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RESEARCHARTICLEOpenAccessCommunicationpatternsinapsychotherapyfollowingtraumaticbraininjury:AquantitativecasestudybasedonsymbolicdynamicsPaulERapp1*,ChristopherJCellucci2,AdeleMKGilpin3,4,MiguelAJiménez-Montao5andKathrynEKorslund6AbstractBackground:Theroleofpsychotherapyinthetreatmentoftraumaticbraininjuryisreceivingincreasedattention.
Theevaluationofpsychotherapywiththesepatientshasbeenconductedlargelyintheabsenceofquantitativedataconcerningthetherapyitself.
Quantitativemethodsforcharacterizingthesequence-sensitivestructureofpatient-therapistcommunicationarenowbeingdevelopedwiththeobjectiveofimprovingtheeffectivenessofpsychotherapyfollowingtraumaticbraininjury.
Methods:Thecontentofthreetherapysessiontranscripts(sessionswereseparatedbyfourmonths)obtainedfromapatientwithahistoryofseveralmotorvehicleaccidentswhowasreceivingdialecticalbehaviortherapywasscoredandanalyzedusingmethodsderivedfromthemathematicaltheoryofsymbolicdynamics.
Results:Theanalysisofsymbolfrequencieswaslargelyuninformative.
Whenrepeatedtripleswereexaminedamarkedpatternofchangeincontentwasobservedoverthethreesessions.
ThecontextfreegrammarcomplexityandtheLempel-Zivcomplexitywerecalculatedforeachtherapysession.
Forbothmeasures,therateofcomplexitygeneration,expressedasbitsperminute,increasedlongitudinallyduringthecourseoftherapy.
Thebetween-sessionincreasesincomplexitygenerationratesareconsistentwithcalculationsofmutualinformation.
Takentogethertheseresultsindicatethattherewasaquantifiableincreaseinthevariabilityofpatient-therapistverbalbehaviorduringthecourseoftherapy.
Comparisonofcomplexityvaluesagainstvaluesobtainedfromequiprobablerandomsurrogatesestablishedthepresenceofanonrandomstructureinpatient-therapistdialog(P=.
002).
Conclusions:Whilerecognizingthatonlylimitedconclusionscanbebasedonacasehistory,itcanbenotedthatthesequantitativeobservationsareconsistentwithqualitativeclinicalobservationsofincreasesintheflexibilityofdiscourseduringtherapy.
Theseprocedurescanbeofparticularvalueintheexaminationoftherapiesfollowingtraumaticbraininjurybecause,insomepresentations,thesetherapiesarecomplicatedbydeficitsthatresultinsubtledistortionsoflanguagethatproducesignificantpost-injurysocialimpairment.
Independentlyofthemathematicalanalysisappliedtotheinvestigationoftherapy-generatedsymbolsequences,ourexperiencesuggeststhattheprocedurespresentedhereareofvalueintrainingtherapists.
Keywords:traumaticbraininjury,psychotherapy,psychoanalysis,complexity,mutualinformation,entropy,infor-mationtheory,symbolicdynamics*Correspondence:prapp@usuhs.
mil1DepartmentofMilitaryandEmergencyMedicine,UniformedServicesUniversity,4301JonesBridgeRoad,Bethesda,MD20814,USAFulllistofauthorinformationisavailableattheendofthearticleRappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/1192011Rappetal;licenseeBioMedCentralLtd.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense(http://creativecommons.
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BackgroundTraumaticbraininjuryisasignificantcauseofacuteandlong-termdisability.
Neurobehavioralsequelaeencom-passcognitive,socialandpsychiatricdomains.
Majordepressivedisorderisthemostprevalentpsychiatricdis-orderfollowingtraumaticbraininjuryregardlessoftheseverityoftheinjury[1-9].
Estimatesofprevalencearehighlyvaried.
Iverson,etal.
[10]reviewedsixstudiesofdepressionfollowingtraumaticbraininjuryandfoundreportsofprevalencerangingfrom12%to44%.
Whileprevalenceratesareuncertain,acriticalconclusioncanbemade.
Thetreatmentofneuropsychiatricdisordersfollowingtraumaticbraininjuryisasignificantclinicalneedthatpresentsuniqueclinicalchallenges.
Ascommonlyconceptualized,theclinicalresponsetotraumaticbraininjuryhasfourcomponents:neuropro-tection(preserveinjuredneurons),plasticmodification(reconstructneuralnetworkswithsurvivingneuronsbypromotingdendriticarborizationandsynaptogenesis),neurogenesis(stimulatethematurationofnewneuronsfromstemcellpopulations),andneurointegration(facili-tatetheintegrationofnewlyformedneuronsintothecentralnervoussystem).
Itisincreasinglyrecognized,however,thatpsychotherapyisanimportantcomple-menttothisneurologicalresponse.
Cope[11]hasarguedthat"themajorityofrecoveringsurvivorsofTBIarenowseenaspotentiallybenefitingfromsomeformofpsy-chotherapeutic/rehabilitationtreatment.
"Nonetheless,mostindividualsexperiencingaheadinjurydonotreceivepsychotherapy.
InareviewoftheearlyhistoryofpsychotherapyfollowingTBI,Prigatano[12]addressedthequestion,"WhyhastheroleofpsychotherapeuticinterventionsintherehabilitationcareofTBIpatientsgoneunrecognized"Hesuggeststhat"theanswerseemstolieintheassumptionthatTBIpatientscouldnotbene-fitfrompsychotherapybecauseoftheirpermanentcogni-tive,linguisticandaffectivedisturbances.
"WhilethisargumentmightbeadvancedwhenconsideringsevereTBI,itdoesnotseemplausibleincasesofmildTBI.
ButisitevenapplicableinthecaseofsevereTBIResultsreportedbyBen-Yishayetal.
[13]andbyEzrachi,etal.
[14]indicatethatpsychotherapyfollowingmoderateorsevereTBIhasapositiveeffectonpost-injuryemployment.
Whilepsychotherapyisthepreferredapproachtothetreatmentofmooddisordersfollowingtraumaticbraininjury[1,2,15-17]thereislimitedresearchtohelpguidetheselectionofthespecifictherapeuticmethod[18,19].
Theheterogeneityofthispopulationdemandsavariedresponse.
Inpart,theappropriatetherapywillbedeter-minedbythephysicalinjury,particularlytheresidualneu-rologicalandcognitivedeficits.
IndividualswithTBImaybenefitfromtreatmentsthattakepost-injurycognitivedistortionsintoaccount[20-22].
Thechoiceoftherapyshouldalsoberesponsivetopre-injurypsychopathology[23,24].
Thereisanemergingliteraturedetailingthebene-fitsofcognitivebehaviortherapyacrossavarietyofmedi-calpatientswithacquiredbraininjuriesofvariousseveritiescomorbidwithmooddisorders[15-18,25,26].
Psychodynamicpsychotherapyhasalsobeenconsidered.
Whilecognitivedeficitsfollowingheadinjurycanlimittheindividual'sabilitytoprofitfrompsychodynamicpsy-chotherapy,thisisnotinvariablythecase.
AsLewisandRosenberg[27]observedinapaperdescribingpsychoana-lyticpsychotherapyfollowingbraininjury,"theoverridingprinciplethatguidessuchpsychotherapeuticworkisthatacquiredbrainlesionsdonotablatethepatient'spsycheorunconscious.
"Theseauthorshaveidentifiedfivecriteriathatcanhelpidentifycandidatesforpsychoanalyticpsy-chotherapyfollowingbraininjury.
(1.
)Thepatientmustbemotivatedtoenterandremainintherapy.
(2.
)Patientswhohavehadatleastonepositivesignificantrelationshipinthepastarebetterabletoformatherapeuticalliance.
(3.
)Patientswhohavehadprevioussuccessesresultingfromactiveeffortaremorelikelytobenefitfromindivi-dualtherapy.
(4.
)Patientsinextremepsychologicaldistressmayrequireamoresupportiveintervention,includinghospitalization,beforeinitiatingpsychoanalytictherapy.
(5.
)Thedegreeandformofbraininjurycanaffecttheappropriatenessofanalytictreatment.
Patientswithsignifi-cantexpressiveorreceptivelanguagedeficitsarenotappropriatecandidates.
Inadditiontooutliningthepoten-tialbenefitsofapsychodynamicallyorientedtherapyforappropriatelyselectedpatients,LewisandRosenbergmaketwopointsthataregenericallyapplicabletotheconsidera-tionofpsychotherapyfollowingtraumaticbraininjury.
First,unaddressedpsychologicalproblemscanbeanimpe-dimenttomeaningfulparticipationinphysical,cognitiveandoccupationalrehabilitation,thusprovidinganaddi-tionalargumentforincludingpsychotherapyinthetreat-mentofsomepresentationsoftraumaticbraininjury.
