Comparison of recurrence quantification methods for the analysis of temporal and spatial chaos
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Comparison of recurrence quantification methods for the analysis of temporal and spatial chaos
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Comparisonofrecurrencequantificationmethodsfortheanalysisoftemporalandspatialchaos
ChiaraMocenni ,AngeloFacchini,AntonioVicino
DipartimentodiIngegneriadell’InformazioneandCentroperloStudiodeiSistemiComplessi,UniversitàdiSiena,ViaRoma,56,53100-Siena,Italyarticleinfoabstract
Acomparativestudyoftherecurrencepropertiesoftimeseriesandtwo-dimensional
spatialdataisperformedbymeansofRecurrenceQuantificationAnalysis.Therecent
extensiontodistributeddataofmethodsbasedonrecurrencesrevealsnewinsights
improvingtheperformancesoftheapproachfortheanalysisofcomplexspatialpatterns.
Indeed,themeasuresdeterminismandentropyprovidesignificantinformationaboutthe
smallandlargescalecharacterizationofthepatternsallowingforabetterconnectionto
thephysicalpropertiesofthespatialsystemunderinvestigation.
©2010ElsevierLtd.Allrightsreserved.Articlehistory:Received9November2009Receivedinrevisedform25March2010Accepted13April2010Keywords:RecurrencePlotsGeneralizedRQA
Temporalandspatialchaos
1.Introduction
Chaoticphenomena,bothtemporalandspatial,areubiquitousinphysical,chemical,andbiologicalfieldsandhavebeentheobjectofintensiveinvestigationinthelastdecades.Thisextraordinaryefforthasproducedawidevarietyofapproachesbasedonnonlinearsystemstheoryandbifurcations.Basically,onecandistinguishbetweenchaosintheevolutionofatemporalvariableandchaosinthespatio-temporalevolutionofdistributedsystems,wherespatiallyextendedsystemsshowthespontaneousemergenceofspatialpatternslikeTuringstructures,travelingandspiralwaves,andturbulence
[1,2].
Inthecaseofthetemporalevolutionofchaoticsystems,whenthemodelofthesystemunderinvestigationisknown,techniquesbasedonnonlinearanalysis,bifurcationtheoryandcontinuationmethodsallowforanappropriatecharacterizationofnonlinearphenomena[3].Ontheotherhand,whenonlythetemporalvariationofsystemstatecanberecorded,apowerfulapproachistheanalysisoftimeseriesunderthepointofviewofdynamicalsystems,thesocalledembeddingtechnique.Thisapproachhasbeendevelopedintheframeworkofnonlineartimeseriesanalysisandconsistsinthereconstructionofthephasespacetrajectorystartingfromtheavailableobservations[4,5].
Whendealingwithspatiallydistributedsystems,GiererandMeinhardt[6]showedthatthecrucialconditionforpatternformationislocalself-enhancement(short-rangeactivatingeffect)andlong-rangeinhibition(depletionextendingoverawiderrange).Forexample,inthecaseofbiologicalpatterns,themodelingapproachisessentiallyrelatedtothepresenceofmultiscalephenomena[7].Inthiscasemostmodelsaredescribedbyasetofpartialdifferentialequationsofparabolictypeaccountingforreactionanddiffusionprocesses.Suchequationshavebeenwidelystudiedand,althoughanalyticalsolutionsarenotalwayseasilyavailable,themechanismsofpatternformationarewellknownfromthemathematicalpointofview[2].
Oneimportantopenproblemariseswhenstudyinganunknowndynamicalspatio-temporalsystemofwhich,onlypartialinformationisavailable,e.g.theobservationofoneorfewspatialstatevariables(oracombinationofthem)areknownandfewdatapointsareavailable.Inthiscasetheproblemofspace-statereconstructionandmodelidentificationofaspatio-temporaldynamicalsystemhasbeeninvestigatedintheframeworkoflatticedynamicalsystemsin[8],whileamethodCorrespondingauthor.
E-mailaddress:mocenni@dii.unisi.it(C.Mocenni).
