomvariabledenedonaprobabilityspacet;F;Pucorrespondingtoeachtimesample.Amultivariatespatialdistributionofsizencanbethendenedbasedontheserandomvariables.DenotingUttXi;iPKuwithKt1;:::;nu,weexpressthecumulativedistributionfunctionintermsofthejointdistribution: Fpx1;:::;xnqPpX1x1;:::;Xnxnq(1{2) Anexpressionforthejointcumulativedistributionwouldbe: FX1;:::;Xnpx1;:::;xnqx1:::xnfX1;:::;Xnpu1;:::;unqdu1:::dun(1{3) WecanthenwriteeachmarginaldistributionfXipxiqas: fXipxiq:::fX1;:::;Xi1;Xi)]TJ /F10 5.978 Tf 5.58 0 Td[(1;:::Xnpx1;:::;xi1;xi)]TJ /F8 7.97 Tf 6.58 0 Td[(1:::;xnqdx1:::dxi1dxi)]TJ /F8 7.97 Tf 6.58 0 Td[(1:::dxn(1{4) Thejointdistributionisn-dimensional.Sincenisatleast64fordense-arrayEEG(weuse129and257EEGsettingsinourcase),workingwiththejointdistributionissimply 28

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outofquestion.MostapproachesusedintheEEGanalysisliteratureworkonindividualchannels.Thattranslatesintousingtheindividualdistributionsunderlyingeachchannel.Suchapproachwouldbejustiedonlyifthejointdistributioncouldbewrittenasaproductofthemarginals,i.e.iftheconsideredrandomvariableswereindependent.Therefore,workingwithindividualchannelsinthehopethatmarginaldistributionswouldprovidealltheintrinsicinformationoftherecordingsisaverystrongassumptionandamajorsimplication,onlyvalidunderindependence,whichofcourseisalsounrealistic. Toovercomethisproblem,weproposetoworkwiththejointbivariatedistributionsofchannelsovertime.Fornchannels,thereare)]TJ /F5 7.97 Tf 5.48 -4.5 Td[(n2npn1q 2bivariatedistributionscorrespondingtoeachdistinctpairofchannels.Onecanarguethatlookingattheproblemfromabivariateperspectiveonlyexploitstheinformationprovidedbythepairwisejointdistributionsandstilldoesnotexploittothefullextenttheavailableinformation.Althoughthisistrue,wearguethatconsideringthebivariatedistributionsapproximatestheanalysistowardsabetterexplorationoftheavailableinformation.Ausefulwaytoexploitthesebivariatedistributionsandestimatethecouplingbetweenthepairwisechannelsisusingmeasuresofstatisticaldependence,butthesemeasuresshouldquantifythe2Djointdistribution,notsomeofitsmomentsascrosscorrelation.Pairwisemodelsoeracrucialsimplifyingassumptioninthatthenumberofpairsisquadraticinthenumberofelements,notexponential. Toelaboratemoreonhowgoodwecandescribeneuralsystemsbasedonlyonknowinghowpairsofelementsinteract,werefertotheworkof Roudietal. [ 2009 ],whoanalyzedtheperformanceofquantitativedescriptorsderivedfrompairwiseanalysisinthecontextofspiketrains.Theauthorstrytoanswerthequestionofwhethertheecacyofpairwisemodelsinapproximatingthetruejointdistributionforsubsetsofbiologicalsystemscontainingasmallnumberofelementscanbeaswellgeneralizedtothewholesystem(thatincludesamuchhighernumberofelements).Aquantitative,genericandmostimportantlysystem-agnosticanswerisprovidedintermsofacrossoverpoint 29

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(determinedbyaspecicnumberofelements),beyondwhichsmallsystemscanbesaidtopossesspredictivepowerforlargesystems.Withinthisspirit,usingahighnumberofelementsinourpairwiseanalysisshouldtendtogiveabetterdescriptorforthewholesystem.Thisspecicaspectwillhencemotivateourcomputationalapproach. 1.2.3MeasuresofDependence 1.2.3.1Brainnetworksandfunctionalconnectivity Previousworkoncognitionpresentsevidencethatareasthatarecoactiveduringcognitivetaskscanbeaswellinterdependenti.e.functionallyconnected,formingacognitivecontrolnetwork[ Bressler 1995 ; ColeandSchneider 2007 ].Basedonthisobservation,itisonlynaturaltoexplorethefunctionalconnectivityasatooltoquantifyacognitivestate.Inthisdissertation,westudythisparticularaspectofcognitiveneuroscience,withastrongfocusondiscriminatingtwocognitivestatesinstatisticaltermssolelyfromtheassociatedfunctionalconnectivityacrossbrainregions.Toelaborate,wespecicallyproposetostudywhetheritisfeasible,startingfromrecordedEEG,todistinguishthebrainresponsetofacialversusnon-facialvisualstimuliandoutlinethewitnesseddierenceinactivatedregions. Thisapproachwouldbemotivated,aspreviouslydiscussed,bythefactthatstimuliwithhigherobject-basedcomplexityorspecicsemanticcontentundergofacilitatedvisualcorticalprocessing[ Bradleyetal. 2003 ; Keiletal. 2009 ; PalermoandRhodes 2007 ].Inspiteofthefactthatmanyaspectsoftheneuralmechanismscontrollingthisfacilitationremainunknown,muchofthepresentevidenceseemstopointtowardstheamygdaloidcomplexandtheparieto-frontalcortexasoriginsofre-entrantmodulationintolower-tiervisualareas,whenperceivingbiologicallysignicantstimuli[ Damasio 1998 ; Davis 1998 ].Accordingtothishypothesis,visualperceptionandattentiontorelevantstimuliinvolvecommunicationbetweentheoccipitalandfrontalcorticesmediatedbysubcorticalstructures[ Baizeretal. 1993 ; Langetal. 1997 ]. TodorovandEngell [ 2008 ]havealsosuggestedthatnovelfacesareautomaticallyevaluatedbytheamygdalaaccordingto 30

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ageneralvalencedimensioncausingtheactivationofafaceresponsivenetworkintheoccipitalandtemporalcortices,whereasthecontributionofthefusiformgyrusandsuperiortemporalsulcusinprocessingfacialstructureshasbeenstudiedin Iariaetal. [ 2008 ]; Sergentetal. [ 1992 ]andothers.Therefore,weexploitthefactthatanincreasedfunctionalconnectivityinandamongthesecognitiveareasisapreciseindicationofthepresenceoffacialvisualstimulationthanastandardobject. Functionalbrainnetworkscanbeexploredusinggraphtheory[ BullmoreandSporns 2009 ; Spornsetal. 2004 ]throughstandardstepsasillustratedinFigure 1-4 3. 1.2.3.2Whymeasuresofdependence? Asmentionedintheprevioussection,comingupwithamodelforfunctionalconnectivityshouldbebasedonameasurethatestimatesthecouplingbetweendierentchannels.Nomatterhowcomplextheprocessingdoneontherecordedwaveformsis,itisveryhardtoassesshowtheserecordingsrelatetoeachotherwithoutameasurethatestimatesthecoupling,andthisisexactlythemotivationforusingmeasuresofdependence.MeasuresofdependencerepresentacommonwaytoextractmeaningfulinformationthatcanlinkdierentEEGrecordingsbyassigningscalarvaluestorepresentthedegreeofcouplingbetweendierentrecordings.Suchrecordingsmaybetakenfromthesamesubjectatdierenttrials,pointsintimeorspaceorfromdierentsubjects. 1.2.3.3Previouswork NoconsensusexistsonanoptimalwaytoquantifycouplingamongEEGrecordings.Theunderlyingreasonforthisliesinthemultifariousnatureofcoupling.Thelattercanbecategorizedaslinearornonlinear,symmetricornonsymmetric,synchronousorasynchronous,transientornontransient.ThisiswhytheEEGliteraturecontainsmanyapproachestoquantifydependency.Linearmeasuresofdependencemainlyinclude 3Thisgurehasbeenreprintedfrom BullmoreandSporns [ 2009 ]withslightmodications. 31

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Figure1-4. Stepsinvolvedinmodellingfunctionalbrainnetworks:(1)deneasetofnetworknodes,(2)estimateacontinuousmeasureofassociationbetweennodes,(3)produceabinaryadjacencymatrixorundirectedgraphand(4)calculatethenetworkparametersofinterestinthisgraphicalmodelofabrainnetwork. cross-correlationandcoherenceapproachesandhavebeendiscussedby Corsi-Cabreraetal. [ 1996 ], Barcaroetal. [ 1986 ], Barlow [ 1973 ], GuevaraaandCorsi-Cabrera [ 1966 ], Schindleretal. [ 2007 ], Davidetal. [ 2004 ]and Carter [ 1993 ].Nonlinearcorrelationmeasureshavefurtherbeeninvestigatedby FernandesdeLimaetal. [ 1990 ], Pijnetal. [ 1992 ]and Allenetal. [ 1992 ].Ontheotherhand,nonlinearapproachesincludemutualinformation(MI)thathavebeendiscussedby Roulston [ 1999 ], Naetal. [ 2002 ], Honeyetal. [ 2007 ]and Jeongetal. [ 2001 ],measuresofgeneralized[ Arnholdetal. 1999 ; Montez 32

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etal. 2006 ; Schietal. 1996 ; Stametal. 1999 ]andphase[ Lachauxetal. 1999 ; Tassetal. 1998 ]associationandmeasuresofsynchrony[ Cavanaghetal. 2009 ; Gruberetal. 2001 ; Lachauxetal. 1999 ]. Cognitiveprocessingusuallyrequiresintegrationofinformationprocessedsimultaneouslyinspatiallydistinctareasofthebrain[ Montezetal. 2006 ].Ifwemodelbyafunctiontheinuencetwodierentareasofthebrainexerciseoneachother,thatfunctionwillbemorelikelyconsistoflinearandnonlinearterms.Dependingonwhichofthetermsisdominating,measuresoflineardependencemaynotbeabletocatchtheconcernedinterdependenciesamongthesignals.Hencenonlinearmeasuresofdependencemaybemoresuitabletocharacterizefunctionalcoupling.Theperformanceachievedbythesemeasureshasbeenthetopicofacomparativepaperby QuianQuirogaetal. [ 2002 ],inwhichphasesynchronization,crosscorrelationandthecoherencefunctiongivequalitativelyequivalentresults,withmoresensitivityobservedfornonlinearmeasures.Concerningmutualinformation,thefollowingconclusionsweredrawn:(i)theperformanceofMIwasdependentontheestimatorusedandtheembeddingtechnique(therstfactortoagreaterextent),(ii)MIdoesnotproducerobustestimatesofsynchronization,especiallywhentheamountofavailabledataislimitedand(iii)thatthebestMIestimatorintermsofperformanceachievedisak-nearestneighborsestimator[ NicolaouandNasuto 2005 ; QuianQuirogaetal. 2002 ]. 1.2.3.4Shortcomingsincurrentmethods Correlationandcoherencebasedmethods:Asdiscussed,thesearemeasuresoflineardependence,anditisastrongassumptionthattwobrainregionscommunicatethroughalinearchannel.Thesemethodsdisregardanynonlinearcouplingbetweendierentrecordings. Grangercausalitybasedmethods:Grangercausalityisamethodthatcanbeusedintimeorfrequencydomainstoestablishthedirectionalityoflineardependencerelationships(moreaboutitinChapter 3 ).Grangercausalitycanbeformulatedusinga 33

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linearvectorautoregressive(VAR)modelingoftimeseries.Thefundamentalassumptionsofthelatterarestability,stationarity,andgaussiandistributionoftheerrorterm.EEGcouldbeconsideredstationaryontimeintervalsof1secondasshownby BernasconiandKonig [ 1999 ]butsuchassumptionremainsquestionabledependingonthecontextused.Moreover,numericalexperimentsconductedby Kammerdiner [ 2008 ]indicatethatthestabilityconditionofvectorautoregressivemodelisoftenviolatedinapplicationtotheEEGdata. Phasesynchronybasedmethods:Asapre-requisitetoestimatingphasesynchronybetweentwogiventimeseries,thephaseofthesignalsatthefrequencyofinterestneedstobedeterminedrst.ThemostpopularstandardapproachesforphaseestimationaretheHilberttransformandtheMorletwaveletapproach[ Lachauxetal. 2002 ].Thedisadvantagebroughtbytherstistheassumptionofnarrowbandnessinthesignalsunderconsideration,whichmightnotalwaysbefullled(ifthatassumptionisbroken,instantaneousquantitiesmightnowbewelldened).Asareminder,theinstantaneousphaseisdenedasthephaseoftheanalyticsignal,inturndenedasthesumofthesignalwithitsHilberttransform.Forasignalwptq,theHilberttransformHpwptqqcanbeexpressedas: Hpwptqq1 wptq1 t1 P.V.wptq td(1{5) whereP.V.referstotheCauchyprincipalvalue.Hencetheestimatedphasebecomes: wptqtan1Hpwptqq wptq(1{6) Ontheotherhand,theMorletwavelet(alsoknownastheGaborfunction)introducestwoparametersasreectedbyEquation 1{7 ,theratedecayandthecenterfrequency 34

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wandimposesanon-uniformtime-frequencyresolutiontradeoontheextractedphaseestimates,i.e.high(low)timeresolutionatlow(high)frequencies[ Aviyente 2009 ]. ^p!qc1 4e1 2p!q2e1 2!2(1{7) MutualInformationbasedmethods:MIisapowerfulmethodtocapturenonlinearstructureandamorerigorousquanticationofassociationbetweentwosignals.Howeverithasthreemaindrawbacks: Forallnitesequences,thereisasystematicoverestimationofthemutualinformation[ Herzeletal. 1994 ]. ItisdiculttochoosethebestestimatorforMIforaspeciccase,especiallywhenthesamplesizeissmallandthedimensionalityishigh. Estimatingmutualinformationinvolvesatleastonefreeparameter,whichaddsmoreuncertaintytotheanalysis. 1.2.3.5Proposedmethods Totackletheseproblems,wemainlyadoptageneralizedassociationapproach.Typicaladvantagesofsuchmethodisthatitisparameter-free,andthatitprovidesanintuitiveunderstandingofdependenceinthecontextofrealizations.Theseareimportantwhenweonlyhaveafewsamples{asituationwheretraditionalestimatorsbecomehighlybiased,thusloosingtheirmeaning.WeelaboratemoreabouttheseconceptsinChapter 3 .Therefore,themethodsdescribedinthisthesiswillcontributein: 1. Introducingtwonovelmeasuresofdependencetousewithtimeseries.Therstisbasedongeneralassociation,cancapturenonlinearassociationandhasanestimatorthatdoesnotinvolvefreeparameters.Thesecondisbasedonanordinalsymbolicdynamicstomeasuretimeseriescomplexity.Althoughitsestimatorinvolvesonefreeparameter,weshowitisapowerfulmethodtoinferrelationshipsintheconsidereddata.Tothebestofourknowledge,usingcomplexitymeasuresinthecontextof 35

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dependencehasnotbeenpreviouslysuggestedintheliterature.ThisisdiscussedinSection 3.2 2. Approachingtheaboveconceptsfromagraphtheoreticalperspectivetoanalyzebrainfunctionalnetworks.Severalgraphtheoreticalmetricswillbeusedtoquantifythedynamicsofconnectivitiesacrosstime.Chapter 4 containsfurtherelaborationonthedynamicdependencegraphandthemaincharacteristicsderivedfromit. 3. Classifyingtrialsinasupervisedfashionusinginformationtheoreticalconceptsappliedonthedependencematrices.MoredetailsontheadoptedapproachareavailableinChapter 5 1.3DissertationOrganization Therestofthedissertationisorganizedasfollows.InChapter 2 ,wegoovertheexperimentalsetting,describeindetailthesignalprocessingapproach,andanalyzethefrequencycomponentsoftheobtainedsignals,processedaccordingtoarobustlteringschemethatreducesthesensitivityofthedependencebackendtobandpassingparameters.InChapter 3 ,weexaminecurrentapproachesintheliteraturethatcanbeusedtoanalyzedependencebetweendierentelectrodemeasurementsandinferassociationsbetweenthecorrespondingchannels.Besidesthesemeasures,wealsodescribenovelordinal-basedmeasuresthatcanbeusedtoquantifydependenceamongEEGchannels.InChapter 4 ,weoutlinetheprocedurefollowedstartingfromtheextracteddependencevaluestotheconstructionofthedynamicdependencegraphs.Chapter 5 discussestheuseofinformationtheoreticconceptsinaclassicationcontexttoestimatewhethertwodependencematricesbelongornottothesamecondition.Chapter 6 providesanoverviewofresultsfromthepreviouslymentionedperspectivesbasedonastandardformulationoftheproblem,andChapter 7 oersdiscussionandfutureworkperspectives. 36

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CHAPTER2EXPERIMENTALSETUPANDSIGNALPROCESSING 2.1Methodology 2.1.1ExperimentalSetting Theexperimentincludedsixparticipants.Allweremalegraduatestudentsintheirearlytomid-twentieswhogavewrittenconsentpriortotakingpartinthestudy.Thesubjectshadnormalvisionandnofamilyhistoryofepilepsyanddidnotreportanypsychotherapeutichistory.TheprocedurewasapprovedbytheinstitutionalreviewboardoftheUniversityofFlorida.Theprimarygoalofthisdissertationistoconcentrateonthemethodologicalaspectsoftheresearchprocessandvalidatethedesignedalgorithmswithrealhumandata,whilebeingabletoassessvariabilityacrossdierentsubjects.Thechosennumberofsubjectswasconsideredsucienttoadequatelycharacterizethecognitivebehavioramongarelativelysmallbutmeaningfulpopulation. 2.1.1.1Stimuliandrecordings Twodataacquisitionsystemsfrom ElectricGeodesicsInc. [ 2007 ]wereusedtorecordtheelectrocorticalactivityofthe6participants:(i)a128-channelHydro-CellGeodesicSensorNet(HCGSN)and(ii)a256-channelGSN200v.2.1net(GSN-200).Threesubjectsusedthe128-channelmontageandtheothersthe256-channelmontage.Afterapplyingthenetsandsettingelectrodeimpedancesbelow50kaccordingtothemanufacturer'srecommendationforthehigh-impedanceampliers,theexperimentalsessionproceededwithcontinuousrecordingsusingCzasrecordingreference.AnillustrationofoneofthedataacquisitionsystemsusedcanbeseeninFigure 2-1 1. Epochsof400mspriortostimulusonsetand4200msafteronsetwereextractedforatotalof4:6secondsrecordings.Subsequenttothebaselinesegment,animageshowinganeutralhumanfacewaspresentedtothesubjectona17"monitorwithaverticalrefresh 1Thisgurehasbeenreprintedfrompage31of ElectricGeodesicsInc. [ 2007 ]. 37

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Figure2-1. 128-channelHCGSNmontageused.ManufacturerisElectricGeodesicsIncorporation(RiverfrontResearchPark,1600MillraceDrive,Suite307,Eugene,OR97403,USA). rateof70Hz.ThesameprocedurewasrepeatedwithacontrolstimulusshowingaGaborpatch,i.e.apatternofstripes,wherepatchesarecalculatedontheyusingpixelsofthefacepicture.Luminanceforbothpictureswassettovaryfromnear-zeroto52cd:m2andbothwerematchedformeanluminance(i.e.9:7cd:m2),averagecontrast(i.e.50%),andmeanspatialfrequency(i.e.4cpd)toprecludesystematicdierenceswithrespecttotheseparameters.Figure 2-2 showsthetwotypesofstimuliusedintheexperiment. ThestimuliickeredatafrequencyFo17:5HzandatotalofNt15orNt20trialswereperformed.Therhythmicstimulationratewasdesignedtoevokeanoscillatoryelectrocorticalresponseatthedrivingfrequency(steady-statevisualevokedpotential).Fromthispointonward,werefertothefacialstimulusas\Face"andtheGaborstimulusas\Gaborpatch".Duringtheexperiment,eachsubjectwasaskedtoxatethevisual 38

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AFacestimulus BGaborpatchstimulus Figure2-2. ThetwotypesofstimuliusedtoinstigatessVEPs.Bothwerematchedformeanluminance,averagecontrastandmeanspatialfrequency. stimuliandattentivelymaintaingazeonthepictures,nottoblinkandavoidasmuchaspossiblemovementsoftheeyeandthehead.EEGdatawascollectedfromtherecordingsoftheNc129orNc257electrodes,atadigitizationrateofFs1000Hz.Inatypicalscenario,wewouldperformoineartifactrejection(thatmightincludesomevisualinspectionsteps)todetectindividualchannelartifactsandinterpolatepotentialsatthesechannelsviasphericalsplines.Thiswasnotperformedinourcasesinceitwasnotclearhowitwouldaectcomputeddependenciesinsensorspace. 2.1.1.2Data Therecordingateachchannellocationiwascollectedasatimeseriesxpi;kqforfaceandypi;kqforGaborpathpertrialk.Asaresult,thewholedatacanberepresentedastwospatio-temporaldatamatricesXpkqandYpkqPRNcNs: Xpkqxp1;kq...xpNc;kqandYpkqyp1;kq...ypNc;kq; 39

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whereNcandNsdenotethenumberofchannelsandsampleddatapointsperrecording,respectively.Nc129forthe128-channelHCGSNandNc257forthe256-channelGSN-200network(thisissincethereferenceelectrodeisalsocounted).ForFs1000andadownsamplingfactord1,Ns4600.Forlateruse,wealsodeneNr400tobethenumberofsamplesforthebaselinesegmentandforagivenvectora,thenotationatn;mudenotesthesubvectortan;:::;amu.Figures 2-3 and 2-4 2showthelocationsofthecollectedrecordingsforbothnets. Figure2-3. Sensormapcorrespondingtothe128-channelHCGSNsettinginFigure 2-1 2.1.1.3Channelsubsets Tosimplifythelaterinterpretationofcomputedresultsoverallpossiblepairsofrecordings,wefurtherproposetogroupchannelsintosubsets.Astandardwayofdoingsowouldbetousetheinternational10-20system,asystemthatemploysmeasurementsofcraniallandmarksforlocatingelectrodesonthescalp[ Homanetal. 1987 ].Figure 2-5 2Thisgurehasbeenreprintedfrompage125of ElectricGeodesicsInc. [ 2007 ]. 40

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Figure2-4. Sensormapcorrespondingtoa256-channelsGSNmontage. illustratesthe10-20systemandFigure 2-6 groupstheelectrodes(showninFigure 2-3 )accordingtoa10-20scheme. 2.1.2SignalProcessingOverview AvarietyofneuralactivitiesisrecordedatthesametimewiththessVEP.Thisincludesmuscleactivity(EMG),electrocardiacactivity(ECG)andelectroocularactivity(EOG).Addedtothatcomeenvironmentalnoisefromthepowersupply,theinstrumentationandstimulusartefact[ Novaketal. 2004 ].Hence,mostEEGsignalprocessingstepsincludeartefactremovalalgorithms.Ourexperimentalsettingprovidesanadvantageinthatthedurationofeachtrialdoesnotexceed5seconds.ThisallowedaskingthesubjectnottoblinkandhenceunderminedtheeectofEOGs.Italsohelpedkeepingmuscleactivitytoaminimum. 41

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Figure2-5. The10-20systemisaninternationallyrecognizedsystemtodescribethelocationofEEGelectrodesintermsofthedistancebetweentheinionandthenasion.Fstandsforfrontal,Cforcentral,Pforparietal,Tfortemporal,OforoccipitalandAforearlobereference.cImmramaInstituteP.O.Box16604Tampa,FL33687-6604. Figure2-6. SensorsinFigure 2-3 groupedaccordingtoa10-20scheme. Toovercometheeectoflinenoise,typicalnotchlterswheredevelopedatthenoisycomponentsaswillbeexplainedindetailinSection 2.2 andAppendix C .Onthe 42