Second,thepatient'salteredexperienceofselfshouldnotbeviewedasanentirelyneurologicalsymptom.
Braininju-rieshavepsychologicalmeaning.
"Althoughsuchdisruptions(braininjury)cansignifi-cantlyaffectthepatient'sself-esteem,andoftenrepre-sentamajorfocusforfamilywork,theymayrepresentamorebasicandprofounddisturbanceinthepatients'senseofself.
Thatis,beyondtheirdifficultiesinper-formingsocialroles,patientsalsostrugglewiththemorefundamentalquestionofwhotheyare;thebraininjuryappearstodisruptseverelytheirpreviouslyacquiredself-imageandsenseofself[28].
Thus,aprimarytaskofpsychotherapyistohelpthepatientconsolidateanewsenseofselfthatsuccessfullyRappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page2of28incorporatesarealisticappraisalofstrengthsandweak-nesses"[27].
Inpresentationswherethisalterationofsenseofselfisasignificantelementoftheclinicalpresentationandthepatienthassufficientegodevelopmenttotolerateaninsightdirectedtherapy,apsychodynamicallyinformedtherapyisindicated.
OnreviewingpsychotherapiesappropriateforTBIpatients,Folzer[29]madethefollowingobservation,"If'immature'defensesandcopingpatternsareremovedtooearly,thetherapistmayprecipitateacatastrophe.
Insteadofdirectlyconfrontingthepatient,thetherapistcanintro-ducethefocusonrealitygradually.
"Thiswouldargueforasupportivetherapy[30]insteadofinsight-orientedther-apy.
Thereisnot,however,astrictdivisionbetweentheseformsoftherapy.
AsWerman[30]observed,"Althoughinthefollowingpagesthesetwoformsoftreatment(suppor-tivetherapyandinsight-orientedtherapy)arecomparedasiftheywerenotonlydifferentfromeachotherbutvir-tuallydichotomousintheiraimsandtechniques,inrealitytheyrarelyexistinpureforms.
Typically,overaperiodoftime,mostpatientsinsupportivepsychotherapygainsomeinsightintotheirbehavior;similarlyitisdifficulttoconceiveofacourseofinsight-orientedpsychotherapyinwhichsomesupportivemeasuresarenotutilized.
"Psychotherapyfollowingtraumaticbraininjuryshouldnotnecessarilybelimitedtoindividualtherapy.
SeveralauthorshaveemphasizedthevalueofgrouptherapywithTBIpatients[29,31,32],andfamilyinvolvementintherapycanbeparticularlyimportant[12,23].
ThediscussionofpsychotherapywithTBIpatientsandindeedpsychotherapyingeneralhasbeenconductedlargelyintheabsenceofquantitativedataconcerningthetherapyitself.
Whilestandardizedinstrumentsforasses-singbaselinesymptomsandtreatmentoutcomesareincreasinglybeingusedinclinicalresearch[33],theseinstrumentsdonotquantifythefinestructureofthether-apeuticinteraction.
Thiscontributioncontinuesthedevel-opmentofquantitativemethodsforthecharacterizationofpatient-therapistcommunicationwiththelongtermobjectiveofimprovingtheeffectivenessofpsychotherapyfollowingtraumaticbraininjury.
Communicationbetweenpatientsandtherapistduringpsychotherapyhasmanycomponentsincludingposture,eyecontact,verbaltone,verbalproduction(thenumberofwordsexchangedirre-spectiveoftheirmeaning)andthemanifestcontentofthecommunication.
Alloftheseinteractionscanbeexaminedquantitatively[34,35].
Forexamplenon-verbalcommuni-cationinthetherapist-patientinteractionhasbeenana-lyzedbyYaynal-Reymond,etal.
[36]andbyMertenandSchwab[37]usingaformofquantificationdevelopedbyMagnusson[38,39].
Whileallcomponentsofpatient-therapistcommunicationareimportant,thispaperfocusesoncontentanalysis.
Usingmethodsofsymbolicdynamicsthisinvestigationextendspreviousanalysesofthefrequencyofcontentbyquantifyingthetemporallydepen-dent,sequence-sensitivestructureofthedialog.
Aslong-termgoals,thequestionsaddressedinthisresearchprogramfollowthoseenumeratedinRapp,etal.
[40].
1.
Aretherenonrandompatternsinthesequentialstructureofpatient-therapistcommunication2.
Dothesepatterns,shouldtheyexist,changeduringthecourseoftherapy3.
Dochangesinthepatternsofpatient-therapistcommunicationcorrelatewiththeclinicallyperceivedsuccessorfailureofthetherapy4.
CanthistypeofanalysisidentifymoreeffectiveformsoftherapistinterventionThiscasestudyislimitedtoanexaminationofthefirstthreequestionsinthreetherapysessionsrecordedfromonepatient.
Generalizedconclusionscannotthereforebemade.
Thelimitedresultsdo,however,indicatethatthereisanonrandomstructureinpatient-therapistcommunica-tionintheseprotocols.
Additionally,quantifiablestruc-tureschangedduringthecourseoftherapyinamannerthatcorrelatedwiththeclinicallyperceivedsuccessofthetherapy.
Quantitativeinvestigationsofpatient-therapistcommunication:PriorResearchAfirstapproachtoquantitativecontentanalysisisthedeterminationofwordfrequency.
AnearlyeffortwasElec-tronicVerbalAnalysis[41]measuringthefrequencyofanxietyrelatedwords.
Inasubsequentstudy,Pennebaker,etal.
[42,43]recordedthefrequencyof2800wordsthatwereplacedintosevencategories,andHart[44]analyzedpoliticaltextswithalibraryof10,000wordsinfiveclasseswithapproximatelysevencategoriesineachclass.
Thelimitationsoftheseanalysesareclear.
Wordfrequencyisinsensitivetocontext.
Arandomlyshuffledtextwillpro-ducedthesamewordcounts.
AsFastandFunder[45]observe,forexample,thephrase"Iamnothappy"maybescoredaspositiveemotionalcontent.
Severalinvestigatorshavedevelopedmethodsthatmovebeyondwordfrequencytoexaminemeaning.
ApioneerinthiseffortwasHartvigDahlwhoseinvestiga-tionofthecaseofMrs.
Canalyzed1,114psychoanalyticsessionswiththesamepatient[46-48].
Inthe1974study[48],entriesinathreethousandworddictionarywereassignedtoonedenotativecategoryandtooneormoreconnotativecategories.
Afactoranalysiswasusedtoidentifygroupsofrelatedwords,anditwasshownthatthesegroupswererelatedtothemespresentinthetran-script.
In1978Dahl,etal.
[49]publishedanapplicationoflinguisticanalysisinpsychotherapythatisintermedi-atetoanalysisofwordcountandtheanalysisofsequen-tialstructurebasedonsymbolicdynamicspresentedinRappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page3of28thenextsection.
Inthisstudy,theanalysiswaslimitedtoanexaminationofthetherapist'sinterventions.
Thispro-videsaninstructiveandvaluablealternativetotheprac-ticeofconsideringonlythepatient'sspeech.
Eachinterventionbythetherapistwasclassifiedbytypeandratedonscalesdesignedtoassesscountertransferencemanifestations,includinghostility,seductiveness,approval,disapprovalandassertionofauthority.
Aquali-tativelinguisticanalysisbasedonChomsky'smodeloftransformationalgrammar[50,51]wasalsoimplemented.
Dahlandhiscolleagueshypothesizedthat"aspeakerhasavailableavarietyofsyntacticoptions,andtheparticularsyntacticstructurewhichhechoosesreflects,amongotherthings,theinventoryofwishesthatheismotivatedbothtoconcealandtoexpress.
"Theanalysisofexamplespresentedinthispapershowsoccasionsofextraposition,pseudocleftconstruction,syntacticambiguityandlexicalambiguityconsistentwiththishypothesis.
IntheGottschalk-Gleseranalysisprocedure[52,53],thegrammaticalclauseistheunitofanalysis.
Contentisscoredonsevenscales.
Inadditiontothestudyofpsy-chotherapy,Gottschalk-Gleserconstantanalysishasbeenappliedinmedicalpsychology[54-58].
GBSoft-waremarketsasoftwareproduct,PCAD2000,thatappliesaGottschalk-Glesercontentanalysistomachinereadabletext.
Inadditiontoderivingscoresonsevenscales,theprogramoffersaneuropsychiatricclassifica-tionbasedontheDSM-IV.
Langsandcolleagues[40,59]analyzedeachelementoftherapytranscriptsonfourteendimensions.
Theresultisacontentmatrixoffourteencolumns.
