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DecodingMLBPitchSequencingStrategiesviaDirectedGraphEmbeddings

AbstractThispaperpresentsanovelanalysisofpitchsequencinginMajorLeagueBaseball(MLB).Byleveraginghigh-resolutionpitchtrackingdatafrom~3.6millionpitchesacrossthe2015-2019seasons,thisworkintroducesdirectedgraph(network)embeddingsthatsuccessfullymapshort-andlong-termpatternsinpitchsequences.Thisquantitativeapproachtopitchsequencingcapturestheintuitionthatpitchersexhibitasenseoflong-termmemorywhenonthemoundthatisnotadequatelyrepresentedbyindividualpitchselectionorsimplepitch-to-pitchcorrelation.Byinterpretingthegraphembeddingsasforwarddependencies,thereiscompellingevidencethatpitchersconstructtheirsequencesviastrategiccomponents,denoted“setup”and“knockout”pitches.Model-basedclusteringonthegraphembeddingssuggestthatMLBpitchsequencescanbegroupedintoafinitecollectionofuniversalpatternswithrespecttobothpitch-typeandzoneselection.Exploratorydataanalysisofthesesequenceclustersindicatethatapitcher’ssequencingstrategiesaredistinct—thoughnotinseparable—fromtheiravailablepitcharsenal;thatis,inadditiontopitchselection,pitchorderingisasignificantcomponentofpitcherdecision-making.Moreover,pitchersexhibitpatternsorstylesinhowtheysequencetheirpitchesoverthecourseofasinglematchup,game,season,andcareer.Ultimately,thispaperintroducesananalyticalframeworktostudyandvisualizeMLBpitchsequenceswithpotentialapplicationsinmatchuppreparation,playerevaluation,andplayerdevelopment.1. IntroductionEachmomentinabaseballgameisanextensionofthebattlebetweenpitcherandbatter.Thisconflictperhapsmostcloselyresemblesthatofahigh-performancemixedmartialartsfight,wherethereisimmensepressure—andcorrespondingincentive—forpitcherstooptimizetheirapproachagainsteachbattertheyface.Inthismatchup,apitcher’sarsenalcanbeimaginedasasetoftechniquesthateachpitcherhasattheirdisposaltodefeattheiropponent.However,apitcher’spitcharsenalalone(i.e.,themovestheyhaveavailable)isanincompleterepresentationofapitcher’saggregatestyle.Bytreatingboththeselectionandorderingofbaseballpitches,pitchsequencingismuchmoreinformativeofpitcherdecision-makingbytakingintoaccounthowpitcherslearn,buildupon,andordertheirpitches.

Amajoranalyticschallengeinsabermetricsiscapturingdifferencesbetweenpitcherswithrespecttonotonlytheirperformanceonthefieldbutalsothestylestheyexhibitandthestrategiestheydeploy.Althougheachat-batbetweenapitcherandbatterisultimatelyeitherwonorlostusinganapproximatelyuniquecollectionofpitches,patternsmayemergewhereinsetsofpitchsequencesrecuracrossvariousin-gamesettings.Ifpitcherssharepitcharsenalsorfacesimilarbatter-matchupsituations,itisalsoexpectedthatsetsofpitchsequencesmanifestacrossmanyindividualsandacrossseasons.Conversely,pitcherscanalsodistinguishthemselvesfromothersintheirrolethatsharetheirpitcharsenalthroughdifferencesintheirpitchsequencing.Thisisadvantageoustothepitchersinceitseparatesthemfromtheirpeersandiscriticalforteamstodiversifytalentintheirroster.

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However,preexistinganalyticalresearchinpitchsequencingcurrentlylimitsequenceanalysistopitchpairsviapitch-to-pitchcorrelation;thatis,apitchsequenceisdefinedexclusivelyintermsofconsecutivepitches[3].Thismemorylessapproachislikelynotreflectiveofhowpitcherspursuedecision-makingsinceitconstrainspitcherstoveryshort-termlearning.Theseearlymethodsglossoversomeofthecrucialqualitiesthatmakepitchsequencingascaptivatingasitisandgenerallyfailtoprovideinsightsastothestrategicmotivationsbehindpitchsequenceselection.

Thispaperintroducesagraph-basedsolutiontocapturethecomplexityofpatternsthatarelatentinpitchsequences.Unlikepriorstatisticalapproachesthatfocusprimarilyonpitch-levelpredictabilityorpitchpairings,thismethodsuccessfullyembedsshort-andlong-termpatternsinpitchsequencesinafinite-dimensionalfeaturespace.Eachgraphembedding,givenitsinterpretationasaforwarddependencybetweentwopitches,alsopreservesthedirected(i.e.,ordered)natureofbaseballpitchsequences.Fromthispropertyoftheembeddings,thispaperpresentsasuiteoftoolsthatcanbevaluableinastrategicsettingwithinateamfront-officeorplayerdevelopmentunit.Inordertomorecloselyanalyzehowpitchersconstructtheirsequences,theanalysisinthispaperseparatelyconsiderspitchsequencesbasedonpitch-typeconstitutionandzones.Throughmodel-basedclustering,thispaperidentifiesacollectionofpitchsequencingstrategiesthatarecommonthroughouttheMLBandvaryintheirconstitution,ordering,andentropy.Comparativeanalysisbetweensequenceclustersprovideinsightintocorestructuralelementsofpitchsequencingstrategy.Inparticular,thisworkfindsevidencefor“setup”and“knockout”pitchesbydrawingfromtheconceptofsinksingraphtheory.Withrespecttohowplayersselectpitchsequences,itisclearfromtheseclustersthatpitcherswhosharepitcharsenalscanalsodifferdrasticallyinhowtheyorderthosepitchesingamesituations.Applicationsofthisworktobaseballoperations,analytics,andscoutingaresuggestedandexploredthroughoutthepaper.

2. MethodologyThefollowingsectionswilldescribethedataandstatisticalmethodologythatunderliesthefindings.Section2.1providesanoverviewofthepaper’sprimarydataset(s)anddiscussesthemodel-specificimplicationsofdataquality.Section2.2.1introducesSequenceGraphTransform(SGT),thegraphembeddingalgorithmthatencodeseachpitchsequenceintoafeaturespace.Furthermore,Section2.2.2detailsthespecificmathematicalpropertiesofSGTthatmakeitattractiveforpracticalapplicationtothisproblem.Finally,Section2.3specifiestheclusteringtechniquethatisappliedtodetermineoptimalgroupingsofpitchsequenceswithoutrelyingonpre-existinglabelsorassumptions.