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otherhand,sincethepresentedstimulihaveaconstantickeringrate,andgiventhenatureofssVEPsthatuctuateatthisspecicfrequency[ Regan 1989 ],itisonlynaturaltoconsiderthatmostrelevantinformationintherecordedsignalsisincludedwithinafrequencybandcenteredattheickeringrate.Suchbandcanbeextractedwiththeuseofabandpasslter.Thelterdesigniscontrolledbytwofactors:theorderandthequalityfactor.Thelatterisdenedastheratioofthecenterfrequencytothebandwidth(RefertoFigure 2-7 ).Settingtheseparametersisaddressedinthenextsection.Atthisstage,itisimportanttokeepinmindthatanysignalprocessingstepshouldnotdistortpossiblesolutionstotheproblemwearetackling.Again,ourgoalistodeterminewhetherchannelrecordingsaredependent.GiventhatanytwoEEGrecordingscomewithanativedelay(becauseoftheirspatiallocation),weneedtomakesurethatanyadoptedlteringapproachdoesnotdistorttheintrinsicdelayinformation.Onewaytodosoistouselinear-phaselters.Linear-phaseltersaredesiredherebecausetheyhaveconstantgroupdelays,i.e.dierentfrequencycomponentshaveequaldelaytimes.Tofurtherelaborate,nodistortionisintroducedduetothetimedelayoffrequenciesrelativetooneanother. Anotherpointtokeepinmindistheultimatepurposeoftheexperimentatthebackend,whichisdiscriminatingtwocognitivestates.Itishencerequiredtoanalyzetheeectofthesignalprocessingapproachontheclassicationperformance.Thusweaddressthebandpassingproblemfromthatperspectiveandtrytond,inSection 2.2.3.3 thesetofparametersthatmaximizestheseparabilitybetweenconditions. 2.2EEGSignalProcessing 2.2.1RawData TherecordedEEGsignalsrevealthepresenceofstrongnoisesat60-Hzanditsoddharmonics(thirdandfth).Ourlteringapproachistwo-fold:rst,wewanttocleanthenoisycomponentsofthesignalandbandpassittoextractthefrequencyrangeofinterestfordependenceanalysisandsecond,wewantthechosensetoflteringparameterstominimizethedistortioninthesignalwithrespecttomeasuresofdependence.In 43

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Figure2-7. Standardparametersofabandpasslter.Thequalityfactorisdenedastheratioofthecenterfrequencytothebandwidth,i.e.Qfo |f2f1|. fact,sincethelteredoutputisaconvolutionoftheinputEEGwiththelterimpulseresponse,onewouldliketoavoidasmuchaspossibleringingthatwillimpactnegativelytheassessmentofchanneldependences. 2.2.2CurrentSourceDensities SeveralfactorsinterferewiththeprocessofgettingahighqualityrepresentationofbrainactivitywhenrecordinganEEGsignal.Theimpactofsomeoftheseartifactscanbeattenuatedlikethesubject'sheadmotionandeyeblinking,whereasothersareuncontrollablesuchastheelectricalactivityofsomemuscles,electrocardiogramsandespeciallytheeectofvolumeconduction.Toreducetheeectofthelatter,weestimatethestrengthofextracellularcurrentgeneratorsunderlyingtherecordedscalppotentials.Thisisdoneusingasphericalmodeloftheheadbasedonalinearvolumeconductionassumption.Suchmodelincludesfourlayers(scalp,skull,cerebro-spinaluidandbrain)andisconvenientsinceitallowsderivationofanalyticresultswithlowmarginoferror[ Wieringa 1993 ].Asaresult,weusecurrentsourcedensity(CSD)measuresthatroughly 44

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approximatethelocationsofthecurrentsourcesandsinksinareference-freefashion[ Nicholson 1973 ; TenkeandKayser 2005 ].Toderivethesemeasures,wefollowtheprocedureoutlinedby Junghoferetal. [ 1997 ]wherethecalculationoftheCSDortheequivalent\Laplacian"methodisusefultoreducethespatiallowpasslteringimpactcausedbythevolumeconductionpropertyofthetissue,uidsandskull[ Codispotietal. 2006a ; b ].Thismethodprimarilyreliesonasucientspatialsamplingaswellasanadequatesignaltonoiseratio[ Junghoferetal. 1997 ].TheserequirementsweremetforthecurrentstudywithERPmeasurementsperformedatatleast129electrodelocationswithmultipletrialspercondition.MoredetailsabouttheCSDtransformcanbefoundinAppendix A 2.2.3SignalProcessing Figure 2-8 showstherecordedsignalfromoneofthechannelsbothintimeandfrequencydomains,correspondingtoafacecondition.Fourclearlydiscerniblespikescanbeseenat17:60;60;180and300HzwhenanalyzingtheFFTofthetimeseriesconsistingofpre-andpost-stimulusportionspN1tNtk1xp72;iqt1;NruandN1tNtk1xp72;kqtNr)]TJ /F1 11.955 Tf -430.18 -23.91 Td[(1;Nsuq.ThedierenceintheestimatedFointhegrapharisesfromthediscrepancyintheanalogdisplayfrequencyofthemonitor. 2.2.3.1Notchlters TocleantheCSDsignalfromthelinenoiseartifactualcomponent,weusenotchlters.Thecenterfrequenciestoattenuateareatf160Hz,f2180Hzandf3300Hz.Thequalityfactor,denedforalterasQfc{4f(wherefcdenotesthelter'scenterfrequencyand4fthewidthofthefrequencybandat3dB)ischosentobeQ120fortherstnotchlter.Moreover,sincewewishthethreelterstohavethesame3-dBbandwidth4f,wesetQ23Q160andQ35Q1100.WeuseButterworthltersbecausetheyhaveamorelinearphaseresponseinthepassbandthanaChebyshevTypeIorTypeIIlteroranellipticlterandshowlessringing(RefertoAppendix B ). 45

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Figure2-8. OriginalCSDsignalatchannellocation72averagedoverthe15trials.(a)Firstsubplotshowsthesignalintimeforatotaldurationof800mssplitinhalfbetweenbaselineandpost-stimulusonset.(b)Secondsubplotshowsthefrequencycomponentsofthewholetimeseries.Notethepeakatthe17:5ickeringfrequencyandthenoisespikesat60,180and300Hz. 2.2.3.2Bandpassing Whencomputingthedependencebetweenchannellocations,thefrequencybandaroundFoisofparticularrelevancetoussinceitcarriesinformationabouttheresponsetothevisualstimuliateachrecordingsite;henceasuitablebandpassingschemeisneededtoextracttheickeringfrequencyband.Sincewedesireconstantgroupandphasedelaysfortheestimationofdependence,alinear-phaselterisappropriate.Moreover,asweproposetostudytheeectofdierentlteringschemeswithrespecttothedependencebackend,weproceedusingasimplenon-interactivemethodthatisoptimalwithrespecttothesquareerrorcriterion,namelythelinear-phaseleast-squareFIRlter.Oneoftheadvantagesofsuchapproachisthatitissuitableforarbitrarydesiredamplituderesponse.Forsuchlter,theproblemcanbestatedasfollows: hN;Qminh||hD||2(2{1) 46

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wherehN;QrhpnqsdenotestheimpulseresponseofaTypeII(evensymmetry)lteroflengthN)]TJ /F1 11.955 Tf 9.76 0 Td[(1(Neven),DrDpwrqsisalengthRvectorcontainingtheidealresponseatasetoffrequenciestwru.DimplicitlydenesthequalityfactorQandthecenterfrequencyfc.Rowpof)]TJ /F1 11.955 Tf 11.98 0 Td[(isgivenby: )]TJ /F5 7.97 Tf 8.08 4.94 Td[(pr1;2cospwpq;:::;2cospwppN1qqswherepPr0;:::;N1s(2{2) Theexpressionof)]TJ /F1 11.955 Tf 11.98 0 Td[(inEquation 2{2 canbeobtainedbyexpressingthelteras: HpwrqhN;Qp0q)]TJ /F1 11.955 Tf 18.96 0 Td[(2N{2n0hN;QpnqcospwrnqwithrPr0;:::;N1s(2{3) wheretheabovestepfollowsfromtheevensymmetrypropertyandshiftingbyN{2samples.Theproblemreducestoaleast-squareoptimizationthatcanbeformulatedby: ^hN;QargminhN;Q||hN;QD||(2{4) Hence:^hN;Qrp)]TJ /F5 7.97 Tf 8.08 4.94 Td[(T)]TJ /F4 11.955 Tf 8.08 0 Td[(q1)]TJ /F5 7.97 Tf 8.08 4.94 Td[(TsD (2{5) Theobtainedestimateisusedtogeneratebandpassltershavingdierentordersandqualityfactors. 2.2.3.3Testing Itiseasytoobservethatdecreasingthelter'sbandwidth,oralternativelyincreasingitsqualityfactor,increasesthedependencebetweentwochannelsoflteredsignals,since 47

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boththesesignalsbecomeclosetoasinglefrequencysinusoidwithaspecicfrequencyi.e.theickeringfrequency.3 Ontheotherhand,lterswithlowqualityfactorsaremoreaectedbynoisebecauseofthelargerbandwidth,andasaresult,dependencevaluesbetweenthelteredchannelsdrop.Therefore,toensurerobustnessitshouldbeestablishedthatthereexistsasucientlylargesetofparametervalueswherethedependencescapturedamongthelteredsignalsaremeaningfulandstable.Moreoversinceourgoalistomaximizetheseparabilitybetweenthetwoconditions,faceandGaborpatch,weconsiderittobeacriteriatojudgetherobustnessoftheeectofltervariation. Wehencecomputepairwisedependencevaluespercondition,pertrialandperlter,andthenapplystatisticalteststoseehoweectivelywecandierentiatethetwoconditionsforeachlter.Thestatisticaltestusedforthispurposeisthetwo-sampleKolmogorov-Smirnov(KS)test,whichisanon-parametrictesttocomparetwosamplevectors.TheKStesttriestoestimatethedistancebetweentheempiricaldistributionfunctionsofthetwosetsofsamples.ItisdescribedinmoredetailsinChapter 6 .ThecompletetestingprocedureisdetailedinAlgorithm 1 forameasureofdependencethatwillbeaddressedinmoredetailsinChapter 3 (Section 3.2 ):thegeneralizedmeasureofassociation(GMA). Impactoflterparametersandbackend:FollowingtheproceduredescribedinAlgorithm 1 ,weobtaintheplotshowninFigure 2-9 .Wepresenttheresultsforonly3dierentlterordersoutof20between10and300,sincetheeectofthelterorderontheshapeofthecurveisminimal.Wehaveevaluatedthecurvefor250qualityfactorsintherange0.1and175.Sincethereadingsofanytwopairofchannelsisaectedbythedelayinsignalpropagation,weembedthesignalinm8dimensionbeforecomputingthe 3Besides,lterswithhigherqualityfactorsinducebetterstopbandattenuationbutexhibitmoreringing. 48

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Algorithm1:TestingProcedurewithDierentFilters Input: OisavectorofMlterorders,fisavectorofNlterbandwidths,istheembeddingdimension Output: AnMNmatrixRofpairwiseKStestdecisionsperlterorderperqualityfactor 1 begin 2formPt1;:::;Mudo 3fornPt1;:::;Nudo 4ComputehOpmq;fpnqasin( 2{5 ). 5foriPt1:::Ncudo 6forjPt1:::Ncudo 7forcondPtGabor;Faceudo 8forkPt1:::Tudo 9-ComputeYpkqpcondqm;n;i;pqhOpmq;fpnqXpkqpcondqi;pqand 10Ypkqpcondqm;n;j;pqhOpmq;fpnqXpkqpcondqj;pq,wheredenotes 11convolution. 12-EmbedlteredseriesYpkqpcondqm;n;i;pqandYpkqpcondqm;n;j;pqin 13dimensiontoget1and2 14-SetDpkqpcondqm;n;i;jGMAp1;2q 15end 16end 17ComputeRm;nasijKSDpq;Facem;n;i;j;Dpq;Gaborm;n;i;j. 18end 19end 20end 21end 22end GMAvalues.m8correspondsto32ms.Moredetailsareincludedin[ Fadlallahetal. 2011 ]. Thecurvestartswithrelativelylowvaluescorrespondingtoqualityfactorsintherange(Q0:4),thenstaysalmoststableintherangeQPr0:6;1:5s,andthendecreasesagain.Thisisinlinewithourexpectationssinceforalowqualityfactor,thedependencelevelisreducedduetothepresenceofnoiseandthemethodloosesdiscriminability,whereasontheotherhand,forahighqualityfactorthedependencevaluesareusuallyhigh,andthereforethemethodagainperformworse.Thestableperformanceofthe 49

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Figure2-9. PlotshowingthechangeinthenumberofchannelsforwhichthetwoconditionsarecorrectlydiscriminatedbytheKStestasthebandpasslterqualityfactorincreases.Testswereperformedfor3dierentlterordersandGMAwasusedasdependencemeasure.Thecurveincreaseswiththequalityfactor,reachesaplateauandthendecreasesasthequalityfactorgoesup,clearlyshowingaregionwheretheestimateddependencevaluesarerobust. methodintherangeQPr0:6;1:5scanbejustiedsinceforthisrangeofqualityfactorsthebandwidthrangebecomesr11;29s.Itispossiblethatthisparticularbandwidthrangecoversthemodulationoftheickeringfrequencywell,thusextractingtheessentialinformationfordiscriminatingthestimuli. Beforeproceedingfurther,wecheckthedependencevaluesfortwopairsofchannelsthatweexpecttobehighlyandlessdependent,respectively.Figure 2-10 showstheGMAvaluesbetweenapairofchannelsintheoccipitalarea(nearO2),andapairhavingonechannelintheoccipital(nearO2)andtheotherinthefrontal(nearFp1)area.Asexpected,weobservethatGMAincreaseswiththequalityfactorforthechannelslikelytobedependentsincetheeectofnoisefadeswiththereductionofthepassband.Forthechannelslesslikelytobedependent,thereductionofthenoiselevelismaskedbythedierencesbetweenthetwosignalsinthepassband. 50

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Figure2-10. GMAvs.qualityfactorplotfor3dierentlterordersfortwochannelshaving(a)highdependenceand(b)lowdependence. Results:TheobtainedssVEPsignalafterlteringisshowninFigure 2-11 .Acleardierenceinfrequencyspectracanbenoticedbetweenthebaselineandpost-stimulusportions.InlinewithourexpectationofhigherpoweratFoafterstimulusinception,therstFFTmagnitudeplotshowspeaksinthehigheralpha-rangewhereasthesecondhasastrongerandmoremarkedpeakattheickeringfrequency. Figure2-11. FilteredCSDsignalatchannellocation72averagedoverthe15trials.(a)Signalintimepriorandafterstimulusonset.(b)Frequencycomponentsofthebaselinesegment.(c)Frequencycomponentsofthepost-stimulussegment. 51

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2.3AnalysisintheFrequency-Domain WerstanalyzethelteredsignalsinthefrequencydomainoverthesensorspacetoassessthelteringimpactandcharacterizetheparticipantsactivityintheFofrequencyrange.Theanalysisisrestrictedtothefrequencydomainandisbasedonallrecordedtimesamples,thuspresentinglimitationsintimeresolution. 2.3.1Power Powerisvisualizedfordierentfrequencybandsbetween16and18:9Hz.Forbothconditions,thepowerina0:5HzbandincludingFoclearlydominatesthoseintheotherrangesandismostlyconcentratedintherightoccipito-temporalregions.ThisisshowninFigure 2-12 wherepowerwasaveragedoveralltrialsandplottedbysensorlocation. Figure2-12. FFTpowerinelectrodespaceforlteredsignalsaveragedovertrials.UpperrowcorrespondstothefaceconditionandlowerrowcorrespondstotheGaborpatchcondition.Bothconditionsshowdistinctivelyhighpowerintheickeringfrequencybandmostlyconcentratedintherightoccipito-temporalregion. ThevariabilitypertrialsoftheFFTpowerattheickeringfrequencyisshowninFigure 2-13 .Thecomputedvariancepertrialisinlinewiththepowerdistributionplot(Figure 2-12 )thatshowsmorelocalizationofthepowerat17:5Hzforthefacecondition.Figure 2-13 (b)illustratestwodierentlineartsbetweenthevariancevectorsquantileswithstandardnormalquantilesforthetwoconditions. 52

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Figure2-13. Assessingpowervariabilitypertrialattheickeringpeak.(a)BoxplotforvectorsrepresentingthevarianceofthepoweratFoperconditionacrossalltrials.Edgesoftheboxcorrespondtothe25thand75thpercentile.(b)Plotofeachvector'squantileswithrespecttostandardnormalquantilesillustratethedierenceindistributionsbetweenthetwoconditions. 2.3.2Phase LookingatthepoweroftheFFTsignalisnotenoughtoinferinformationaboutthedynamicsofthesignalinsensorspaceatFoandthusdescribethetemporalcharacteristicsofthedata.Aknowledgeofthephaseisnecessarytoextractsuchinformation.FortheFFTSpfqofatimeseriesspnq,thephaseisdenedaspfqtan1r=tSpfqu{
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Figure2-14. Powerdistributionsforthetwoconditions.Errorbarsrepresentvariabilitywithtrials.TheKSstatisticobtainedis0:2891withap-valueof0:000316.TheKSstatisticisdiscussedinChapter 6 Figure2-15. FFTphaseinelectrodespaceforlteredsignalsaveragedovertrials.LeftmapcorrespondstothefaceconditionandrightmaptotheGaborpatchcondition.Regionswithlightercolorscorrespondtolowerphasevalues.Phasewrappingmakesitdiculttodrawacorrectinterpretationintermsoftimedelaysbetweendierentchannels. Thephaseplotsdonotprovideanyinfointherangeofinterest.Evenifweassumeotherwise,thetimeresolutionprovidedviathisapproachisverylimited.Thus,inordertoquantifythedependencebetweenelectrodelocationsinresponsetothevisualstimuli,weapplydependencemeasuresontimewindowsofthelteredsignals.Suchapproach 54

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bypassestheneedtoextracttimedelaysfromthesignal'sphase.Besides,itperformsbetterthanaclassierbasedonlyonpowerdiscrimination(asillustratedinFigure 2-14 ).Wehencetackletheproblemfromatime-domainperspective,usingdirectlythetimeseriesprocessedbythelinear-phaseFIRlterpreviouslydiscussedtoprovideabetterdiscriminationbetweenthetwoconditions.Anotherreasontomotivatethisapproachisthedicultytoinferpatternsthatdierentiatethetwoconditions(FaceandGabor)fromFFTdescriptorsasshowninFigure 2-16 ,forallsubjects. Figure2-16. Averagepowerat17:50:3Hzforeachchannelacrossthesixsubjects.ThetimeserieswereaveragedovertrialspriortocomputingtheFFTpower.129EEGrecordingshavebeencollectedfromsubjects1,2and3and257recordingshavebeencollectedfromsubjects4,5and6.Itisdiculttoinferpatternsthatdierentiatethetwoconditions(FaceandGabor)fromFFTdescriptors. 55

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CHAPTER3MEASURESOFDEPENDENCE Inthischapter,wedescribemeasuresofdependencethatcanbeappliedinthegeneralcontextofneuraldataandthespeciccontextofthisdissertation.Afunctionapproachtomeasuresofdependencehasbeenproposedby Renyi [ 1959 ],whoproposedsevenpostulatestodeneameasureofdependencebetweentworandomvariables.Westartbyoutliningtraditionalmeasuresofdependencesuchascorrelation,mutualinformation,phasesynchronyandstandardmeasuresofassociation,beforediscussingmeasuresofcausality,afamilyofmeasuresthatprovidesdirectionalityinformationbesidesdependence.Wethenmovetodiscusstwonovelmeasuresofdependence:therstisbasedonthenotionofgeneralizedassociationandthesecondonmeasuresofsignalcomplexity. 3.1MeasuresofDependenceintheLiterature 3.1.1Dependence Dependencecanbesimplyinterpretedastheabsenceofindependence.Manyapplicationsinengineering,statistics,econometrics,geophysics,biologyandmedicinerequire,whenevertworandomvariablesarenotindependent,todeterminethedegreeofdependencebetweenthetworandomvariables.Inthissection,wegiveabriefsketchofthemostusedmeasuresofdependence.Threecorrelationmeasuresandtwoinformationtheoreticmeasuresarepresented,inadditiontoGrangercausality,phasesynchrony,andcoherence-basedmeasures.Notethatthelattercanbecategorizedasfrequency-basedmeasuresbutarestillreportedherebecauseoftheirrelativelypopularuseinneuralcontexts. 3.1.1.1Correlation Correlationtriestomeasuretherelationbetweentwoormorevariablesandisprobablythemostwidespreadusedmeasureofdependence.Thecommonunderstandingofcorrelationreferstolinearcorrelation,thatcanbeestimatedusingPearson'scorrelationcoecient.Othermeasuresofcorrelationhowevertranscendthisrestriction. 56

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Pearson'sr:Pearson'scorrelation[ Pearson 1896 ]isobtainedfortwovariablesXtxiuni1andYtyiuni1bydividingthecovarianceofthetwovariablesbytheproductoftheirstandarddeviations.Itcanbedenedas: CXYcorrpX;YqcovpX;Yq XYErpXXqpYYqs XY(3{1) whereintheaboveequation,thecovariancebetweenXandYisdenedascovpX;YqEpXErXsqpYErYsq.Pearson'scorrelationrangesbetween-1and+1.Therstreectsaperfectnegativelinearrelationshipwhereastheseconddescribesaperfectpositivelinearrelationship.Acorrelationof0isobtainedifXandYarestatisticallyindependent,butcanalsobeobtainedifXandYareuncorrelatedbutnotindependent.Replacingtheexpectedvalueoperatorbythesamplemean,weobtainthesamplecorrelationcoecientthatcanbedenedas: rXYni1pxixqpyiyq pn1qsxsyni1pxixqpyiyq c ni1pxixq2ni1pyiyq2(3{2) Spearman'srho:Spearmanrankcorrelation(orSpearman'srho)isanonparametricmeasureofthedegreeofassociationbetweentwovariablesXandY.ItcanbeobtainedbyapplyingPearson'scorrelationcoecientonranksoftheinputvariables.MoredetailsaboutthepowerofSpearman'srhohavebeendiscussedby SiegelandCastellan [ 1988 ].ForthetwovariablesXandY,itcanbedenedas: 16d2i npn21q(3{3) wheredirepresentsthedierencebetweentheranksofthetwoobservationsxiandyi: diranktxiuranktyiu(3{4) 57

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Kendall'stau:Kendall'srankcorrelation(orKendall'stau)addressessomeofSpearman'srankcorrelationinsensitivitiestospecialkindsofdependence.IncontrasttoSpearman'srhothatproceedswithmeasuringthedierenceintheranksofeverypairofobservations,Kendall'scorrelationmeasuresinanon-parametricfashionthedegreeofassociationbetweentwovariablesintermsoftheoccurrenceprobabilityofconcordantanddiscordantpairs.ItisgenerallyreferredtoasKendall'stauandcanbedenedas: XYNcNd 1 2npn1q(3{5) wherendenotesthetimeserieslengthandNc,Ndreferrespectivelytothenumberofconcordantanddiscordantpairs,inturndenedfortwovariablesXandYas: 1. Nccorrespondstoeitherofthefollowingcaseswhere: txixjandyiyju txixjandyiyju 2. Ndcorrespondstoallothercaseswhere: txixjandyiyju txixjandyiyju Discussion:Spearman'srhoandKendall'staucancapturemonotonedependenceratherthanjustlineardependence[ Nelsen 2002 ].Thefollowinginequalitycapturestherelationshipbetweenthetwo[ SiegelandCastellan 1988 ]: 13Kendal's2Spearman's1(3{6) Ontheotherhand,Kendall'stauandSpearman'srhomeantwodierentthings.ThelattercanbethoughtofastheregularPearsonproduct-momentcorrelationcoecientappliedontheranks,whereasKendall'staurepresentsaprobability.Itisspecicallythe 58

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dierencebetweentheprobabilitiesthattheobserveddatasharethesameorderorhavedierentorders[ HillandLewicki 2007 ]. 3.1.1.2Mutualinformation Mutualinformationhasbeenchampionedbymanyresearchersasameasureofdependence,mainlybecauseoftheinuentialworkof Shannon [ 1948 ]inproposingmathematicalinterpretationsforconceptsofuncertaintyandinformation. Denition:AssumingthatXandYhaveajointprobabilitydistributionPXYpx;yq,themutualinformationbetweenthetwovariablesisdenedas: IpX;Yqx;yPXYpx;yqlogPXYpx;yq PXpxqPYpyqEPXYlogPXY PXPY (3{7) whereEXrepresentstheexpectedvalueoperatoroverdistributionXandPXpxqandPYpyqdenotethemarginaldistributions:PXpxqyPXYpx;yqandPypyqxPXYpx;yq.Analternativedenitionofmutualinformationusingthedenitionoftheentropyandconditionalentropyquantitiesis: IpX;YqHpXqHpX|Yq(3{8) Mutualinformationcanbethenseenasthereductioninavariable'suncertaintywhenthesecondvariableisknown.InEquation 3{8 ,themarginalentropyHpXq(resp.HpYq)andconditionalentropyHpY|Xq(resp.HpX|Yq)canberespectivelyexpressedas: HpXqxPXPXpxqlogPXpxq(3{9) and: HpY|XqxPX;yPYPXYpx;yqlogPXpxq PXYpx;yq:(3{10) 59