Theanalysisincludedcalculationsofthefrequencyofeachentry,Shannoninformationofeachcolumnandthecontextfreegrammarcomplexity(Jiménez-Montao,[60]describedinthenextsectionofthispaperandinAppen-dixOne).
Inthe1991study[40],twoone-hourprotocolsobtainedfromthesamepatientwithdifferenttherapistwereanalyzedbythisprocedure.
Onetherapistwasaclassicallytrainedpsychoanalyst.
TheothertherapistusedacommunicativeapproachdevelopedbyLangs[61].
Themostnotabledifferencesbetweenthetwopro-tocolswerethefrequencyofscoresforthevariablechar-acterizingthesphereofreference(1=therapyrelated,2=situationsoutsideoftherapy,3=referencetotherapyandsituationsoutsideoftherapy,4=unclear).
Inthecaseoftheanalyst,90%ofthematerialreferredtositua-tionsoutsideoftherapyandlessthan1%referredtother-apyrelatedissues.
Inthecaseofthecommunicativetherapist,20%ofthematerialfocusedonthetherapeuticsituation.
Giventhefocusonthepatient-therapistrela-tionshipincommunicativepsychotherapy,thisobserva-tionisconsistentwiththerapistexpectations.
StilesVerbalModeAnalysis[62-64]couldbedescribedasastatementclassificationmethod.
Theunitanalyzedisan"utterance"(definedpresently).
Eachunitiscodedintooneofeightclassesbyasequenceofthreeforced-choicequestions.
Eightverbalresponsemodesresult.
Theanalysiscontinueswithacalculationofthefrequencyofeachclass.
VerbalModeAnalysisisconsideredatgreaterlengthintheDiscussionsectionofthispaper.
Investigatorshavealsoexaminedthenarrativespeechofclinicalpopulationsusingsymbolicdynamics.
Incon-trastwiththeresearchdescribedabove,thesestudiesdonotexaminepatienttherapistcommunication.
Rathertheyexaminethesequence-sensitivestructureofcontin-uousnarrativeselicitedbythequestion,"Canyoutellmethestoryofyourlife"[65,66]oranarrativepro-ducedbyaparticipantinresponsetoarequesttorecallthecontentofastorythattheyhavejustread[67].
TheLeroy,etal.
[67]studyinvestigatedthesequence-sensitivestructureofarecallnarrativepresentedbyschi-zophrenicpatients.
FollowingKintschandVanDijk[68,69],theparticipant'snarrativewastreatedasasequenceofpropositions.
TheKintschandVanDijkdefi-nitionofapropositionistheminimalsemanticunitthatcanbeeithertrueorfalse.
Propositionswereclassifiedasmacro-propositionsthatspecifythetopicofdiscourseormicro-propositionsthatprovidedetails.
Macro-proposi-tionswereassignedthesymbol"M,"andmicro-proposi-tionswereassignedthesymbol"m.
"thenarrativesamplewasthusrecastasanorderedsequenceofM'sandm's.
Entropy,Lempel-Zivcomplexityandthefirstordertransi-tionmatrixwerecalculated.
Comparisonswithsurrogatedataestablishedthepresenceofasequence-sensitivenon-randomstructureinthedata.
Theglobalcomplexityofrecalldidnotdifferforcontrolandschizophrenicpartici-pants.
Therewas,however,adifferenceinthetransitionstructure.
Thereweremoremicro-propositionstomicro-propositiontransitionsinschizophrenicnarratives.
InDoba,etal.
[65]autobiographicalspeechofanorexicswasparsedinto5secondepochs.
Eachepochwasassignedoneoffoursymbolscorrespondingtonegativeemotion,positiveemotion,neutralemotionandsilence.
Inadditiontodistribution-determinedmeasures,theLempel-Zivcomplexityandthefirstordertransitionmatrixwereexamined.
Complexitycalculationswithsurrogatedataestablishedthepresenceofanon-randomsequentialstructureinthenarratives.
Inanorexics,dynamicalmea-suresidentifiedrecurrentcyclesbetweenexpressionsofnegativeemotionandsilencethatwerelessprominentinthecontrolpopulation.
Inasubsequentstudy[66],thesametranscriptswereanalyzedwithadifferentscoringsystem.
Fivesymbolswereused(familyrelations,socialrelations,loverelations,self-referenceandsilence).
Calcu-lationofmutualinformationwiththeoriginalsymbolsequencesandsurrogatedatasetsagainestablishedthepresenceofanon-randomdynamicalstructureintheRappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page4of28narrative.
Theexaminationofthesummedprobabilitycurrents,ameasurederivedfromthefirstordertransitionmatrix,demonstratedthatthenarrativesofanorexicsareclosertostatisticalequilibriumthanthenarrativesofcontrols.
MethodsPatientHistoryInthisstudy,wedescribetheanalysisofthreetherapysessions(eachseparatedbyfourmonths)conductedwiththesamepatient(female,32yearsofage)bythesametherapist(female).
Thepatienthadexperiencedseveraltraumaticbraininjuresinasequenceofmotorvehicleaccidentstwoyearspriortotheinitiationoftherapy.
Thepatientwasreferredbyherpsychiatristfortargetedpsychotherapytreatmentofpre-existing,non-suicidalself-injuryandsevereemotionaldysregulation.
Neurologicalexaminationestablishedtheabsenceofresidualneurologicaldeficitspriortotheinitiationoftherapy.
Theaccidenthistorywas,however,deemedtobepsychologicallysignificantandhadacontinuingnegativeimpactonthepatient'srelationshipwithherpartner.
Thepatientreceivedweeklyindividualoutpati-enttherapyandgroupdeliveredtraininginbehavioralskills.
Theanalyzedsessionswerefromtheindividualtherapycomponent.
Eachsessionwassixtyminuteslong.
AtthetimeofinitiationoftreatmentthepatientmetDSM-IVdiagnosticcriteriaforborderlinepersonal-itydisorder.
ThisdiagnosiswasconfirmedwithaSCID-II(StructuredClinicalInterviewforDiagnosis)assess-ment.
ThepatientwasindialecticalbehaviortherapyfollowingthemethodsdevelopedbyLinehan[70,71].
Treatmentwasongoingbetweenthesessionscoded.
InstitutionalReviewBoardandtheparticipant'sinformedconsentwereobtainedpriortoinitiationofthestudy.
Therapysessionswerevideotapedforsubse-quentanalysis.
AnassessmentbasedontheDSM-IVwasrepeatedattheendoftreatmentatwhichtimethepatientnolongermetclinicalcriteriaforadiagnosisofborderlinepersonalitydisorder.
Selfreportratingsofmisery,depressedmoodandanxietywereimproved.
Indicesthatbroughtthepatienttotreatment,frequentsuicidalideationandrepeatedself-injury,werenolongerpresentandwerenotpresentatpost-treatmentfollow-upsixmonthsaftertheterminationoftherapy.
RestatementoftheProtocolasaSymbolSequenceThereareseveralpossibleproceduresforparsingather-apyprotocolpriortorestatementasasymbolsequence.
Onepossibilityistosetafixedtimeintervalandcodethecontentofthatinterval.
ThiswastheprocedurefollowedbyDoba,etal.
[65,66]whoused5secondintervalsintheiranalysisofautobiographicalspeech.
Whilehavingtheadvantageofexplicitspecification,thisprocedurehasthedisadvantageofbeingnonresponsivetothevaryingpaceofnaturaldialog.
Weimplementedherethemorecommonpractice,followingStiles[62-64,72]ofparsingtheprotocolintonaturalspeechelements.
Theseele-mentsarecalledutterancesinthetechnicalliterature.
AsdefinedbyStiles,etal.
[72]"Thecodingunitforbothformsandintentistheutterance,definedasanindepen-dentclause,nonrestrictivedependentclause,multiplepredicate,ortermofacknowledgment,evaluationoraddress.
"AdetailedpresentationofthedefinitionofanutterancewhichincludesexamplesisgiveninChapter8ofStiles'book"DescribingTalk"[62].
EachunitoftheprotocolwasassignedoneormoresymbolsusingthescoringsystemshowninTable1.
Theprotocolwasthusreducedtoasequenceofsymbolsdrawnformatwenty-twosymbolalphabet(Therapist:A,B,C,.
.
.
K,Patient:a,b,c,.
.
.
k)asshowninTable1.
Thissymbolsetwaschosentoemphasizeelementsthatareprominentinapsychotherapyofborderlinepersonalitydisorderbasedondialecticalbehaviortherapy[70,71].
Patientandtherapistcontentwasscoredforallthreeses-sions.
Inthispreliminarycasestudyparsingintoutter-ancesandsymbolassignmentwasaccomplishedbythecollectivedecisionofthreeinvestigators.
Itisrecognizedthatamoresystematicinvestigationwillrequireindepen-dentassessmentandaquantitativetestofinter-raterreliability.