2.1 DataItisnosecretthatbaseballhasexperiencedatechnologicalrevolution,largelythankstoanexplosionofreliablehigh-resolutiondata.Allin-gamedata—includingallmeasurementsonpitches,idtracking,andcatcherdatathatareincludedinthispaper—arepubliclyavailablethroughtheMLBStatcastAPI[2].Thepitchingdataincludes3,595,944pitchesfromthe2015-2019MLBseasonsthatareprovidedalongside40featuresthatdescribepitch-levelcharacteristicsandinformationthatcontextualizeswheneachpitchwasthrown.Inaggregate,1523distinctpitchersand1894distinctbattersarerepresentedinthedata.Thisdataset—orashortenedformofit—hasbeenusedextensivelyinrecentlypublishedbaseballanalyticsresearch[4].

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Sincethispaperisprimarilyconcernedwithsequentialdata,itisnecessarytorelyondiscretepitch-classificationlabels.Theuniquepitch-typeandzonelabelsintheMLBdatasetserveasthealphabetthattheembeddingalgorithmwilllearnfrom.Theaccuracyandprecisionofthepitch-classificationlabelsinthisdatasetisthuscriticaltothesuccessofthisresearch.Thismethodologyassumeshomogeneitywithinpitch-typeswithrespecttopitch-levelcharacteristics,thoughitisclearinFigure1thatthisassumptionhasshortcomings.Itisimportanttonotethatpitchclassificationitselfisanon-trivialproblem.Currently,in-houseMLBmodelsperformreal-timepitch-classificationfornearly750,000pitcheseachseasonusingpatentedneuralnetworks[2].Irrespectiveofthepitch-classificationlabelsthatareutilizedinthisanalysis,independentresearcherscaneasilyadaptthisworkusingtheirownpitchclassificationsystemsordatasets.

Figure1:PitchCharacteristicPairplot

Severaladjustmentstothedatasetaremadetoimprovethequalityofmodeling.At-batswithincompletemeasurement/labeldataarediscardedfromthedatasettominimizetheinadvertenteffectsofnoiseorcalibrationissues.Severalfeaturesarealsoaddedtotherawdataset.Toavoidlook-aheadbiasinpredictivemodeling,rollingstatisticsareusedtoreportpitcherandbatterperformancesandtrackchangesinplayeroutcomesovertime[4].Additionally,tocomparethesequencingtendenciesofpitchersinvariousrolesandcontexts,“starter”designationsforeachpitcherareinferredfromtheinformationavailableinthefirstat-batintopofthefirstinningandthefirstat-batinthebottomofthefirstinning[4].

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2.2.1SequenceGraphTransform–OverviewandIntuitionThispaperappliesagraph-basedfeatureextractionmethodtoencodeforpatternsinMLBpitchsequences.Thismethod—SequenceGraphTransform—achievesthegoalofrepresentingthecomplexityofpitchsequencingintermsofbothpitchconstitutionandordering.

Asequencecanbedefinedasasetofdiscreteitemsthatarestructuredinanorderedseries[5].Sequencesareoneofthemostcommondatatypesinnatural,physical,andcomputationalsettings.Forexample,sequencingminingtechniqueshavefoundvaluebioinformaticsandcomputationalgenomics.Pitchsequencesnaturallyfollowthissequentialstructure;boththeorderandconstitutionofapitchsequencingareimportantindisruptingabatter’scalibrationtothepitcher’sstrategies.However,sincepitchsequencesrepresentunstructureddata—definedbyarbitrarilyplacedalphabetsofanarbitrarylength—theirmappingintoaEuclideanspaceisnon-trivial.Moreover,sincepitchsequencesgenerallyextendbeyondtwo-pitches,thisanalysisrequiresafeatureextractionmethodthatspecifieslong-termdependencies(i.e.,theeffectofdistantelementsinasequenceoneachother)inadditiontoshort-termdependencies(i.e.,theeffectofelementsinasequencethatoccurinshortsuccessionofeachother).

ThispaperutilizesSequenceGraphTransform(SGT),anembeddingfunctionthatextractspatternsinsequencesandmapsthemintoafinite-dimensionalEuclideanspace.TherearemultipleadvantagesthatareimpliedinthefunctionandmakeSGTparticularlyattractivetotheproblemofpitchsequencing.SGToperatesbyquantifyingtheobservablepatternsinasequencebyscanningthepositionsofeachitemrelativetootheritems.Theoutputtedembeddingscanbeinterpretedasadirectedgraph,whereeachalphabet(e.g.,pitch-type)becomesanodeandadirectedconnectionbetweentwonodesexplainstheirassociation.Inordertomaximizetheutilityofsequencingmininginbaseballpitchsequenceanalysis,itisimportanttohaveasequence-levelmeasureof(dis)similaritybetweenvarioussequences.Similarityisgivenexplicitlybythedot-productbetweentwoseparateembeddingswherehigherresultsindicatehighersimilarity.Additionally,SGTprovidesasolutiontoextractlength-insensitivepatternsinpitches.Asafeatureembeddingtool,SGTisparticularlyvaluablebecauseitprovidesamachine-interpretablerepresentationofacomplexdata-type.Thegraphinterpretationofeachembeddingallowsforpitch-typestoformnodesandthedirectedconnectionbetweennodesasasignalofthestrengthoftheirassociation.Graphembeddingsalsoenableunsupervisedlearningandclusteringforpatternanalysis.Moreover,eachgraphembeddingprovidedbySGTisdirected;thatis,boththeconstitutionandtheorderingdynamicsofeachpitchsequencearemaintained.[5]

Figure2:SGTApplicationtoPitchSequences

AlthoughthereareembeddingalternativestoSGT,theseexistingmethodseitherfailtoaccountforlong-termpatterns,producefalsepositives,orsubstantiallyincreasecomputation.Forexample,if

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eachsequenceistransformedintoastring,thereareseveralmetricswhichdefinethedistancebetweentwouniquepatterns(e.g.,editdistance,hammingdistance).However,thesemetricsimproperlyaccountforthecomplexitiesofpitchsequencesanddonotoffertheadvantageofrepresentingpitchsequencesintermsofforwarddependencies.MeasuresdrawnfrominformationtheorysuchasShannonentropy,thoughinformativeofthelevelof“uncertainty”or“randomness”inasequence,alsodonotprovideenoughgranularitytoexplainsequencecharacteristics,namelypitchconstitutionandordering[6].