0895-7177/$–seefrontmatter©2010ElsevierLtd.Allrightsreserved.
doi:10.1016/j.mcm.2010.04.008
2C.Mocennietal./MathematicalandComputerModelling()–
forspatialforecastingfromsinglesnapshotshasbeenproposedin[9].Insuchcasesonehastocopewiththeproblemofunderstandingthedynamicsofasystembyusingonlyalimitednumberofdata.Infact,forrealsystems,theequationsdescribingthesystemdynamicsareoftennotknownandtheproblemofpatternformationandanalysismaybetackledbyreconstructinginformationabouttheunderlyingdynamicalsystembyasetofmeasurementsoravailabledata.Insomecases,findingthevalueofsomeparametersofthesystembysolvingsuitableinverseproblemsispossiblebyestimatingstatisticalmodelsfromthedata:forexample,wavelengthandspeedoftravelingwavesinecologicalmodelshavebeenestimatedin[10].
Amethodologyforidentifying,analyzingandclassifyingcomplexpatterns,suchasTuringandturbulentpatterns,hasbeenproposedin[11,12]bytheauthorsofthepresentpaper.ThemethodconsistsintheextensionoftheRecurrencePlottoatwo-dimensionalspace[13]andtheRecurrenceQuantificationAnalysis(RQA)[14],usuallyappliedforthestudyofnonlineartimeseries.
TheaimofthispaperistogiveamoreformaldefinitionoftheabovemethodbymeansofaconsistentinterpretationoftherecurrencemeasuresDETandENTwhencomputedforspatiallydistributedsystems.Inparticular,lookingatthestatisticalpropertiesoftherecurrences,wecharacterizethemeasuresusedforidentifyingthepatternandanalyzethecorrelationbetweenthemathematicalformulationandtheirphysicalmeaning.Finally,acomparisonofsuchmeasureswiththeanalogoustimeseriescaseisperformedshowingthattheconceptsoftimerecurrencemethodsarenoteasilyextendabletohigher-dimensionaldatasets.
Thepaperisstructuredasfollows:Section2describesthemethodsbasedonrecurrencesbothfortemporalandspatialdata,introducingtheRecurrencePlot(RP),RecurrenceQuantificationAnalysis(RQA)andtheirapplicationstobiologicalsystems.InSection3wepresentthepropertiesofthestructuresintheRPandexplainthedifferentmeaningsinone-andtwo-dimensionalcases,focusingonthecharacteristicsofDETandENTmeasures.InSection4westateourconclusions.
2.Recurrencebasedmethods
TheconceptofrecurrenceisstrictlyrelatedtothetemporalevolutionofcomplexdynamicalsystemsandwasinitiallyintroducedbyPoincaré[15]fordynamicalsystems(thethreebodyproblem)andbyKac[16]fordiscretestochasticsystems.Inparticular,Poincaréstated:‘‘Inthiscase,neglectingsomeexceptionaltrajectories,theoccurrenceofwhichisinfinitelyimprobable,itcanbeshown,thatthesystemrecursinfinitelymanytimesascloseasonewishestoitsinitialstate’’.Fortimeseries,theconceptofrecurrencewasintroducedbyEckmannbymeansoftheRecurrencePlot,avisualtooldesignedtodisplayrecurringpatternsandtoinvestigatenonstationarypatterns[13].Inrecentyears,RPsfoundawiderangeofapplicationsinthetimeseriesanalysisofnonstationaryphenomena,suchasbiologicalsystems[17–20],speechanalysis
[21,22],financialtimeseries[23],andearthsciences[24,25].ThepopularityofRPsliesinthefactthattheirstructureisvisuallyappealingandallowsfortheinvestigationofhigh-dimensionaldynamicsbylookingatasimpletwo-dimensionalplot.BymeansoftheRecurrenceQuantificationAnalysis[26],theRPcanbeusedasatoolfortheexplorationofbifurcationphenomenaandchangesinthedynamicsalsoinnonstationaryandshorttimeseries.Someauthorshavealsorelatedthequantificationmeasurestoinvariantsofthephasespace,likethecorrelationdimensionandthelargestLyapunovexponent[27].
InthefollowingsubsectionweprovidebasicnotionsonRPfortimeseries(foradeepertreatmentthereaderisreferredto[14]).