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Estimation:Estimatingmutualinformationisachallengingproblem.Inthisdissertation,weusetheestimatorproposedby Kraskovetal. [ 2004 ].Unlikeconventionalmutualinformationestimatorsbasedonbinningorkerneldensityestimation(KDE),thismethodreliesonentropyestimatesusingthek-nearestneighbour(kNN)algorithm.Theexpressionoftheestimatorisshownbelow: ^IpX;Yq pkq'rp pnx)]TJ /F1 11.955 Tf 11.76 0 Td[(1q)]TJ /F3 11.955 Tf 18.97 0 Td[( pny)]TJ /F1 11.955 Tf 11.76 0 Td[(1qqs)]TJ /F3 11.955 Tf 27.54 0 Td[( pnq(3{11) where'r:sx:yN1Ni1Er:piqs, pxqisthedigammafunctionandnepiqforarandomvariableErepresentsthenumberofpointsejwhosedistancefromeiisstrictlylessthanacertaindistancei,inturncomputedasthedierencebetweenzi(Zdenotingthemaximumnormoverthespaceconsistingofthetworandomvariables)anditskthneighbour.Inthisdissertation,weuseavalueofk4,afterseveraltrials.AnotherwayofwritingthisestimateofmutualinformationisshowninEquation 3{12 ^IpX;Yq pkq)]TJ /F3 11.955 Tf 18.97 0 Td[( pNqx pnxq)]TJ /F3 11.955 Tf 18.97 0 Td[( pnyqy(3{12) 3.1.2MeasuresofCausality Causalinterpretationsofrandomvariableshaverecentlybecomeanareaofincreasedfocus[ Edwards 2000 ; Patterson 2006 ].Howevermostcommonapplicationsusingcausalityliewithinthecontextoftimeseries,andthiswillbetheassumptionalongthelinesofthefollowingdenitions. 3.1.2.1Whatiscausality? Deningcausalityhasbeenacontroversialissueforcenturies.Eventoday,thereisstillnoclearanswertohowcausalityisdened.Narrowingdownthespanofthequestion,wemightnddenitionsofcausalitythatsatisfy(ornot)theinterestsofscientistsinthatarea. 60

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InSignalProcessing,acausalSystemisasystemwheretheoutputdependsonpast/currentinputsbutnotfutureinputs.i.e.theoutputyptoqonlydependsontheinputxptqforvaluesoftto.Similarly,adiscrete-timesystemissaidtobecausalif,foranygivenm,theoutputsequenceattheindexnmdependsonlyontheinputsequencefornm.Causalityisamandatoryfactorinthedesignandimplementationofreal-timeDSPsystemsforanobviousreason:allreal-timesystemsmustbecausalastheydonothavefutureinputsavailabletothem[ Patterson 2006 ].Inthismanuscript,wedenotecausalitybyC,i.e.XCYmeansXcausesY,orXandYarerelatedthroughacausalsystem.CausalityistransitiveinthesensethatifECFandFCGaretrue,thenECGistrueaswell.Inotherterms,feedingtheoutputofcausalSystemAtotheinputofcausalSystemBmeansthattheinputofAcausestheoutputofB,oralternativelythatthecombinedsystemABiscausal. 3.1.2.2Cross-spectralanalysis Itiscrucialformanyapplicationstoknowhowtwotimeseriesarerelated.Cross-Spectralmethodsessentiallyrepresentanextensionofspectralanalysistotime-series,tryingtodeterminethecorrelationsbetweentwotime-series.Theyareparticularlyusefultodescriberelationshipsamongvariableswhenoneiscausingtheothers.Ithasbeenusedforseveralapplicationsincludingcommunicationsengineering,econometrics,geologyandclimatology. Thecross-spectrumfunctionisnoneotherthantheFouriertransformofthecross-covariancefunction.Itishenceacomplexvaluedfunctionthatprovidesinformationaboutthedistributionoftwoprocessesactivityacrosstheirfrequencycomponents.Inthefollowing,letxypqand)]TJ /F5 7.97 Tf 30.07 -1.8 Td[(xypfqdenoterespectivelythecross-covariancefunctionoftwotimeseriesXandYandtheircross-spectrumfunction. )]TJ /F5 7.97 Tf 7.31 -1.79 Td[(xypfqFtxyupfq8xypqe2if(3{13) 61

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Coherence:Withtheabovedenitionofthecross-spectrum,wecandenethecoherencefunctioninananaloguewaytotheconventionalcorrelationcoecient,i.e.astheratioofthesquaredmagnitudeofthecross-spectraldensityandtheproductoftheauto-spectraldensities.Thisfunctioncanbeseenasanindicatorofhowgoodthecontributionofxptqisinpredictingyptqusinganoptimumlinearleastsquaresfunction.Theadvantagecoherencehasovercorrelationconsistsinbeingafunctionoffrequency,whichallowstotrackcorrelationsacrossfrequencybands.Coherenceachievesvaluesbetween0and1andworksbestunderstationarityassumption. Cxy|)]TJ /F5 7.97 Tf 7.31 -1.8 Td[(xy|2 )]TJ /F5 7.97 Tf 7.31 -1.79 Td[(xx)]TJ /F5 7.97 Tf 7.32 -1.79 Td[(yy(3{14) Fromthecross-spectrum,wecanalsodeneaphaseinformationindicatorxypfqateachfrequencyfas: xypfqtan1=t)]TJ /F5 7.97 Tf 7.31 -1.8 Td[(xypfqu
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Blinowska [ 1991 ].In2001, BaccalaandSameshima [ 2001 ]introducedtheconceptofpartialdirectedcoherence(PDC).PDCprovidesafrequency-domainrepresentationofmultivariateprocesses.PDCisbasedontheconceptsofGrangercausality(tobediscussedshortly),DCandDTF. Applications:Spectralmethodshavebeenusedby Scheeringaetal. [ 2011 ]toanalyzehowtheblood-oxygen-level-dependent(BOLD)signalinhumansisrelatedtoneuronalsynchronizationacrossdierentfrequencybandswhenperformingaspeciccognitivetask,by Thatcheretal. [ 1986 ]toexplaindierencesinconnectivitybetweendierentbrainregionsbycomputingEEGcoherenceasafunctionofdierentinterelectrodedistances,andby Tuckeretal. [ 1986 ]tomeasurethedegreeoffunctionalconnectednessamongrighthemisphereregions. ApplicationsofDConEEGdatahavebeenproposedby WangandTakigawa [ 1992 ], Wangetal. [ 1992 ], Inouyeetal. [ 1995 ]and Korzeniewska [ 2003 ].ContributionshavealsobeenmadeusingDTF[ Astoletal. 2004 ; Babilonietal. 2005 ; Kaminskietal. 2001 ]andPDC[ Astoletal. 2006 ; Schelteretal. 2006 ]. 3.1.2.3Grangercausality Manyreal-worldsituationspresentatwo-waycausality.Insuchcaseswherefeedbackoccurs,usingcoherenceandphasediagramsmightnotbeuseful.Grangercausality[ Granger 1969 ]addressesthisshortcomingofearlyspectralmethodsandhasbeenperhapsthemostwidely-establishedmeansofidentifyingcausalrelationshipsbetweentimesseries[ Hesseetal. 2003 ]. Intuition:TheideabehindGrangerCausalitycanbeunderstoodfromthefollowingillustrations: 1. Given :X,YandW,IftheforecastofXt)]TJ /F8 7.97 Tf 6.59 0 Td[(1givenXt;YtandWtisfoundtobemoreaccuratethantheforecastofXt)]TJ /F8 7.97 Tf 6.58 0 Td[(1givenXtandWt,wededucethatYthasinformationusefulinforecastingXt)]TJ /F1 11.955 Tf 11.9 0 Td[(1thatcannotbededucedfromXtandWt.Inthiscase,wesaythatYtGranger-causesXt)]TJ /F8 7.97 Tf 6.58 0 Td[(1pXGCYqunderthebelowconditions: 63

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(a) YtoccursbeforeXt)]TJ /F8 7.97 Tf 6.59 0 Td[(1 (b) YtcontainsinformationusefulinforecastingXt)]TJ /F8 7.97 Tf 6.59 0 Td[(1notavailablefromothervariables. 2. AvariableXGranger-CausesYifYcanbebetterpredictedusingthehistoriesofbothXandYthanitcanusingthehistoryofYalone. 3. XdoesnotGranger-CauseYpX|{GCYqifandonlyifthepredictionofYbasedontheuniverseUofpredictorsisnotbetterthanthepredictionofYbasedonthesetofpredictorsUX,i.e.ontheuniversewithXomitted. Formulation:Mathematically,letourinformationsetbedenotedbyIttXt;Yt;Zt;Wtutxt;yt;zt;wt;xt1;yt1;zt1;wt1;:::;x1;y1;z1;w1u.OthervariablescanbecontainedinItaswell. Denition.ytissaidnottobeGrangerCausedbyxtwithrespecttotheinformationsetItif: Eryt)]TJ /F5 7.97 Tf 6.59 0 Td[(h|ItsEryt)]TJ /F5 7.97 Tf 6.58 0 Td[(h|ItXts@h:(3{16) Equivalently,XtGCYtifthelinearpredictorofyt)]TJ /F5 7.97 Tf 6.59 0 Td[(hbasedonIthasdierentexpectedvaluethantheoptimallinearpredictorofyt)]TJ /F5 7.97 Tf 6.58 0 Td[(hbasedonyt;yt1;:::;y1foranyh. Shortcomings:ThemainshortcomingofGrangercausalityisthatitcanonlygiveinformationaboutlinearfeaturesofsignals.Henceitquantieslinearcausalityinthemean.Also,Grangercausalitydoesnotimplytruecausality.Toillustratethispoint,considertwoprocessesdrivenbyacommonthirdatdierentlags.Insuchcase,thealternativehypothesismightstillbenotrejected. Applications: Seth [ 2010 ]recommendsthefollowingmeasurestobeperformedpriortoapplyingGrangerCausalityonEEGdata: 1. Artifactreduction. 2. Downsampling. 3. Re-referencingtoacommonsource. 4. Subtractionofpre-stimulusbaseline. 64

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Othermeasuresofcausalitybasedoninformationtheoreticconceptshavealsobeenproposed.Directedinformation[ Quinnetal. 2011 ]hasbeensuggestedforexampleasaninformationtheoreticquantitytoinferrobustmeasuresofstatisticalcausalityfromspiketrainpopulations. 3.1.3MeasuresofSynchrony Althoughnoconsensusexistsonthedenitionofsynchronization,itisratheracceptabletoformulatephasesynchronyasthedegreeofphaselockingbetweenoscillators.Toillustratetheidealcase,letusassumewehavetwooscillatorswithphases1and2atfrequencyf.1and2usuallyvarywithtimebuttofurthersimplify,weconsideracasewheretheyareconstant.Wesaythattheoscillatorsaresynchronizedatfifitispossibletondtwointegersnandmthatverifytherelationn1m2const[ Rosenblumetal. 2010 ].Phasesynchronizationofchaoticoscillatorscanbewitnessedinseveralcomplexsystemssuchasthehumanbrain,especiallywhenitperformscognitiveprocesses.Asaresult,measuresofsynchronyhavebeenwidelyusedinavarietyofdisciplinesincludingneuralscienceandcognitivepsychology. 3.1.3.1Phase-lockingstatistics Quantifyingfrequency-specicsynchronizationbetweentwoneuroelectricrecordingshasbeenrstsuggestedby Lachauxetal. [ 1999 ].Assumewehavetwotimeseriesxptqandyptq,therststeptowardscomputingthephasesynchronybetweenthetwoseriesisextractingtheirinstantaneousphasesxptqandyptq.Twomainmethodsareusedforthatpurpose:therstusestheanalyticsignalandtheHilberttransform: xptqarctan~xptq xptq(3{17) where: ~xptq1 P.V.t1t1xpt1q tt1dt1(3{18) 65

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In 3{18 ,P.V.referstotheCauchyprincipalvalueorprincipalvalueintegral.Suchrepresentationofthephaseispossiblesincetheanalyticsignalcanbewrittenas: xaptqxptq)]TJ /F3 11.955 Tf 18.96 0 Td[(i~xptqAxeixptq(3{19) Wavelet-basedtechniquesconstituteanotherclassofphasesynchronyestimationmethods. Lachauxetal. [ 1999 ]suggesttoconvolutethesignalrstwithaGaborwaveletGpt;fq,centeredatfrequencyf,whichisdenedattimetas: Gpt;fqexpt2 22texppj2ftq(3{20) wheretisdenedasaconstantmultipleof1{f.Wethencomputethequantitypt;nqtodenethephasedierencebetweenrecordingsitesiandj: pt;nqipt;nqjpt;nq(3{21) wherenreferstothetrialindex,andpt;nqargpxptqGpt;fqqisthephaseofthesignal'sconvolutionwiththewavelet.Thephaselockingvalue(P.L.V.)canbethendenedas: P.L.V.ptq1 NNn1exppjpt;nqq(3{22) AsimilarapproachbasedontheMorletwaveletwasdiscussedby Gruberetal. [ 2001 ]. 3.1.3.2Meanphasecoherence Anothermeasureofphasesynchronizationbasedonthedistancebetweenphasevaluesisproposedby Mormannetal. [ 2000 ].Tocomputethemeanphasecoherence,theinstantaneousphaseisrstextractedfromapairofsignalsusingtheHilberttransform. 66

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Theestimatedvaluesarethenprojectedtoaunitcircleandthemeanphasecoherenceisreturnedasafunctionalofthedistancesbetweentheinstantaneousphases. M.P.C.1 NN1k1expjpxpk{Fsqypk{Fsqq(3{23) whereFsdenotesthesamplingfrequencyandNthenumberofsamplesinthetimeseries.Analternativewayofwritingthemeanphasecoherenceis: M.P.C.b Ercospxptqyptqqs2)]TJ /F16 11.955 Tf 11.76 0 Td[(Ersinpxptqyptqqs2(3{24) whereEr:sdenotestheexpectationoperator. 3.2NovelMeasuresofDependence Inthesecondpartofthischapter,wediscusstworecentlyproposedmeasuresthatcanbeusedtoinferdependencebetweentwotimeseries.Therststartsbygeneralizingtheconceptofassociationfortimeseriesandthesecondisbasedonsymbolicdynamics.Beforediscussingthesemeasures,weintroducesomepreliminaryconcepts. 3.2.1PreliminaryConcepts 3.2.1.1Dependence Renyi [ 1959 ]proposedsevenaxiomsapropermeasureofdependenceQpX;Yqshouldsatisfy,whereXandYrepresenttworandomvariables[ Prncipe 2010 ].AccordingtoRenyi,ifQpX;Yqisameasureofdependence,thenQpX;Yqis: 1. WelldenedoverXandY. 2. SymmetricorQpX;YqQpY;Xq1. 3. Boundedor0|QpX;Yq|1. 1Ameasureofdependencecanbeallowedtobeasymmetrictoinfercausalrelationshipsi.e.eectiveconnectivitiesamongdata. 67

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4. Zeroforindependentrandomvariables,i.e.QpX;Yq0iX&Yareindependent. 5. Oneifarandomvariableisafunctionoftheother,i.e.QpX;Yq1ifYfpXq. 6. InvariantorQpX;YqQpfpXq;gpYqqforanyfunctionsf;g. 7. QpX;YqpX;YqforX;YNp0;2q,wheredenotesthecorrelationcoecient. Unlikeotherapproachestodenedependence,thesepostulatesleadtothenotionof\functional"dependence. 3.2.1.2Association Therearetwomainstreammethodstoestimatedependencebetweentworandomvariables.Therststartsbyconstructingafunctionoftherandomvariablesundercertaincriteriathentriestobuildaconsistentestimatorforthatmeasure[ Kruskal 1958 ].Thesecondstartsfromthesamples,ndsawaytocaptureattributesfromthesesamples,thenformulatesameasureforwhichtheproposedmethodisanestimator[ Seth 2011 ].Workingwithsamplescircumventsthedicultytondasuitableandconsistentestimatorwhenstartingwiththerandomvariablesthemselves.Thetradeohereisbetweentwothings.Therstistohaveaclearandwellunderstoodconceptofdependenceforwhichitisdiculttondanestimatorsatisfyingthedierentproperties.Thesecondistostartfromtherealizationsanddenethemeasureinabuild-upfashion.Inmostpracticalsituations,weonlyhaveaccesstoanitenumberofrealizations,andbecauseofthislimitation,understandingthepropertiesofanestimatorcanbeseenasmoreimportantthanthemeasure.Theconceptofassociationprovidesaclearunderstandingofdependencebothinthecontextofrandomvariablesandrealizationssinceittreatsrealizationsasaprobabilitylawbyassigningequalmassesovertherealizationvalues[ Whitt 1976 ].Howevertheconceptofassociationisonlyrestrictedtorandomvariablesassumingvaluesontherealline.ExtendingtheconceptofassociationfromRtometricspaceshasbeenproposedby Seth [ 2011 ]sincesuchspacesareoftenencounteredinpracticalproblems. 68

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3.2.1.3Generalizedmeasureofassociation Motivation:Asdiscussedinthebeginningofthischapter,both Spearman [ 1904 ]and Kendall [ 1938 ]generalizedtheconceptofcorrelationtocapturemonotoneinsteadofjustlineardependence.Tosatisfythedesiredpropertiesofanestimatorofdependence,theconceptofassociationisextendedfromRtometricspaces.Inspiredbytheideaofassociation,andgivenrealizationsptxi;yiuni1qfromapairofrandomvariablespX;Yq,YcanbesaidtobeassociatedtoXifcloserealizationpairsofY(ortyi;yju)areassociatedwithcloserealizationpairsofX(ortxi;xju).Closenessinthiscontextisdenedintermsofthedistancemetricsofthespaceswheretherealizationsliein(XandY).Suchapproachtodependenceisparticularlyappealingsinceitisintuitive,anddoesnotrequirethedomainoftherandomvariablestohaveanorder.Inaddition,itallowsrealizationstoexistinmoreabstractdomainssuchasvectorsorpointprocesses[ Sethetal. 2010 ],andtherefore,denesdependenceirrespectiveofthenatureofthesedomains. Thecorrespondingestimatorisreferredtobythetermgeneralizedmeasureofassoci-ation(orGMAforshort).GMAgeneralizestheconceptofassociationbyconsideringthedistancebetweenrealizationsratherthantheirabsolutelocations,retainscertaindesiredinvariancepropertiesofameasureofassociation,andonlyrequiresthattheybedenedinametricspace.Ithastheadvantageofbeingaparameter-freemethod.Fromatimecomplexityperspective,itrunsinOpn2qintheworstcasescenario.Inthiswork,GMAisappliedonrealvaluedsequences.AdetailedformulationofGMAisprovidedinthenextsection. Formulation:Nowthattheconceptisroughlydetermined,thenextstepwouldbetodesignanestimatorthatsatisesthedesiredproperties.Thiscanbedonebylettingtheestimatoronlyconsidertherelativepositionsoftherealizationswithrespecttoeachotherratherthantheirabsolutelocations.Again,thisisinspiredbytheapproachconsideredbyKendallorSpearmancomparedtotheapproachconsideredbyPearson.We 69

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rstconsiderasimpleillustrationofthatbyperformingthefollowingtwostepsforallindexesiPt1;:::;nu: 1. FindxjclosesttoxiintermsofX. 2. FindtherankriofyjintermsofY Theranksriinthesecondstepcanbeobtainedviari#tj:ji;Ypyj;yiqYpyj;yiqu.ThenotationZdenotestheassociatedmetricofthespaceZ.Thenextstepwouldbetoconsiderthecomputedri'sasrealizationsofarandomvariablewerefertoastherankvariableR.UnderindependenceofXandY,Rshouldhaveauniformdistribution.Ontheotherextreme,ifthetwovariableshaveaperfectlinearrelationship,thePDFofRwillassignaprobabilityof1whenri1.Inanintermediatescenario,theri'swouldbeskewedtowardsalowervaluetowards1.HencewecansaythatthemoretheskewnessofthedistributionofR,themorerandomvariablesXandYaredependent.ThenalstepindependenceestimationiscapturingtheskewnessofRbysimplycomputingtheareaunderitsCDFandnormalizingbypn1q.ThismeasureiscalledgeneralizedmeasureofassociationorGMA[ Seth 2011 ]andthevaluesitestimatesrangefromnear0:5forindependentrandomvariablesand1fordependentones.Thesetofpointsxjclosesttoxiintermsoftheassociatedmetricxcanbedenedasfollows: jargminjixpxi;xjq(3{25) Algorithm 2 describesindetailthestepsinvolvedincomputingGMAbetweentwotimeseries.InAlgorithm 2 ,thespreadoftheranksiscomputedtoaddresscaseswheretwoormorerealizationssharethesamedistancefromathirdone.Uponcompletionofthealgorithm,PpRrq1andhencecorrespondstoavalidPDF.AnalternativewayofformulatingGMAwouldbe: GMA1 n1n1r1pnrqPpRrq;(3{26) 70

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Algorithm2:GeneralizedMeasureofAssociation(GMA) Input: Realizationspairstxi;yiuni1assumingvaluesinthejointspaceXY Output: EstimateddependencedPr0:5:1s 1 begin 2Initialization:AssignPpRrq0@rPt1;:::;pn1qu 3foriPt1:::nudo 4-Findxj(jPJ)closesttoxi,equivalentlyjargminjixpxi;xjq, 5wherexdenotestheassociatedmetricofspaceX. 6-ForalljPJ,ndthespreadofranks,i.e.ri;maxandri;minofyj 7intermsofysuchthat: 8ri;max#tj:ji;ypyj;yiqypyj;yiqu 9ri;min#tj:ji;ypyj;yiqypyj;yiqu 10-Forallrankvaluesri;minrri;max,assign: 11PpRrqPpRrq+1{|J|{pri;maxri;minq{n 12end 13-ComputeCastheempiricalCDFoftr1;:::;rnu. 14-ReturndastheareaunderCnormalizedbypn1q 15end wherePpRrqisdenedasPpRrq#ti:riru{n;andrepresentstheempiricalprobabilityoftherankvariable.GMAsatisesthepropertiesofameasureofdependencei.e.itisupperandlowerbounded,invariantunderageneralsetoftransformations(likerotationandscaling)andcanbeasymmetric.GMAassumesvaluesbetween0:5and1.Beingparameter-free,itenjoysauniquecomputationaladvantageoverotherapproaches.AsimpleillustrationofGMAovertherealizationsoftworandomvariablescanbeshowninFigure 3-1 3.2.1.4Timeseriescomplexityandpermutationentropy Motivation:Allofthepreviouslymentionedmeasuresofdependencelookateithertherealizationsoftherandomvariablesortheirrankstomeasuretheinvolveddependence.Thisapproachcanbesometimesvulnerabletonoiseandoutliers.Moreover,itispronetoconfuserealdependencebetweenthetimeserieswithtemporaldependenceresidingineach.Weproposetocomputedependencebasedonderivedfunctionsthatmeasuretimeseriescomplexity.Theadditionofanintermediate\layer"priorto 71

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Figure3-1. SimpleillustrationofGMA.SupposeXr1;4;5sandYr1;7;3s.Redarrowsareusedtoshowtheclosestpointwithrespecttothex-axisandbluearrowsareusedtoshowtheclosestpointwithrespecttothey-axis.HerePpRank1q1{3andPpRank2q2{3.henceGMA=NormalizedAreaunderCDF=p1{3)]TJ /F15 9.963 Tf 9.96 0 Td[(1q{20:67. dependencecomputationmightnotseemusefulatrst,howeverworkingwithfunctionsthatencompasscomplexityinformationabouttheoriginaltimeseriesisappealingsinceithelpsovercomesensitivitytonoiseandoutliersaswillbeexplainedinthefollowingsections.Thisapproachentailsatradeobetweenaddingafreeparameter(ascomparedtomostinformationtheoreticalestimators)androbustnesstonoise/sensitivenesstoabruptchanges.Oneparticularaspectthatneedsscrutinyhereistoassesshowmuchinformationaboutthedataispreservedusingthesecomplexitymeasures.Inthecaseofpermutationentropyforinstance(willbedescribedindetailshortly),thecomplexityvalueisestimatedastheentropyderivedfromtheprobabilitydistributionofextractedordinalpatterns.Althoughsuchapproachisabletodescribethenonlineartrendsintimeseries,thelostamplitudeinformationwhencollectingordinalpatternshindersthereconstructionoftheinitialdatadistribution.Itisactuallyhardtothinkofanypreprocessingfunctionthatmapstheinputdataintoasymbolicspacethatpermitstheeasyextractionofcomplexity 72