Thefollowinggivesanexampleofeachcon-tenttype.
Acknowledging:"Thankyouforremindingmeofthat.
"Information(requesting/providing):"I'vehadthatcarfortwoyears.
"RequestforValidation:"WasIwrongtothinkthatway"Table1ProtocolScoringProcedureTherapistPatientContentAaAcknowledgingBbInformation(Requesting/Providing)CcRequestforValidationDdValidatingEeEmotionalDischargeFfComplaintGgTransitional/elicitationHhProblemPresentationIiBehavioralAnalysis/EducationalJjReflectiveKkIrreverentAsymbolisassignedtospecificcontentelements.
Uppercasesymbolswereusedwhenthetherapistwasspeaking,andlowercasesymbolswereusedwhenthepatientwasspeaking.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page5of28Validating:"Everyonefeelsthatwayfromtimetotime.
"EmotionalDischarge"Never!
Never!
Never!
"Complaint:"Mychildrenneverlistentome.
"Transitional/Elicitation:"IwantedtoremembertotellyouaboutlastSaturday.
"ProblemPresentation:"Myhusbandlosthisjob.
"BehavioralAnalysis/Educational:"Doyouthinkhewouldresponddifferentlyifyoutelephonedfirst"Reflective:"Youseemtobesayingthatyouwouldn'tlikethat.
"Irreverent"Wellhecertainlyfailedthattime!
"Table2.
showsthedistributionfrequencyofeachsym-bolinthealphabetforallthreesessions.
Thedistributioncomputedusingallsessionsisunremarkable.
Thethera-pist'scontributionsconsistprimarilyofacknowledgments,elicitationsandproblempresentations.
Thehighfrequencyofpatientcomplaintsandemotionaldischargesisconsis-tentwithadiagnosisofborderlinepersonalitydisorder.
Thesymbolfrequencydistributionwasalsocalculatedforeachsessionwithaviewtodeterminingiflongitudi-nalchangesinsymbolfrequenciescouldofferinsightsintothepatient-therapistinteraction.
Wedefineaconsistentchangeasoneinwhichthefrequencyofappearanceofasymboleitherincreasesordecreasesoverallthreesessions.
Inthecaseofthepatient,onlyonevariableshowedaconsistentpattern;thefrequencyofpatientacknowledgmentsdecreased.
ThedecreasefromSession1toSession2was,however,minimal.
Otherwise,theonlyconsistentpatternswereseenintherapistbeha-vior.
Thefrequencyofeducationalinterventionsdecreased,andthefrequencyofreflectiveinterventionsincreased.
Thefrequencyofvalidatinginterventionsfromthetherapistdecreasedoverthethreesessions.
Thispos-siblyreflectsthegrowingconfidencethatbothpartici-pantshadinthetherapeuticrelationship.
Asidefromdescribingpredictablechangesintherapistbehavior,theanalysisofsymbolfrequencieswaslargelyuninformative.
Thisissignificanttothepresentinvesti-gationbecauseitsuggeststheneedformeasuresthatquantifysequentialbehavior.
ResultsAnalysisofRepeatedPairsThemostelementaryformofsequentialanalysisistheanalysisofrepeatedpairs.
TheresultsfromthisanalysisTable2SymbolFrequencyDistributionContentSymbolFrequencyAllSessionsFirstSessionSecondSessionThirdSessionP:BehavioralAnalysis/Educationali.
1184.
1073.
1475.
1010P:Acknowledginga.
0988.
0978.
0947.
1024T:AcknowledgingA.
0899.
1041.
0638.
1038P:Information(Requesting/Providing)b.
0830.
0726.
0692.
0982P:Complaintf.
0798.
0915.
0692.
0827T:Transitional/ElicitationG.
0766.
0726.
0893.
0687T:ProblemPresentationH.
0735.
0536.
0820.
0757P:EmotionalDischargee.
0697.
0820.
0820.
0547T:BehavioralAnalysis/EducationalI.
0659.
0820.
0729.
0533T:ValidatingD.
0532.
0915.
0474.
0407P:RequestforValidationc.
0532.
0599.
0455.
0561T:ReflectiveJ.
0450.
0221.
0492.
0519P:ProblemPresentationh.
0317.
0221.
0346.
0337T:Information(Requesting/Providing)B.
0298.
0221.
0328.
0309T:IrreverentK.
0108.
0032.
0000.
0224P:Transitional/Elicitationg.
0108.
0032.
0182.
0084P:Validatingd.
0051.
0126.
0018.
0042P:Irreverentk.
0044.
0000.
0000.
0098T:RequestforValidationC.
0006.
0000.
0000.
0014T:EmotionalDischargeE.
0000.
0000.
0000.
0000T:ComplaintF.
0000.
0000.
0000.
0000P:Reflectivej.
0000.
0000.
0000.
0000Thetableshowstherank-orderedfrequencyofeachsymbolinthesymbolicreduction.
Theexpectationfrequencyis.
0455.
TheletterTinthefirstcolumndenotestherapistcontributionsandtheletterPindicatespatientcontributions.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page6of28aftercombiningallthreetherapysessionsareshowninTable3.
Theexpectationfrequencyofarepeatedpairisp=.
0021.
Ninerepeatedpairsappearwithafrequencythatisatleastoneorderofmagnitudegreaterthantheexpectationfrequency.
Mostoftherepeatedpairsareassociatedwithwhatmightbedescribedasthemechanicsoftherapy:requesting,presentingandacknowledginginformation.
Asinthecaseofsinglesymbolfrequencies,patientcomplaintsandemotionaldischargesappearfrequentlyaselementsinrepeatedpairs.
AnalysisofRepeatedTriplesWhenrepeatedtriplesareexaminedamarkedpatternofchangeincontentisseenoverthethreesessions.
InamessageoflengthLMthereareLM-2triples.
Nonethelessthere,areveryfewrepeatedtriplesintheclinicaldata.
DuringVisitOneninetriplesappearmorethan1%ofthetime.
DuringVisitTwoonlytwotriplesappearinmorethan1%ofthesample,andinVisitThree,fourtriplesappearatafrequencyexceeding1%(Table4).
Thereisachangeinthecontentofrepeatedtriplesoverthethreesessions.
Inthefirstsessionthemostfre-quentlyobservedtripleisarequestforvalidationbythepatientfollowedbyanemotionaldischargefollowedbyacomplaint.
Thesethreecodingelementsappearpromi-nentlyintheotherrepeatedtriplesobservedduringthefirstsession.
Bythesecondsession,whichoccurredfourmonthsafterthefirstsession,behavioralanalysisbythetherapistandacknowledgmentofthesecommunicationsbythepatientarethemostfrequentlyobservedtriples.
Thispatternisconsistentwithclinicalexpectations.
Intheearlysessions,thepatient-therapistrelationshipisconstructedbythetherapist'snonjudgmentalacceptanceofthepatient'scomplaints,emotionaldischargesandneedforvalidation.
Thisisparticularlytrueinthecourseofborderlinepersonalitydisorder.
Theworkoftherapy,implementedbybehavioralanalysisandeducation,beginsaftertheconstructionofthetherapeuticalliance.
ContextFreeGrammarComplexityWhileseveralmethodscanbeusedtocharacterizeasymbolsequence,weconsiderfirstmeasuresofcom-plexity.
Quantitativemeasuresofcomplexitycanbemostreadilyintroducedbyconsideringanexplicitexample.
Considertwomessages,thatistwosymbolsequences,M1andM2.
M1=AAAAAAAABBBBBBBBCCCCCCCCDDDDDDDDM2=BCBADBCADBBDAAADAADDBCCCCDCACBBDItshouldbenotedthatbothmessageshavethesamesymbolfrequency,eightappearancesofeachsymbol.
Theyareindistinguishablewithdistribution-determinedmeasures,forexampleShannoninformation,butM2ismorecomplexthanM1inourconventionalunderstand-ingoftheterm.
Thereareseveralmethodsforquantify-ingthecomplexityofsymbolsequences.
Ataxonomyofcomplexitymeasureshasbeenpublished[73].
Inthefirstinstance,weconsiderthecontextfreegrammarcomplexityintroducedbyJiménez-Montao[60](adescriptionisgiveninAppendixOne).
Consistentwithourqualitativeexpectations,itisfoundthatthatgram-marcomplexityofM1is20bitsandthecomplexityofM2is27bits.
Thecomplexityofanobservedsymbolsequenceisoftenexpressedinbits/unittimebydividingthecom-plexityofthemessagebytheperiodofobservation[74].
TheresultsfromthethreetherapysessionsareshowninFigure1.
Complexitygenerationisseentoincreaseacrossthethreesessions.