Ifpitchsequencesdonotcorrespondintermsofbothshortandlongpatterns,SGTembeddingswillnotproduceahighsimilaritymatch.WhilethepitchsequencesFF-CU-FF,FF-CU-FF-FF-CU-FF,andFF-SL-FF-CU-FF-SL-FF-SL-SLallshareasubsequenceinFF-CU-FF,SGTdistinguishesthefirsttwoexamplesfromthirdgiventheoverall,long-termdifferencesinthesequencepattern.Conversely,traditionalsubsequencematchingmethodswouldproduceasimilaritymatchbetweenallthreesub-strings[5].Giventheapplicationsetting,SGTispreferablesincetraditionalsubsequencingmatchingmethodswouldproduceafalsepositive.Finally,SGTiswidelyapplicableacrossmultipledomainsanddoesnotexhibitsettingbias.TheSGTalgorithmispubliclyavailable,anditslowcomputationalrequirementsallowittoberuninlocalenvironments[5].2.2.2SequenceGraphTransform–DefinitionSupposeasetofsequences𝑆iscontainedinadataset.Anyindividualsequenceinthedatasetcanberepresentedby𝑠 ∈ 𝑆,wheresisconstitutedbyanalphabet𝒱.Forthepurposesofpitchsequencing,alphabet𝒱eithercontainsacollectionofpitch-typelabelsordiscretezone-locationlabels.Eachsequence𝑠hasatleastoneinstanceofoneelementfromthealphabet𝒱,thoughitisnotnecessaryforeveryelementinalphabet𝒱toberepresentedineachset𝑠.Inbaseballterms,apitchsequenceisdefinedbyatleastonepitchthatisthrowninanat-bat;moreover,duetopitcharsenalconstraints,strategicdecision-makingmotivations,andin-gamefeasibility,itisnotexpectedthatapitcherthrowseverypitch-typeinasinglesequence.Thelengthofaspecificsequence𝑠isdenotedby𝐿(").Foreachsequence,𝑠$ willrepresenttheelementthatoccursatposition𝑙where𝑙 = 1,… , 𝐿(")and𝑠$ ∈ 𝒱.𝜑%(𝑑)isafunctionthattakesadistanceasinputand𝜅asatuninghyper-parameter.Thefeaturesforsequence𝑠areextractedintheformofassociationsbetweenalphabetinstances.Eachassociationisdenotedas𝜓&'

("),where𝑢, 𝑣 ∈ 𝒱correspondtospecificalphabetinstancesand𝜓correspondstoahelperfunctionof𝜑.

SinceSGTconsiderstherelativepositionsofinstancestoformeachfeature,𝜑3𝑑(𝑙,𝑚)5quantifiestheinformationthatisextractedfromtherelativepositionsoftwoinstances,where𝑙, 𝑚arethepositionsoftwodistinctinstancesand𝑑(𝑙,𝑚)isacorrespondingdistanceoutput.Specifically,𝜑3𝑑(𝑙,𝑚)5denotestheeffectofthefirstinstanceonthelaterinstance.

ToapplySGTonasetofsequences𝑆,thefollowingconditionsmustholdon𝜑:(1)Thefunctionoutputisstrictlygreaterthan0suchthat𝜑%(𝑑) > 0; ∀𝜅 > 0, 𝑑 > 0;(2)Thefunctionstrictlydecreaseswith𝑑suchthat (

()𝜑%(𝑑) < 0;and(3)thefunctionstrictlydecreaseswith𝜅suchthat

((*𝜑%(𝑑) < 0.ThefirstconditionisdesignedtomaintaintheinterpretabilityofeachSGT

embedding,whereasthesecondconditionmaintainsthatcloserneighborshavehighercorrespondingdependencies,andthelastconditionmaintainsthattheeffectofdistantneighborscanbeeffectivelytuned.

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SGTusesanexponentialfunctionfor𝜑sinceitsatisfiesallthreeconditions.Usingthisapproach,thedistancebetweentwodistancescanbetakensimplyas𝑑(𝑙,𝑚) = |𝑚 − 𝑙|.

𝜑*3𝑑(𝑙,𝑚)5 = 𝑒+%(,+$), ∀𝜅 > 0, 𝑑 > 0 (1)

Sincepitchersthrowmultiplepitchesinasequence,therearelikelytobeseveralinstancesofalphabetpairs(𝑢, 𝑣).TheSGTalgorithmisinitiallyconcernedwithdetermininghowmanyofeachalphabetpairexistsinthesample.Eachalphabetpairisultimatelystoredina|𝒱| × |𝒱|asymmetricmatrixΛ.Λ&'willcontaineveryalphabetpair(𝑢, 𝑣)thatiscontainedaspecificsequence𝑠suchthatthe𝑣’spositioncanalwaysbeinferredtobeafter𝑢ineachinstancepair.

Aftercomputing𝜑foreach(𝑢, 𝑣)pairinstancethatiscontainedinasequence,theassociationfeature𝜓&'

(")isdefinedasthenormalizedaggregationofallinstances.Sinceallanalysisconductedinthispaperisconcernedwithlength-insensitiveproblems,theeffectoflengthiscontrolledforbynormalizing|Λ&'|,thesizeofthesetΛ&'orthenumberofunique(𝑢, 𝑣)pairs,withthesequencelength𝐿(")asshowninEq.2.

𝜑&'(𝑠) = ∑ .!"($!%)∀(%,$))*+(,)

|0*+(")|/2(,) (2)

TheSGTfeaturerepresentationofanentiresequencecanberepresentedastheaggregationofΨ(𝑠) = [𝜓&'(𝑠)], 𝑢, 𝑣 ∈ 𝒱.ThefeaturesΨ(")canbeinterpretedasadirectedgraphwithedgeweights𝜓andnodesin𝒱foreachsequence𝑠inthedataset.