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quantierswhileallowinganexactretrievaloftheinitialdataproperties.WehenceproceedaccordingtoamethodthatretainsamplitudewhilestillexploitingtheadvantageofusingordinalpatternstoquantifythenonlinearbehavioroftimeseriesandrefertothisapproachasWeighted-PermutationEntropy(WPE) Timeseriescomplexity:Thereislittleconsensusonthedenitionofasignal'scomplexity.Amongthedierentapproaches,entropy-basedonesareinspiredbyeithernonlineardynamics[ Pincus 1991 ]orsymbolicdynamics[ BandtandPompe 2002 ; Kurthsetal. 1996 ].Permutationentropy(PE)hasbeenrecentlysuggestedasacomplexitymeasurebasedoncomparingneighboringvaluesofeachpointandmappingthemtoordinalpatterns[ BandtandPompe 2002 ].Usingordinaldescriptorsishelpfulinthesensethatitaddsimmunitytolargeartifactsoccurringwithlowfrequencies.PEisapplicableforregular,chaotic,noisyorreal-worldtimeseriesandhasbeenemployedinthecontextofneural[ Lietal. ],electroencephalographic(EEG)[ Bruzzoetal. 2008 ; Caoetal. 2004 ; Lietal. 2008 ; 2007 ],electrocardiographic(ECG)[ GraandKaczkowska 2012 ; Zhangetal. 2008 ]andstockmarkettimeseries[ Zuninoetal. 2009 ]. Permutationentropy:ConsiderthetimeseriestxtuTt1anditstime-delayembeddingrepresentationxm;jtxj;xj)]TJ /F5 7.97 Tf 6.58 0 Td[(;:::;xjpm1quforj1;2;:::;Tpm1q,wheremanddenoterespectivelytheembeddingdimensionandtimedelay.TocomputePE,eachoftheNTpm1qsubvectorsisassignedasinglemotifoutofm!possibleones(representingalluniqueorderingsofmdierentrealnumbers).PEisthendenedastheShannonentropyofthem!distinctsymbolstm;ium!i1,denotedas: Hpm;qi:m;iPppm;iqlnppm;iq(3{27)ppm;iqisdenedas: ppm;iq}tj:jN;typepxm;jqm;iu} N(3{28) 73

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wheretypep:qdenotesthemapfrompatternspacetosymbolspaceand}:}denotesthecardinalityofaset.Analternativewayofwritingppm;iqis: ppm;iqjN1u:typepuqipxm;jq jN1u:typepuqPpxm;jq(3{29) where1ApuqdenotestheindicatorfunctionofsetAdenedas1Apuq1ifuPAand1Apuq0ifuRA.PEassumesvaluesbetweenintheranger0;lnm!sandisinvariantundernonlinearmonotonictransformations. 3.2.2GeneralizedAssociationforTimeSeries 3.2.2.1Motivation Typically,whenconsideringrealizationsfromtwotimeseries,thenearestneighborinamplitudeforagivenpointissimplythenearestintime,howeverthisdoesnotrevealdependencestructure.Toovercomethisobstacle,weproposetomodifytheGMAroutinebydecreasingtheeectoftemporalstructureintheinputtimeseries.Again,theleadingincentivebehindthisisthatapairofrealizationsfromeachtimeserieswillmostprobablybeveryclosetothepair(s)correspondingtotheclosestintime. 3.2.2.2Proposedalgorithm Foreachrealizationinthetimeseries,wedismisstherealizationswithinaneighboringwindowtodiscarddependencepertainingtotimestructure.Onlypointsfallingoutsidethatwindowwouldbeaccountedfor.Thelattermightincludepointsrevisitedbythetimeseriesastimepasses.Thechoiceofthewindowlengthisnotastraightforwardtask.Wesuggesttouseawindowsizeintrinsictotheinputdomainanddeterminedbythezero-crossingoftheautocorrelationfunction(ACF)foreachinputtimeseries.Ifnosuchcrossingexists,wechoosealagcorrespondingtotherstminimumoftheACF,orits1{edecayifitdoesnotachieveone.Theaimistodecreasethecorrelationovertimeasmuchaspossibleandhenceavoidmisinterpretinghighintrinsicassociationwithineachtimeseriesforhighvaluesofinterdependence.Thischoiceworkswellinourcontextalthough 74

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otherchoicesarepossible.Theadvantagebroughtbysuchsettingistokeepthemethodparameter-free.TheupdatedalgorithmisoutlinedinAlgorithm 3 andFigure 3-2 showsanillustrativeexample: Algorithm3:TimeSeriesGeneralizedMeasureofAssociation(TGMA) Input: Bivariatetimeseriestxt;ytunt1assumingvaluesinthejointspaceXY Output: EstimateddependencedPr0:5:1s begin Initialization:-AssignPpRrq0@rPt1;:::;pn1qu-LetxandydenotetheACFsofxandy,andlet0lx;lyn.-Setlxasfollows:lx$&%lagatrstzero-crossingofxifxhasazero-crossing.lagatrstminimumofxifxhasnozero-crossing.lagat1{edecayofxifxhasnozero-crossing&ismonot.-Setthevalueoflyinasimilarfashionwithrespecttoy.foriPt1:::nudo a.Findxj(jPJ),wherejsatises: jargmin|ji|maxplx;lyqxpxi;xjqb.ForalljPJ,ndthespreadofranks,i.e.ri;maxandri;minofyjintermsofysuchthat:ri;max#tj:ji;ypyj;yiqypyj;yiquri;min#tj:ji;ypyj;yiqypyj;yiquc.Forallrankvaluesri;minrri;max,assign:PpRrqPpRrq+1{|J|{pri;maxri;minq{nend-ComputeCastheempiricalCDFoftr1;:::;rnu.-ReturndastheareaunderCnormalizedbypn1qend AthoroughdiscussionofTGMAwithinacomparativeapproachtoKendall'stauappliedoncopulagenerateddataandrealdatacanbefoundin[ Fadlallahetal. 2012a ]. 75

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Figure3-2. GMAvs.TGMAandeectoftemporalstructure.Top:TwosignalsXandYandtheirjointscatterplot.GMAandTGMAgivecloseassociationestimates.Middle:DataismodulatedusingacarriersignalhenceinducingsignicanttemporalstructuretowardswhichTGMAshowslesssensitivity.Bottom:Sameprocedurefortwodierentsignals. 3.2.3Weighted-PermutationEntropyBasedDependence 3.2.3.1Motivation ThemainshortcominginthedenitionofPEprovidedinSection 3.2.1.4 residesinthefactthatnoinformationbesidestheorderstructureisretainedwhenextractingtheordinalpatternsforeachtimeseries.Thismaybeinconvenientforthefollowingreasons:(i)mosttimeserieshaveinformationintheamplitudethatmightbelostwhensolelyextractingtheordinalstructure(ii)ordinalpatternswheretheamplitudedierencesbetweenthetimeseriespointsaregreaterthanothersshouldnotcontributesimilarlytothenalPEvalueand(iii)ordinalpatternsresultingfromsmalluctuationsinthetimeseriescanbeduetotheeectofnoiseandshouldnotbeweighteduniformlytowardsthenalvalueofPE.Figure 3-3 showshowthesameordinalpatterncanoriginatefromdierentmdimensionalvectors. WehencesuggestamodicationthataltersthewayPEhandlesthepatternsextractedfromagivensignal.Thenewschemebettertracksabruptchangesinthe 76

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Figure3-3. Twoexamplesofpossiblem-dimensionalvectorscorrespondingtothesamemotif.Thevalueofmusedis3. signalandassignslesscomplexitytosegmentsthatexhibitregularityoraresubjecttonoiseeects. 3.2.3.2Weighted-permutationentropy Tocounterweighttheaforementionedshortcomings,weextendthecurrentPEproceduretoincorporatesignicantinformationfromthetimeserieswhenretrievingtheordinalpatterns.Themainmotivationliesinsavingusefulamplitudeinformationcarriedbythesignal.Werefertothisprocedureasweighted-permutationentropy(WPE)andsummarizeitinthefollowingsteps.First,theweightedrelativefrequenciesforeachmotifarecalculatedasfollows: pwpm;iqjN1u:typepuqipxm;jq:wj jN1u:typepuqPpxm;jq:wj(3{30) WPEisthencomputedas: Hwpm;qi:m;iPpwpm;iqlnpwpm;iq(3{31) 77

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Notethatwhenwj@jNand0,WPEreducestoPE.Itisalsointerestingtohighlightthedierencebetweenthedenitionofweightedentropyinthiscontextandpreviousonessuggestedintheliterature.Weightedentropy,denedasHwekwkpklnpk,hasbeensuggestedasavarianttoentropythatusesaprobabilisticexperimentwhoseelementaryeventsarecharacterizedbyweightswk[ Guiasu 1971 ].WPEontheotherhand,extendstheconceptofPEwhilekeepingthesameShannon'sentropyexpressionreectedbyEquation 3{31 ,henceweightsareaddedpriortocomputingtheppm;iq.Thechoiceofweightvalueswiisequivalenttoselectingaspecic(orcombinationof)feature(s)fromeachvectorxm;j.Suchfeaturesmaydieraccordingtothecontextused.Notethattherelationipwpm;iq1stillholds.Inthismanuscript,weusethevarianceorenergyofeachneighborsvectorxm;jtocomputetheweights.Letxm;jdenotethearithmeticmeanofxm;jor: xm;j1 mmk1xjpk)]TJ /F8 7.97 Tf 6.59 0 Td[(1q(3{32) Wecanhenceexpresseachweightvaluesas: wj1 mmk1)]TJ /F3 11.955 Tf 5.48 -9.81 Td[(xjpk1qxm;j2(3{33) Themotivationbehindthissettingistospecicallycounteractthelimitationsdiscussedintheprevioussection,i.e.weightdierentlyneighboringvectorshavingthesameordinalpatternsbutdierentamplitudevariations.Inthisway,WPEcanbealsousedtodetectabruptchangesinnoisyormulticomponentsignals.Themodiedppiqcanbethenthoughtofastheproportionofvarianceaccountedforbyeachmotif.TheabovedenitionofWPEretainsmostofPE'spropertiesandisinvariantunderanelineartransformations.WPEhoweverpresentsaspecicity,givenitincorporatesamplitudeinformationanddemonstratesmorerobustnesstonoise.AmorethoroughdiscussionofWPEcanbefoundin[ Fadlallahetal. 2013a ]. 78

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3.2.3.3WPE-baseddependence Generallyspeaking,everytimeseriesisassignedascalarvalueforPEandWPE.Weproposetoslideawindowoneachtimeseriestogenerate\complexity"curveswhoselengthisdeterminedaccordingtothechosenwindowlength.Oncewehavethesecurves,wecanapplyanymeasureofassociationsuchthatSpearman'srho,Kendall'stauorGMAtocapturethemonotonicityofthedependencebetweenthetwocurves.MoreelaborationontheuseofWPEasanovelcomplexitymeasurefortimeseries,thatincorporatesamplitudeinformationcanbefoundinAppendix D 79

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CHAPTER4THEDYNAMICDEPENDENCEGRAPH Graphtheoreticalmethodshavebeenalreadyproposedasatooltoanalyzestructural,functionalandeectivebrainconnectivity[ BullmoreandSporns 2009 ],whereverticescorrespondtobrainregionsorneurons,andedgesrepresentsynapsesorpathsofpronouncedstatisticalassociationbetweenneuralelements.Dependencegraphshaveprovedtheirusefulnessindescribingdependencerelationsbetweenrandomvariablesandtimeseries[ Lauritzen 1996 ].Inthischapter,weshowhowtousegraphtheoreticalconceptstotackletheproblemformulatedinSection 1.1.6 4.1GraphModel WemodeltheelectrodesnetworkasacompleteundirectedgraphGpV;Eq,whereVisthesetofverticesandEisthesetofedges[ Fadlallahetal. 2013b ].Foreachedgeeijbetweentwoverticesiandj,weassignavaluemijrepresentingthedependencebetweentheprocessedsignalsrecordedattheelectrodeslocations.Notethatforcorrelation,mijexistsintheintervalr1;1s,whereasforGMAitliesintheintervalr0:5;1s.Weassumethatthegraphisundirectedsinceourgoalistoquantifydependenciesbetweenbrainregions,whichdoesnotaccountfordirection.Torelaxthisassumption,measuresofcausalityandasubsequentdirectedgraphcanbeconsideredforassessingeectiveconnectivity.ThegraphGisfurthertransformedintoanincompletegraphbydiscardingvaluesfallingbelowapredenedstatisticalthreshold(moredetailsinSection 4.3 ).Asaresult,andafteraveragingovertrials,anNcNcNwadjacencymatrixMcdisgeneratedforeachdependencemeasuredandconditionc,whereNwreferstothenumberofframesortimewindows. Choosingthedependencemeasureiscrucialforourapproach.Correlationcanbeusedinthisregard,butsinceitonlycapturessecondorderinteractions,itperformsratherpoorlyfortimeseriesfeaturinghigherorderinteractions.Athoroughdiscussionofpossible 80

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measuresofdependence(orcausality)thatcanbeusedtoweightthegraphedgescanbefoundinChapter 3 4.2GraphTheoreticalConcepts 4.2.1BasicNotations TheneighborhoodofavertexvPVisthesetNvofallvertices(orneighbors)connectedtov,i.eNvtr:ervPEormrv0u.Thedegreeofvconsistsofthenumberofverticesthatareincidenttov,i.e.nv|Nv|,where|:|denotesthecardinalityofaset.Apathfromvertexrtovertexsisasequenceofverticesandedgesthatbeginswithrandendswiths,withanedgeconnectingeachvertexwiththesucceedingone.Thedistancedr;sbetweenrandsistheminimumlengthofanypathconnectingthetwovertices.Weherebypresentsomegraphtheoreticalmeasuresofinterestforouranalysis. 4.2.2NodeClusteringCoecient Theclusteringcoecientisafrequentlyusedmeasuretocharacterizethelocalandglobalstructureofunweightedgraphs[ Boccalettietal. 2006 ].Theclusteringcoecientmeasurestheextenttowhichnodesinagraphtendtoclustertogether.Itisdenedforanodeiinthegraphas: CCi|tejk:vj;vkPNi;ejkPEu| nipni1q(4{1) whereni|Ni|.Theclusteringcoecientassumesvaluesbetween0and1.Acommonpracticeistoaveragetheclusteringcoecientsofallnodesinthegraph: CC1 |V||V|i1Ci(4{2) Theclusteringcoecientofavertexcanbeinterpretedintermsofitstendencytopromoteconnectionsamongitsneighborhood,andcanthereforebeconsideredasanindicatorofinformationowindynamicnetworks[ Lopez-Fernandezetal. 2006 ]. 81

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4.2.3MeasuresofCentrality Avertex'simportanceinagraphcanbequantiedusingseveralmeasures,includingmeasuresofcentrality. BetweennessCentrality:Thebetweennesscentralityofaverteximeasuresthenumberofshortestpathstraversingthatvertex.Itwasrstproposedby Anthonisse [ 1971 ]andhasbeenlaterusedinseveralcontexts[ Barthelemy 2004 ; Freeman 1977 ; Gohetal. 2003 ].Itcanbedenedas: BijikNo.shortestpathsfromjtokviai No.shortestpathsfromjtok(4{3) Alternatively: Bijikijk jk(4{4) where,inEquation 4{4 ,ijkdenotesthenumberofshortestpathsfromjtokincludingi.Whenthebetweennesscentralityofavertexishigh,thevertexismorelikelytobeanintermediatecommunicationnodeinthegraph.Suchverticescanbeseenasoccupyingthe\structuralholes"inthenetwork[ Burt 1995 ; Lopez-Fernandezetal. 2006 ]. SubgraphCentrality:Thesubgraphcentralityofavertexicanbedenedastheweightedsumofclosedwalkshavingdierentlengths,startingandendingati. Si8k0ki k!(4{5) whereinEquation 4{5 ,kireferstothekthlocalspectralmoment,whichdenesthenumberofclosedwalksoflengthk,startingandendingoni.kiiscomputedusingtheithdiagonalentryofthekthpowerofthegraphadjacencymatrix. ki)]TJ /F7 11.955 Tf 5.48 -9.8 Td[(Mdkii(4{6) 82

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Dueofspacelimitations,thecomputationaldetailspertainingtosubgraphcentralityareomittedandhavebeenthoroughlydiscussedby EstradaandRodriguez-Velazquez [ 2005 ]. ClosenessCentrality:Theclosenesscentralityofavertexiisdenedastheinverseofthesumofdistancesbetweeniandallothervertices. Ci1 jPVdi;j(4{7) 4.2.4LocalEciency ThelocaleciencyofavertexicanbedenedintermsofthesumofinversedistancesbetweentheverticesinNi.Thelocaleciencymainlymeasurestheeciencyincommunicationbetweenthedirectneighborsofi,whenthenodeitselfisremoved. Eloci1 nipni1qj;kPNi1 dj;k(4{8) 4.2.5ConnectedComponents Aconnectedcomponentofanundirectedgraphisamaximalsubgraphinwhichanytwoverticesareconnectedtoeachotherbypaths.TheconceptisgraphicallyillustratedinFigure 4-1 Figure4-1. Anundirectedgraphconsistingofthreeconnectedcomponents. 83

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4.2.6GraphVisualization Visualizingthepairwisevaluesofdependencefor129or257sensorscanbeachallengingtask.Toincreasetheuserfriendlinessofthegraphdisplays,wemaptheEEGsensorstothecircumferenceofacircle,inawaythatpreservesthemainstructuresofthescalpgeography.ThisisillustratedinFigure 4-2 Figure4-2. Thestandard10-20conguration(RefertoFigures 2-5 and 2-6 )isusedtomaptheEEGsensorstothecircumferenceofacircle.Thecirclestillretainsthemainstructuresofthescalpgeographyinthesensethatlowerverticescorrespondtotheoccipitalareas,whereasthehigheronescorrespondtofrontalareas. 4.3WeightsComputations Foreachmeasureofdependence,wedeneMdasthedierencebetweenmatricespertainingtoeachcondition,i.e.MdMFdMGd.Itisthenpossibletovisualizehowpairwisedependencesvaryinsensorspaceacrosstime.Theresultingmatricesarerst 84

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processedforstatisticalsignicancebydiscardingallvaluesfallingwithin2standarddeviationsoftheirmeans. 4.3.1Time-DelayEmbedding Priortocomputingsomemeasuresofdependence(likemutualinformationorGMA),weperformatime-delayembeddingforeachtimeseriesxpi;kqandypi;kq.Themotivationbehindthisistoworkonareconstructedstate-spaceversionofthedynamicalsystemthatwouldaccountforpropagationdelays,especiallybetweenneighboringchannels.Wefurthersuggestamethodbasedonsyntheticdatatoanalyzetheembeddingparameterswhenusingthesemeasures(Section 4.3.2 ).Again,foreachtimeseries,wedeneanembeddingpi;kqjpm;qas: pi;kqjpm;qxpi;kqj;xpi;kqj)]TJ /F5 7.97 Tf 6.59 0 Td[(;:::;xpi;kqjpm1q(4{9) forj1;2;:::;pNsNrqpm1q,whereintheaboveequation,andmdenotethetimedelayandembeddingdimensionandpi;kqjpm;qthejthreconstructedvectorwithtimedelayandembeddingdimensionm.Similarembeddingisperformedforeachypi;kq. Inourcomputations,weusedtimewindowsof114mscorrespondingto114samplesgivend1andFs1000Hz.TherstdurationcorrespondstotwocyclesofasinusoidwithfrequencyFoandcoversroughlythepropagationtimefromtheoccipitaltothefrontalcortices.Timeserieswereembeddedin8dimensionstoaccountforthepropagationdelayamongneighboringchannelswhencomputingcertainmeasuresofdependencesuchasmutualinformation. 4.3.2ImpactofFreeParameters 4.3.2.1Embeddingdimension Embeddingintimeisacrucialstepbecauseofthelimitationofthenumberofsamples.Thechosenvaluefortheembeddingdimensionshouldnotbetoohighforcomputationalconstraintsandnottoolowbecausethisunderminesthegoalofcapturingthedelayeddependence.Toestimate,simulationswereperformedontwosynthetically 85

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generatednoisytimeseries.Eachconsistedofatwo-periodsinusoidoscillatingatafrequencyFo,correspondingto114samples,withadditivewhitegaussiannoise(AWGN)inthephaseandamplitude.Oneofthetimeserieswasshiftedby25sampleswithrespecttotheotherandourgoalwastoestimatewhichembeddingdimension(ortimelagforcorrelation)achievesalevelofdependenceequaltotheonerecordedbetweenthetwonon-shiftedtimeseries.Specically,s1ptqS1sinp2Fot)]TJ /F2 11.955 Tf 12.83 0 Td[(Np1;21qq)]TJ /F2 11.955 Tf 25.65 0 Td[(Np2;22qands2ptqS2sinp2Fopt)]TJ /F3 11.955 Tf 12.98 0 Td[(s1s2q)]TJ /F2 11.955 Tf 21.41 0 Td[(Np1;21qq)]TJ /F2 11.955 Tf 25.96 0 Td[(Np2;22qwheres1s225,t1{For1;:::;p114)]TJ /F3 11.955 Tf 12.35 0 Td[(s1s2qs.S1;S2werechosentobe5and3and1;2;1;2wererespectivelysetto0:1,0:15,0:3and0:4.Figure 4-3 (candd)showsthatforGMAandMI,anembeddingdimensioncloseto8achievesthedesiredlevelofdependence.Besides,asanticipated,atimelagof25sampleswasneededtomaximizethecorrelationbetweenthetwotimeseries.Hence,tomaximizecorrelationwhencomputingdependenceonthelteredtimeseries,weiterateoverlagscorrespondingtooneperiodofthesignalor54samplesandwhencomputingabsolutecorrelation,wesimplifytheproblembylookingatlagsbetween1and29samples(theequivalentofahalf-periodaccountingforoutofphaseoscillations). 4.3.2.2Timedelay Weusethesamesimulationsettingdescribedabovetoanalyzetheimpactofthetimedelay.Valuesforrangingbetween1and5wereutilizedtoembedthetwotimeseries.Figure 4-3 (e)showsthattheimpactofisnotasimportantasthatoftheembeddingdimensionsincethedependencecurvesdonotchangemuchwhenincreasingthetimedelayvalue. 4.3.2.3Timewindow Timewindowsof114and456sampleswereusedforcomputingdependencevalues.Whiletherstoneachievesahighertimeresolution,thesecondhasmoresamplesandyieldsamorereliableestimatorofthejointdensity. 86

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4.3.2.4Timecomplexity ThetimecomplexityforthemutualinformationestimatorisOpn2qassumingastraightforwardneighboursearchimplementation.GMAalsorunsinOpn2qsincearankcomputationstepisperformedforeachpointofthetimeseries.Inoursimulations,GMAwasnoticeablyfasterthanmutualinformationbutobviouslybothperformedslowerthanPearson'scorrelation. Figure4-3. (a)Syntheticdata:twonoisytimeseriess1ptqands2ptqoscillatingatthesamefrequencywithdierentamplitudes.s2ptqhasbeendelayedby25samples.(b)Scatterplotofthetwotimeseries.(c)MIvaluefordierentembeddingdimensions.RedlinecorrespondstotheMIvaluecomputedbetweennon-shiftedversionsofthetimeseries.(d)GMAandcorrelationvaluesfordierentembeddingdimensionsorlags.Blue(red)linecorrespondstoexpectedGMA(correlation)value.(e)PlottingGMAvaluesfordierenttimedelaysdoesnotaltertheshapeofthecurve. 87

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CHAPTER5INFORMATIONTHEORETICCLASSIFICATIONOFDEPENDENCEMATRICES Inthischapter,wediscusshowinformationtheoreticalconceptscanbeusedtoclassifycognitiveconditionsfromdependencematrices.Werstdescribethestepsinvolvedinestimatingentropy-likequantitiesusingpositivedeniteandinnitelydivisiblematrices(RefertoAppendix E ).Wethenproceedtocreatinganullhypothesisofdistributionequalitybasedontheestimatedentropyvalues.Thisnullhypothesiscanbeusedinclassicationcontexttodeterminewhetheragiventrialbelongsornottoaspeciccondition,usingamaximumlikelihood(ML)ormaximumaposteoriprobability(MAP)approach. 5.1PreliminaryConcepts Usinginformationtheoreticquantitiesasteststatisticsisachallengingproblem.OperationalquantitiesinInformationTheoryaretypicallydenedintermsofprobabilitylawsunderlyingthedata,whichrepresentsaseriouslimitationsincethelatterarerarely,ifever,knowninmostreal-worldstatisticallearningsettings.Insuchscenarios,theonlysourceofinformationiswhatcanbederivedfromtheobservedsamples;thereisaneedfornovelmethodsthatcanbeemployedasawayofbypassingthedensityestimationproblem. Acurrenteortinourlabfocusesonderivingentropy-likequantitiesdirectlyfromempiricaldata[ Sanchez-GiraldoandPrincipe 2013 ].Thisderivationmethodusespositivedenitematricesassuitabledescriptorsofdataanddoesnotassumethatprobabilitiesofeventsareknownorhavebeenestimated.Assuch,thedesignedfunctionalusestherepresentationalpowerofpositivedenitematricesandistypicallycomputedusingGrammatricesconstructedfrompairwisecomputationsbetweendatasamples.AquantitysimilartomutualinformationcanthenbederivedusingtheaxiomaticcharacterizationofRenyi'sentropyandtheHadamardproductofmatrices. 88