(Theprocedureusedtoesti-matetheuncertaintiesofthesecomplexityvaluesisout-linedinAppendixOne).
Thisresultisconsistentwiththeincreaseinthenum-berofsymbolsgeneratedinthethreesessions(NDATA=317,549,713respectively).
WhileanyobservationbasedTable3HighFrequencyRepeatedPairsFirstElementofPairSecondElementofPairFrequencyT:BehavioralAnalysis/EducationalP:Acknowledging.
0279T:ProblemPresentationP:Acknowledging.
0260P:AcknowledgingT:Behavioralanalysis/Educational.
0247P:Information(Requesting/Providing)T:Acknowledging.
0241T:AcknowledgingP:Information(Requesting/Providing).
0234T:ProblemPresentationP:BehavioralAnalysis/Educational.
0234P:EmotionalDischargeP:Complaint.
0228T:Transitional/ElicitationP:BehavioralAnalysis/Educational.
0222P:ComplaintT:Acknowledging.
0209MostFrequentlyObservedRepeatedPairs.
Analyzedoverallthreesessions,thefrequenciesofninerepeatedpairsexceedtheexpectationfrequencyof.
0021byatleastoneorderofmagnitude.
Pdenotesthepatient,andTdenotesthetherapist.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page7of28onasinglecasehistorymustbestatedcircumspectly,theincreaseinthefrequencyofsubjecttransitionwhichisreflectedintheincreaseinNDATAoverthethreesessionsisconsistentwithqualitativeclinicalobservationswithborderlinepatients.
Aspatientsprogressintherapyonecan,insomeinstances,observeadecreasedperseverationintopicandagreaterflexibilityofdiscourse.
ThisresultisconsistentwiththequantitativeresultsofMcDaniel,etal.
[75]whofoundcorrelationsbetweenrateofimprovementandanestimateofthenumberofpatientutterances.
TheresultseenhereisalsoconsistentwiththeWinefield,etal.
[76]quantitativecharacterizationofapsychodynamicallyorientedpsychotherapywhichshoweddecreasingasymmetryinpatient/therapistverbalbehaviorduringthecourseoftreatment.
Thisdecreaseinasymmetrywasevidencedbyincreasedtherapistspeechactivity.
Increasedparticipationbythetherapistwouldresultinanincreaseinpatient-to-therapisttransitionsinthesymboltranscript,anincreaseinNDATA,andanincreaseincomplexitygeneration.
Itisalsoamatterofinteresttodeterminethestabilityofcomplexitywithinasession.
Thiswasdonebydeter-miningcomplexitygenerationforeachquartersession.
AvisualinspectionofFigure2suggeststhatthereisasomewhatgreaterwithin-sessionvariationinthethirdsession.
Thisisconsistentwithourunderstandingofanincreaseincomplexitygenerationduringthecourseofasuccessfultherapy.
Itisimportanttomakeadistinctionbetweenthecomplexityofamessageandtheintrinsicdynamicalcomplexityofthesystemthatgeneratedthemessage.
TheintrinsiccomplexityofthegeneratorcanbeTable4RepeatedTriplesAppearingataFrequencyExceeding1%VisitOneTripleFrequencyContentSymbol1ContentSymbol2ContentSymbol3cef.
022P:RequestValidationP:EmotionalDischargeP:ComplaintaIa.
019P:AcknowledgingT:BehavioralAnalysis/EducationalP:AcknowledgingIaI.
016T:BehavioralAnalysis/EducationalP:AcknowledgingT:BehavioralAnalysis/EducationalbfA.
016P:Information(Requesting/Providing)P:ComplaintT:Acknowledgingefc.
013P:EmotionalDischargeP:ComplaintP:RequestforValidationfAb.
013P:ComplaintT:AcknowledgingP:InformationHaI.
013T:ProblemPresentationP:AcknowledgingT:BehavioralAnalysis/EducationalbAb.
013P:Information(Requesting/Providing)T:AcknowledgingP:Information(Requesting/Providing)fce.
013P:ComplaintP:RequestValidationP:EmotionalDischargeVisitTwoTripleFrequencyContentSymbol1ContentSymbol2ContentSymbol3aIa.
015P:AcknowledgingT:BehavioralAnalysis/EducationalP:AcknowledgingIai.
011T:BehavioralAnalysis/EducationalP:AcknowledgingP:BehavioralAnalysis/EducationalVisitThreeTripleFrequencyContentSymbol1ContentSymbol2ContentSymbol3bAb.
018P:Information(Requesting/Providing)T:AcknowledgingP:Information(Requesting/Providing)IaI.
017T:BehavioralAnalysis/EducationalP:AcknowledgingT:BehavioralAnalysis/EducationalAbA.
011T:AcknowledgingP:Information(Requesting/Providing)T:AcknowledgingaIa.
011P:AcknowledgingT:BehavioralAnalysis/EducationalP:AcknowledgingRepeatedTriplesAppearingataFrequencyExceeding1%.
Resultsarepresentedseparatelyforeachsession.
Pdenotesthepatient.
Tdenotesthetherapist.
123024681012GrammarComplexityforThreeTherapySessionsGrammarComplexity(bits/minute)TherapySessionFigure1Complexitygenerationinthreepsychotherapysessions.
Thecontextfreegrammarcomplexityofthesymbolicreductionofeachsessionwasnormalizedagainstthedurationofthesessiontodeterminecomplexitygenerationinbits/minute.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page8of28estimatedbycomparingthecomplexityofthemessageagainstthecomplexityofrandommessagesofequallengthgeneratedwiththesamesymbolset.
Theresultisthenormalizedcomplexity.
MathematicalproceduresforconstructingthisnormalizationareoutlinedinAppendixOne.
Thenormalizedcomplexityisdimen-sionlessandvariesbetweenavalueclosetozeroforaconstantsymbolsequence(onesymbolrepeatedthroughouttheentiremessage)andavalueofoneforarandomsequence.
Examplesgivingintermediatevaluesofnormalizedcomplexityareshownintheappendix.
Thenormalizedgrammarcomplexityofthethreetherapysessionsis.
765±.
033,.
758±.
015and.
763±.
017.
Thereisnosignificantchangeinthenor-malizedgrammarcomplexitywhichsuggeststhat,atleastinthistherapy,grammarcomplexitydidnotdetectchangesintheunderlyingdynamicalprocess.
ThecontrastbetweentheconsistencyofnormalizedcomplexityandtheincreaseincomplexityperunittimeisconsideredintheDiscussionsectionofthispaper.
Anexaminationofthenormalizedcomplexityforeachquarterofasessionallowsanexaminationofthestatio-narityoftheunderlyingdynamicalprocess(Figure3).
Theresultsaredisplayedon[0,1],thedefinedrangeofnormal-izedcomplexity.
Therearenosignificantwithin-sessionorbetween-sessiondifferenceswhenquartersessionsareanalyzed.
Acomparisonofthecomplexityvaluesobtainedwiththeoriginaltherapysymbolsequenceandcomplexityvaluesobtainedfromrandommessagesofthesamelengthmakesitpossibletoaddressthefollowingnullhypothesis:Asassessedbythiscomplexitymeasure,thesequen-tialstructureoftheoriginalmessageisindistinguish-ablefromthesequentialstructureofanequi-probable,randomsequenceofthesamelengthcon-structedfromthesamesymbolalphabet.
Severalstatisticaltestsofthenullhypothesishavebeenconsidered(AppendixOne).
WeuseheretheMonteCarloprobabilityofthenullhypothesis.
PNULL=NumberValues≤CORIG1+NSURRNSURRisthenumberofcomparisonrandommessages(calledsurrogates)computed.
Thenumberofcomplex-ityvaluestestedinthenumeratorincludesthecomplex-ityoftheoriginalsymbolsequenceaswellasthecomplexityvaluesobtainedwithsurrogates,ensuringthatthenumeratorhasavalueofatleastone.
Inthe123434567891011121314GrammarComplexityforThreeTherapySessionsGrammarComplexity(bits/minute)QuarterofSessionAnalyzedFigure2Withinsessioncomplexitygenerationforthreetherapysessions.
Grammarcomplexitygeneration(bits/minute)wasdeterminedseparatelyforeachquarterofeachsession.
ThetopcurvecorrespondstoSessionThree.
ThebottomcurvecorrespondstoSessionOne.
123400.
20.
40.
60.
81NormalizedGrammarComplexityforThreeTherapySessionsNormalizedGrammarComplexityQuarterofSessionAnalyzedFigure3Normalizedgrammarcomplexityforeachquarterofeachtherapysession.
Normalizedcomplexityisdefinedon[0,1].
ThegreenlinecorrespondstoSessionOne,thebluelinetoSessionTwoandtheredlinetoSessionThree.