2.1.RecurrencePlotfortimeseriesanalysis
TheRecurrencePlotisatwo-dimensionalbinarydiagramrepresentingtherecurrencesthatoccurinanm-dimensionalphasespacewithinanarbitrarilydefinedthresholdεatdifferenttimesi,j.TheRPiseasilyexpressedasatwo-dimensionalsquarematrixwithonesandzerosrepresentingtheoccurrence(ones)ornot(zeros)ofstates xiand xjofthesystem:
Rij=Θ(ε xi xj ), xi∈Rm,i,j=1,...,N,(1)whereNisthenumberofmeasuredstates xi,Θ(·)isthestepfunction,and · isanorm.Inthegraphicalrepresentation,eachnon-zeroentryofRi,jismarkedbyablackdotintheposition(i,j).Sinceanystateisrecurrentwithitself,theRPmatrixfulfillsRi,i=1andhenceitcontainsthediagonalLineofIdentity(LOI).TocomputeanRP,http://wendang.chazidian.comuallythel∞normisused,becauseitisindependentofthephasespacedimensionandnorescalingofεisrequired.Furthermore,specialattentionmustbepaidtothechoiceofthethresholdε.Althoughthereisnotageneralrulefortheestimationofε,http://wendang.chazidian.comually,εischosenasapercentageofthediameterofthereconstructedtrajectoryinthephasespace,notgreaterthan10%[14].
AnRPischaracterizedbytypicalpatterns,whosestructureishelpfulforunderstandingtheunderlyingdynamicsofthetimeseries.Thesepatternscanbeclassifiedaccordingtotwofeatures:typologyandtextures.TypologycatchestheglobalappearanceoftheRP,andallowsforafirstunderstandingoftheRP.Ahomogeneousdistributionofpointsisusuallyassociatedwithstationarystochasticprocesses,e.g.Gaussianoruniformwhitenoise.Periodicstructures,likelongdiagonallinesparalleltotheLOIindicateperiodicbehaviors,whiledriftsinthestructureoftherecurrencesareoftenduetoaslowvariationofsomeparameteroftheunderlyingsystemandwhiteareasorbandsindicatenonstationarityandabrupt
C.Mocennietal./MathematicalandComputerModelling(–
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ab
d
ef
Fig.1.(Coloronline)ExampleofRecurrencePlots.(a)and(b)Uniformwhitenoise;(c)and(d)periodicsignal;(e)and(f)thelogisticmap.StructuresinRPs(b),(d),(f)reflectthenatureofthesignal:Noisemainlyproducesisolatedpointsandfewshortlines;PeriodicsignalsproducelonglinesparalleltotheLOIspacedbytheperiodofthesignal;Chaoticsignalsproducecomplicatedpatternsmadeofisolatedpoints,shortandlonglines,indicatingacomplexstructureoftherecurrences.
changesinthedynamics.Recently,curvedmacrostructureshavebeenrelatedtoverysmallfrequencyvariationsinperiodicsignals[28].
ThetexturescatchthelocalstructuresformingthepatternsintheRP.Theymaybe:(a)Singlepoints,ifthestatedoesnotpersistforalongtime;(b)Diagonallinesoflengthl,indicatingthatthetrajectoryvisitsthesameportionofthephasespaceatdifferenttimes;(c)Verticalandhorizontallines,indicatingthatthestatechangesveryslowlyintime.Becauseofthescreenresolutionandthelengthofthetimeseries,itisdifficulttoanalyzetheRPonlybymeansofvisualinspection(whichisanywayusefultodetect,e.g.simplenonstationarities).Tocopewiththisproblem,theRQAoffersasetofindicatorscomputedonthestructuresoftheRP(isolatedpoints,verticalandhorizontallines).Fig.1showshowtypicalRPslook:(a)–(b)Whitenoise,(c)–(d)periodicsignal,and(e)–(f)chaoticlogisticmap.Asonecansee,theRPofwhitenoisemainlyshowsisolatedblackpointsandfewshortlines,whilelongdiagonallinesaretypicalofperiodicsignals.Chaoticsystemsarecharacterizedbythedistributionofdiagonallinesofdifferentlengths.
