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SIGNALIDENTIFICATIONANDFORECASTINGINNONSTATIONARY TIMESERIESDATA By i ^ DENG-SHANSHIAU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2001 Copyright2001 by Deng-ShanShiau Idedicatethisworktomyfamily. ACKNOWLEDGMENTS Aschairmanofmycommittee,Dr. MarkC.K.Yangguidedmethroughout mydissertation. Withouthisencouragementandsupport,thisworkwouldnever havebeencompleted. Iextendtohimmysinceregratitudeandwillremember himalways. IwouldalsoliketothankDr. PejaverV.Rao,Dr. RandyL.Carter, Dr. SamuelS.Wu,andDr. PanosM.Pardalosforservingonmydissertation committee. Dr. J.ChrisSaekellares,directoroftheBrainDynamicsLaboratoryatthe UniversityofFlorida,andDr. LeonD.lasemidis,associateprofessoratArizona StateUniversity,supportedandencouragedmethroughoutmyyearsatthe UniversityofFlorida. Ioffermysincerethankstothem. Iwishtoexpressmyspecialthankstomyfamily: mywifeShu-Jen,forher loveandpatience;andtoourdaughter,Marie,forbeingagloriousjoytous. Iam gratefultomyparentsandmother-in-lawfortheirencouragement. Finally,Iwouldliketothankallmycolleaguesandfriendsfortheir assistance. IV TABLEOFCONTENTS page ACKNOWLEDGMENTS iv LISTOFTABLES vii LISTOFFIGURES viii ABSTRACT x CHAPTERS 1 INTRODUCTION 1 1.1 ResearchScopeinTimeSeriesAnalysis 1 1.2 MotivationandPurposeofResearch 2 1.3 Outline 3 2 LITERATUREREVIEW 4 2.1 QuantificationofRegularityforTimeSeries: Approximate Entropy(ApEn) 4 2.2 AnalysisandForecastinginNonstationaryTimeSeries ... 6 2.2.1 ARJMAModels 6 2.2.2 TrendRemovingProcess 7 2.2.3 ARARMAModels 8 2.2.4 KalmanFilter 9 2.2.5 DynamicalStochasticApproximationMethod 10 2.2.6 AdaptiveForecastingMethod 12 2.2.7 WhittlePseudo-MaximumLikelihoodEstimation ... 13 2.3 AnalysisofWolf’sSunspotTimeSeries 13 3 PATTERNMATCHREGULARITYSCORES:ACOMPARISON WITHAPPROXIMATEENTROPY 16 3.1 ConceptofPatternMatch 16 3.2 PatternMatchRegularityScores 18 3.3 Example: AComparisonWithApEn 20 3.4 SummaryandDiscussion 30 4 PATTERNMATCHSIGNALIDENTIFICATION(PMSI)ALGO- RITHM 32 V 4.1 Introduction 32 4.2 PatternMatchSignalIdentification(PMSI)Algorithm.... 33 4.3 AnalyticProofforthePatternIdentifiabilitybyPMSIAlgo- rithm 38 4.4 ApplicationsofPMSIAlgorithm 67 4.4.1 SimulationStudies 67 4.4.2 ApplicationonEEGTimeSeries 72 4.4.3 ApplicationonSunspotTimeSeries 76 4.5 SummaryandDiscussion 78 5 FORECASTING 81 5.1 Method 81 5.2 ForecastinginEEGTimeSeries 82 5.3 ForecastinginWolf’sSunspotTimeSeries 88 5.4 SummaryandDiscussion 92 6 CONCLUSION 94 REFERENCES 97 BIOGRAPHICALSKETCH 100 vi , , LISTOFTABLES Table page 4.1 MinimumrequiredPeforidentifyinganypatternofembeddedsignals fordifferentselectionofliineachgivenIanda 59 4.2 Minimumrequiredvalueofpgforidentifyinganypatternofembed- dedsignalswithoptimalselectionofliinTable4.1 65 4.3 Resultsofthe20simulations(n=40,000ineachsimulaiton)fortest- ingthepatternidentifiabilitybythePMSIalgorithm 71 4.4 Patternidentificationanalysisina30-minuteEECtimeseries 73 4.5 Patternidentificationanalysisinsmoothedsunspottimeseries 78 5.1 AR(li)modelsfittedinthelearningperiodofanEEGtimeseriesfor Zi=5,6,... 10,whereu\=Ui~p,andpisthemeanvalueofU. . 83 5.2 Regressionequationsofthenext6pointsregressonthepreviousZi pointswhenthesubsequencesarepatternZi-matchedidentifiedby thePMSIalgorithminthelearningperiodofanEEGtimeseriesU. 84 5.3 Averagepredictionerrorsforthenext6pointsinthepredictionpe- riodofanEEGtimeseriesbyregressionequations,AR{li)andop- timalARmodelsfittedinthelearningperiod 87 5.4 Regressionequationsofthenextsixthpointregressonthepreviousli pointswhenthesubsequencesarepattern/i-matchedidentifiedby thePMSIalgorithminthelearningperiodofthesmoothedsunspot timeseriesU 89 5.5 AR{li)modelsfittedinthelearningperiodofsmoothedmonthlysunspot timeseriesforZi =6,... 