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Inthischapter,weproposeusingtheseinformationtheoreticestimatorsinthecontextofcognitivetaskanalysis.Dependencematricesarerstconstructedfromcollectedandprocessedelectroencephalographic(EEG)data.Wethenusematrixentropyestimatorstoobtainscalardescriptorsofthesematricesandbasedonthesequantities,estimatemutualinformationbetweenanytwomatrices,inordertodeterminewhethertheybelongtothesameordistinctcognitiveconditions. Therestofthechapterisorganizedasfollows.InSection 5.2 ,webrieydiscussthemathematicalfoundationsoftheinformationtheoreticframework.Section 5.3 describesthepreprocessingdoneonthedependencematricesandinSection 5.4 ,wedescribethecomputationalstepsinvolvedincomputingnetworkspecicquantitiesfromthederiveddependencematrices.Sections 5.5 and 5.6 provideanoverviewoftheresultsandformulatetheproblemasaclassicationproblem,andSection 5.7 oersdiscussionandconcludingremarks. 5.2EstimatingEntropy-LikeQuantitieswithPositiveDeniteMatrices 5.2.1MatrixEntropyEstimator LetMndenotethealgebraofnnmatricesoverrealnumbers.Considertheset)]TJ /F5 7.97 Tf 0 -7.3 Td[(nofpositivedenitematricesinMnforwhichtrpAq1,andletAP)]TJ /F5 7.97 Tf 0 -7.3 Td[(nandBP)]TJ /F5 7.97 Tf 0 -7.3 Td[(n.WhentrpAqtrpBq1,thefunctional: SpAq1 1log2ptrpAqq;(5{1) satisesthefollowingsetofconditions(referto[ Sanchez-Giraldoetal. 2012 ]forproofs): 1. SpPAPqSpAqforanyorthonormalPPMn. 2. SppAqisacontinuousfunctionfor0p1. 3. Sp1 nIqlog2n(entropyisexhaustive). 4. SpAbBqSpAq)]TJ /F3 11.955 Tf 18.96 0 Td[(SpBq. 89

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5. IfABBA0,thenforthestrictlymonotonicandcontinuousfunctiongpxq2p1qxfor1and0,wehavethat: SptA)-222(p1tqBqg1ptgpSpAqq)]TJ -67.27 -26.89 Td[(p1tqgpSpBqqq:(5{2) ThecomputedSpAqcanbeseenasananalogousquantitytoRenyi'sentropy,althoughnoestimationofPDFswasundertaken.Thisispossiblebecausethereisanaxiomaticformulationofentropy,thereforethegoalofthisdevelopmentistocreateoperatorsinreproducingkernelHilbertspaces(RKHSs)thatdisplaythesamefunctionalproperties.Thenextnaturalstepwouldbethentodeneaquantitysimilartomutualinformation,andtoachievethat,atermthatrepresentsjointentropyneedstobederived. 5.2.2MatricesJointEntropyEstimatorusingHadamardProduct TheHadamardproductbetweenmatricespqcanbeusedtodeneaquantitythatisanalogoustojointentropyasfollows: SAB trpABq1 1log2trAB trpABq(5{3) UsingEquation 5{3 ,theexpressiondeningmutualinformationcanbethenwrittenas: IpA;BqSpAq)]TJ /F3 11.955 Tf 18.97 0 Td[(SpBqSAB trpABq;(5{4) whereinEquation 5{4 ,AandBarepositivesemidenitematriceswithnonnegativeentriesandunittracesuchthatAii1 nforalli1;:::;n.Noticethat,whendenedthisway,theabovequantityisnonnegativeandsatises:SpAqIpA;Aq: 5.3TheDependenceMatrices Intheremainderofthischapter,andformathematicalnotationbrevity,werefertothefaceconditionbythesuperscriptFandtheGaborpatchconditionbythesuperscript 90

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G.Threerank-basednonparametricmeasuresofstatisticaldependencewereusedinthecalculations.Thersttwoaretraditionalmeasuresofcorrelationincludedforthesakeofcomparison:Spearman'srhoandKendall'stau.Thethirdisanovelmeasureofassociationthatcancapturenonlinearinteractionswhileattenuatingtheeectoftemporaldependence:TimeSeriesGeneralizedMeasureofAssociation(TGMA).Typically,whenconsideringrealizationsfromtwotimeseries,thenearestneighborinamplitudeforagivenpointissimplythenearestintime.Importantly,thistemporalcontiguitydoesnotrevealthedependencestructureinthedata.TGMAtriestoaddressthisproblembyreducingthecontributionofneighborsthatareverycloseintime.Thealgorithm'scomputationalstepshavebeenalreadyoutlinedinChapter 3 andAlgorithm 3 andathoroughdiscussioncanbefoundin Fadlallahetal. [ 2012a ].AnimportantadvantageoftheTGMAapproachtoassociationthatisnotexploitedinthisdissertationisthatitallowscomputingdependencebetweenvariableswithdierentmetricspaces. WesymmetrizethevaluesofTGMAbyaveragingforwardandbackwarddependencemeasures.Thereforeforeachpairofchannelsiandj,weassignavaluemijrepresentingthedependencebetweentheprocessedsignalsrecordedattheelectrodeslocations.ForSpearman'srhoandKendall'stau,mijexistsintheintervalr0;1s,whereasforTGMAitliesintheintervalr0:5;1s.Asaresult,andafteraveragingovertrials,anNcNcNwdependencematrixMcd;kisgeneratedforeachdependencemeasured,trialkandconditionc,whereNc129orNc257dependingonthedataacquisitionsystemandNw39referstothenumberofframesortimewindowsoverwhichcomputationswerecarriedout. 5.4ComputationalSteps LetMFd;kandMGd;kbethedependencematricescorrespondingtothetwoconditionsF(neutralface)andG(Gaborpatch).Forsimplicity,withoutlossofgenerality,weaveragethematricesoverthetimewindowsdimension.Weperformtwosetsofcomputations.Intherst,weaveragethematricesoveralltrialsandthencomputetheircorrespondingIp:;:qvalueasdenedinEquation 5{4 .Inthesecond,weperformthe 91

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samecomputations,thistimeusingallpossiblecombinationsoftrialsandusingbothconditionstoserveasasurrogatestatistic.Computationsareperformedformatricesbelongingtothesameanddierentconditions.Thelogicunderlyingthisparadigmistoverifywhetherwedoobtaindierentdistributionswhenusingmatricesbelongingtodierentvisualperceptionconditions.Thekeystepsareoutlinedbelow: 1. LinearlyscaletheTGMAmatricestothesamerangeofKendall'stauandSpearman'srho.Whilethisstepisnotmandatory,itisstilldesirabletoprecludeanysystematicdierencesduetotherangeoftheinputdata.Atthesametime,italsoenablesperformanceevaluationacrossdierentmeasuresofdependence.Scalingthematricescanbedoneusing: McTGMA;k2McTGMA;k1: 2. Ensurepositivedenitenessofthematrices(bycheckingtheeigenvaluesorattemptingaCholeskydecomposition)andtheirinnitedivisibility.Tomotivatethisstep,itisusefultoremindthatthefunctionalSdenedinEquation 5{1 isappliedonGrammatricesconstructedfrompairwiseevaluationsofapositivedenitekernel.However,thematricesofdependencedonotnecessarilyrepresentGrammatricesandmightnotbeinnitelydivisible.Thisisbecausetheconditionni0nj0ijd2pxi;xjq0;foranyPRn)]TJ /F8 7.97 Tf 6.58 0 Td[(1withni0i0mightnotalwaysbemetwhendpxi;xjqisderivedfromtheassociationbetweenthetimeseriesxiandxj.NotethatTGMA(andmeasuresofabsolutecorrelation)canbeeasilyframedintheformofasadistancemeasureusing: McTGMA;k1McTGMA;k(5{5) 92

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ThedenitioninEquation 5{5 doesnotnecessarilysatisfythefourthcharacteristicofadistancefunction1,whichisthetriangleinequality,andasaresultdoesnotnecessarilyentailadistance.Therefore,wecheckthatexpprd2pxi;xjqqdenesapositivedenitematrixbecausematricesderivedfromsuchfunctionswouldbeinnitelydivisible(RefertoAppendix E ).Thiscanbeseenaswellfromadierentperspective.Proposition E.2.1 statesthatBlogAforaninnitelydivisiblematrixAandanegativedenitematrixB.SoAexpCforthepositivedenitematrixCB,andthisisinlinewiththecomputationofexpprd2pxi;xjqq. 3. Normalizethematricestracesto1,asperthesuggestionof Sanchez-GiraldoandPrincipe [ 2013 ].Thiscanbedoneusing: Mcd;kpi;jqMcd;kpi;jq b Mcd;kpi;iqb Mcd;kpj;jq: 4. Setthevalueofto1:01asperthesuggestionof Sanchez-GiraldoandPrincipe [ 2013 ].Subsequently,foreachmeasureofdependence,performthefollowingsetsofcomputations: [Usingaveragedtrials] (a) AveragematricesovertrialstoobtainMFd&MGd. (b) Computethefollowingquantity: QFGavgIpMFd;MGdq:(5{6) 1AfunctionddenedonXisadistanceifforx;y;zPX,thebelowconditionsaremet: (1)dpx;yq0(2)dpx;yq0ifxy(3)dpx;yqdpy;xq(4)dpx;zqdpx;yq)]TJ /F3 11.955 Tf 18.97 0 Td[(dpy;zq 93

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[Usingallthetrials] (a) Foralldierentpairsoftrialsmandnbelongingtothesamecondition(sayF),compute: QFm;nIpMFd;m;MFd;nq:(5{7) (b) Repeatthesameforpairsoftrialsbelongingtotwodierentconditions,andcompute QFGm;nIpMFd;m;MGd;nq:(5{8) 5. CheckwhichdependencemeasureachievestheleastvalueofQFGavg. 6. ComparethesurrogatedistributionsseparabilityofQFGandQFforeachmeasureofdependence. 5.5Results AsmentionedinSection 2.1 ,thedimensionalityofthedependencematricesvariedfrom129129to257257betweentherstandlastparticipants,whichmeansthateachdependencematrixconsistedof12964or32896entries.Atotalof40trialswereperformedperparticipant,equallydividedbetweenthetwoconditions(20trialsforFand20trialsforG).Figure 5-1 showstheobtainedmeanvaluesofQFGavg(asexpressedinEquation 5{6 )foreachofthethreemeasures,withthecorrespondingvarianceacrossparticipants. WethengeneratethedistributionofIpA;Bq,whenAandBcorrespondtotrialsfromthesamecondition,ordierentconditions.Inbothcases,allpossiblecombinationsoftrialsareconsidered.WhenAandBcorrespondtothesamecondition,thetotalnumberofpossiblecombinationsis2019380,becausethereisnoneedtocomputeIpA;Aq,andwhenAandBbelongtodierentconditions,thetotalnumberofpossiblecombinationsis2020400.Figure 5-2 showstheobtainedhistogramsofQFGm;nandQFm;n(asexpressedinEquations 5{7 and 5{8 )foreachofthethreemeasures.Theobtained 94

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Figure5-1. ComputedvaluesofQFGavgforTGMA,Kendall'stauandSpearman'srho.Foreachparticipantandcondition,thedependencematriceswereaveragedovertrialsandtimewindows.Theerrorbarscorrespondtoonestandarddeviationofthemeanacrossparticipants. distributions(Figure 5-2 )arecongruentwithourexpectations,sinceweanticipatedthattwomatricesbelongingtodierentconditionswouldachievealesserlevelofmutualinformation.Thisisexactlywhatweobserve.Thereasonwhyweinitiallyexpectedalowermutualinformationfordierentconditionsisthatmutualinformationcanbeinterpretedastheinformationgainfromassumingindependencetoknowingthejointdistribution.Fordierentconditions,nosubstantialinformationgainisachievedbetweenassumingindependenceandknowingthejointdistributionbecauseitiseasiertoexpressthelatterasaproductofthemarginals,ascomparedtoacaseinwhichthematricesbelongtothesamecondition,wheresuchanassumptionwouldbetotallymisleading. 95

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Figure5-2. DistributionsofQFGm;nandQFm;nforTGMA,Kendall'stauandSpearman'srho.Eachdistributioncorrespondstomutualinformationestimatesfromtrialsbelongingthesame(orseparate)conditions.380estimatesofmutualinformation(alluniquecombinationsof20facetrials)wereobtainedfortrialsbelongingtothesamecondition(greendistributions).400estimatesofmutualinformation(alldistinctcombinationsof20facetrialsand20Gaborpatchtrials)wereobtainedfortrialsbelongingtodierentconditions(reddistributions). 5.6Classication Asimilarapproachtotheonedescribedintheprevioussectioncanbeadoptedforsupervisedclassication.Wecantrainwithdatagatheredfrom5participantstogenerate 96

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distributionsofentropyestimatesforthetwoconditions,thenusethe6thparticipantfortesting.With40trialspersubject,thenumberoftrainingmatrixsamplesamountsto100pereachcondition.Fortherst5subjects,welabeleachdependencematrixwithitscondition(afteraveragingovertimewindows)andthereforeobtainonedistributionofentropyvaluespercondition.The20trialsbelongingtothe6thsubjectarethenassignedlabelsbasedonwheretheyfallinthedistributions.TheclassicationresultsusingthisapproachareshownintheResultsChapter(Chapter 6 ,Table 6-5 ). 5.7Conclusions Inthischapter,wedemonstratedthatitispossibletouseinformationtheoreticconceptstocharacterizedistinctvisualcognitivestatesbyusingdependencematricesextractedfromprocessedhumanEEGdata.Dependencebetweenpairwisechannelswascomputedwithtwotraditional(Kendall'stauandSpearman'srho)andonenovel(TGMA)rank-basedestimator.Entropy-likequantitieswerethenestimatedfromthedependencematrices.PositivedenitenessenablesdeninganaloguestoRenyi'smarginalandjointentropies,andthereforeprovidesameanstoestimatemutualinformationbetweenmatrices.Twosetsofcomputationswereperformed:intherst,weaveragedovertrialsandcalculatedthecorrespondingmutualinformationbetweenmatricesbelongingtotwodierentconditionsandinthesecond,weusedalltrialstogeneratedistributionsofconditionequalityandinequality.Computationswerecarriedoutusingacarefullyselectedvalueofthefreeparameter(1:01),asperthesuggestionof Sanchez-GiraldoandPrincipe [ 2013 ].Tryingothervaluesdidnotseemtosignicantlyaltertheresults. ResultsrevealedthatalowervalueofmutualinformationisachievedacrossparticipantsforTGMA.AllmethodsgeneratedseparabledistributionsbetweenthetwovisualperceptualconditionswithoverallhigherdiscriminativepowerobservedfortheTGMAalgorithm. 97

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CHAPTER6COMPUTATIONSANDRESULTS 6.1OverviewandComputationalSettings Thischapterdelvesintomoredetailsaboutthecomputationalproceduresandpresentsthemainresults.Theresultspresentationwillbeorganizedasfollows: 1. Resultswithrespecttoareferencechannel:Inthissection,dependencevaluescomputedwithrespecttoareferencechannelareprojectedontosensorspaceanddisplayedoverthescalp. 2. Resultsusingallthechannels:Inthissection,allpairwisevaluesofdependenceareevaluated,thenvisualizedinagraphthatevolveswithtime.Tofurtherunderstandthenetworkstructureinthegraphs,weextracttwotypesoffeaturesdescribingthegraphcharacteristics: (a) Localdescriptorsofthenodesusinggraphtheoreticalquantities.Theextractedfeaturesincludenodedegree,clusteringcoecient,betweennesscentrality,subgraphcentrality,andlocaleciency,andareagainprojectedontothescalp. (b) Globaldescriptorsofthegraphstructures.Estimatesofmatrixentropiesormutualinformationbetweendependencematricesbelongingthethesameordierentconditions,canbeusedtogeneratedistributionsthatarelaterusedforclassication. 3. Consistencyresults:Inthissection,wecomputeCronbach'salphaperconditionforalltheparticipants.Togetabetterideaaboutconsistencyacrosssubjects,Cronbach'salphaisalsocomputedafterconcatenatingtrialsfromallsubjects. 4. Performanceresults:Inthissection,weassessthemethodsperformanceusingtwoapproaches: (a) ApplyingtheKStestforallsubjects.Todoso,weaveragethematricesovertimewindowsandtrials,percondition. 98

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(b) Learningfromthedataandevaluatingtheclassicationperformance.Wetrainwith5subjects(100trialspercondition)andclassifythetrialspertainingtothe6thsubject(20trialspercondition). Dependenciesarecomputedpertrialforeverypairofchannels.Foreachdependencemeasureandcondition,anNcNcNtNwmatrixwasconstructed,whereNwdenotesthenumberoftimewindowsoverwhichdependenceiscomputed.Ford1andatimewindowlengthof114ms,Nw36.Exceptforcaseswhereweareinterestedinthetimeevolutionofdependence,wechoosetoaveragethematricesofdependenceoverthetimewindows.Mostresultswerevisualizedusingscalpprojections[ EMEGS 2011 ],ordependencegraphs.Thefollowingmeasuresofdependencewereusedtoweighttheedgesofthedependencegraph: 1. Pearson'scorrelationcoecient[ Pearson 1896 ] 2. MutualinformationusingakNNapproach[ Kraskovetal. 2004 ] 3. Spearman'srankcorrelationcoecient[ Spearman 1904 ] 4. Kendall'srankcorrelationcoecient[ Kendall 1938 ] 5. GeneralizedMeasureofAssociation(GMA)[ Seth 2011 ] 6. TimeSeriesGeneralizedMeasureofAssociation(TGMA)[ Fadlallahetal. 2012a ] 7. PermutationEntropy(PE)[ BandtandPompe 2002 ] 8. WeightedPermutationEntropy(WPE)[ Fadlallahetal. 2013a ] 9. PhaseSynchronyorPhaseLockingValue(PLV)[ Wanetal. 2013 ] Absolutecorrelationwastakenintoconsiderationtoachieveabettercomparisonwiththeotherdependencemeasuressincenegativecorrelationvalueswithhighmagnitudeindicateanticorrelationandhencecorrespondtoregionsofstrongstatisticaldependence.Furthermore,formeasures2,4and5,signalswereembeddedinm8dimensionsasperSection 4.3.2 .Toaccountforasimilareectincorrelation,dependencevaluesweremaximizedovertimelagsbetween1and29samples.Asaresult,adependencemapcould 99

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bedrawnforeachchannel,showingsensorlocationsexhibitinghighlevelsofdependence,inheatmapformat. 6.2TheKolmogorov-SmirnovTest Toassessquantitativelythedependencymapsbetweenconditions,weusethetwo-sampleKolmogorov-Smirnov(KS)test,whichisanon-parametrictestthatcomparestwosamplevectors.TheKStesttriestoestimatethedistancebetweentheempiricaldistributionfunctionsofthetwosetsofsamples.Thenullhypothesisisthatbothsamplesaredrawnfromthesamedistribution.Assuming1pxqand2pxqtobethesamplevectors,theKSstatisticcanbecalculatedas:KS1;2maxx|F2pxqF1pxq| (6{1) whereF1pxqandF2pxqdenotetheempiricalcumulativedistributionfunctionsfortheniidobservations,alternativelyFtX1;:::;Xnupxq1 nni1IXix,whereIkdenotestheindicatorfunction. Thenullhypothesisisrejectedatthe-levelifa pn1n2q{pn1)]TJ /F3 11.955 Tf 11.76 0 Td[(n2qKS1;2K,wheren1andn2denotethenumberofsamplesfromeachobservationvectorandKreferstotheKolmogorovdistribution[ Marsagliaetal. 2003 ].Inourcase,n1n2Nt. 6.3Cronbach'salpha Cronbach'salphaisameasureofinternalconsistency,commonlyusedasanestimateofthereliabilityofapsychometrictest.Cronbach'salphawasproposedby Cronbach [ 1951 ]andmaybeseenasanindicativeofreproducibilityofresultsforagivenexperimentalparadigm. Nt Nt11Nti12i 2T(6{2) where,inEquation 6{2 ,Ntreferstothenumberofconsidereditems(hereNt15orNt20,i.e.thenumberoftrials).iandTdenoterespectivelythevarianceofthe 100

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ithitemandthevarianceofthetotalscoreformedbysummingalltheitems[ BlandandAltman 1997 ]. 6.4Results 6.4.1SingleReferenceChannel Forvisualizationpurposes,weselectachannellocationintheoccipito-parietalregion(POz),asshowninFigure 6-1 ,todisplayarepresentativemapoftheobtaineddependencies[ Fadlallahetal. 2012b ].Twocriteriawereconsideredwhenselectingthechannellocation,(i)itsunbiasednesstowardseitheroftherightorleftbrainhemispheresand(ii)itsproximitytotheoccipitalregionswherethesignalisknowntooriginatefrom.Theadvantagesoughtbyselectingthischannelconsistsinprovidinganinsightabouthowtheoccipitalregionofthebrainrelatesintimetothesurroundingoccipito-temporalandparietalareas. Figure 6-2 plotsdependencemapsoverthesensornetworkforcorrelation,mutualinformationandGMA.Figures 6-3 and 6-4 showtheaveragedphase-lockingvaluesandtheirdistributionpercondition,andeachofFigures 6-5 6-6 6-7 and 6-8 showsprojecteddependenciesontherightandleftlateralpartsoftheheadforthetwoconditions.GMAwasusedforFigures 6-6 and 6-7 ;absolutecorrelationandMIwereusedrespectivelyforFigures 6-5 and 6-8 .Givenitsasymmetry,GMAplotswerecomprisedoftworowstoreectbothdirectionsofthemeasure.ThelowerrowcorrespondstodependencevaluescomputedfromallchannellocationstochannelPOzor72. Correlation:Absolutecorrelationshowsconsistencyinlocationsofactiveregionsacrosstimeandconditions.Inadditiontothevisualcortex,theseregionsseemtopointtowardssourcesinthesuperiortemporalsulcusregionofthebrain. GMA:Unlikecorrelation,GMAshowshigherdependenceforthefaceconditionintherightparietal-temporal-occipitalregionneighboringP4(Figures 6-6 and 6-7 ).Thismightbeexplainedbyacommunicationbetweensourcesintheprimaryvisualcortex 101

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Figure6-1. Sensornetworkandchannel72highlightedinred.Thelocationofthiscentralelectrodeisclosetotheoccipitalregionswherethesignalisknowntooriginatefrom. andventralregionsourcesandisconsistentwithengagementofthefusiformorrightoccipito-temporalareas.Aremarkableobservationisthatthisactiveregionseemstobereinforcedwiththedurationofthepresentation.TheaveragedvalueofGMAinthatregionshowsconsistentlyslightincreaseswithtime,noticeableaslongasthestimulusisapplied,whichsuggeststheenforcementofcommunicationbetweensourcesastimepasses. MutualInformation:MItendstoshowamorebalanceddistributionofactiveregionsbetweenthetwobrainhemispheresascomparedtocorrelationandGMA.Somepatternsofrightparietalactivitycanbespottedforthefacecondition,buttheircorrespondingdependencelevelsvaryacrosstimewindows.Similartothetwootherdependencemeasures,aslightlyhigherlevelofdependencewasobservedforthefacecondition. 102

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Figure6-2. Dependencemapsdisplayedinsensorspaceinheatmapformat.Redareasdenoteregionsofhighersynchronizationwiththereferencechannel(POzor72),andblueareasdenotethosethatexhibitlowersynchronizationwiththereferencechannel.Threemeasuresofdependenceareillustrated.Top:GMA,Middle:MIandBottom:Absolutecorrelation. 103