Thecomplexityvaluesobtainedwithrandomnumbers(ablacklineatthetopofthegraph)andwithaconstantsymbolsequencewhereonesymbolisrepeatedthroughoutthemessage(ablacklineatthebottomofthegraph)areshownforcomparison.
Datasetsofthesamesizewereusedinthecomparisoncalculations.
Thenormalizedcomplexityobtainedwithrandomnumbersisapproximatelyone,andthenormalizedcomplexityobtainedwithaconstantsignalisapproximatelyzero.
DetailsofthecomparisoncalculationsaregiveninAppendixOne.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page9of28calculationsinFigures2and3,NSURR=499andCSurro-gate>CORIGinallcases.
ThenullhypothesisisrejectedwithPNULL=.
002;thatis,thesequentialstructureofpatient-therapistcommunicationinthesesessionsasscoredbythisprocedureandassessedbythismetricisnonrandom.
Lempel-ZivComplexityTheresultsobtainedwithgrammarcomplexitywereconfirmedbycalculationsofLempel-Zivcomplexity([77]describedinAppendixTwo).
Lempel-Zivcomplex-ityandgrammarcomplexityareinthesametaxonomicgroupofcomplexitymeasures(randomnessfinding,nonprobabilistic,modelbased).
ThevaluesobtainedwithLempel-Zivcomplexityarenotthesameasthoseobtainedwiththegrammarcomplexity,butthetwomeasuresshowthesamesensitivitytorandomnessinasymbolsequence.
TheLempel-ZivresultsanalogoustothoseobtainedwithgrammarcomplexityareshowninFigure4.
Asinthecaseofgrammarcomplexitythereisanincreaseincomplexitygenerationoverthethreesessions.
Thewithin-sessionvariabilityofLempel-Zivcomplex-ity(Figure5)showsthesamepatternthatwasobservedwithgrammarcomplexity.
Thewithin-sessionvariabilityisgreaterinSessionThree.
Lempel-Zivcomplexitycanalsobenormalizedbycomparisonswithrandomsurrogatesymbolstringspro-videdthatthecomplexityofthesurrogateisalsodeter-minedwiththeLempel-Zivalgorithm.
ThenormalizedLempel-Zivcomplexityforthethreesessionsis.
765±.
033,.
758±.
015and.
763±.
017respectively.
Incommonwithgrammarcomplexity,nochangeinthegeneratingdynamicalprocesswasdetectedwithLem-pel-Zivcomplexity.
Theseresultsshouldnotbegeneral-izedinappropriately.
Itremainspossiblethatsignificantchangemightbedetectedifadifferentmeasurewasappliedtothesamedata.
Itcanonlybesaidthatnor-malizedgrammarcomplexityandnormalizedLempel-Zivcomplexityfailedtodetectanybetween-sessionchangeswhilechangeswereseenincomplexitygenera-tionrateswithbothmeasures.
Aspreviouslynoted,thebetweensessionconsistencyofnormalizedcomplexityandtheincreaseincomplexityperunittimeisconsid-eredintheDiscussionsection.
Thewithin-sessionnor-malizedcomplexitywasalsocomputedwiththeLempel-Zivalgorithm(Figure6).
Asinthecaseofgrammarcomplexity,nosignificantwithin-sessionchangeswereseeninthenormalizedcomplexity.
Asbefore,thesurrogatenullhypothesisofrandomstructurewasrejectedbyLempel-ZivcomplexitywithPNULL=.
002(NSURR=499)inallcases.
Itcanagainbeconcludedthatpatient-therapistcommunicationhasnonrandomstructure.
MutualInformationConsidertwosimultaneouslyobservedsymbolsetsMes-sageA=(A1,A2,.
.
.
.
.
AN)andMessageB=(B1,B2,.
.
.
.
.
BN)constructedfromthesamealphabetofNaelements.
PA(I)istheprobabilityoftheappearanceofSymbolIinMessageAPB(J)istheprobabilityoftheappearanceofSymbolJinMessageB.
PAB(I,J)istheprobabilitythatSymbolIappearsinMessageAandSymbolJappearsin12300.
511.
522.
533.
544.
55LempelZivComplexityforThreeTherapySessionsLempelZivComplexity(bits/minute)TherapySessionFigure4Complexitygenerationinthreepsychotherapysessions.
TheLempel-Zivcomplexityofthesymbolicreductionofeachsessionwasnormalizedagainstthedurationofthesessiontodeterminecomplexitygenerationinbits/minute.
123422.
533.
544.
555.
566.
57LempelZivComplexityforThreeTherapySessionsLempelZivComplexity(bits/minute)QuarterofSessionAnalyzedFigure5Withinsessioncomplexitygenerationforthreetherapysessions.
Lempel-Zivcomplexitygeneration(bits/minute)wasdeterminedseparatelyforeachquarterofeachsession.
ThetopcurvecorrespondstoSessionThree.
ThebottomcurvecorrespondstoSessionOne.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page10of28MessageBatthesametime.
Theaveragemutualinfor-mationofMessagesAandBistheaveragenumberofbitsofMessageBthatcanbepredictedbymeasuringMessageA.
ItisdenotedbyI(A,B).
Itcanbeshown[78]thatI(A,B)=NαI=1NαJ=1PAB(I,J)log2PAB(I,J)PA(I)PB(J)MutualinformationissymmetricalI(A,B)=I(B,A).
Also,iftwoprocessesarestatisticallyindependentthenPAB(I,J)=PA(I)PB(J),andI(A,B)=0.
ThespecialcasewhereMessageAandMessageBarethesame,I(A,A),iscalledself-information.
Inthisapplication,weexaminetheabilityofames-sagetopredictitsownfuture.
WedefineI(AI,AI+1)asthemutualinformationobservedwhenMessageA=(A1,A2,.
.
.
.
.
AN-1)andMessageB=(A2,A3,.
.
.
.
.
AN).
Thiscanbegeneralizedtoconsiderlargertemporaldisplace-ments.
I(AI,AI+K)iscalculatedbysettingMessageA=(A1,A2,.
.
.
.
.
AN-K)andMessageB=(AK,AK+1,.
.
.
.
.
AN).
Thetimeshiftedself-informationisanonlinearmeasureoftemporaldecorrelation.
ExplanatoryexamplesaregiveninCellucci,etal[79].
IfamessagehasstrongtemporalpredictabilitythenI(AI,AI+K)remainshighasKisincreased.
Iftheprocessgeneratingamessageisdyna-micallydisordered,thenI(AI,AI+K)decreasesrapidlyasKincreases.
MutualinformationforthecaseK=1hasbeenappliedtotheexaminationofthesequence-sensitivestructureofnarrativecomponentsintheautobiographi-calspeechofanorexicadolescentsandcontrols[66].
TheseinvestigatorsfoundthatI(AI,AI+1)issignificantlylowerinpatients.
TheyalsocomparedI(AI,AI+1)valuesobtainedwiththeirdataagainstthevaluesobtainedwithrandomshufflesurrogatesandfoundthatsurrogatesdecorrelatefasterthantheoriginalsymbolsequenceindicatingthepresenceofnon-randomstructureintheoriginalsymbolsequence.
Figure7showsmutualinformationI(AI,AI+K)asafunctionofthetemporalshiftparameterKforthethreetherapysessions.
ThemutualinformationmeasuredinthefirstsessiondecorrelatesmoreslowlythanmutualinformationobtainedwithSessionTwoandThreeindi-catingahigherdegreeofpredictabilityinSessionOne.
ThisisconsistentwiththepreviousobservationofalowercomplexitygenerationrateinSessionOne.
Theseparationofmutualinformationfunctionsbetweenthefirstandsecondsessionandthefirstandthirdsessionissignificant(P0.
Hnquantifiestheaverageamountofinformationcontainedinasubstringoflengthn,andthereforeismonotoneincreasinginn.
Therelatedcon-ditionalentropies,hn,aregivenbyhn=Hn+1Hnhnistheaverageamountofinformationneededtopredictthenextsymbolinasubstringifthefirstnsym-bolsareknown,givinghn≥hn+1.
ThevaluesofHnandhnasafunctionofordernforthethreetherapysessionsareshowninFigure8.
Ateachorder,thevaluesofHnobtainedinthethirdses-sionaregreaterthanthevaluesobtainedinthesecondsessionwhicharegreaterthanthevaluesobtainedinthefirstsession.
Thisresultisconsistentwiththepre-viouslypresentedrateofcomplexitygeneration(Session3>Session2>Session1)andwiththeobservationthatmutualinformation,whichisrelatedtoentropy,decorr-elatesfasterinthelatersessions.
Thebetween-sessionseparationofconditionalentropyislessmarked,buttheconditionalentropyofSession3isgreaterthanthatofSession1atallordersofn.