2.2.Extensiontospatiallydistributedsystems
Inthecaseofad-dimensionaldataset,theRPisspecifiedby[29]:
R x x , =Θ(ε )(2)where =i1,i2,...,idisthed-dimensionalcoordinatevectorand x istheassociatedphasespacevector.ThisRP,calledGeneralizedRecurrencePlot,accountsforrecurrencesbetweenthed-dimensionalstatevectorsandpresentsaLOIreplacedbyalinearmanifoldofdimensiondforwhichR = . , =1,
Weconsidertwo-dimensionalspatiallydistributedsystemsatacertain(fixed)timeoftheirevolution.Inthisparticularcased=2andthediscretizedsolutionsofthecorrespondingdistributedsystemcanbevisualizedasanimage,i.e.atwo-dimensionalobjectcomposedofscalarvalues,forwhichtheGRPreads:
Ri1,i2,j1,j2=Θ(ε xi1,i2 xj1,j2 )ik,jk∈Nx(·,·)∈R.(3)Eachblackdotrepresentsaspatialrecurrencebetweentwopixels,andeverypixelisidentifiedbyitscoordinates(i1,i2),beingi1andi2therowandthecolumnindex,respectively.Inthiscase,theRecurrencePlotisfour-dimensionalandtheLOIisgeneralizedbyatwo-dimensionalidentityplane,definedbysettingi1=j1andi2=j2.
4C.Mocennietal./MathematicalandComputerModelling()–
Avisualinspectionofthefour-dimensionalRPispossibleonlybyprojectionsinthreeortwodimensions.Althoughthisispossible(seee.g.[29],page548),relevantinformationishardtoextract,andonemustcopewiththefactthatGeneralizedRecurrencePlotslosetheirvisualappeal.Despitethisdrawback,wecanstilltalkofRecurrencePlots,sincethefour-dimensionalspaceisfilledbyblackdotsassociatedwithspatialrecurrencesbetweentheimage’spixels.Furthermore,RQAcanbeeasilygeneralizedasdescribedinSection3.TheGRPhasbeenexploitedbyMarwanfortheassessmentofthetrabecularbonestructuresunderdifferenthealthconditions[29],andrecentlybytheauthorsofthispaperforthecharacterizationofspatiallydistributedsystems[11]andinvestigationofpatternformationinchemicalsystems[12].Afurtherapplicationofthismethodtotheanalysisofbifurcationsinspatiallydistributedsystemshasbeenrecentlyproposedin[30].
2.3.RecurrenceQuantificationAnalysis
TheRPwasinitiallyintroducedasavisualtool.Thisapproachisclearlynotappropriatewhenanalyzinglargedatasets,forwhich,thevisualizationoftheRPonthescreenisaffectedbythescreenresolution.Tocopewiththisproblem,ZbilutandWebber[26]introducedthequantificationofthestructurespresentintheRP.
FocusingonisolatedpointsandlinesparalleltotheLOI,onecandefineseveralindicators,themostimportantofwhicharetheRecurrenceRateRR,theDeterminismDET,andtheEntropyENT.TheyarecomputedonthebasisofthedistributionofthediagonallineslengthP(l).1
TheRRisdefinedas:
RR=N1
N2Ri,j=N1
i,jN2
l=1lP(l),(4)
inthecaseoftimeseries,and
RR=1
N4N
i1,i2,j1,j2Ri1,i2,j1,j2=N1 N4l=1lP(l),(5)
inthecaseofspatialdata.TherecurrenceRaterepresentsthefractionofrecurrentpointswithrespecttothetotalnumberofpossiblerecurrences.ItisadensitymeasureoftheRP.
TheDeterminism,definedas:
N lP(l)
DET=l=lmin
N
l=1,(6)lP(l)
isthefractionofrecurrentpointsformingdiagonalstructureswithaminimumlengthlminwithrespecttoalltherecurrences.Thereisnotaspecificguidelineforchoosinglmin:fortimeseriesitusuallycorrespondstothefirstminimumoftheautocorrelationfunction;forspatialdataitreferstotheminimumlengthscaletobeconsideredforthepatternanalysis.Choosingtoolargeortoosmallvaluesoflminwillintroducebiasesintotherecurrencemeasures,e.g.lmin=1yieldsDET=100%.Toourbestknowledge,avalueoflmin=4isusuallyanappropriatechoice.DETprovidesameasureofthepredictabilityofthesystem,becauseitaccountsforthediagonalstructuresintheRP.HighvaluesofDETmeanthattherecurrencepointsaremainlyorganizedindiagonallines.