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PhaseSynchrony:Figures 6-3 and 6-4 showtheprojectionofphase-lockingvaluesoverthescalpandthePLVsdistributionpercondition.PhasesynchronywasevaluatedonSubject1andallthetimeseriessampleswereusedinthecomputations.Phasesynchronyseemstooutperformtheotherdependencemeasuresindiscriminability.Thishowevercomesattheexpenseofthereducedtimeresolution.Theoretically,weexpectsignalsrecordedforthefacestimulustobemoresynchronizedthanthoseoftheGaborpatchcondition,sincemorestructureswouldbeengagedwhenrespondingtoafacialvisualstimulus.SincethecumulativedistributionfunctionofthePLVsfortheGaborpatchconditionincreasesfasterthanthatforfaceconditionatthesmallerPLVrange,andsloweratthelargerPLVrange,thePLVsbetweenchannelsfortheGaborpatchstimulusconditionismainlydistributedatasmallnumericrange.Thus,thesignalsrecordedfortheGaborpatchconditionarelesssynchronizedthanthoseforthefacestimulus,whichconrmsourexpectations. DependenceacrossTimeWindows:AsubsetofelectrodescorrespondingtotheregionneighboringelectrodelocationP4(asshowninFigure 6-9 .a)wasselectedtostudythevariationsofourdependencemeasuresacrosstimewindows.GMAvaluesexhibitlowvariabilitywithslightincreasesastimeows,whichmightbeexplainedbyhabituationeectsduetopresentingthesamevisualobject.Ontheotherhand,absolutecorrelationandMIshowmoreuctuationsanddonotpresentthesameincreasingpattern. Table6-1. Two-sampleKolmogorov-SmirnovtestresultsforPEandWPE.Asinglesubjectwasconsideredintheanalysis. KSTestPEWPE NullhypothesisrejectionFalseTruepvalue0.5080.009Teststatistic0.1010.202 104

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Figure6-3. Theaveragedphase-lockingvaluesbetweenthereferencechannelandalltheotherchannelsfor(a)thefacecondition,and(b)theGaborpatchcondition.ThewholetimeserieswereusedtocalculatethePLVs,resultinginareducedtimeresolution.The Figure6-4. Theempiricalcumulativedistributionsoftheaveragedphase-lockingvaluesforthefaceandGaborpatchconditions.Acleardistinctionbetweenthetwoconditionscanbeobserved.Thiscomeshoweverattheexpenseofasacricedtimeresolution. PEandWPE:AprecursorforausefulusageofPE(WPE)withintheabovecontextistoassigna\complexity"curveforeachrecordedsignal,correspondingtoanarrayofPE(WPE)valuescomputedoveragiventimewindow(114msinthiscase).Wecanthencomputethedependencebetweenthedierentchannelsbysimplyapplyingcorrelationonthesecurves.Intuitively,thisimpliesusingalinearmeasureofdependence 105

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AFL-CC BFR-CC CGL-CC DGR-CC Figure6-5. Usingabsolutecorrelationtoweightgraphconnectionsforchannel72.Firsttwosubplots(aandb)showinterpolatedcorrelationmeasuresoverrightandleft(RandL)headsurfaceforthefacecondition(F)andsubsequentsubplots(candd)exhibitthesamefortheGaborpatchcondition(G).TheactiveregionsaresimilarforthetwocasesandvariationswithtimearestudiedinFigure 6-9 .AstatisticalassessmentofthediscriminatoryperformancebetweenthetworegionsisconductedinSection 6.2 tomeasurehowclosethecomplexityoftwotimeseriesare.Inoursimulations,weselectSpearman'srhoasameasureofstatisticaldependencebetweenthedierentPE(WPE)curves.InFigure 6-10 ,theobtainedcorrelationvaluesaremappedontothecorrespondinglocationsonthehumanscalp.InthecaseofWPE(Figures 6-10 .Aand 6-10 .B),moreactivitycanbespottedinlocationsthatseemtopointtowardssourcesintheoccipito-parieto-temporalareaoftherightbrainhemisphere.Thisoutcomealignswiththeresultsobtainedin[ Fadlallahetal. 2012b ],which,aspreviouslymentioned,indicatehigheractivityinthatspecicregion.Ontheotherhand,PEtendstoshowactivitylocalizedinrightposteriorareas. Kendall'stauandTGMA:Figure 6-12 showstheobtaineddependencemapswhenusingKendall'stauandTimeSeriesGMA(TGMA)onEEGdataforthefacecondition.Bothmeasuresindicatehighercouplingforthisconditionbetweenoccipitalsitesandthetemporal-parietal-occipitalsitesneighboringelectrodeP4.UsingthedistributionsinFigure 6-13 (obtainedviasurrogatedatagenerationbypermutationofsamples),thoselocationswouldcorrespondtoregionsexhibitingstatisticallysignicantdependencewithrespecttothereferenceelectrode.Thenumberofstatisticallysignicantpairwiselinks 106

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AFL-GMAf BFR-GMAf CGL-GMAf DGR-GMAf EFL-GMAb FFR-GMAb GGL-GMAb HGR-GMAb Figure6-6. SameprocedureasinFigure 6-5 appliedforGMA.TheupperrowcorrespondstoforwardcomputationsofGMA,i.e.fromtheselectedchanneltotheotherswhiletheloweroneshowsthebackwardcomputations.Moredependencecanbeseenforthetwoconditionsintherighthemisphere.Thenoticeableactiveregionforthefaceconditionseemstocorrespondtomedialoccipitotemporalstructuresandmightreecttheactivityofthefusiformfacearea.Theasymmetryofthemeasuredoesnotsignicantlyaectthelocationsoftheactiveregions. canbeseeninFigure 6-14 .Thesmallsignicancelevelusedis0:039%andhasbeendeterminedbyBonferroni'scorrectioncriterionformultiplecomparisonswherewedividethefamilywiseerrorrateof5%bythenumberofperformedcomparisons(129inthiscase).Withthismethod,TGMAhas51signicantlinksfortheFaceconditionand32fortheGaborcondition.ForKendall'stau,thenumbersarerespectively90and72.Thenumberofcommonlinksreturnedbybothmethodsis41,correspondingto81%ofthetotalTGMAlinks. 107

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AFL-GMAf BFR-GMAf CGL-GMAf DGR-GMAf EFL-GMAb FFR-GMAb GGL-GMAb HGR-GMAb Figure6-7. Sameprocedureasinthepreviousgure(Figure 6-6 )appliedonthesecondsubject.Thesimilarityinobtainedresultssupportsthevalidityoftheanalyticprocedureacrossmultipleindividuals. AFL-MI BFR-MI CGL-MI DGR-MI Figure6-8. SameprocedureasinFigure 6-7 appliedformutualinformation.UnlikecorrelationandGMA,mutualinformationshowslessdiscriminabilitybetweenthetwoconditionsandrelativelysimilarlevelsofdependencebetweenthetwobrainhemispheres. 108

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ASubset BAverageddependenceovertimewindows Figure6-9. (A)Electrodesubsetselectedtotrackthevariationsacrosstimewindowsofdependencevaluespertainingtotheactiveregion.(B)GMAvaluesshowmorestabilityacrosstimethanabsolutecorrelationandMI. AFL-WPE BFR-WPE CFL-PE DFR-PE Figure6-10. UsingSpearman'scorrelationtoweightgraphconnectionsforchannel72.Firsttwosubplots(aandb)showinterpolatedcorrelationmeasuresoverrightandleft(RandL)headsurfaceforthefacecondition(F)whenusingWPEandsubsequentsubplots(candd)exhibitthesamewhenusingPE.AstatisticalassessmentofthediscriminatoryperformancebetweenthetwoconditionscanbeseeninFigure 6-11 andTable 6-1 6.4.2AllChannels 6.4.2.1Displayingthedynamicdependencegraph Aninteractivedisplayofchanneldependenciesevolvingovertimeisgeneratedforeachdependencemeasureandcondition.Everydynamicdependencegraphconsistsof36frames.Forconvenience,wepresentatotalof12snapshots,eachrepresentingtheaverage 109

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Figure6-11. Empiricalcumulativedistributionfunctions(CDFs)perconditionforPE(left)andWPE(left).Datafromonesubjectwasusedtogeneratethisgure.PEshowsindiscernabledistributionsofdependence,unlikeWPE,wherethecurvesaremoredistinguishable. ATGMA-L BTGMA-R CKendall-L DKendall-R Figure6-12. Firsttwosubplots(aandb)showinterpolatedTGMAmeasuresoverrightandleft(RandL)headsurfacefortheFaceconditionandsubsequentsubplots(candd)exhibitthesamewhenusingKendall'stau. ofthreeframes.Furtherextractionofgraphtheoreticalmeasuresfromtheshowngraphscanbeseeninthenextsection(Section 6.4.2.2 ).Figures 6-15 6-16 6-17 and 6-18 displaydynamicdependencegraphsforTGMAandabsolutecorrelation,forthetwoconditions.Ineachgure,12frameswiththeircorrespondingtimestampsareshown.TGMAshowsmorepronouncedlocalandlong-rangedependenciesforthefaceconditionthanfortheGaborpatchcondition.Absolutecorrelation,ontheotherhand,seemstohavealotofspuriousconnectionsforthetwoconditions. 110

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Figure6-13. GeneratingnullhypothesesofuncorrelatednessforTGMAandKendall'stauusingEEGdata.Thetoprowshowsthenullhypothesesforuncorrelatednessgeneratedbyrandomlypermutingtimeseriessamplesandthedistributionoftheobtainedp-values,computedwithrespecttothegenerateddistributions.Thebottomrowplotstheobtainedp-valueperchannelperconditionforTGMA/Kendall'stau. 6.4.2.2Localdescriptorsusinggraphtheoreticalmeasures TofurthercharacterizethemainnetworkstructuresinvolvedindiscriminatingthefaceandGaborpatchconditionsinthedependencegraphs,wecomputegraphtheoreticalquantitiespernode,usingthedierenceofthefaceandGaborpatchdependencematrices,andthenstatisticallyprocessingthematricestodiscardvaluesfallingwithintwostandarddeviationsofthemean.Tworepresentativemeasuresofdependenceareusedwhenextractinggraphtheoreticalquantitiesfromdependencematricescomputedusingallthechannels:therstisthetime-seriesgeneralizedmeasureofassociation(TGMA)[ Fadlallahetal. 2012a ],andthesecondisSpearman'srho[ Spearman 1904 ],anonparametricmeasureofcorrelation.Foreachmeasureofdependence,wecomputevenodequantities,namelythedegree,clusteringcoecient,betweennesscentrality,subgraphcentrality 111

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Figure6-14. ThedashedlinecorrespondstothesignicancelevelofTGMAandthesolidlinecorrespondstothesignicancelevelofKendall'stau.TGMAhas51signicantlinksforthefaceconditionand32fortheGaborpatchcondition.ForKendall'stau,thenumbersarerespectively90and72,whichreectsthediscriminatorypowerofTGMA. andlocaleciency.Computationsaremadeperwindowsoftimecorrespondingto114ms,toachieveabettertimeresolutionandallowtrackinganytime-varyingactivity.Alternatively,thewholetimeseriescanbeusedtoobtainmeanassessmentvalues.Theobtainedvaluesarevisualizedinsensorspacetoidentifytheactiveregionsinvolved.Figure 6-19 showstheresultingplots(averagedoverthetimewindows)forGMAandFigure 6-20 thoseforSpearman'srho. InFigure 6-21 ,weshowhowbetweennesscentralitychangeswithtime.Eachofthedisplayedsubplotscorrespondstocomputationsextractedfrom1secondofdata.Wecanobserveaconsistencyintheactiveregionsacrosstime,especiallytowardsthelatertimewindows,whichsuggeststhereinforcementofcommunicationbetweensourcesastimepasses. Figure 6-22 showstheconnectedcomponentscorrespondingtoeachcondition.Thesizeofthecorrespondingconnectedcomponentwasplottedforeachchannel.Thesizeofthemainconnectedcomponentforthe\Face"conditionissubstantiallyhigherthanthat 112

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114ms 456ms 798ms 1140ms 1482ms 1824ms 2166ms 2508ms 2850ms 3192ms 3534ms 3876ms Figure6-15. DynamicdependencegraphusingTGMA(Facecondition).DatafromSubject3isused.Thedependencematriceswereaveragedovertrialsandprocessedastodiscardallvaluesfallingwithintwostandarddeviationsofthemean.12frameswiththeircorrespondingtimesareshown.Morepronouncedlocalconnectivitypatternscanbeobservedforthefacecondition,withmoreinvolvementoftheCz,C4andF4clusters.TofurthercharacterizethemainnetworkstructuresinvolvedindiscriminatingthefaceandGaborpatchconditions,graphtheoreticalquantitiesarecomputedpernode,usingthedierenceofthefaceandGaborpatchdependencematrices(RefertoFigure 6-19 ). ofthe\Gaborpatch"condition.Figure 6-23 mapsthedierentinactiveregionstothesensorspace. 113

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114ms 456ms 798ms 1140ms 1482ms 1824ms 2166ms 2508ms 2850ms 3192ms 3534ms 3876ms Figure6-16. DynamicdependencegraphusingTGMA(Gaborpatchcondition).DatafromSubject3isused.Thedependencematriceswereaveragedovertrialsandprocessedastodiscardallvaluesfallingwithintwostandarddeviationsofthemean.12frameswiththeircorrespondingtimesareshown.Morepronouncedlocalconnectivitypatternscanbeobservedforthefacecondition,withmoreinvolvementoftheCz,C4andF4clusters.TofurthercharacterizethemainnetworkstructuresinvolvedindiscriminatingthefaceandGaborpatchconditions,graphtheoreticalquantitiesarecomputedpernode,usingthedierenceofthefaceandGaborpatchdependencematrices(RefertoFigure 6-19 ). 6.4.2.3Globaldescriptorsusinginformationtheoreticconcepts DistributionsofmutualinformationbetweenmatricesbelongingtothesameordierentconditionsweregeneratedandareshowninFigure 5-2 114

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114ms 456ms 798ms 1140ms 1482ms 1824ms 2166ms 2508ms 2850ms 3192ms 3534ms 3876ms Figure6-17. Dynamicdependencegraphusingabsolutecorrelation(Facecondition).DatafromSubject3isused.Thedependencematriceswereaveragedovertrialsandprocessedastodiscardallvaluesfallingwithintwostandarddeviationsofthemean.12frameswiththeircorrespondingtimesareshown.TofurthercharacterizethemainnetworkstructuresinvolvedindiscriminatingthefaceandGaborpatchconditions,graphtheoreticalquantitiesarecomputedpernode,usingthedierenceofthefaceandGaborpatchdependencematrices(RefertoFigure 6-20 ). 6.4.3InternalConsistency WeuseCronbach'salphaasameasureofinternalconsistency.Sincetherecordingsdidnotallhavethesamenumberofchannels,weroughlyalignourdatabymappingeveryelectrodeiinthe257-settingtoitsri{2scounterpartinthe129-setting.Sincesuchassignmentofelectrodesisnotaccurateanddoesnotnecessarilycoincidewiththeexactlocationsonthescalp,thecomputedvaluesofCronbach'salphaareapproximateandcan 115

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114ms 456ms 798ms 1140ms 1482ms 1824ms 2166ms 2508ms 2850ms 3192ms 3534ms 3876ms Figure6-18. Dynamicdependencegraphusingabsolutecorrelation(Gaborpatchcondition).DatafromSubject3isused.Thedependencematriceswereaveragedovertrialsandprocessedastodiscardallvaluesfallingwithintwostandarddeviationsofthemean.12frameswiththeircorrespondingtimesareshown.TofurthercharacterizethemainnetworkstructuresinvolvedindiscriminatingthefaceandGaborpatchconditions,graphtheoreticalquantitiesarecomputedpernode,usingthedierenceofthefaceandGaborpatchdependencematrices(RefertoFigure 6-20 ). bethoughtofaslowerboundsofconsistency(thisissincewehypothesizethatanaccuratemappingwillmostprobablyincreasetheobservedconsistency).Cronbach'salphawascomputedperconditionperdependencemeasure. Twosetsofcomputationswereperformed.Intherst,thedependencematriceswereaveragedoverchannelsandtimewindowsforeachsubjectandcondition,resultingintwoNcNtmatricespersubject(Table 6-2 ).Inthesecond,thedependencematriceswere 116

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(a) (b) (c) (d) (e) Figure6-19. SeveralmeasuresextractedfromtwodependencegraphsconstructedusingTGMA.First,wecomputethedierencebetweenthetwodependencematricescorrespondingtothefaceandGaborpatchconditions(averagedovertrials),anddiscardthevaluesfallingwithin2standarddeviationsofthemean.Wethenusegraphtheoreticalmeasurestocharacterizetheimportanceofeachnode.Themeasuresincludefromlefttoright:(a)thenodedegree,(b)thenodeclusteringcoecient,(c)thenodebetweennesscentrality,(d)thenodesubgraphcentrality,and(e)thenodelocaleciency.Computationswereperformedusingwindowsof114samplesandplotswereaveragedover36windows. averagedoverchannelsandtimewindowsthenconcatenatedfromdierentsubjectstoformtwoNcNtmatrices.TheresultsforthisapproachareshowninTable 6-3 Table6-2. Cronbach'salphapersubjectcomputedfordierentdependencemeasuresandconditions.Thedependencematriceswereaveragedovertimewindowsandtheconsistencywascomputedacrosstrials. DependencemeasureFaceGaborpatch TGMA0.93360.9322Kendall'stau0.96720.9671Spearman'srho0.96350.9634 6.4.4PerformanceAnalysis 6.4.4.1TheKStest Figure 6-24 showstheempiricaldistributionsperconditionacrossthesensornetworkforeachoftheconsidereddependencemeasuresandthecorrespondingvalueoftheKSstatistic.Distributionsforabsolutecorrelationandmutualinformationaremoredistinct 117

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(a) (b) (c) (d) (e) Figure6-20. SeveralmeasuresextractedfromtwodependencegraphsconstructedusingSpearman'srho.First,wecomputethedierencebetweenthetwodependencematricescorrespondingtothefaceandGaborpatchconditions(averagedovertrials),anddiscardthevaluesfallingwithin2standarddeviationsofthemean.Wethenusegraphtheoreticalmeasurestocharacterizetheimportanceofeachnode.Themeasuresincludefromlefttoright:(a)thenodedegree,(b)thenodeclusteringcoecient,(c)thenodebetweennesscentrality,(d)thenodesubgraphcentrality,and(e)thenodelocaleciency.Computationswereperformedusingwindowsof114samplesandplotswereaveragedover36windows.Again,computationswereperformedusingwindowsof114samplesandplotswereaveragedover36windows. Figure6-21. Thebetweennesscentralitypernodeoverwindowsof1secofdata.First,wecomputethedierencebetweenthetwoTGMAdependencematricescorrespondingtothefaceandGaborpatchconditions,anddiscardthevaluesfallingwithin2standarddeviationsofthemean.Every9timewindowswereaveragedtoyieldaresolutionofapproximately1second.Wethencomputethebetweennesscentralitypernodeandprojectittothescalp. 118

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Figure6-22. ThesizeoftheconnectedcomponentsperchannelforthefaceandGaborpatchconditions.ThedependencemeasureusedisTGMAandthethirdsubjectwasused.Thedependencematriceswereaveragedovertrialsandtimewindows,thenstatisticallyprocessedtodiscardvaluesfallingwithintwostandarddeviationsofthemean.Theconnectedcomponentswerethenextractedfromtheresultingadjacencymatrixandthesizeoftheconnectedcomponentateachnodeisdisplayed. thancorrelationbutshowlessdiscriminabilitythanGMA.Lookingatatimewindowof114samples,theteststatisticobtainedforGMAis0:9125,comparedto0:3441and0:4844formutualinformationandabsolutecorrelation,asshowninFigure 6-25 .AllmeasuresperformedbetterthanthepowerclassicationschemeinFigure 2-12 .Resultsweresimilarwhenusingalongertimewindowwitha15%dierenceforGMA,whencomparingtoasmallertimewindow.Therangeofvaluesdropsforallmeasuresofdependencemeasuresbutthiscomesattheexpenseofreducedtimeresolution.Table 6-4 showstheKStestresultsobtainedpersubject. 119

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Figure6-23. ThecardinalityofthedierenceinconnectedcomponentssizesshowninFigure 6-22 mappedtosensorspace.Eachnodedisplaysthedierencebetweenthesizesofthelargestconnectedcomponentsincludingthenode,pertainingtoeachcondition. Table6-3. Cronbach'salphaacrosssubjectscomputedfordierentdependencemeasuresandconditions.Thedependencematriceswereaveragedovertimewindowsandconcatenatedfromallparticipants. MeasureConditionSubjectsSubj1Subj2Subj3Subj4Subj5Subj6 TGMAFace0.93750.92310.97290.92650.90680.9664Gabor0.92210.91680.97280.92870.91620.9473Kendall'stauFace0.92050.94970.97680.96210.89140.9408Gabor0.90320.95580.97760.96490.90250.9328Spearman'srhoFace0.90340.95090.97440.95700.87040.9332Gabor0.87770.95660.97570.95990.88900.9255 6.4.4.2Classicationresults Wetrainwithdatagatheredfrom5participantstogeneratedistributionsofentropyestimatesforthetwoconditions,thenusethe6thparticipantfortesting.Thereforewehave100trainingmatrixsamplespercondition,or200intotal.Theapproachissupervisedinthatthelabelsforthetrainingmatricesareknown.The20trialsbelonging 120

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Figure6-24. Empiricalcumulativedistributionfunctions(CDFs)computedfromdependencevaluesacrossthe129electrodesandaveragedovertrialsfor:(a)Pearson'scorrelation,(b)AbsolutePearson'scorrelation,(c)mutualinformationand(d)GMA.Timewindowusedis114samplesor114ms.Theerrorbarsshowthevariabilitywithtimewindows(38windowsinthiscase). Figure6-25. Teststatisticforthetwo-sampleKStestappliedforeachofthefourdependencemeasures.Pointsinredwereobtainedwhenusingtimewindowsof114samplesandthoseinbluewhenusing456samples.ThedottedgreenlinerepresentstheKSstatisticobtainedusingpowerdistributions. 121

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Table6-4. KStestresultsforallsubjects.Thedependencematriceswereaveragedovertrialsandtimewindows. MeasureKStestSubjectsSubj1Subj2Subj3Subj4Subj5Subj6 TGMAKSresult111101KSpvalue0.04870.00580.06130.00470.83020.0047KSstatistic0.39730.20930.17750.15180.15450.1518Kendall'stauKSresult100101KSpvalue0.00010.61120.50750.00130.98860.0156KSstatistic0.42640.09320.10080.16730.03890.1362Spearman'srhoKSresult100101KSpvalue0.00010.71750.50750.00130.96960.0044KSstatistic0.46510.08530.10080.16730.04280.1556 tothe6thsubjectarethenassignedlabelsbasedonwheretheyfallinthedistributions.TheclassicationresultsusingthisapproachareshowninTable 6-5 Table6-5. Cronbach'salphaacrosssubjectscomputedfordierentdependencemeasuresandconditions.Thedependencematriceswereaveragedovertimewindowsandconcatenatedfromallparticipants. ClassicationrateforSubject6perconditionDependencemeasureConditionClassicationrate(rst2windows|lasttwo) TGMAFace32/40(rsttwo)|30/40(lasttwo)Gabor30/40(rsttwo)|29/40(lasttwo)Kendall'stauFace25/40(rsttwo)|26/40(lasttwo)Gabor23/40(rsttwo)|23/40(lasttwo)Spearman'srhoFace27/40(rsttwo)|26/40(lasttwo)Gabor25/40(rsttwo)|27/40(lasttwo) 122

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6.5Discussion Resultsshowactiveregionsintheoccipito-parietalpartofthebrainforbothconditionswithagreaterdependencybetweenoccipitalandinferotemporalsitesforthefacestimulus.Thisalignswithpreviousevidencesuggestingre-entrantorganizationoftheventralvisualsystem,showingheightenedre-entrywhenviewingmeaningfulorsalientstimuli.ThepatternofactiveregionscouldbetracedacrossthedierentmeasuresofdependencewithabetterclassicationachievedbyGMA/TGMA.Moreover,theslowervariationofthedependencelevelastimepassesseemstopointtowardshabituationeectsduetopresentingthesamevisualobject.Classicationusingtwotimewindows(114and456samples)performedbetterthanthesimplepowerclassierasshowninFigure 6-25 .Moreover,agoodinternalconsistencywasobservedsinceallsubjectshadvaluesofCronbach'salphanearorabove0:9. 123

