ARISTOTLE UNIVERSITY OF THESSALONIKI DEPARTMENT OF MATHEMATICS

201 Pages·2009·3.33 MB·English

ARISTOTLE UNIVERSITY OF THESSALONIKI
DEPARTMENT OF MATHEMATICS, PHYSICS
and COMPUTATIONAL SCIENCES
FACULTY OF TECHNOLOGY
ANGELIKI D. PAPANA
B.Sc. in Mathematics
NONLINEAR STATISTICAL ANALYSIS OF
BIOLOGICAL TIME SERIES
PhD THESIS
Thessaloniki,2009
ANGELIKID.PAPANA
NONLINEARSTATISTICALANALYSISOFBIOLOGICALTIMESERIES
PhDTHESIS
SubmittedtoDepartmentofMathematics,
PhysicsandComputationalSciences,
FacultyofTechnology
Dayoforalexamination: 24June,2009
Adjudicatecommittee
Ass. ProfessorD.Kugiumtzis,Supervisor
Ass. ProfessorG.Zioutas,Memberoftheconsultivecommittee
ProfessorA.Bountis,Memberoftheconsultivecommittee
ResearcherA’A.Provata,Examiner
ProfessorC.Moyssiadis,Examiner
Asso. ProfessorN.Maglaveras,Examiner
Ass. ProfessorI.Rekanos,Examiner
i
Acknowledgements
This research project is implemented within the framework of the ”Reinforce-
ment Programme of Human Research Manpower” (PENED) and is co-financed
at 90% jointly by E.U.-European Social Fund (75%) and the Greek Ministry of
Development-GSRT(25%)andat10%byRikshospitalet,Norway.
ii
(cid:176)c ANGELIKID.PAPANA
(cid:176)c A.U.T
NONLINEARSTATISTICALANALYSISOFBIOLOGICALTIMESERIES
ISBN
”The approval of this PhD Thesis from the Department of Mathematics, Physics
andComputationalSciencesofAristotleUniversityofThessalonikidoesnotimply
acceptanceoftheopinionofthewriters”(Regulation5343/1932,Article202,par.
2).
iii
Contents
1 Introduction 2
2 Physiology 7
2.1 EEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 ExtracranialEEG . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 IntracranialEEG . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 ArtifactsinEEG . . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 ExtracranialvsintracranialEEG . . . . . . . . . . . . . . 10
2.1.5 EEGbands . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.6 EEGpreprocessing . . . . . . . . . . . . . . . . . . . . . 11
2.2 MEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 BrainStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 EEGDatafortheEvaluationoftheNonlinearMeasures . . . . . . 13
3 Methodology 15
3.1 DynamicalSystemsandChaosTheory . . . . . . . . . . . . . . . 15
3.1.1 Dynamicalsystems . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Nonlineartimeseriesanalysis . . . . . . . . . . . . . . . 18
3.2 Statisticalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Surrogatedatatest . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Trenddetection . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 Statisticaltestsforparameters . . . . . . . . . . . . . . . 23
3.2.4 ROCcurves . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 MutualInformation 27
4.1 MutualInformation: DefinitionandProperties . . . . . . . . . . . 28
4.2 MutualInformationEstimators . . . . . . . . . . . . . . . . . . . 29
4.2.1 Binningestimators . . . . . . . . . . . . . . . . . . . . . 30
4.2.2 k-nearestneighborsestimator . . . . . . . . . . . . . . . 32
4.2.3 Kernelestimator . . . . . . . . . . . . . . . . . . . . . . 33
4.3 ApplicationsofMutualInformation . . . . . . . . . . . . . . . . 35
4.3.1 Independencetest . . . . . . . . . . . . . . . . . . . . . . 36
4.3.2 Nonlinearitytest . . . . . . . . . . . . . . . . . . . . . . 41
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4.3.3 Detectionofdynamicalchanges . . . . . . . . . . . . . . 45
4.4 EvaluationofMutualInformationEstimators . . . . . . . . . . . 50
5 EvaluationofUnivariateCorrelationandInformationMeasuresinDe-
tectingDynamicalChanges 73
5.1 CorrelationandInformationmeasures . . . . . . . . . . . . . . . 73
5.1.1 Lineardecorrelationtimeordecaytime . . . . . . . . . . 73
5.1.2 Nonlineardecorrelationtime . . . . . . . . . . . . . . . . 74
5.1.3 Declinationfromnormality. . . . . . . . . . . . . . . . . 74
5.2 Entropymeasures . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.1 Shannonentropyontwovariables . . . . . . . . . . . . . 76
5.2.2 Shannon entropy on variables from recurrence quantifica-
tionanalysis . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.3 Tsallisentropy . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.4 Sampleentropy . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.5 Permutationentropy . . . . . . . . . . . . . . . . . . . . 79
5.3 ApplicationsofInformationMeasures . . . . . . . . . . . . . . . 79
5.3.1 Evaluation of three types of correlation measures in dis-
criminatingregimesofdynamicalsystems . . . . . . . . . 79
5.3.2 Evaluation of information measures in detecting dynami-
calchanges . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 ApplicationsofUnivariateInformationMeasuresonEEG 93
6.1 Short-termPredictionofEpilepticSeizuresfromPreictalEEGRecords
andDiscriminationofDifferentBrainStatesusingStatisticalTests 94
6.1.1 Results for the short-term trend detection on late preictal
EEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.2 Results for the discrimination between early and late pre-
ictalstates . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Evaluation of a large set of Correlation and Entropy Measures in