Asinthecaseofmutualinformation,theresultsoftheseentropycalculationsmustbeviewedwithcare.
AsimpleanalysisindicatesthatlengtheffectswillcauseasignificantdeteriorationinanestimateofHnasnincreases,ifoneusestheequationforHngivenabove.
AmessageofNsymbolscontainsN-(n-1)n-words.
Aspreviouslynotedthenumberofpossiblen-wordsintheabsenceofrestrictiverulesis(Na)n.
Thusthenumberofpossiblen-wordsincreasesexponentiallywithordern,whilethenumberofwordsactuallypresentislimitedbyN.
Letμ(n)ibetheexpectationvalueofthenumberofappearancesofthei-thn-wordforthecaseofanequi-probabledistribution.
μ(n)i=p(n)iN=N/Nmax=N/(Nα)nThecalculationofHnusingthepreviousequationiswarrantedinthecaseofgoodstatisticswhichisobtainedwhenμ(n)iisontheorderoften[87].
InthepresentanalysisNa=22,andthesmallestvalueofNisobtainedinSession1whereN=317.
Thecriterionμ(n)i≥10issatisfiedforn=1whereHnforSession1,themeanvalueofcomplexitycalculatedfromallpossiblesubstringsoflengthLM/2,ands1/2thestandarddevia-tionofthatmean.
Expressedasafraction,uncertaintyiss1/2/,andΔCORIGisgivenbyCORIG=CORIGσ1/2Thisprocedureis,however,computationallyinsuppor-tableforlongermessages.
SupposeLM=8000.
Thispro-cedureforestimatingwouldrequireaveraging4000valuesofcomplexitycalculatedfromstringsoflengthLM/2=4000.
Wehaveadoptedtheprocedureofcalculatingfrom100stringsoflengthLM/2.
TheyareselectedrandomlyfromthesetofallpossibleLM/2substrings.
IncaseswhereLMiscal-culatedfromallpossiblesubstringsoflengthLM/2.
Aqualitativeunderstandingofthecomplexityofasymbolssequencecanbeobtainedbyapplyingthesemeasurestosymbolsequencesgeneratedbystandardsystemsthatarecommonlyexaminedindynamicalsys-temstheory.
Fiveexamplesareconsideredhere:acon-stantsequence(thesamesymbolisrepeated),sequencesgeneratedbytheRsslerandLorenzsystems(boththreedimensionalordinarydifferentialequations),theHénonsystem(atwodimensionaldifferenceequation)andarandomnumbergenerator.
Thetechnicalspecifi-cationsofthesystemsaregiveninAppendixThree.
TheRssler,Lorenz,Hénonandrandomdataareexpressedasrealvariables.
Inordertoapplyasymbolicdynamics-basedmeasureofcomplexity,itisnecessarytoprojectthesedatasetstoadiscretesymbolset.
Thereareseveralpossibleproceduresfordoingthis.
Radhak-rishnan,etal.
[165]usedK-meansclustering.
Whileconceptuallyattractive,theresultsofK-meansclusteringcanbeverysensitivetoinitialconditions.
BradleyandFayyad[166]addressedthissensitivitybyconstructingaK-meansalgorithmthatproducesarefinedinitialcondi-tionthatimprovedperformance.
Insofarasweknow,thismethodhasnotbeenappliedtotheproblemofconvertingrealdatatosymbolicdata.
AnalternativeapproachhasbeenpublishedbyHirata,etal.
[167]whoapproximateageneratingpartitionfromatimeseriesusingtessellations.
Thisisacomputationallydemandingprocedureandtherearepracticalissuesconcerningthesensitivityofthepartitionontheinitialization.
Steuer,etal[86]recommendusingthepartitionthatmaximizesRappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page18of28entropy.
Inthepresentexamples,thecontinuousvari-abletimeseriesispartitionedaboutthemedian.
Inthisprocess,themedianiscomputedfromtheoriginaltimeseries.
Arealvariableisreplacedbysymbol'0'ifitislessthanthemedianandbysymbol'1'ifitisgreaterthanorequaltothemedian.
Thechoiceofthemedianratherthanthemeaniscriticaltothisprocess.
False-positiveindicationsofdeterministicstructureinrandomdatacanresultifthemeanisused[168].
Thepartition-ingprocessisdepictedinFigure9.
(Itshouldbenotedthatinthepresentpaper,theconsiderationofpartition-ingprotocolonlyappliestothedidacticexamplespre-sentedintheappendices.
Thepsychotherapydataaresymbolicandpartitioningisnotrequired).
ThegrammarcomplexityvaluescomputedfromonethousandelementsymbolsequencesgeneratedbythesemodelsystemsareshowninFigure10.
Theresultsareseentobeconsistentwithourqualitativeunderstandingofcomplexity.
Theconstantsequencegivesthelowestvalue,andtherandomnumbergeneratorproducesthelargestvalue.
TheorderingRsslerlessthanLorenz,lessthanHénonisalsoconsistentwithexpectationsbasedonavisualexaminationofthetimeseriesintheleftcol-umnofFigure9.
Acriticaldistinctionmustbemadebetweenthecom-plexityofamessage,CORIGandtheintrinsiccomplexityoftheprocessthatgeneratedthemessage.
Thevalueofgrammarcomplexitywilldependontwofactors,thecomplexityofthedynamicalprocessgeneratingthesymbolsequenceandthelengthofthesymbolsequence.
ThisisseenintheupperpanelofFigure11wheregrammarcomplexityisplottedasafunctionofthelengthofthedataset.
Theorderingofcomplexityvaluesseenwith1000elementsequencesinFigure2ispreserved(random>Hénon>Lorenz>Rssler>con-stant)andthevaluesincreasewiththesizeofthedataset.
Itisthereforenecessarytofindaneffectivenormal-izationofcomplexityvaluesthatallowscomparisonofintrinsiccomplexitieswithoutthecomplicationofdatasetsize.
ItmightbesupposedthatdividingCORIGbythelengthofthemessageisanacceptablesolution.
Ithasbeenshownthatthisisnotthecase[169].
AneffectivenormalizationofCORIGcanbeachievedbycomparingitagainstthevaluesofcomplexityobtainedfromrandomequiprobablemessagesofthesamelength.
LetNabethesizeofthesymbolalphabet(thenumberofdistinctsymbolsavailableformessageconstruction,intheseexamplesNa=2).
NaisnotmessagelengthLM.
Anequiprobablesurrogateisonewhereeachsymbolappearswithprobability1/Na.
Letdenotetheaveragevalueofcomplexityobtainedfromrandomequi-probablesurrogatesoflengthLM(thesubscriptsdenotesasurrogate).
ThenormalizedcomplexityisdefinedbyCN=CORIG/CNrangesfromclosetozeroformessagescontainingasinglerepeatedsymboltoclosetooneformessagesgeneratedbyrandomprocesses.
AsoutlinedinRapp[170],CNcannotbeformedbynormalizingagainstrandomshufflesurrogates.
ConsiderthecaseofamessagethatconsistsofasinglerepeatedsymbolselectedfromanalphabetofsizeNa>1.
(SequencesinamessagespaceofNa=1consistofasinglesymbolandonlydifferbylength.
Trivially,theircomplexityisthenumberofbitsrequiredtoencodelengthLM.
)AneffectivenormalizationshouldgivealowvalueofCNtoarepeatedsymbolmessage.
Supposesur-rogateswereformedbyarandomshuffle.
Sincethemessagecontainsonlyonesymbol,theyallhavethesamevalueofcomplexity.
Inthiscase,CORIG=andhenceCN=1,whichisthecomplexityofarandommessage.
IfinsteadsurrogatesareequiprobableonNa,Na>1,thenisgreaterthanCORIGandCNhasalowvalue.
Alowvalueofcomplexityisexpectedforaconstantsequence.
TheuncertaintyinCN,ΔCN,canbeestimatedbythefollowingargumentCN=CORIG/(CN)2=CNCORIG2(CORIG)2+CN2()2(CN)2=12(CORIG)2+CORIG22()2TheestimationofΔCORIGhasbeendiscussed.
isthemeancomplexitycomputedfromadistributionofequiprobablesurrogates.
Δisthestandarddevia-tionofthatdistribution.
ComparisonofCORIGandthecomplexityvaluesobtainedwithsurrogatesmakesitpossibletoaddressthefollowingsurrogatenullhypothesis:Asassessedbythiscomplexitymeasure,thesequen-tialstructureoftheoriginalmessageisindistinguish-ablefromthesequentialstructureofanequi-probable,randomsequenceofthesamelengthcon-structedfromthesamesymbolalphabet.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page19of280501001502002501050510ContinuousVariableRossler05010015020025000.
51BinaryPartition050100150200250201001020Lorenz05010015020025000.
5105010015020025010.
500.