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CHAPTER7CONCLUSIONSANDFUTUREDIRECTIONS 7.1SummaryandDiscussion Inthisdissertation,weexaminedssVEPsevokedbyickeringstimulishowingahumanfaceandGaborpatches.Dependenciesbetweenchannellocationswerecomputedintimewindowsof114msforseveraltraditionalandnovelmeasuresofdependence. AsuitablepreprocessingframeworkwasdesignedtoanalyzedependenceincurrentCSDtimeseriesextractedfromrecordedEEG.Theframeworkisrobustinthatdesiredoutputachievesminimalsensitivitywithrespecttofreeparameters.Themainparametertobeconsideredwhenlteringisthequalityfactor.Graphandinformationtheoreticalframeworksweredesignedtoexplorethediscriminabilitybetweenconditions.Theperformancewasstudiedintermsofdiscriminatingbetweenthetwoconditions.Resultsshowconsistencyintheactivebrainregionsovercycles.MoreactivityisvisibleintherighthemisphereofthebrainfortheFacecondition,nearP4.Ahighlevelofdependenceisnoticedbetweentheoccipitalandrightparieto-temporo-occipitalsensorlocations.Thismightbeexplainedbyacommunicationbetweensourcesintheprimaryvisualcortexandventralregionsources.Thisisalsoconsistentwithengagementofthefusiformorrightoccipito-temporalareas.Proposedmeasuresofassociationshowedmorediscriminabilityforthetwostimulithanstandardmeasures. EstimatesofsourcelocationsunderlyingthessVEPactivitystronglysuggestthatmostscalp-recordedssVEPsignalsoriginatefromlower-tiervisualcortex.Thisisinlinewiththendingsby Russoetal. [ 2007 ]and WieserandKeil [ 2011 ]andthosein Mulleretal. [ 1997 ]thatlocalizethevisualssVEPinposterioroccipitalandventraloccipitalcortex. Russoetal. [ 2007 ]furthersuggestthatsimplessVEPmodelsconsistingofoneortwosourcesintheoccipitalregionofthebrainareunabletoexplainthessVEPsmagnitudeandphase,whichfavorsthehypothesisofadistributedsourcesystemwithenhancedcommunicationbetweentheoccipital/parieto-occipitalregionandsources 124

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inventralregions.Potentialsourcessuggestedbythisstudyconcurwithsourcesofface-specicresponsesfoundby Samsetal. [ 1997 ]. 7.2FutureWork Futureworkwouldproceedinoneormoreofthefollowingdirections: 1. Firstweproposetoincorporatetoouranalysismeasuresofsignalregularityorcomplexity.Thiscanbedoneaccordingtotwoways: (a) Adoptingaslidingwindowanalysisusingmeasuresofsignalcomplexity.Suchapproachcanyielddescriptorsofdatathatkeeptemporaldependencetoaminimum,whichfacilitatestheusageofmeasuresofdependenceatasecondstage. (b) Derivingnovelmeasuresofassociationusingcorrentropyandentropyestimators.Suchapproachwoulduseagainmeasuresofcomplexity,combinedeitherwithnearestneighborsorordinalranks,inordertoquantifydirectlydependencebetweentwovariables. 2. Improvingthequalityofthestatisticaltestsused.ThisissincetheKStestissensitivetodeviationsintheempiricalCDFfunctionsofthetwosamplesbeingcompared.WecaneitherchoosetosetcondenceintervalsoftheKStestusingabootstrappingapproachorgobeyondtheKStesttogeneratethenullhypothesisofdistributionequalitybyusingaresamplingapproach.Thebootstrapapproachwouldproceedaccordingtothefollowingsteps: (a) Drawingaresampleofacertainsize(sayhalfthenumberoftrials)withreplacementfromtherstsampleandaseparateresampleofthesamesizefromthesecondsample. (b) Choosingastatisticthatcomparesthetworesamples(canbeKSorMann-Whitney). (c) Repeatingthisresamplingprocesshundredsoftimes. (d) Constructingthebootstrapdistributionofthestatistic. 125

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3. Adaptingthemethodologyforusewithdierentdatasets.Examplesincludepresentingdierentpictureswithvaryingemotionalcontextandcheckingtheabilitytodiscriminatebetweentheneutralandmorethreateningoremotionallyinvolvingpictures. Althoughwedidnotstudydirectionalconnectivitiesinthiscontext,thelocationofactiveregionswitnessedforthefacestimulusisconsistentwithndingssuggestingre-entrantmodulationofearlyvisualcortex,originatingfromhighertiersandenteringlowertiersofvisualcortex[ Keiletal. 2009 ].Ontheotherhand,tofurtherelaborateontheobtainedresults,itwouldbeinterestingtoinfermoreinformationaboutthenatureofcommunicationbetweentheoccipito-parietalcortexandtheinferotemporalandwhetheritinvolvesintermediatesources.Thiscanbedonebystudyingthetimedelaypropertiesoftheobtainedsignalsoverasetofchannels.Anothergraphtheoreticalapproachwouldbetoworkdirectlyontheadjacencymatrixviagraphmatchingprocedures,usingaprocedurethatautomaticallymatchestwographsdisplayingdependencestructure.Previousexamplesintheliteraturewheregraphcomparisonandmatchingmethodswereappliedincludethepapersby Conteetal. [ 2004 ]and Zeguraetal. [ 1997 ]. 126

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APPENDIXACURRENTSOURCEDENSITY A.1BasicConcepts CurrentsourcedensityorCSDmeasuresinareference-freefashionthestrengthofextracellularcurrentgeneratorsunderlyingtherecordedrawEEG.ThecomputationofCSDisbasedonalinearvolumeconductionmodel[ Nicholson 1973 ; NicholsonandFreeman 1975 ].ToderivetheCSDmeasure,westartfromtheverywellknowOhm'slaw: JE(A{1) whereJdenotesthecurrentowdensity,Ethecorrespondingelectriceld,andtheconductivityofthemedium.NotethatEcanbewrittenas: ErV(A{2) whereVdenotesthescalarvalueoftheelectricpotential(orvoltage)andrdenotesthegradientoperator.UsingEquation A{1 and A{5 andapplyingthedivergenceoperatoronbothsidesoftheequation. IdrJrprVq(A{3) Hence: IdrprVqr2VV(A{4) 127

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ThusCSDisproportionaltothesecondspatialderivative(orLaplacian)ofthemeasuredeldpotential[ TenkeandKayser 2005 ].InCartesiancoordinates,theLaplaceoperatorcanbedenedas: fB2f Bx2)]TJ 12.95 8.09 Td[(B2f By2)]TJ 12.95 8.09 Td[(B2f Bz2(A{5) TheSurfaceLaplacian(SL)canbethenestimatedusingtwodierentapproaches,alocaloneandaglobalone.Themainlocalapproachwasproposedby Hjorth [ 1975 ],inwhichtheSLisestimateddirectlyatselectedsites.Themethodsimplyproceedsbycomputingthedierencebetweenthepotentialateachelectrodesiteandtheaveragepotentialofthenearestfourneighbors.Suchmethodassumethatdistancesseparatingelectrodesareequal,justliketheanglesbuiltbytheelectrodesconguration[ Tandonnetetal. 2005 ]. Ontheotherhand,theglobalapproachusesallelectrodestocomputetheSL.ItproceedsbyconstructinganinterpolationfunctiontorepresentthepotentialdistributionsandestimatingtheSLbasedonthatfunction.Mostglobaltechniquesarebasedonsplineinterpolationmethodsasdoneby NunezandCutillo [ 1995 ], Perrinetal. [ 1989 ]and Babilonietal. [ 2001 ]. A.2ComputingtheCSD A.2.1ForwardandInverseProblems LetPpr;tqbetheelectricdipolemomentperunitvolume.ScalppotentialmaybeexpressedasavolumeintegralofPpr;tqovertheentirebrainprovidedPpr;tqisdenedgenerallyratherthanincolumnarterms[ SrinivasanandNunez 2007 ].Fortheimportantcaseofdominantcorticalsources,scalppotentialmaybeapproximatedbythefollowingintegraloverthecorticalvolume,whereGpr;r1qreferstoaGreen'sfunctioncontaining 128

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allgeometricandconductiveinformationabouttheheadvolumeconductorandweightstheintegralaccordingly[ SrinivasanandNunez 2007 ]. VSpr;tqGpr;r1qPpr1;tqdpr1q(A{6) ItmightbeusefulheretoremindthattheforwardprobleminEEGconsistsintuningtheheadmodelandthenusingthecorrespondingGreen'sfunction,performthecalculationsinEquation A{6 foranassumedsourcedistribution.Ontheotherhand,theinverseproblemconsistsofstartingwiththerecordedscalppotentialdistributionVSpr;tqundersomeconstraintsonPpr;tqinordertoretrievethebestsourcedistributionPpr;tq.Theinverseproblemhasnouniquesolution,i.e.foranygivendistributionoftheelectricalpotentialonthesurfaceofthehead,thereexistsaninnitenumberofpossiblesourcecongurationsinsidethehead[ Helmholtz 1858 ; Junghoferetal. 1997 ]. A.2.2CSDandCorticalMapping Thecomputationofthecorticalmappingandthecurrentsourcedensity(CSD)canbeobtainedthroughthesameprocedurewhenusingsphericalsplineinterpolation.Modelingthevolumeconductorasaspherically-shapedisotropicvolumeconsistingoffourlayers(scalp,skull,cerebro-spinaluidorCSFandbrain),dierentconductivitiesiareassignedtoeachlayer.Radialsourcesareassumedinthederivation. CunandCohen [ 1979 ]expressthepotentialcausedbyasingleradialcurrentdipole,locatedonthesurfaceofthesphereas: Vp;qp8n1Ponpcosq:)]TJ /F5 7.97 Tf 7.31 4.94 Td[(scalppnq(A{7) where)]TJ /F5 7.97 Tf 41.13 4.34 Td[(scalppnqisgivenby: )]TJ /F5 7.97 Tf 7.31 4.93 Td[(scalppnq1 44R2p2n)]TJ /F1 11.955 Tf 11.76 0 Td[(1q4:fn1:pcdq2n)]TJ /F8 7.97 Tf 6.58 0 Td[(1 )]TJ /F4 11.955 Tf 7.32 0 Td[(pnq(A{8) 129

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whereintheaboveequations: b,candddescribeaproportionalrelationtothecorrespondingshellradius. )]TJ /F4 11.955 Tf 7.32 0 Td[(pnqfpb;c;d;iq. anddenotethepolarandazimuthanglesinsphericalcoordinates. PndenotesaLegendrepolynomialofordern. Thesolutioncanbecastasaneigenvalueproblemwhenwritteninmatrixnotation: Vp:Gscalpij(A{9) whereGscalpijinfn1Pn:pcosijq:)]TJ /F5 7.97 Tf 7.32 4.34 Td[(scalppnqandijreferstotheangleformedbythepairofelectrodesiandj(i;jP1;:::;N). Itisimportanttonotethatcastingtheproblemintheaboveformatisusefulandecient.ThisissincethetermGscalpijonlydependsonthemodelparametersandisindependentofthemeasurements,i.e.itcanbecomputedandstoredthenappliedtodierentmeasurementmatrices. Atthisstage,theideaistointerpolatethescalppotentialandusetheresulttocomputetheCSD.ThiscanbedoneusinganimportantpropertyoftheLaplacianinsphericalcoordinates,i.e. ;Ymlp;qlpl)]TJ /F1 11.955 Tf 11.76 0 Td[(1qYmlp;q(A{10) thatresultsinhavingthesphericalLaplacianofaLegendrepolynomialequaltoanintegermultipleofthesamefunction,orPnnpn)]TJ /F1 11.955 Tf 12.63 0 Td[(1qPn.HencewecanwriteCscalpasCscalp8n1Pnpcosikq:n:pn)]TJ /F1 11.955 Tf 11.79 0 Td[(1q:)]TJ /F5 7.97 Tf 7.31 4.34 Td[(scalppnq.AsimilarexpressionisderivedfortheCSDofthecorticalpotentialdistribution. 130

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AgureillustratingthepotentialversusCSDmapscanbeseeninFigure A-1 1 FigureA-1. Left:Potentialmapversusright:CSDmapforavisualevokedpotential(VEP)recording.MoreactivitylocalizationcanbeseenintheCSDmap. 1Thegurehasbeenreprintedfrom[ VandeWassenberg 2008 ]. 131

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APPENDIXBLINEAR-PHASEFIRFILTERDESIGN Aleast-squareslinear-phaselterdesignisasimpleandoptimal(withrespecttothesquareerrorcriterion)methodtodesignFIRlters. LetRpfqandDpfqdenotetheobtainedanddesiredamplituderesponsesand)]TJ /F4 11.955 Tf 305.75 0 Td[(pfqbeapositiveweightingfunctionwhosevalueschangeaccordingtothepass,transitionandstopbands.Ourgoalistondhpnqsuchthatef1{2f0)]TJ /F4 11.955 Tf 7.31 0 Td[(pfq:rRpfqDpfqs2dfisminimized. AssumingaTypeIlter,wehave: RpfqMn0rpnq:cosp2nfq(B{1) whereintheaboveequationtherelationbetweenhpnqandrpnqisknown.Thecoecientsofrpnqcanbehenceretrievedbytakingthederivativeoftheintegraldeninge.Thiswouldgive: Mn0rpnqf1{2f0)]TJ /F4 11.955 Tf 7.32 0 Td[(pfqcosp2nfqcosp2kfqdff1{2f0)]TJ /F4 11.955 Tf 7.32 0 Td[(pfqDpfqcosp2kfqdf(B{2) Thiscanbewrittenasrwhere: pkq1 f1{2f0)]TJ /F4 11.955 Tf 7.32 0 Td[(pfqDpfqcosp2kfqdf(B{3) and: pk;nq1 f1{2f0)]TJ /F4 11.955 Tf 7.32 0 Td[(pfqcosp2nfqcosp2kfqdf(B{4) Hencenally,rpnqcanbeobtainedasrpnq1. 132

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APPENDIXCSECOND-ORDERBUTTERWORTHFILTERS Themagnituderesponseofsecond-orderButterworthltercanbeexpressedas: |Hpjwq|1 ? 1)]TJ /F3 11.955 Tf 11.76 0 Td[(w4(C{1) Wecanthenobtaintheminimumphasetransferfunctionfromthemagnituderesponseas: Hpsq1 1)]TJ 11.76 9.97 Td[(? 2s)]TJ /F3 11.955 Tf 11.76 0 Td[(s2(C{2) Thecorrespondingamplituderesponsefunctioncanbethendeducedas: Hpjwq1 p1w2q)]TJ /F3 11.955 Tf 18.97 0 Td[(jp? 2wq(C{3) Thephaseresponsewouldthenbe: pwqtan1? 2w 1w2(C{4) Andthegroupdelay: pwq? 2p1)]TJ /F3 11.955 Tf 11.76 0 Td[(w2q 1)]TJ /F3 11.955 Tf 11.76 0 Td[(w4(C{5) 133

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APPENDIXDWEIGHTED-PERMUTATIONENTROPY Anadequatetestingschemetoweighted-permutationentropy(WPE)wouldincludespikydatabecauseitposesachallengetoasimplemotifcountapproachandexhibitssuddenchanges.SimulationswereperformedonbothsyntheticdataandEEGdata. D.1SyntheticData Asarstmotivation,wesuggesttoanalyzethebehaviorofPEandWPEinpresenceofanimpulsiveandnoisysignal.Figure D-1 .ashows1000samplesofasignalconsistingofanimpulseandadditivewhiteGaussiannoise(AWGN)withzeromeanandunitvariance.Windowsof80samplesslidby10sampleswereusedandresultswereaveragedover10simulations.AremarkabledropinthevalueofWPEisnoticedintheimpulseregion.NomarkedchangecanbeobservedinthecaseofPEforthesameregion. Asnextstep,wetryatrainofGaussian-modulatedsinusoidalpulseswithdecayingamplitudes.Thevalueofwassetto1.Slidingwindowsof50sampleswithincrementsof10sampleswereusedandmwassetto3.Again,thesignalwascorruptedbyAWGNandsimulationswererunacrossdierentvariancelevels.Figure D-2 showsthevariationsofthesignal'sentropyforfourdierentmethods.TheperformanceofPEandWPEiscomparedtotwoothermethodsfromtheliterature,namelyapproximateentropyorApEn[ Chonetal. 2009 ; Pincus 1991 ]andthecompositePEindexorCPEI[ Olofsenetal. 2008 ].Inthefollowing,wegiveabriefdescriptionofeach. ApproximateEntropy(ApEnorAE):Approximateentropyisameasurethatquantiestheregularityorpredictabilityofatimeseries.Itisdenedwithrespecttoafreeparameterrasfollows: Hamprqm)]TJ /F8 7.97 Tf 6.59 0 Td[(1prq(D{1) 134

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FigureD-1. (Coloronline)PEversusWPEinthecaseofanimpulse.(a)ImpulsewithadditivewhiteGaussiannoisewithzeromeanandunitvariance.(b)ComputedPEandWPEvalueswithwindowsof80samplesslidby10samples.AremarkabledropinthevalueofWPEisnoticedintheimpulseregionforwhichPEvaluesdonotshowanymarkedchange. wheremprqisdenedas: mprq1 Npm1qNpm1qi1lnCmiprq(D{2) andCmiprqisdenedusingtheHeavysidefunctionpuqp1foru0;0otherwiseqandadistancemeasuredist: CmiprqNpm1qj1)]TJ /F3 11.955 Tf 5.48 -9.8 Td[(rdistpXm;i;Xm;jq Npm1q(D{3) 135

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Herethevalueofrissettobe0:2timesthedatastandarddeviationasperthethoroughdiscussionin[ Chonetal. 2009 ].Thedistancemeasureweuseisthesamesuggestedin[ Pincus 1991 ]andcanbeformulatedas: distpxm;i;xm;jqmaxk1;:::;m|xipk1qxjpk1q| CompositePEindex(CPEI):ThecompositePEindex(CPEI)isanalterationofpermutationentropythatdierentiatesbetweenthetypesofpatterns.Itiscalculatedasthesumoftwopermutationentropiescorrespondingtomotifshavingdierentdelayswherethelatter(denotedasinthisdissertation)isdeterminedbywhetherthemotifismonotonicallydecreasingorincreasing.CPEI,whichwedenotebyHiinthisdissertation,respondsrapidlytochangesinEEGpatternsandcanbedenedasfollows[ Olofsenetal. 2008 ]: Hi1 lnpm!)]TJ /F1 11.955 Tf 11.76 0 Td[(1qHpm;1q)]TJ /F3 11.955 Tf 18.96 0 Td[(Hpm;2q 2(D{4) ThenormalizationdenominatorinEquation D{4 consistsoftheoriginalnumberofmotifsinadditiontoanewlyintroducedmotiftoaccountforties(tiesdescribecaseswherenegligibledierencesinamplitudeoccurwithinamotif).Asasidenote,theaveragingstepperformedinthatequationishighlyapproximativebecauseofthelackofindependencybetweenmotifsatdierentdelays. ItisnoticeablethatWPEconsistentlydropsforportionsofthesignalshowingpulses.Thisisdesiredbecauseofthelessercomplexityoftheseregionsandexpectedbecauseoftheirimmunitytonoise.Hereweassumethattheinformationcontainedintheexaminedsignalsisamplitude-dependent.SuchresultsmeetourexpectationssinceWPEisclearlyabletodierentiatebetweenburstyandstagnantregionsofthepulsetrain.Inotherwords,usingthevariancecontributestoweakeningthenoiseeectsandassigningmore 136

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FigureD-2. (Coloronline)Dierententropymeasures(PE,WPE,CPEIandAE)appliedonaGaussian-modulatedsinusoidaltrainwithafrequencyof10kHz,apulserepetitionfrequencyof1kHzandanamplitudeattenuationrateof0:9.InitialsignalwascorruptedbyadditivewhiteGaussiannoise(AWGN)havingmean0andvariance20:2.Thesamplingratewas50kHzandcomputationsuseda50-sampleslidingwindowwithincrementsof10samples.TherecordedSNRwasof4:8dB. 137

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weighttotheregularspikypatternscorrespondingtoahigheramountofinformation,whichresultsineasierpredictabilityandlesscomplexity.Itisimportanttonotetwothings:(1)thecontributionofpatternswithhighervariancetowardsthevalueofWPEdominatesthoseofpatternswithlesservariancewhichhighlightsthepowerfulnessofthemethodindetectingabruptchangesintheinputsignaland(2)thefactthatWPEiscomputedwithinaspecictimewindowexplainswhyWPEvaluescorrespondingtoimpulsivesegmentsofthesignaldonotdecreaseinspiteofthedecreasingamplitudesofthespikes(thenormalizationeectin(4)takesplacewithineachwindow).WealsoplotinFigure D-3 thevaluesofPEandWPEfordierentlevelsofsignal-to-noiseratio(SNR).Asanticipated,bothentropymeasuresdecreasewiththeincreaseoftheSNRsincetheeectofnoisecontributingtomorecomplexitybecomeslesssignicant.WPEdecreasesatahigherpacethanPE,whichreectsabetterrobustnesstonoise.Asanalnoteonthissection,wepointoutthattraditionalmethodslikezero-crossingspikedetectiontechniquesmightbeusefulforthepurposeofthissimulation,howeverthesoughtgoalwastodemonstrate,usingsyntheticdatatheabilityofWPEtodiscriminatebetweenregimesofdata. FigureD-3. (Coloronline)NormalizedPEandWPEvaluesfordierentSNRlevels.ThesignalusedisthesameasinFigure D-2 138

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D.2Single-ChannelEEGDataAnalysis InFigure D-4 ,thesamecomparisonsareperformedforasampleEEGrecordingprocessedasin[ Fadlallahetal. 2011 ].Highpasslteringwasfurtherappliedonthesignalbecauseweareinterestedinremovingverylowfrequencycomponents.ItcanbeseenthatWPElocatestheregionswhereabruptchangesoccurintheinitialsignalmoreaccuratelythantheothermethods,whichisinlinewithouroriginalexpectations.ThesameisreectedinFigure D-5 thatshowsaprocessedEEGportioncorrespondingtoanotherchannel.Ourresultsshowthatincreasingmbeyond4aectstherunningtimewithoutsignicantlychangingtheobtainedentropies.Thisisinlinewiththendingsin[ Lehnertz 2007 ]wheretheparameterselectionproblemhasbeenaddressedand[ Olofsenetal. 2008 ].Forsituationswheretheeectofmismorepronounced,therunningtimeissuecanbeaddressedbyspeedinguptheslidingofthewindowasthisentailsahighernumberofaectedpatternsateachinstance. D.3EpilepsyDetection Setting:NextweproposetoapplyWPEforepilepsydetection.Weusethesamedataas Quirogaetal. [ 1997 ; 2002 ],inwhichtonic-clonicseizuresofasubjectwererecordedwithascalprightcentralelectrode(locatednearC4inastandard1020montage).Therecordingconsistedof3minutes,includingaround1minuteofpre-seizuretimeand20secondsofpost-seizureactivity.Asamplingrateof102:4Hzwasusedtocollectthesignal. Discussion:Wecomputeddierentmeasuresofentropyonwindowsof50samplesofdataslidby5samples(Figure D-6 .b).TheobtainedcurvesarefurthersmoothedinFigure D-6 .cusingamovingaveragelteroflength35samples.Thecommencementofepilepticactivityintherecordedsignalinducesnoticeablechangesforallentropymeasures,in 139

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FigureD-4. (Coloronline)AnalysisperformedonlteredEEGdatasampledat1000Hzandprocessedasin[ Fadlallahetal. 2011 ].WPEoutperformsotherentropymeasuresinlocationregimentsexhibitingabruptchangesinthesignal.Thewindowlengthusedforthisplotwas114withanoverlapof2samples. particularforWPEthatexhibitsasignicantjumpinvalue.Thisisfurtherquantiedbycomputingtheratioofaveragemeasuredentropiesofepilepticandnon-epilepticsegments(Table D-1 ),whichshowsamorepronounceddierencebetweenbothportionsforWPE.Thelatterachievesalmosttwicebetterdiscriminabilitybetweenthetwoportionsofthesignal,i.e.42%betterthanthenextclosestmeasure(CPEI). 140

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FigureD-5. (Coloronline)SameprocedureappliedonaprocessedEEGportioncorrespondingtoanotherchannel.WPEmirrorsbestthesharpchangeinthesignalnoticeablebeforet850ms.Thewindowsizeusedwas200withanoverlapof2samplesateachiteration. TableD-1. Ratioofaveragemeasuredentropybetweenepilepticandnon-epilepticsegments. MeasureRatio1 PE1.27WPE1.85CPEI1.30AE0.57 141

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FigureD-6. (Coloronline)Dierententropy-basedmeasuresappliedonepilepticEEG.(a)EEGrecordingofanepilepticsubject.Therecording,sampledat102:4Hzcontainsapproximatelyoneminuteofpre-seizureactivityand20secondsofpost-seizureactivity.(b)Dierentmeasuresofentropycomputedusingaslidingwindowof50sampleswith5samplesoverlap.(c)Smoothedentropymeasurescurvesobtainedbyapplyingamovingaveragelteroflength35samples. 142