SGTcontainsasingletuningparameter𝜅.𝜅modulatestheextenttowhichlong-termdependenciesarecapturedineachembedding.Asmallvalueof𝜅preserveslonger-termdependencies.Sincetheaveragepitchsequenceisapproximately5pitchesattheat-batlevel,asmall𝜅 = 1(defaultvalue)isselected.[5]

2.3Model-BasedClustering–GaussianMixtureModelsUnsupervisedlearningviamodel-basedclusteringcanbeusedtodiscoverpatternsinat-batpitchsequences.ThispaperreliesonGaussianMixtureModels(GMMs),amodel-basedclusteringtechniquethatusesanexpectation-maximization(EM)algorithm,tofindclustersthatcapturethecommonpatternsthatmanifestandrecurinpitchsequences.Giventhereisnosetofintuitionthatinformshownumberoftheseclustersorpatternsexists,thisanalysisrequiresanapproachthatself-determinesanoptimalnumberoftargetclustersKfromthegraphembeddings.

Otherpopularalgorithms,suchashierarchicalclustering,arealsoabletoproduceKwithoutuserlabels,usingmeasuresofin-clusterandout-of-clusterdistanceandvariance[6].However,currenthierarchicalclusteringalgorithmsscaleinO(𝑛3)timeandoftenproduceunstableresults[6].Thiscomputationalhurdleisespeciallyrelevantwhendealingwithahighvolumeofdataacrossmultipleseasons.

Toassessthepotentialinstabilityofanyresults,itisdesirabletoobtainaprobabilityestimatethatasequencebelongstotheirassignedcluster.Additionally,GMMscanframeK-selectionasamodel-selectionproblemusinglikelihood-basedmeasuressuchasBayesianInformationCriterion(BIC),whichbenefitsinterpretability[7].

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AGaussianMixtureisafunctionthattreatsseveralGaussians,eachidentifiedby𝑘 ∈ {1,… , 𝐾}.EachGaussiankincludesamean𝜇thatdefinesitscenter,acovarianceΣthatdefinesitswidth,andamixingprobability𝜋thatdefinesthesizeoftheGaussianfunction,asshownbyFigure2[8].

Eq.3explicitlydefinestheGaussianMixtureModelLikelihoodFunctionusedinthisapproach.TheGaussianMixtureModelLikelihood-FunctioncomputesaMaximum-LikelihoodEstimate(MLE)tofindtheoptimaldistributionthatunderliesthedataset.

𝐿(𝜃|𝑋4, … , 𝑋5) = ∏ ∑ 𝜋%𝑁(𝑥6; 𝜇% , 𝜎7)8%94

5694 (3)

AllmodelingandassociateddataanalysiswereconductedinPythoninaJupyterNotebookenvironment.TheGaussianMixtureModelsandExpectationMaximizationalgorithmwerebuiltandimplementedfromscratch.3 Pitch-Type&ZoneSequencingStrategiesDuringMLBgameplay,pitchersengageinadvanceddecision-making.Pitchersareincentivizednotonlytopromoteunpredictabilityintheirsequencingdecisionsbutalsoadapttothestrengthsandweaknessesofthebattertheymatchupwith.Ineachofthesesubsequentanalyses,itisassumedthatapitcheriscapableof“callingtheirownpitches;”thatis,pitchersarelargelyresponsibleforwhatpitchestheythrow.Theintuitiontosupportthisassumptionisstrong—pitchersdonotcompletelyredesigntheirpitchingstyleswhenevertheircatcherorumpirechanges,forexample.Generally,pitchsequencingcanbedistilledintotwodistinctcomponents:Pitch-typeselectionandzone (location) selection. Inorder to fitMLBpitchingdata to the required input structureof theSequenceGraphTransformalgorithm,pitchsequencesarerepresentedasorderedlistsofpitch-typelabelsanddiscretezone-location labels, respectively (SeeSection2.2.1).Fromtheseembeddings,unsupervisedlearning(GaussianMixtureModels)isusedtoclusterpitch-typeandzonesequencesindependently (See Section 2.3). Then, each at-bat sequence is assigned with two cluster labels(pitch-typeandzone)andispairedwithrelevantin-gameinformation(e.g.,score,inning,resultofthesequence).This analytical design has multiple advantages: (1) It is possible to discern which sequencingstrategiesarecommoninavarietyofin-gamecontexts;(2)Thecovarianceofpitch-typeandzone-labelscanbeanalyzed,andthecomplexitybetweenpitch-typeselectionstrategiesandzoneselectionstrategies can be compared; and (3) Sequencing styles across players and time periods can besystematicallyevaluated.3.1.1Pitch-TypeSequenceClustersModeling results from pitch-type sequences show that there are many diverse groups of pitchsequencesat theat-bat level thatare thrownin theMLB.Sincetheoptimalnumberofclusters isselected on Bayesian Information Criterion (BIC), the number of pitch-type sequence clusters isinformativeoftheaggregatecomplexityofpitch-typesequencingstrategies.Eachsequenceclustervariesintheirpitch-typeconstitutionandordering(SeeTable1;Notethatonlythetopthreepitch-typesbyrelativefrequencyaredisplayedinTable1foreachclusterforthesakeofclarity).The fitted Gaussian Mixture Model supplies a probabilistic distribution for each sequence’sassignment to a pattern cluster. The overwhelming majority of at-bat sequences (99.87%) are

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assignedtoclusterswithahighprobability(greaterthanorequalto99%assignmentprobability).The0.13%ofat-batsthatareassignedtoaclusterwithlessthananassociated99%probabilityallhave an assignment probability greater than 50% and have a mean assignment probability of86.37%.Withinthis0.13%ofat-bats,itisimportanttonotethereisnoexplicitpatternastowhichsequencesseemdifficulttoassigntoacluster.Giventhisdistribution,itisclearthatmostsequencesmapverystrongly toa singleclusterand that themodel-results canbe interpreted in termsofaprimaryclusterassignment.FromTable1, it isclearthatpitchersrelyonavarietyofpitchsequencesacrossmatchups.Sinceclusteringtakesintoaccountallpitchsequencesforpitchersthatmeetanat-batminimum,themodelresultsareexpectedlycorrelatedwithwhicharsenalsaremostpopularintheMLB(SeeFigure3,Table1).Forexample,cluster3containspitchsequencesthatincludethreeofthemostpopularpitch-typesintheMLB,soitisexpectedthatthisclusterisalsorelativelypopularamongpitchers.