DiscriminatingPreictalStatesusingStatisticalTests . . . . . . . . 101
6.3 Evaluation of Information and Complexity Measures in Discrimi-
natingStatesusingROCCurves . . . . . . . . . . . . . . . . . . 107
6.3.1 Apilotstudy . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.2 Alargescalestudy . . . . . . . . . . . . . . . . . . . . . 108
7 Evaluation of Causality Measures in Detecting the Direction of Infor-
mationFlow 115
7.1 CausalityMeasures . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1.1 Statespacemeasures . . . . . . . . . . . . . . . . . . . . 118
7.1.2 Synchronizationmeasures . . . . . . . . . . . . . . . . . 121
7.1.3 Informationmeasures. . . . . . . . . . . . . . . . . . . . 122
7.1.4 Relationshipsofcausalitymeasures . . . . . . . . . . . . 124
7.2 EvaluationofCausalityMeasuresonknownsystems . . . . . . . 124
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7.2.1 SimulationSet-Up . . . . . . . . . . . . . . . . . . . . . 124
7.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.3 A pilot study: effectiveness of the causality measures on
EEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3 ImprovingStatisticalSignificanceofCausalityMeasures . . . . . 131
7.3.1 ModificationsofCausalityMeasures. . . . . . . . . . . . 132
7.4 EvaluationofImprovedCausalitymeasuresonknownsystems . . 139
7.4.1 SetUp . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.3 Quantitative results from the evaluation of the causality
measures . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5 EvaluationofImprovedCausalityMeasuresonEEG . . . . . . . 168
7.5.1 SetUp . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 170
8 Conclusions 172
8.1 GeneralOverview . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.2 Suggestionsforfuturework. . . . . . . . . . . . . . . . . . . . . 174
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Abstract
In the present thesis, methods of non-linear time series analysis and dynamical
systemsanalysishavebeencombinedwithstatisticalmethods,andstatisticalmea-
sures with high discriminating power directly estimated on time series have been
studiedanddeveloped. Specifically,existingunivariateandmultivariatelinearand
nonlinearmeasureshavebeenthoroughlyreviewed,newnonlinearmeasureshave
been defined here and existing methods have been extended or modified to gain
statistical significance and be more effective on different applications. All the in-
vestigated measures were first tested on simulation studies and were then applied
onrealdata. Electroencephalogram(EEG)recordingsfromepilepticpatientshave
beenconsideredinordertoevaluatetheeffectivenessofthemeasurestodetectdy-
namicalchangesinthebrainactivityofthepatientsatdifferentstates,e.g. inorder
todiscriminatebetweenEEGrecordsmanyhoursbeforetheseizureonsetandEEG
records from one hour before the seizure onset. The prediction of the onset of an
epilepticseizureandthedetectionofthedynamicalchangesofthebraindynamics
just before the seizure onset were investigated in the frame of univariate time se-
riesanalysis. Theperformanceoftheunivariatemeasureswasquitepromisingand
therefore the study was extended beyond the univariate case. Thus, the existence
ofinterdependenciesbetweendifferentbrainareasandtheidentificationofthedi-
rectionofthecausaleffectsamongthebrainareaswasinvestigatedintheframeof
multivariatetimeseriesanalysis.
Thestudyfocusedoninformationmeasures,astheyaremodel-free,computa-
tionallyefficientand do not require anyprior knowledgeof the distributionof the
data. TheevaluationoftheinformationmeasureswasfirstassessedbyMonteCarlo
simulationsonwell-knowndynamicalsystemsinordertoexaminetheirdiscrimi-
natingpowerandsignificance. Dynamicalcharacteristicsofthesystemsvariedby
changingtheparametersoftheequationofthesystems. Thediscriminatingability
ofthemeasureswasalsoassessedintermsofthetimeserieslength. Thestochas-
ticity of the systems was also a factor that was examined as it can be controlled
by variation of the level of noise added in the observations of the systems. The
variation of the complexity or stochasticity of a system is considered to simulate
thedifferentstatesofthebrainactivityandthusoftheEEGsignal.
Mutual information is an essential tool in nonlinear time series analysis and
thereforewasthoroughlyreviewedandtestedondifferentapplications,e.g. inde-
tectingdynamicalchangesofsystems. Differentestimatorsofmutualinformation
havebeencomparedandtheselectionoftheirparameterswasinvestigatedinorder
tobe optimized. The promising performance ofmutual information led to a more
comprehensive study which included also entropy measures (e.g. Shannon, Tsal-
lis, Permutation). This investigation has led to the extension and modification of
manyexistingmeasures. Thestatisticalsignificanceandpowerofthemeasuresin
discriminationtaskswereassessedusingdifferentstatisticaltests(t-test,Wilcoxon
ranksumtest,ROCcurves,surrogatedatatest). Itwasalsoexaminedwhetherthe
discriminatingpowerofthemeasurescanbeimprovedifthetimeserieswerefirst
transformedtohaveuniformornormalmarginaldistribution.