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APPENDIXEINFINITELYDIVISIBLEFUNCTIONS Thefollowingsectionisanextractfrom[ Sanchez-GiraldoandPrincipe 2013 ]thatcontainsmoreelaborationaboutinnitelydivisiblefunctions. E.1NegativeDeniteFunctionsandHilbertianMetrics LetMpX;dqbeaseparablemetricspace.AnecessaryandsucientconditionforMtobeembeddableinaHilbertspaceHisthatforanysettxiuXofn)]TJ /F1 11.955 Tf 12.26 0 Td[(1points,ni;j1ijpd2px0;xiq)]TJ /F3 11.955 Tf 18.97 0 Td[(d2px0;xjqd2pxi;xjqq0;foranyPRn.Thisconditionisequivalenttoni;j0ijd2pxi;xjq0;foranyPRn)]TJ /F8 7.97 Tf 6.59 0 Td[(1,suchthatni0i0.Thisconditionisknownasnegativedeniteness.Interestingly,theaboveconditionimpliesthatexpprd2pxi;xjqqispositivedeniteinXforallr0.Indeed,matricesderivedfromfunctionssatisfyingtheabovepropertyconformaspecialclassofmatricesknowasinnitelydivisible. E.2InnitelyDivisibleMatrices AccordingtotheSchurproducttheoremA0impliesAnAAA0foranypositiveintegern.DoestheaboveholdifwetotakefractionalpowersofA?Inotherwords,isthematrixA1 m0foranypositiveintegerm?Thisquestionleadstotheconceptofinnitelydivisiblematrices.AnonnegativematrixAissaidtobeinnitelydivisibleifAr0foreverynonnegativer.Innitelydivisiblematricesareintimatelyrelatedtonegativedenitenessaswecanseefromthefollowingproposition PropositionE.2.1. IfAisinnitelydivisible,thenthematrixBijlogAijisnegativedenite. FromthisfactitispossibletorelateinnitelydivisiblematriceswithisometricembeddingintoHilbertspaces.Ifweconstructthematrix: DijBij1 2pBii)]TJ /F3 11.955 Tf 11.75 0 Td[(Bjjq;(E{1) 143

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usingthematrixBfromproposition E.2.1 .ThereexistsaHilbertspaceHandamappingsuchthat: Dij}piqpjq}2H:(E{2) Moreover,noticethatifAispositivedeniteAisnegativedeniteandexpAijisinnitelydivisible.Inasimilarway,wecanconstructthematrix: DijAij)]TJ /F1 11.955 Tf 12.95 8.09 Td[(1 2pAii)]TJ /F3 11.955 Tf 11.76 0 Td[(Ajjq;(E{3) withthesameproperty( E{2 ).Thisrelationbetween( E{1 )and( E{3 )suggestsanormalizationofinnitelydivisiblematriceswithnon-zerodiagonalelementsthatcanbeformalizedinthefollowingtheorem. TheoremE.2.1. LetXbeanonemptyset,andletd1andd2betwometricsonit,suchthatforanysettxiuni1,ni;j1ijd2`pxi;xjq0,foranyPRn,andni1i0,istruefor`1;2.ConsiderthematricesAp`qijexpd2`pxi;xjqandtheirnormalizations^Ap`q,denedas: ^Ap`qijAp`qij b Ap`qiib Ap`qjj:(E{4) Then,if^Ap1q^Ap2qforanynitesettxiuni1X,thereexistisometricallyisomorphicHilbertspacesH1andH2,thatcontaintheHilbertspaceembeddingsofthemetricspacespX;d`q,`1;2.Moreover,^Ap`qareinnitelydivisible. AbeautifulobservationfromTheorem E.2.1 isthattheproposednormalizationprocedureforinnitelydivisiblematricescanbethoughtofasndingthemaximumentropymatrixamongallmatricesforwhichtheHilbertspaceembeddingsareisometricallyisomorphic. 144

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BIOGRAPHICALSKETCH BilalFadlallahreceivedthebachelor'sdegree(BE)incomputerandcommunicationsengineeringandthemaster'sdegree(ME)inelectricalandcomputerengineeringfromtheAmericanUniversityofBeirut,Lebanon,in2006and2008respectively.HejoinedtheComputationalNeuroEngineeringLaboratoryattheUniversityofFloridain2010,workingoninferringfunctionaldependenciesinthehumanbrainfromEEGdata.HisPh.D.workmostlyliesattheintersectionofseveralelds,includingsignalprocessing,cognitiveandcomputationalneuroscienceandmachinelearning.Bilalhasfouryearsofindustryexperienceindesigningandtestingsystemssoftware,atMurexSystemswhereheleadtheMurexLimitsControllerteam,andMicrosoftwherehecontributedtotheWindowsFundamentalsteam.HeistherecipientoftheUniversityofFloridaCerticateofOutstandingAcademicAchievementandafellowoftheLebaneseNationalCouncilforScienticResearch. 162



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Fair Medium Access in 802.11 based Wireless Ad-Hoc Networks Brahim Bensaou Centre for Wireless Communications National University of Singapore 20 Science Park Road, #02-34/37 Singapore 117476 brahim@cwc.nus.edu.sg AbstmctThe Medium Access Control (MAC) protocol through which mobile stations can share a common broadcast channel is essential in an ad-hoc network. Due to the existence of hidden terminal problem, partiallyconnected network topology and lack of central administration, existing popular MAC protocols like IEEE 802.11 Distributed Foundation Wireless Medium Access Control (DFWMAC) [l] may lead to capture effects which means that some stations grab the shared channel and other stations suffer from starvation. This is also known as the fairness problem. This paper reviews some related work in the literature and proposes a general approach to address the problem. This paper borrows the idea of fair queueing from wireline networks and defines the fairness index for ad-hoc network to quantify the fairness, so that the goal of achieving fairness becomes equivalent to minimizing the fairness index. Then this paper proposes a different backoff scheme for IEEE 802.11 DFWMAC, instead of the original binary exponential backoff scheme. Simulation results show that the new backoff scheme can achieve far better fairness without loss of simplicity. I. INTRODUCTION An ad-hoc network is a dynamic multi-hop wireless network that is established by a group of mobile stations without the aid of any pre-existing network infrastructure or centralized administration. It can be installed quickly in emergency or some other special situations and is self-configurable, which makes it very attractive in both civilian and military applications [2]. An efficient medium access control (MAC) protocol through which mobile stations can share a common broadcast channel is essential in an ad-hoc network because the medium or channel is a scarce resource. Due to the limited transmission range of mobile stations, multiple transmitters within range of the same receiver may not know one anothers transmissions, and thus are in effect hidden from one another. When these transmitters transmit to the same receiver at around the same time, they do not re0-7803-6534-8/00/$10.00 0 2000 IEEE Yu Wang, Chi Chung KO Department of Electrical Engineering National University of Singapore 10 Kent Ridge Crescent Singapore 119260 engp8843-elekocc@nus. edu .sg alize that their transmissions collide at the receiver. This is the so-called hidden terminal problem [3] which is known to degrade throughput significantly. To address the hidden terminal problem, various distributed MAC protocols were proposed in the literature [l], [4], [5], [6]. Among them, IEEE 802.11 Distributed Foundation Wireless Medium Access Control (DFWMAC) is a proposed standard for wireless adhoc and infrastructure LANs and is commonly used in testbeds of wireless ad-hoc networks for research in routing for example [2]. DFWMAC is a kind of Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) protocols and provides basic and RTS/CTS access method. The basic access method includes only exchange of data packet and acknowledgment packet between a pair of source and destination stations. The RTS/CTS access method is used to combat the hidden terminal problem and requires additional handshake, namely short Request-to-Send (RTS) and Clear-to-Send (CTS) packets between a pair of source and destination stations before actual data packet transmission. RTS and CTS packets include a field called Network Allocation Vector (NAV). It is used to inform stations who overhear the RTS and/or the CTS packets how long they should defer access to the channel. Although the RTS/CTS access method can alleviate hidden terminal problem leading to an increased throughput [7], DFWMAC still suffer from the fairness problem which was first investigated in MACAW [4] (short for Multiple Access with Collision Avoidance for Wireless) which is another protocol proposed for wireless LANs. In MACAW, additional control packets and a different backoff algorithm named Multiplicative Increase and Linear Decrease (MILD) with a backoff copy scheme are used to increase throughput and alleviate fairness prob99

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Fig. 1. Sample network lem. In addition, per stream fairness is introduced in MACAW. It means that each stream that originates from either the same station or different stations should be treated equally and given equal share of the channel capacity. This is different from per station fairness which accords channel capacity to individual stations instead of individual streams. For multiple streams that originate from a station, MACAW keeps separate queues for each stream and runs backoff algorithms independently for each stream. However, MACAW still left some problems unsolved. For example, in the configuration shown in Fig. 1, station 1 has load for station 2 and station 3 has load for station 4. When the load increases to a certain degree, station 3 will capture the channel and station 1 will suffer severe degradation in throughput. DFWMAC faces the same problem as well because it uses a binary exponential backoff which always favors the last succeeding station. This has been pointed out in MACAW. Another problem for MACAW is that backoff copy scheme only works when congestion is homogeneous which is not necessarily the case in ad-hoc networks. For more details about these problems, readers can refer to MACAW paper More recently, Ozugur et. a1 [8] proposed a pijpersistent CSMA based backoff algorithm. This paper first defined the fairness index to be the ratio of maximum link throughput to minimum link throughput. Then it proposed that each station calculates a link access probability pij for each of its links based on the number of connections from itself and its neighbors(connection based), or based on the average contention period of its and other stations individual links(time based). Whenever its backoff period ends, station i will send RTS packet to j with probability pij or back off again with probability 1 pij. The proposed scheme relies on periodic broadcast packets in the time-based approach or on aperiodic broadcast packets in the connection-based approach whenever the network topology changes. This paper also investigated the effects of combination of contention window [41. Backoff timer is generated from the uniform distribution which exchange with either connection-based or time-based approach. However, none of the schemes can achieve the best results for all network configurations investigated in [B] and sometimes the best results are achieved when these schemes are in fact not used. In addition, broadcast packets are unreliable to disseminate information to neighbors. As the RTS/CTS access method cannot be used and no acknowledgment packet can be sent in this case, no one can ensure if broadcast packets can be delivered to all the sending stations neighbors, which makes the performance of this method tightly coupled to the successful dissemination of the information in the network. In the ongoing research work of Vaidya and Bahl [9], they identified the difficulties in defining fairness itself in multi-hop networks and defined a Generalized Resource Sharing (GPS) algorithm which needs further investigation as it includes sorting flows which requires global information. In addition, a distributed fair scheduling algorithm is also proposed to achieve fairness on local area networks (one hop) and its performance was evaluated. The goal of this paper is to address the fairness problem in multi-hop ad-hoc networks with a general and more practicable approach. This paper will present preliminary results as this is still ongoing work. Section 2 first defines new metrics for measuring fairness and then proposes a different backoff scheme for the DFWMAC protocol. Section 3 evaluates the performance of the proposed scheme and compares its performance to those obtained from the original backoff scheme using several ad-hoc network configurations. Section 4 concludes this paper. 11. ESTIMATION-BASED FAIR MEDIUM ACCESS In this paper, we want to address the fairness problem in a general approach. We define fairness in the sense of fair queueing as defined in [lo]. To facilitate the discussion, we introduce the following notation: di : a pre-defined fair share that station i should receive. Normally, it should be determined at admission control, i.e. when the node joins the ad-hoc network, and can be readjusted for example when a node becomes a router. How to choose this parameter, how to do admission control and how to adjust the parameter, are still open research problems. Wi : The actual throughput achieved by station i; Li : Station is offered load. A fair MAC protocol should have the following properties. When stations offered load to the channel is much lower than the channel capacity, each stations request for ranges from 0 to the size of the current contention window. 100

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transmission should be met. This means that for any station i, Wi = Li. When stations offered load exceeds the channel capacity, each station should be able to get its fair share of the channel, i.e, proportional1 to 4. This means that for any station i and j, 2 = ~. This is just for ideal situations. In reality, -we want to bound the value of 17 by the smallest possible value. Instead of working with absolute values, we define the fairness index, FI, to be: wi wj wi wj 4i 4j 4i 4j FI = ma~{Vi,j : ~ux(-, -)/min(-, -)} Therefore, our goal becomes the design of a distributed MAC protocol that can minimizes FI and thus achieve fairness for all the stations in an ad-hoc network. There comes the problem of how to choose $i for any station i. As mentioned previously, this is an admission control problem which is beyond the scope of this paper, however, in situations where the ad-hoc network is open to everyone without admission control, which can happen in situations where all the stations are trusted and known not to misbehave, the following approach can be used. If each station is considered to be a greedy source and wants to get the same share as all other stations as a whole, then it can just set 4 = 0.5 regardless of the number of its neighbors. As to any station, say i, it requests the same share as all the others in its vicinity. These stations have a total share of 4, = 1 = 0.5, which equals to this stations share +i. This can be interpreted as a per-station fairness. If a station has two active links (or streams in MACAWS terminology), which can happen when a station acts as a router in an ad-hoc network, it can set 4i to meet : which shows simply that the station (router) wants to obtain two times as much share of bandwidth as other stations to function as a router properly. This can be interpreted as MACAWS pet-stream fairness. To achieve the fairness goal, we propose a different backoff scheme. In this scheme, each station will estimate its share and other stations share of the channel and then adjust the contention window accordingly. We use the following notations in fair share estimation algorithm: W,i : The estimated share of the estimating station itself; We, : The estimated share of other stations; Ttype : Time to transmit a packet of type type. Algorithm 1 shows how estimation works. The basic idea is that from the point of view of station i, it sees that it is sharing the channel with a group of belligerent stations who are competing with it for channel access. Thus we have the notion of me, and the others. Stations estimate dynamically what throughput they get and what throughput others get, and then adjust their contention window according to the fairness index defined. In other words the contention window is adjusted in order to equalize the throughput obtained by the different stations. A station can estimate roughly how much bandwidth others obtain by looking at the packets in its vicinity. For example (the details can be seen in the algorithm) an RTS packet that station i sends leads to an increase of its obtained throughput since it used the channel, a received RTS means others are trying to obtain the channel and thus it increases others obtained throughput, etc. Algorithm 1 Fair share estimation switch (received packet type) { case RTS: if (destID != localID) We, += TTts else {send CTS packet; Weo += (Trts + Tcts)} case CTS: if (destID != localID) We, += (Trts + Tcts) else {send DATA packet; wei += (Trts Tcts + Tdata)} case DATA: if (destID != localID) else {send ACK packet; weo += (Th Tcts Tdata) wei + = (T~ts + Tcts + Tdata f Tack)} case ACK: if (destID != localID) weo += (Th + Tcts + Tdata + Tack) else {wei += (Trts + Tcts Tdata + Tack)} 1 Whenever sending an RTS packet, Wei += TTts In this algorithm, RTS and CTS packets transmission is counted towards the estimated share because RTS and CTS packets are used as a channel reservation scheme and consume channel resource as well. With this estimation, we modify the binary exponential backoff scheme used in DFWMAC. We define the estimated fairness index to be: FI, = (?)/( 2) and the IO1

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adjustment of contention window is shown in Algorithm 2. Algorithm 2 Contention window adiustment switch (He) { case >C: case (l/C, C): case <1/C: CWneW = min(CWneW x 2, CWMAX) cwnew = CWold CWnew = max(CWold / 2, CWMIN) 1 In Algorithm 2, C is a constant used to adjust the adaptativity of the algorithm. The smaller the value of C, the more aggressively is the contention window size adjusted and vice versa. However, the choice of C is rather limited. For example, if we choose C = 2, stations would not change their contention windows when estimated FI is between (0.5,2) and probability of collision may be high when the number of competing stations is large and load to channel is high. On the other hand, if C is too close to 1, say 1.01, stations may be busy adjusting their contention windows all the time and the algorithm becomes instable. The calculation shows that if a station estimates that it has got more share than it should get, it will double its contention window size until it reaches a maximum value (CWMAX) so that its neighbors can have more chances to recover earlier from backoff procedure and win access to the channel and vice versa. If a station estimates that it has got only its fair share, it will hold onto its current contention window size. 111. SIMULATION RESULTS In our experiments, we investigate some configurations of wireless ad-hoc networks used in MACAW and pijpersistent CSMA. These are the 4-station, 5-station and 6-station scenarios. They are shown in Fig. 2, where arrow lines indicate that there is traffic between stations and dashed lines indicate that the stations are within communication reach of each other but no traffic flows between them. We assume a lMbps ideal channel with zero preamble and processing overhead and a propagation delay of about 6p seconds. We have performed different sets of simulations with OPNET Modeler/Radio and we compare our results with the original IEEE 802.11 DFWMAC protocol2. Table I lists the parameters used to generate the 2We use its specification for Direct Sequence Spread Spectrum (a) 4-station (b) 5-station (c) 6-station Fig. 2. Network configurations simulations results. As we ignore the extra time incurred by hardware and software, the different InterFrame Spaces (IFSs) in IEEE 802.11 are reduced accordingly and they are also shown in Table I. Unless otherwise specified, all stations will use a fair share $J = 0.5. In the 4-station scenario, station 1 and 3 generate Poisson traffic with the same mean rate, and results are shown in Fig. 3. Figure 3 shows that DFWMAC will have serious fairness problem when the offered load is high enough, which can be explained as follows. Most of the time station 1s transmission may coincide with station 3s transmission as they are hidden from each other. Station 2 will not be able to receive station 1s packet due to station 3s concurrent transmission. However, station 4 can when applicable. 102

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RTS I CTS I DATA I ACK 25-bvte I 20-bvte I 500-bvte I 20-bvte CWMIN I CWMAX I C I backoff unit time 31 I 1023 I 1.1 I 6usec DIFS I SIFS I EIFS 12usec I Ousec I 1.3msec TABLE I PROTOCOL CONFIGURATION PARAMETERS still receive station 3s transmission successfully and reply to station 3 thereafter. According to the binary exponential backoff (BEB) scheme used in DFWMAC, a stations contention window size will be doubled after unsuccessful transmission and will return to the minimum value if a data packet is successfully transmitted. Therefore station 3 usually enjoys a much smaller contention window, thus statistically shorter backoff timer than station 1. When the load is high, station 3 will capture the channel eventually. In our backoff scheme, if station 3 overhears a few packets transmitted from station 2 (in this case, either CTS or ACK packet), its estimatition will show that it has obtained more bandwidth share than what it should have and will increase its contention window size3 accordingly. With the ever increase of station 3s contention window, station 1 will get more chances to transmit packets to station 2. In the end, the station 1s throughput can be balanced with station 3s throughput, so this scheme can achieve far better fairness than the BEB. In the 5-station scenario, we investigate two cases. In the first case, each station generates Poisson traffic with the same mean rate. For station 2, 3 and 4, each has two active links to its neighbors. For each packet that is generated, these stations will randomly choose a neighbor as destination. For this case, we consider per-station fairness only and aggregate the two links throughputs as the corresponding stations throughput. The results are shown in Fig. 4. Due to symmetry, we show results for station 1, 2 and 3 only. In this case, edge stations (1 and 5) face less congestion and their packets are easier to get through. As the binary exponential backoff always favors the last succeeding station, the edge stations will get much higher throughput than other stations. Our scheme works much better to achieve fairness because station 1 3Here station 3 does not even need not know the fact that station 1 has packets for station 2. 0.05 0.1 0.2 0.5 1 2 5 lndlvldual natlDnl Onered kad Fig. 3. (a) Station throughput, (b) fairness index versus stations offered load for the 4-station scenario. and 5 will yield the channel to other stations when they estimate that they obtained extra share than what they should get. In the second case, each station generates Poisson traffic for each link with the same mean rate. Therefore, stations 2, 3 and 4, require equal fair share for each of their links. We experiment with two situations. One is that station 2, 3 and 4 still set 4 = 0.5, the other is that these stations set r#~ = 0.67 which indicates that they require two times the share of other stations as they have two active links. The results are shown in Fig. 111. It 103

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1 0.0 0.8 0.7 fo., 5 i 0.5 I 0.4 0.4 5, 10.3i 0.5 1 2 5 10 ldvldual .Ut". c4l.r.d kad 0.05 0.1 0.2 2 5 10 0.5 1 0.05 0.1 0.2 lndlvrtual slsfm's dlere.3 load shows that even if station 2, 3 and 4 do not increase their 4, the modified algorithm can still achieve much better fairness than DFWMAC. When they do increase 4, the fairness can be further improved. In the 6-station scenario, each station generates Poisson traffic with the same mean rate and results are shown in Fig. 6. Due to symmetry, we show the results for station 1 and 3 only. As there may be concurrent transmissions between two pairs of edge stations (station l and 2, station 5 and 6), inner stations 3 and 4 suffer severe degradation in throughput as in the case of original DFWMAC. Our estimation becomes somewhat inaccurate in this case d I 1 0.5 1 5 10 lndlvldual statian'r Onwad b.d 0.05 0.1 0.2 Fig. 4. (a) Station throughput (original algorithm), (b) station throughput (modified algorithm), (c) fairness index versus station offered load for the 5-station scenario. because some of these concurrent transmissions between edge stations may be interpreted as noise by inner stations and will not be counted in fair share estimation. However, our approach can still achieve far better fairness than DFWMAC. All the simulations show that we tradeoff some throughput for fairness. As our approach in fact encourages stations to participate in fair competition, some channel bandwidth is lost due to collisions. IV. CONCLUSION In this paper, we defined the fairness metrics for wireless ad-hoc networks incorporating both the idea of perstation and per-stream (or per-link) fairness and pointed out that the target to achieve fairness is to minimize fairness index. We then proposed a different backoff scheme for IEEE 802.11 DFWMAC, where each station adjusts its contention window according to the estimated share it obtained and other stations. Simulation results show that this scheme can achieve far better fairness than the original backoff scheme of DFWMAC, despite the fact that it sacrifies some throughput. As this scheme does not assume any knowledge about the network's topology and thus does not require broadcast packets to disseminate information to other stations, it is very simple and can be easily overlaid in the existing DFWMAC protocol. 104

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0.0 0.8,0. 0.ffi 0.1 0.2 0.5 1 2 5 10 IndMdW IMe der4 bad -eLk& 1-r2 Lhk 2->I + Lnk2->3 + Link 2->2 0.5 -eLhk 1->2 LN 2->I + Lk&2-%3 + LM3->2 B ; io.3 e 0.2 I 0.5 1 2 5 10 IMwl Ms dwd bad los 0.1 0.2 0.5 -8 Lnk I->2 -+ Link 2-1 + Link 243 + LN3->2 0.41 1 0.1 0 I 0.05 0.1 0.2 0.5 1 2 5 10 ldivnl IWa dared bad / 2 1 0.05 0.1 0.2 0.5 1 2 5 10 IndMdvnl Ilnki onerad bad Fig. 5. (a) Link throughput (original algorithm), (b) link throughput (modified algorithm, 4 = 0.5 for all), (c) linke throughput (modified algorithm, q?~ = 0.67 for station 2, 3 and 4), (d) fairness index versus station offered load for the 5-station scenario. REFERENCES [l] IEEE Computer Society LAN MAN Standards Committee, ed., IEEE standard for wireless LAN medium access control (MAC) and physical layer (PHY) specifications. IEEE Std 802.11-1997, The Institute of Electrical and Electronics Engineers, New York, 1997. F. A. Tobagi and L. Kleinrock, Packet switching in radio channels: part I1 the hidden terminal problem in carrier sense multiple-access modes and the busy-tone solution, in IEEE Thnsactions on Communications, vol. COM-23, no. 12, [2] http://www.ietf.org/html.charters/manet-charter.html. [3] pp. 1417-1433, 1975. [4] V. Bharghavan, A. Demers, S. Shenker, and L. Zhang, MACAW: A media access protocol for wireless LANs, in Proceedings of ACM SIGCOMM, 1994. P. Karn, MACA a new channel access method for packet radio, in ARRL/CRRL Amateur Radio 9th Computer Networking Conference, pp. 134-140, ARRL, 1990. J. Garcia-Luna-Aceves and C. L. Fullmer, Performance of floor acquisition multiple access in ad-hoc networks, in Proceedings of 3rd IEEE ISCC, 1998. H. S. Chhaya and S. Gupta, Throughput and fairness properties of asynchronous data transfer methods in the IEEE 802.11 MAC protocol, in 6th International Conference on Personal, Indoor and Mobile Radio Communications, 1995. [5] [6] [7] 105

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0.7 I0.B .z 0.5 0.05 0.1 0.2 0.5 1 2 5 10 Individual slalm. oftarad bad Fig. 6. (a) Station throughput, (b) fairness index versus stations offered load for the 6-station scenario. T. Ozugur, M. Naghshineh, P. Kermani, and J. A. Copeland, Fair media access for wireless LANs, in Proc. of IEEE GLOBALCOM, Dec. 1999. N. H. Vaidya and P. Bahl, Fair scheduling in broadcast environments, Tech. Rep. MSR-TR-99-61, Microsoft Research, Dec. 1999. A. K. Parekh and R. G. Gallager, A generalized processor sharing approach to flow control in integrated services networks: the single node case, IEEE/ACM Tkansactions on Networking, vol. 1, pp. 344-357, 1993. 106