Inthecaseoftimeseries,alineoflengthlindicatesthat,forltimesteps,thetrajectoryinthephasespacevisitsthesameregionatdifferenttimes(seealsoFig.2).Inthecaseofspatialdata,alineoflengthlmeansthattwodiagonalregionsoflengthlwithpixelvalueswithinathresholdεarepresentintheimage(seeFig.3).
TheEntropy,definedas:
ENT= N
l=lminp(l)logp(l),p(l)=P(l)N
l=lmin,(7)P(l)
isacomplexitymeasureofthedistributionofthediagonallinesintheRP.ItreferstotheShannonentropywithrespecttotheprobabilityoffindingadiagonallineofexactlylengthl.Forperiodicsignalsoruncorrelatednoisethevalueissmall,whileforchaoticsystemsishigher.Inthecaseofspatiallydistributedsystems,ENTiscorrelatedtothefinestructureoftheimage.ThecomputationofthemeasuresbasedonthediagonallinesandtheirdistributionprovidesvaluableinformationaboutthestructureoftheRP.
1ThedistributionP(l)maybeknownaprioriorcomputednumericallybyconsideringallthelinesofdifferentlengthpresentintheRP.
C.Mocennietal./MathematicalandComputerModelling(–5
Fig.2.(Coloronline)DefinitionofalineoflengthLinthecaseoftimeseries.AlineoflengthLisformedwhentwotrajectoriesvisitthesameregionofthephasespaceandliewithinadistanceεforexactlyLtimesteps.
Fig.3.(Coloronline)Definitionoflinesandisolatedpointsfortwo-dimensionalspatialdata:iandjarethehorizontalandverticallabelsofthepixels.Anisolatedpoint(lineoflengthzero)isformedbythepoints(jm,in)and(jm ,in ),whilealineoflength3isformedby(jk,ih)and(jk ,ih ).
ForadeepertreatmentofRQAmethodswithintheframeworkofnonlineartimeseriesanalysisthereaderisreferredto[14],whilefortheapplicationofRQAtospatialsystems(GRQA)referto[11,29].
2.4.Applicationtobiophysicalsystems
RecurrenceQuantificationAnalysiscanbefruitfullyexploitedwhendealingwithecologicalsystems.Ingeneral,whentryingtomodelthetemporalevolutionofphysico-chemicalvariablesofecologicalsystems,onemustcopewiththefactthattheclassofrealisticmodelsisoftenwideandcomplicated,andtheidentificationoftherightmodeloftenneedssomeaprioriknowledgeaboutthesystem.Furthermore,identificationtechniquesneedtherecordingoftimeseries,which,inthecaseofecologicalsystems,areoftenshort,nonstationaryandcorruptedbynoise.Therefore,providingapre-analysistoolabletocatchvisualandquantitativeinformationcanbecomeveryuseful.
TheproblemofdealingwithsuchtimeseriesiseasilysolvedbyRPsandRQA,asshownforthecaseoftheOrbetelloLagoon[31]:theapplicationofRQArevealedatransitionintheoscillationsofdissolvedoxygenjustbeforetheonsetofananoxiccrisis.Thismethodwasalsoproposedformonitoringregimeshiftsinenvironmentaltimeseries,likelakeeutrophicationandseawateroxygenvariabilityincoastalregions[32].
Physiologyandmedicinearealsofieldswheretheapplicationofrecurrencestrategiesleadtointerestingresults.Inthecaseofelectrocardiogramrecordings,Zbilutetal.[33]showedthatRQAcanbeusefulinthedetectionofstatechangesinthecardiacrhythmwhentraditionaltechniquesfail.Regardingelectroencephalogramrecordings,Marwanfoundchaos–chaostransitionsinmultivariaterecordingsofEventRelatedPotentials[34].
Theapplicationofrecurrencestrategiesinthefieldofspatiallydistributedsystemisrecent,andonlyfewworkscanbefoundintheliterature.Amongthem,GRQAwasusedinthebiomedicalfieldtoassessthetrabecularbonestructureinpatientwithosteoporosis[29],andintheanalysisofTuringpatternformationintheBelousovZhabotinskyreactionperformedinawater–oilmicro-emulsion[35].Inthislastcase,itwasfoundthattheGRQAparametersallowedforthedetectionofthedifferentroutestopatternformation[12].
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