Figure3:Pitch-TypeSequenceClusterbyRelativeFrequency/Density

Withineachpitchsequence,itisobservablethatthereisasplitbetweenfastballtypesandoff-speedpitchesthatarethrown;inotherwords,everysequencepatterninvolvesbothfastballpitchesandoff-speed pitches, though in varying frequencies. These results present significant evidence thatpitch-typemixingisacriticalaspectofallsequencingstrategiesacrossadiversesetofarsenals.Notethatsequenceclusters thatsharepitch-typesathigh frequenciesalsodiffer in theirorderingandentropy(SeeAppendix:Table2,Table3).While it is more difficult to summarize ordering in a table or visualization, Table 2 (Appendix)presentsaside-by-sideexample-basedcomparisonofcluster3,cluster7,andcluster10.Eachat-batexampleyieldsanestimated99%probabilityofmembershiporgreater,which impliesthatthesesequencesareexemplaryoftheidentifiedsequencecluster.Itisobservablethatsequenceclusterswithsimilarfrequenciesofpitch-typesalsodifferinordering.Theorderingcomponent ishighlightedfurtherbywhichpitch-typestendtobethrownearly inasequence versus later in the sequence (Table 2; colored by beginning, middle, and end of thesequence).Finally,itisevidentthateachclusterisapproximatelythesamelength,whichisexpectedsincethegraphembeddingslearnlength-insensitivepatternsinpitchsequences.

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Table1:Pitch-TypeClusterMembership

3.1.2DefiningSetupandKnockoutPitcheswithGraphSinksBy interpreting the graph embeddings as forward dependencies, this analysis finds evidence for“setup”and“knockout”pitchesinvariouspitch-typesequenceclusters.Drawingfromtheconceptofa“sink”ingraphtheory,itispossibletodissecteachpitchsequenceclusterintermsofitsstrategiccomponents.Inoperationsresearchandgraphtheory,asinkisformallydefinedasanodewhichonlyhasincomingflow;thatis,multiplenodesdirecttothesinkbutthesinkdoesnotdirectitselftoanyothernodesinthesystem[9].Thisdefinitioncanbestrictlymaintainedforcertainapplicationsandcontexts,suchas when modeling fluids in a pipe system, currents in an electrical circuit, or even whenapproximatingwebpage importancesuchas inthePageRankalgorithm[10].However,giventhenatureofbaseballpitching,itisunlikelythatpitchersreservetheuseofacertainpitchesonlyfortheendorbeginningofasequence.Thisiscounterintuitivetotheincentiveeachpitcherhastominimizetheirpredictability[3].Instead,bylooseningthisformaldefinitionofgraphsinkstoallowforlimitednon-zerooutflowfromsinknodes,itispossibletoidentifysetupandknockoutpitchesinthecontextofpitchsequencing.Withoutlossofgenerality,eachgraphembedding(A,B),forapitch-typeAandpitch-typeB,providesinformationonthepreferredorderingbetweentwopitches.Ahighvalueinthefeature(A,B)indicatesthatAisfollowedbyasignificantnumberofBsinthesequence.However,itisnotguaranteedthatthisrelationshipholdsfor(B,A).ItispossiblethatBisnotfollowedbyasignificantnumberofAs,eveniftheconverseistrue.Specifically,thefrequencyofpitch-typeAmaynotbedependentonpitch-typeBtotheextentthatthefrequencyofpitch-typeBisonpitch-typeA.In particular, a setup pitch is defined within each sequence cluster as a pitch whose forwarddependencies are generally similar across all associated pitch-pairings and order permutations.Giventhisdefinition,setuppitchesarethusmorecommonrelativetoknockoutpitchesinsequenceconstructionbydesign(SeeTable4).Aknockoutpitchisdefinedwithineachsequenceclusterasapitchthatfollowsmanypitchesbuthaslimitedcorrespondingoutflow.Additionally,aknockoutpitch

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couldfunctionasa localoruniversalsinkwithineachsequenceclustergraph;that is,aknockoutpitchcouldhavealargepositivedifferencebetweenitsinflowandoutflowforasinglespecificpitch(SeeAppendix:Table4–“GreatestDiff.”)orhavepositivedifferencesbetweeninflowandoutflowformultiplesetuppitches.Withineachcluster,boththesetupandknockoutpitchesneededtomeetafrequencythresholdtobeidentifiedassuch—thispreventspitchesthatrarelymanifestwithinasequencefrombeingdesignatedaseitherasetuporknockoutpitchforthesequence.Givengeneraldefinitionalconstraints,itisexpectedthattherearefarmoresetuppitchesandfarfewerknockoutpitches.Thenomenclaturehereisnotintendedtosuggestthatsetuppitchesarenotthrownwiththegoalofachievingafavorableoutcome(e.g.,strikingthebatterout,recordinganout)atthatspecificmoment.Likewise,aknockoutorterminalpitchisnotaninterpretationofaplayer’smosteffectivepitcheswithrespecttoachievingthosesimilarlyfavorableoutcomes.Table4displays theprimarysetupandknockoutpitches thatwere identifiedupon inspectionofforwarddependencies.Itisimportanttonotethatnotallclustershaveknockoutpitches.Specifically,using thedefinitionabove, for these fewclusters, therewerenopitches thathada largeenoughpositivedifferencebetweenitsinflowandoutflowforaspecificpitchorhadconsistentlypositivedifferencesbetween inflowandoutflowformultiplesetuppitches. Instead, forclusterswithoutaknockout pitch, the forward dependencies between pitch-types were approximately equal inmagnitudeirrespectiveoftheexactdirectionbetweenthetwocomparisonfeatures(A,B)and(B,A).

Table4:SetupandKnockoutPitches

Note:“Val.ofDiff.”reportsthedifferenceinthesumofforwarddependenciesfor(A,B)andthesumoftheforwarddependenciesfor(B,A)withineachcluster.PitchBisconsideredaknockoutpitch(i.e.,localsink)givenarelativelylargepositivedifferencebetweenthesevalues.Additionally,apitchBcanbeconsideredaknockoutpitchiftherearemultiplepitchesA,C,D…,nsuchthatthedifferencebetween∑ (𝑖, 𝐵)!

"#$,"&' ≫∑ (𝐵, 𝑖)!

"#$,"&' isalsotrue.