9,whereu[ =Ui~pandpisthemean valueof17 89 5.6 Averagepredictionerrorsforthenextsixthpointintheprediction periodofsmoothedmonthlysunspottimeseriesbyregressionequa- tionsandARmodelsfittedinthelearningperiod 92 vii LISTOFFIGURES Figure page 3.1 Anexampleofgoodpatternmatchbutnotvaluematch 18 3.2 MIX(p)processfordifferentvaluesofp 22 3.3 NormalizedApEnversuspintheMIX(p)processfordifferentvalues ofI,givenr=0.18 24 3.4 NormalizedScorelversuspintheMIX(p)processfordifferentvalues of/,givenr=0.18 25 3.5 NormalizedScore2versuspintheMIX(p)processfordifferentvalues of/,givenr=0.18 26 3.6 NormalizedApEnversuspintheMIX(p)processfordifferentvalues ofr,given/=3 27 3.7 NormalizedScorelversuspintheMIX(p)processfordifferentvalues ofT,given/=3 28 3.8 NormalizedScore2versuspintheMIX(p)processfordifferentvalues ofT,given/=3 29 4.1 Examples(Z=7,l\=4,Z2=3)forpattern^-matches 36 4.2 Estimatedmatchingprobabilities(Pj’s)withlengthZ = 5anda = 0.0(whitenoise),0.4,0.7,1.0(randomwalk) 57 4.3 MinimumrequiredembeddingprobabilityPefordifferentcombina- tionsofZxandI2forgivenlengthofsignal1=7anda 60 4.4 MinimumrequiredembeddingprobabilityPefordifferentcombina- tionsofZiandI2forgivenlengthofsignalZ=8anda 60 4.5 ExamplesofestimatingtherequiredvalueofPe(givenapatternpZj) byplottingpeversusTSj—maxi{TSi)forZ=5anda=0.4,0.8in AE(1)timeseries 62 4.6 ExamplesofestimatingtherequiredvalueofPe(givenapatternpZ^) byplottingpeversusTSj—maxj(TS’j)forZ=7anda=0.4,0.8in A/2(l)timeseries 63 viii 4.7 Examples(I=6,a=0,0.4,0.7,1.0)oftheminimumrequiredvalue ofpefordifferentpatternsofembeddedsignals 64 4.8 MinimumrequiredvalueofPeforidentifyinganypatternofembed- dedsignals,wherelength/=6,7,8,9anda=0,0.1,...,1.0 66 4.9 Fourdifferentpatterns(pt[,ptl,pt\,andpt%)oftheembeddingsig- nalsfortestingthepatternidentifiabilitybythePMSIalgorithm. . 69 4.10 Fourdifferentpatterns(ptj, pt\)oftheembeddingsig- nalsinasimulatedunderlyingAR(1)timeseries(a=0.4) 70 4.11 TenminuteEEGtimeseriesdata 72 4.12 Tenidentifiedsignalsinthis30-minuteEEGtimeseriesusingli = 5,6andI2=& 74 4.13 Tenidentifiedsignalsinthis30-minuteEEGtimeseriesusingli = 7,8andI2=G 75 4.14 Thefirst1500points,from1749to1873,ofthesix-monthsmoothed Wolf’smonthlysunspottimeseries 76 4.15 Fiveidentifiedsignals(/i=6,7,8,9andI2=6)withthesamemost predictablepatternofthesix-monthsmoothedWolf’smonthlysunspot timeseries 77 5.1 Patternfi-matchedsubsequencesandtheir6-stepaheadprediction valuesinthepredictionperiodofanEEGtimeserieswhenli = 5and6 85 5.2 Pattern/i-matchedsubsequencesandtheir6-stepaheadprediction valuesinthepredictionperiodofanEEGtimeserieswhenli = 7and8 86 5.3 Pattern/i-matchedsubsequencesandtheirpredictionvaluesinthe predictionperiodofsmoothedmonthlysunspottimeserieswhen =6and7 90 5.4 Pattern/i-matchedsubsequencesandtheirpredictionvaluesinthe predictionperiodofsmoothedmonthlysunspottimeserieswhen /i=8and9 91 IX AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SIGNALIDENTIFICATIONANDFORECASTINGINNONSTATIONARY TIMESERIESDATA By Deng-ShanShiau December2001 Chair: MarkC.K.Yang MajorDepartment: Statistics Traditionaltimeseriesanalysisfocusesonfindingtheoptimalmodeltofit thedatainalearningperiodandusingthismodeltomakepredictionsinafuture period. However,manypracticalapplications,suchasearthquaketimeseriesor epilepticbrainelectroencephalogram(EEG)timeseries,mayonlycontainafew meaningful,orpredictablepatterns,whichcanbeusedformeaningfulforecasting suchastheoccurrencesofsomespecificeventsfollowingsimilarpatterns. Inthese cases,thetraditionaltimeseriesmodelsuchastheautoregressive{AR)model usuallygivespoorpredictionssincethemodelisconstructedtofittheentire learningperiod,whilethepatternusefulforpredictionmayoccurduringonlya smallportionoftheperiod. Thepurposeofthisresearchistoprovideastatisticalalgorithmtoidentify themostpredictablepatterninagiventimeseriesandtoapplythispatternto makepredictions. Inthisdissertation,weproposethePatternMatchSignalIdentification (PMSI)algorithmtoidentifythemostpredictablepatterninagiventimeseries. Inthisalgorithm,theconceptofthepatternmatchisusedinsteadofthegenerally X

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