51Henon05010015020025000.
5105010015020025000.
51IndexRandom05010015020025000.
51IndexFigure9Partitioningrealdataontoadiscretesymbolset.
Themedianisdeterminedformtheoriginaldata.
Inthisexample,realvariablesarereplacedbythesymbol'0'iftheyarelessthanthemedianandbysymbol'1'iftheyaregreaterthanorequaltothemedian.
Thelocationsofmediansareindicatedbythehorizontalredlines.
Graphsintheleftcolumnshowtherealvariabletimeseries.
Thecorrespondingsymbolsequencesareshownintherightcolumn.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page20of28Severalstatisticaltestsofthenullhypothesishavebeenconsidered[168].
WeuseheretheMonteCarloprobabilityofthenullhypothesis.
PNULL=NumberValues≤CORIG1+NSURRNSURRisthenumberofsurrogatescomputed.
Thenumberofcomplexityvaluestestedinthenumeratorincludesthecomplexityoftheoriginalsymbolsequenceaswellasthecomplexityvaluesobtainedwithsurro-gates,ensuringthatthenumeratorhasavalueofatleastone.
Thisstatisticaltestwaschosenbecauseitisadistribution-agnostictest,thatis,itmakesnoassump-tionsaboutthestructureoftheCSurrogatedistribution.
Surrogateshavearandomstructure,andthegrammarcomplexitygivesthehighestvaluetorandomsequences.
WethereforeexpectthevaluesofCSurrogatetobegreaterthanthevalueofCORIGifanonrandomstructureispre-sentintheoriginalsequence.
ThesmallestvalueofPNULLwillbeobtainedwhenallvaluesofCSurrogatearegreaterthanCORIG.
Thatminimumvalueistherefore1/(1+NSURR).
InthecalculationsinFigures10and11,NSURR=499andCSurrogate>CORIGinallcasesfortheconstantsequence,Rssler,LorenzandHénondata.
ThenullhypothesisisthereforerejectedwithPNULL=.
002.
Asexpected,thenullhypothesisisnotrejectedbysymbolsequencesproducedbyarandomnumbergen-erator.
ForthetencasesinFigure11correspondingtoLM=1000,2000,.
.
.
10,000,theaveragevalueofPNULLobtainedwithrandomdataisPNULL=.
562Barnard[171]andHope[172]haveproposedanon-parametrictestforrejectingthenullhypothesis.
UndertheircriterionthenullhypothesisisrejectedifCORIGHénon>Lorenz>Rssler>constant(Figure12).
AsshowninFigure13,Lempel-Zivcomplexityshowsthesamedependenceonmessagelengththatwasobservedwithgrammarcomplexity.
Thenormalizationwithequi-probablerandomsurrogateswasalsoimple-mentedwithLempel-Zivcomplexity.
Asbefore,thenor-malizedcomplexityisindependentofLM.
Inthesecalculations,499surrogateswerecomputedandthenullhypothesiswasrejectedwithprobabilityPNIULL=.
002inallcaseswiththeexceptionofrandomdatawheretheaveragevalueofPNIULLwas.
450.
Aspreviouslyreported[169]grammarcomplexityandLempel-Zivcomplexityarehighlycorrelated.
Thecom-plexityvaluescomputedwithRssler,Lorenz,HénonandrandomdataforLM=1000,2000,.
.
.
.
10,000ele-mentdatasetswerecompared.
ThePearsonlinearcor-relationcoefficientwasfoundtober=.
998(thesamevaluethatwasobtainedwithdifferentdatain[169]).
12345020406080100120LempelZivComplexityforTestSystemsLempelZivComplexity(bits)ConstantRosslerLorenzHenonRandomFigure12Lempel-Zivcomplexityforonethousandsymbolsequencesgeneratedbymodelsystems.
Symbolsequencesweregeneratedbyabinarypartitionaboutthemedian.
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page23of28Theprobabilityofthenullhypothesisofnocorrelationwaslessthan10-8.
AppendixThree:SpecificationofModelSystemFivemodelsystemsareinvestigatedinthecalculationspresentedinAppendicesOneandTwo.
Theconstantsymbolsequenceisconstructedbyrepeatingonesymbolfortheentirelengthofthedataset.
TheRsslersystem[173]isathreedimensionalsystemofautonomousordinarydifferentialequations.
dx/dt=yzdy/dt=x+aydz/dt=b+xzcza=.
2b=.
4c=5.
7Thedifferentialequationswereintegratedwithasteplengthofh=.
1usingasixthorderRunge-Kutta-Huttaalgorithm[174].
TheLorenzsystem[175,176]isspeci-fiedbydx/dt=σ(xy)dy/dt=xz+rxydz/dt=xybzσ=10b=8/3r=28AsinthecaseoftheRsslerequations,aRunge-Kutta-Huttacalculationwasperformedwithh=.
1.
TheHénonsystem[177,178]isatwodimensionaldifferenceequation.
xt+1=1ax2t+ytyt+1=bxta=1.
4b=0.
30200040006000800010000120000200400600800LempelZivcomplexityandnormalizedcomplexityasafunctionofdatasetsizeLZComplexity(bits)RandomHenonLorenzRosslerConstant02000400060008000100001200000.
51NormalizedComplexityDataSetSizeRandomHenonLorenzConstantRosslerFigure13Lempel-Zivcomplexityandnormalizedcomplexityasafunctionofdatasetsize.
Symbolsequencesweregeneratedfromthepreviouslydefinedmodelsystemsusingabinarypartitionaboutthemedian.
TheupperpanelshowsLempel-Zivcomplexity.
ThelowerpanelshowsthecorrespondingnormalizedcomplexitywherevaluesofLempel-Zivcomplexityarenormalizedagainstrandom,equi-probablesurrogates.
Inthesecalculations499surrogateswereusedtocomputeCN(modifiedfromRapp,2007).
Rappetal.
BMCPsychiatry2011,11:119http://www.
biomedcentral.
com/1471-244X/11/119Page24of28Therandomnumbergenerator[179]produceduni-formlydistributedrandomnumberson[0,1].
ItisbasedonL'Ecuyer'stwo-sequencegenerator[180]andincor-poratesaBays-Durhamshuffle[181].
AcknowledgementsPERwouldliketoacknowledgetrainingreceivedatthePhiladelphiaSchoolofPsychoanalysisandthePhiladelphiaConsultationCenterandspecificallyDr.
StephenDayEllis,President,andDr.
AngelaSandone-Barr,DirectorofClinicalServices.
DiscussionswithDr.
ArnoldFeldmanandwiththemembersoftheEntwurfGruppeareacknowledgedwithgratitude.
TheopinionsandassertionscontainedhereinaretheprivateopinionsoftheauthorsandarenottobeconstruedasofficialorreflectingtheviewsoftheUnitedStatesDepartmentofDefense.
MAJMacknowledgespartialsupportfrom:SistemaNacionaldeInvestigadores;PROMEP,Project:UV-CA-197MEXICO,andUniversidadVeracruzana.
PERwouldliketoacknowledgesupportfromtheTraumaticInjuryResearchProgramoftheUniformedServicesUniversityoftheHealthSciencesandfromtheDefenseMedicalResearchandDevelopmentProgram.
Authordetails1DepartmentofMilitaryandEmergencyMedicine,UniformedServicesUniversity,4301JonesBridgeRoad,Bethesda,MD20814,USA.
2Aquinas,LLC,2014St.
AndrewsDrive,Berwyn,PA19312,USA.
3HuntonandWilliamsLLP,2200PennsylvaniaAve.
NW,Washington,DC20037,USA.
4DepartmentofEpidemiologyandPublicHealth,UniversityofMarylandSchoolofMedicine,HowardHall,Suite200,660W.
RedwoodStreet,Baltimore,MD20201USA.
5FacultaddeFísicaeInteligenciaArtificial,UniversidadVeracruzana,SebastiánCamacho#5,ColCentro,Xalapa,Ver.
91000,Mexico.
6DepartmentofPsychology,UniversityofWashington,Box355915,Seattle,WA,98195,USA.
Authors'contributionsPERwrotethesoftwaretocomputeLempel-Zivcomplexity,softwaretoconfirmcalculationsofcontextfreecomplexity,performedthecomplexitycalculationsandwastheprimaryauthorofthemanuscript.
AMKGsupervisedthestatisticalanalysisoftheresults.
MAJMwrotethesoftwareforgrammarcomplexitycalculationsandcomputedtherepeatedpairsanalysis.
CJCcontributedcalculationsofmutualinformation,n-thorderentropiesandconditionalentropies.
KEKwasthetreatingpsychotherapistandledtheprocessofrestatingthetherapyprotocolsassymbolsequences.
Allauthorshavereadandapprovedthefinalmanuscript.
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
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