Themagnitudeofthedifferencebetweentheforwarddependenciesfromonedirectiontoanotheralso indicate that certain knockout pitches are “stronger” than other knockout pitches betweenclusters.Asshownincluster3,thereisaveryextremedistinctionbetweenthesumoftheforwarddependenciesin(FF,CH)versusthesumoftheforwarddependenciesin(CH,FF).Thisimpliesthatsequencesthatbelongtocluster3rarelyusechangeupsunlessthechangeupcomesattheendofthesequence.Incomparison,thegreatestdifferencebetweenforwarddependenciesforcertainclusterssuchascluster12isfarlowerthanthedifferencethatisdisplayedincluster3.Thisfindingsuggeststhat thechangeup, theknockoutpitch incluster12, isnotexclusivelyusedtowardstheendofa

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sequence.Acrossallsequenceclusters,knockoutpitchestendtobebreaking-ballswhereassetuppitchesarelargelyfastballsorfastballvariations.Whilethisintuitionisassumedastrivialdomainknowledge for teams, coaches, and players, this quantitative assessment of setup and knockoutpitcheswithinthecontextofpitchsequencesislikelynovel.3.1.3Pitch-TypeSequencinginContextThereareseveralsequencingstrategiesandpatternsthatarepotentiallyobservablefromthemodel-results.Forexample,eachsequenceclusterinvolvesacombinationoffastballsandoff-speedpitches,ashighlightedabove.Itisperhapssignificanttonotethattherearemultiplesequencesthatsharepitch-typeconstitutionyetdifferinordering.Thissuggeststhatorderingisanimportantcomponentof pitch-type sequences which alternatives such as simple pitch frequencies and pitch-to-pitchcorrelationdonototherwisecapture.Sinceitisexpectedthatpitchersthrowpitchsequencesthatbelongtodifferentseveralclusters,itisfeasibletoanalyzehowsequenceusagevariesinmultiplein-gamecontextsandmatchups.Figure5&6(Appendix)displaystherelativefrequencyofeachclusterconditionalonpitcher-batterhandednessandstarter/relieverrole,asexamples.3.2.1ZoneSequenceClustersAs with pitch-type sequencing, there are several universal zone sequencing patterns that wereidentifiedbymodel-basedclustering.Pitcherstendtovaryinhowtheyset-upbattersviatheirzonesequences,buttheirusageoftheseclustersisnotuniformlydistributedacrossgameplaysituations(SeeFigure7,Appendix:Figure8&9).Sinceplayershaveallzonesavailabletothem(i.e.,theyarenotinhibitedbypitcharsenal),clusterpopularityrelatestowhichlocationstrategiesarepreferredbyMLBpitchersinsteadofphysicalorarsenalconstraints.Itissignificanttonotethatthemodel-basedclusteringidentifiesfewerclustersforzone-basedsequences thanpitch-typesequences.Additionally,whencomparedbetweeneachother,zone-basedclustersreflectthepitcher’sdesiretonotleavepitchesinthemiddleorhighinthezonetopreventsolid-contactandbatted-ballsthatarehitintheair(Table5).Instead,eachclusterhasahighrepresentationofzonesthatarelowerdowninthestrikezone.Thisultimatelysupportsacasethatpitch-typesequencingismorecomplexinaggregatethanzonesequencing.

Figure7:ZoneSequenceClusterbyRelativeFrequency/Density

Aswithpitch-typesequenceclustering,model-basedclusteringallowsustocalculateaprobabilisticdistributionforeachuniqueat-bat’sassignmenttoazonesequencecluster.Likepitch-typesequenceclusters, theoverwhelmingmajorityofat-batsequences(99.60%)areassignedtoclusterswithahighprobability(greaterthanorequalto99%assignmentprobability).The0.40%ofat-batsthatare

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assigned to a zone clusterwith less than an associated 99% probability all have an assignmentprobabilitygreaterthan50%andhaveameanassignmentprobabilityof88.99%.Aswiththepitch-typeclusters,here isnoexplicitpatternas towhichzonesequences seemdifficult toassign toacluster.Theseresultsindicatethattheprimaryzoneclusterassignmentcanbeuniformlyusedgiventhehighassignmentprobabilities.

Table5:ZoneClusterMembership

3.2.2ZoneSequencinginContextIfzonesequencingstrategiesarerelativelynon-complex,thenitisexpectedthatthedistributionofzone sequence clusters to be relatively consistent between pitchers and across matchups. Inparticular, split summary statistics can easily describe how the usage of zone sequence clustersdiffersagainstvariousright-andleft-handedbattersandbetweenstartersandrelievers.Thetwozone-basedsequenceclusters(clusters4and6)thathaveahigherfrequencyofpitchesuporinthemiddleofthestrikezonetendtobethrowninsimilarcontexts,asshowninFigure8&9(Appendix).Bothcluster4andcluster6arethrownmorebyrelieversthanbystartersincomparisontotheotherzoneclusters,thoughtheabsolutedifferenceinusagebetweenstartersandrelieversisfarlessdrasticthandifferencesthatareobservedinusageofpitch-typeclustersbetweenstartersandrelievers.Additionally,bothclusters4and6are thrownpredominantlywhenpitcherssharehandednesswiththeirbatters.3.3RelationshipbetweenPitch-TypeandZoneSequencingTheChi-Squaretestofindependence(df=104)findsamplestatisticalevidencetosuggestthatthepitch-typeandzone-clustersarenotindependent;thatis,thevalueofoneclustervariablesimpliesachangeintheexpectedprobabilitydistributionoftheother(p<0.001).Giventhatcertainpitch-typescoincidewith location (e.g., breaking-balls aredesigned tobe in lowerzones), it is expected thatclusterswith high frequencies of these types of pitches tend to be associatedwith certain zonesequences.Thus,futureanalysisandworkcanattempttore-clusterpitchlabels(beforesequenceanalysis)by including locationasapitch-level characteristic. Inaggregate,playersdonotexhibitstrongdifferencesinzonesequencingstrategiesabovewhatisexpectedbydifferencesintheirpitch