The results from the simulation studies on known dynamical systems were
required in order to interpret the values of the estimated measures on the EEG
recordingsintermsoftheirstochasticityandcomplexity. Thus,themeasureswere
evaluatedindiscriminatingEEGsignalfrompreictalandinterictalstages. Theef-
fectiveness of the nonlinear measures was compared to that of some linear ones,
as there are contradictory studies on this matter. Based on the frame of univari-
atetimeseriesanalysis,EEGrecordingsfromdifferentchannelswereusedforthe
analysis; channels were chosen based on knowledge of the epileptic focus area or
were randomly selected in cases of generalized seizures to cover all parts of the
brain.
EEG recordings are multivariate signals, and therefore one should also test
the interdependencies between the recordings from the different channels, i.e. the
existence of interactions among the different parts of the brain should be exam-
ined. Here, the direction of the information flow was investigated using bivariate
nonlinear measures. The study focused again on information measures detecting
thedirectionofinterdependenciesbetweeninteractingsystems,howeveralsoother
types of causal measures were included for comparative studies, e.g. state-space
and synchronization measures. The evaluation of the causality measures was ini-
tially assessed on well-known nonlinear systems and results were considered in
ordertoapplythemeasuresonEEGsignals. Thisstudyledusagaininthemodifi-
cationoftheexistingmeasuresandtheextractionofmeasuresthatseemtoimprove
theirperformance.
Thereexistmeasuresthatcandiscriminateamongdifferentdynamicalsystems
anddetectthedynamicalchangesofasystemsandsomeofthemeasuresthathave
been defined here may improved the discriminating power of the original mea-
sureswhentestedonsyntheticdata,howevereventheseimprovedmethodsdonot
perform dramatically better on EEG. EEG signal is very complex and the finite
length of the EEG signal, the measurement artifacts and the interactions between
thedifferentbrainpartsrendertheproblemoftheEEGanalysistobeevenharder.
Thereisstillalongwaytogofortheuseofanymeasureinclinicalpractice. The
main contribution of this work is to assess the statistical significance of the infor-
mationmeasuresforunivariateandbivariatetimeseries,proposemodificationsto
attainbettersignificanceandshowinanobjectiveclinicalsettingtheshortcomings
andpossiblelimitedsuccessofmanyexistingmeasuresfortheproblemofseizure
prediction.
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Publications
Thefindingsofthisthesishavebeenpresentedinconferences,havebeenpublished
orhavebeensubmittedtointernationaljournalsinthefieldofnonlineardynamics,
time series analysis or statistical analysis. The respective references of the papers
inproceedingsandjournalsarepresentedhere.
P1 A. Papana and D. Kugiumtzis. Cumulative Mutual Information Function as a
StatisticfortheTestforNonlinearityonTimeSeries(inGreek). Proceedings
ofthe18thGreekStatisticalConference,Rhodes,pp. 315–325,2005.
P2 D. Kugiumtzis, A. Papana, I. Vlachos and P.G. Larsson. Time series feature
evaluationindiscriminatingpreictalEEGstates. LectureNotesinComputer
Science,Vol. 4345,pp. 298–310,2006.
P3 A. Papana and D. Kugiumtzis. Linear and nonlinear correlation measures of
timeseriesforseizureprediction(inGreek). Proceedingsofthe19thGreek
StatisticalConference,Kastoria,pp. 415–423,2006.
P4 D. Kugiumtzis, I. Vlachos, A. Papana and P.G. Larsson. Assessment of Mea-
suresofScalarTimeSeriesAnalysisinDiscriminatingPreictalStates. NEU-
ROMATHWorkgroupmeetingandWorkshop,Rome,2–4December,2007.
P5 D. Kugiumtzis, I. Vlachos, A. Papana and P.G. Larsson. Assessment of Mea-
sures of Scalar Time Series Analysis in Discriminating Preictal States. In-
ternationalJournalofBioelectromagnetism,Vol. 9(3),pp. 134–145,2007.
P6 A. Papana and D. Kugiumtzis. Evaluation of Histogram-Based Estimators
of Mutual Information in Time Series (in Greek). Proceedings of the 20th
GreekStatisticalConference,Nicosia,Cyprus,pp. 329–336,2007.
P7 A. Papana and D. Kugiumtzis. Evaluation of Mutual Information Estimators
onNonlinearDynamicSystems. NonlinearPhenomenainComplexSystems,
Vol. 11(2),pp. 225–232,2008.
P8 A. Papana and D. Kugiumtzis. Detection of Directionality of Information
Transfer in Nonlinear Dynamical Systems. Topics on Chaotic Systems, Se-
lected Papers from CHAOS 2008 International Conference, 3-6 June, Cha-
nia,Crete,pp. 251–264,WorldScientific.
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