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arsenals.However, it is noteworthywhenpitchersdecide todeviate from their otherwise strongtendencies. This analytical framework allows teams to evaluate game- and matchup-levelcircumstanceswhentheymayanticipateshiftsinpitchsequencingbehavior.3.4ExamplePlayerSequencingComparisonsTheclusteringframeworkenablesplayercomparisononthebasisofpitchsequencing.Whileit isvaluabletoknowwhatsequencesaplayerisaccustomedtothrowing,itisincreasinglyimportanttohaveanunderstandingofhowaplayer’s sequencingbehavior changes invariousgamecontexts.Teamscanachieveanedgeinscouting,playerdevelopment,andmatchuppreparationbyusingthesetoolstodecryptpitchingpatternsandinvestigatemotivationsbehindthesedecisions.Figure 10 displays side-to-side comparison of the pitch-type sequencing strategies used by 4prominent MLB pitchers against left-handed and right-handed batters. There are several keytakeawaysthatareobservablefromthissmallsampleandarelargelyreflectiveofbaseballintuition.Each pitcher relies on various clusters or pitch-type sequencing patterns. Pitchers are largelyincentivized to remainunpredictable andvariance inpitch-type sequencing is one componentoftheirabilitytodeceiveabatter.Therearealsocommonalitiesamongpitcherswhoshowsimilaritiesinsequencepatterndistribution.Forexample,starters(e.g.,BlakeSnell)tendtoshowmorevarianceandmatchupversatilityintheirpitch-typesequencingstrategiesthanChapmanandHader,bothofwhomarehigh-leveragerelievers.Usingapitcher’sclusterdistributionasaproxytheoveralldiversityoftheirsequencingstrategies,astrong negative correlation between sequencing complexity and fastball velocity is detected.Intuitively,thissuggeststhatplayerswhothrowhardcanrelyontheir“stuff”more-sothanplayerswhodonothavethatadvantage.Conversely,pitchersthatcannotthrowinhighvelocitiesneedtotakemorediverseandcomplexapproachestotheirsequencingdecisions.Thisfindingisconsistentwithpriorresearchoneffectivevelocityandpitchsequencingstrategy[11].PlayersseemtopursueuniversalsequencingstrategieswithrespecttozoneselectionasshowninFigure11.Theirzonesequencedecision-makingishighlycorrelatedwiththeirpitcharsenal.Unlikepitch-typeselectionwhereplayersneedtodrawfrompitch-typesintheirpitch-arsenal,anyzoneisavailable toapitcherat agiven time, assuming thatapitcherhasageneral commandover theirpitches.Accordingly,eachpitcherinthissamplethrowsatleastonesequencethatbelongstoeachcluster.Aswithpitch-typesequencing,eachpitcherdisplaysauniqueassortmentofpatternsandbehaviorsintheirzonesequencingstrategies.BlakeSnell(LHP),forexample,takesdivergentapproacheswhenhepitchestoleft-handedandright-handedbatters;specifically,hereliesmuchmoreonsequencecluster6(characterizedbypitchesupinthezone)againstleft-handedbattersandmoreonsequencecluster 1 (mixed locations) and sequence cluster 8 (low locations). This shift is not particularlyobservable inotherpitchers and is visually apparent inFigure6.Otherpitchers, suchasAroldisChapman, Josh Hader, and Clayton Kershaw, depend on very similar zone sequencing strategiesagainstleft-handedandright-handedpitchers.So,whilezonesequencingstrategiesarelesscomplexthan pitch-type sequencing strategies in aggregate (See 3.2.1), players still exhibit significantdifferencesinhowtheyformstrategiesforlocatingtheirpitches.

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Given that both pitch-type and zone sequences are clustered on an at-bat level, player-specificdevelopmentandevolutionacrossvarioustime-spans(e.g.,withinagameorseason)isobservable.Whilesomeplayers’sequencingbehaviorsarerelativelystableacrosstheseasonstheyparticipatedin,otherplayersshowevidenceofshifts inhowtheyapproachsequencingovertime.Thiscanbeattributedtochangesinmatchupstrategy,growthanddevelopmentofpitcharsenals,oreveninjurymitigation and recovery processes. Conversely, pitchers do not seem to exhibit significant time-dependenceintheirzonesequencingstrategies.

Figure10&11:Pitch-Type&ZoneSequencingTendencies–PlayerExamples

3.5AggregateSequence-BasedPlayerSimilarityAfteraggregatingpitch-leveldata,pitchsequencesareanalyzedattheat-batandplayer-careerlevel.Theat-batembeddingsareusedtoperformanalysisongame-varyingqualitiesofpitchsequencing;an example analysis that this paper explores is what pattern clusters are identifiable across allpitchers.Theaggregatepitchersequenceembeddingswillformthebasisforplayersimilaritysearch.Playersimilaritiesareimmenselyusefultoteams,scouts,andfans,andhavealonghistoryinbaseballanalytics dating back to Bill James’ similarity scores [11]. Since player similarity enhances ourcontextualunderstandingofaplayer’stendenciesandbehavior,itisvaluabletodetermineplayersimilarity as measured through sequencing decisions that players have made during in-gamesituations. InthesamewaythataDNAsequencedefinesthebiologicalmakeupofanorganism,apitchsequence—whenaggregatedacrossaplayer’scareer—summarizesthecumulativedecisions

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that the pitcher has previously made. After embedding each player-aggregate sequence, playersimilaritiesarecomputedwiththedotproductofthequeryembeddingwithotherembeddingsinourplayersequencedatabase[5].Foreachplayersequencequery,theplayersequencingembeddingthatproduces the largestdotproductwith thequerywillbe takenas themostsimilarsequence.Whenusing rawembedding outputs, this similaritymethod takes into account diversity in pitcharsenals since pitch-types that are not seen in a specific players sequence will have a forwarddependencythatisinitializedwithazero-value.Usingthedotproduct,pitcherswithdiversepitcharsenals(i.e.,manypitch-typestheycanthrow)tendtoappearfrequentlyassimilaritymatchessincetheyhavethelastnumberofforwarddependenciesinitializedaszero.Thus,itisimportanttonotethat while these similarity matches are correlated with player pitch arsenals, sequence-basedsimilarityalsotakesintotheorderinwhichpitchesarethrown.Sequence-based player similarities are used to begin to explore the relationship between playervalueandsequencing.Drawingfromaplayersimilarityfunctionthatoutputstheclosestplayerforeachplayerinput,itisevidentthatsimilarityinpitch-typesequencingdoesnotcorrelatetoeithersimple(e.g.,K%,ERA,H/9)oradvancedmetricsofplayerperformance(e.g.,FIP,xFIP,tERA)abovewhatisachievedfromrandomassignment.Whiletherearemoreextensivestatisticalmodelsthatcantest thishypothesis further,playerswhomanifestsimilarsequencingstrategiesseemtovaryquitedrasticallyintheirperformance.Thissignalsthatsequencingalonedoesnotpredictaplayer’sperformanceorcompensateforalackofpitching“stuff.”

It is also important to acknowledge that the approach described in this paper should not beinterpretedasameanstoassignvaluetopitchsequencesorsequencingdecisions.Inordertodefinesequencingskillprecisely,researchersshouldstrivetoseparatetheeffectandvalueof individualpitches from the added benefit of a specific ordering. Instead, this research is instrumental inprovidingaframeworktodecodethesequencingstrategiesthatpitchersconsistentlyutilizeforthepurposes of in-game strategy. Instead of applying pitch sequencing similarities for player valuepredictions,playersequencingsimilaritiescanbeutilizedtoassistteamsinmatchuppreparationandpitchersintheirdevelopmentalgoals.

Fromthisgraphembeddingapproach,apitcher’saggregatesequencingdecisionscanberepresentedin total by a directed graph or network. Each graph representation can easily be presented as avisualization(SeeFigure12&13)thatcharacterizesapitcher’ssequencingpatternsacrossmultipleseasons (2016-2019). This visualization can capture the general complexity of each pitcher’spitchingstyle foravarietyofresearchanddevelopmentpurposes.Therelativesizeofeachnodecorrespondstotherelativefrequencyofeachpitch-typewhereastherelativethickness(i.e.,weight)ofeachedgecorrespondstotheforwarddependencybetweentwopitch-types.In order tomaintain visual consistency across all graphs, eachmajor pitch-type is included in apitcher’sgraphvisualizationirrespectiveofwhetherthatpitcherhasthatpitchintheirarsenal.Eachlinkiscoloredaccordingtothedirectoftheoriginatingnode.Thestrongerassociationislaidovertheweakerassociationdirectionally.Therelativesizeofeachnodedenotestheexistenceofthatpitchinaspecificpitcher’sarsenal.Thisconsistency inthevisualizationformatallowsforaqualitativeside-to-side comparison of player network graphs, though the design/aesthetic features arecustomizablebytheuser.Thedatathatunderlieseachnetworkvisualizationcanbelimitedtocertainin-gamescenariosormatchupstoillustratesituationaldecision-making.

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Figure12&13:SequenceGraphNetworksforBlakeSnellandClaytonKershaw(2016-2019)

4 RelatedApplications&FutureWorkIn addition todeveloping anovel analytical framework to studypitch sequencing strategies, thispaperpresentsmultipletoolsthatMLBteamscanuseinmatchuppreparation,scouting,andplayerdevelopment.Sincepitchsequencingisotherwisedifficulttoencodeforusingtraditionalstatisticalmethods, teams can take immediate steps in applying this graph embedding approach to studypitcher styles and behavior. There are many questions to be asked about the motivationssurroundingusageofcertainpitchsequences.Thisframeworkenablesresearcherstoreproducethisapproachintheiranalysisofpitcherbehavior.Furthermore,graphnetworkvisualizations(asshownin Figure 12 & 13) that are based upon player-aggregate sequence data can be helpful incommunicatingapitcher’ssequencingprofileinaninterpretablemanner.Whenmappedinatime-series,teamscanobservehowsequencenetworkschangeovertimeandhowplayersaredevelopingand modifying their existing strategies. This tool can supplement advance scouting preparationagainst other teams, especially after conditionalizing player data based on certain in-game andmatchupscenarios.Additionally,teamscanconsiderhowapitcher’ssequencingchangesinresponsetotheirpreviousoutcomesinanat-batorgame.Thereisastrongstrategiccomponenttothisworkaswell—futureworkmayconsidertestingwhetherpitchsequenceclustersarepredictiveofplayerdecisions,whichwouldimplythatpitchersarevictimstoinformationleakageatthesequencelevel.Finally,stepsaretakentodevelopanovelopen-sourceaggregationdatabasethatallowsindependentresearchersandMLBteamstoperformpitchsequenceanalysisbasedonthisresearch.Functionalityofthistoolkitincludessearchingforspecificpitchsequencesthathaveoccurredoveramulti-yearspan,ability tovisualizenetworksanddownloadpitchsequencinggraphembeddings forspecificpitchers,andaccesstodrop-downcomparisonsandplayer-similaritymatching.

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References[1]Bock,J.(2015).PitchSequenceComplexityandLong-TermPitcherPerformance.Sports,40-55.[2]Sharpe,S.(2020).MLBPitchClassification.https://technology.mlblogs.com/mlb-pitch-classification-64a1e32ee079.[3]Glenn,H.,&ZhaoS. (2017).UsingPitchf/x tomodel thedependenceof strikeout rateon thepredictabilityofpitchsequences.https://content.iospress.com/articles/journal-of-sports-analytics/jsa103.[4] Zhan, J., Gerstner, L., & Polimeni, J. (2020). Measuring the Impact of Robotic Umpires.https://global-uploads.webflow.com/5f1af76ed86d6771ad48324b/5f6a65851d1ac98081a707f0_Zhan_MeasurinM-the-impact-of-robotic-umpires.pdf.[5] Ranjan, C., Ebrahimi S., & Paynabar, K. (2016). Sequence Graph Transform (SGT): A FeatureEmbeddingFunctionforDataMining.arXiv:1608.03533v13.[6]Manning,C.,Raghavan,P.,&Schütze,H.(2008).IntroductiontoInformationRetrieval.ISBN:0521865719.[7]Srihari,S.(2019).MixturesofGaussians.https://cedar.buffalo.edu/~srihari/CSE574/Chap9/Ch9.2-MixturesofGaussians.pdf. [8]Carrasco,O.(2019).GaussianMixtureModelsExplained.TowardsDataScience.towardsdatascience.com/gaussian-mixture-models-explained-6986aaf5a95.[9]Brossard,E.(2010).GraphTheory:NetworkFlow.UniversityofWashington.https://sites.math.washington.edu/~morrow/336_10/papers/elliott.pdf.[10] Agarwal, B., & Khan, M. H. (2013). Analysis of Rank Sink Problem in PageRank Algorithm.InternationalJournalofScientific&EngineeringResearch.https://www.ijser.org/researchpaper/Analysis-of-Rank-Sink-Problem-in-PageRank-Algorithm.pdf.[11]DrivelineBaseball.(2019).CallingtheRightPitch:InvestigatingEffectiveVelocityattheMLBLevel.https://www.drivelinebaseball.com/2019/05/calling-right-pitch-investigating-effective-velocity-mlb-level/.[12]BaseballReference.(2020).SimilarityScores.https://www.baseball-reference.com/about/similarity.shtml.

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AppendixTable2:OrderComparisonbyCluster

Table3:ClusterAverageEntropy

Figure5&6:Pitch-TypeSequenceClusterSplitsbyMatchupType&Role

Figure8&9:ZoneSequenceClusterbyRelativeFrequency/Density