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MixedMembershipMartialArts:Data-DrivenAnalysisofWinningMartialArtsStyles

SeanR.HackettandJohnD.StoreyLewis-SiglerInstituteforIntegrativeGenomicsPrincetonUniversity,Princeton,NJ08544,USA

Abstract

AmajoranalyticschallengeinMixedMartialArts(MMA)iscapturingthedifferencesbetweenfightersthatareessentialforbothestablishingmatchupsandfacilitatingfanunderstanding.Here,wemodel~18,000 fightersasmixturesof10data-definedprototypicalmartialartsstyles,eachwithcharacteristicwaysofwinning.Bybalancingfighter-leveldatawithbroadertrends inMMA, fighterbehaviorcanbepredictedeven for inexperienced fighters.Beyondprovidinganinformativesummaryofafighter'sperformance,styleisamajordeterminantofsuccessinMMA.Thisisreflectedbythefactthatchampionsofthesportconformtoanarrowsubsetofsuccessfulstyles.

1. IntroductionEarlyeventsinMixedMartialArts(MMA)weretoutedasachancetodeterminewhichstylesofpuremartialartsweremosteffectivefordefeatinganopponent[1,2].Theseevents,epitomizedbythemid-1990seventsoftheUltimateFightingChampionship(UFC)andValeTudo,featuredmatchupsofstylisticallydiversemartialartistsrepresentingsportssuchasBoxing,JudoandKarate[1,3].AsthesportofMMAhasevolved,fightershavebecomeincreasinglywell-rounded[1,2].ModernMMAfightersneedtobeeffectivestrikerswhodictatewherethefightoccurs,byeithertakingdowntheiropponentsorforcingthemtofightstanding.

EachMMA fighters skills aredrawn fromamixofmultiplepuremartial arts thatdefinehis/herpersonalizedfightingstyle.WhilemodernMMAfightersaremorestylisticallymixedthantheirpuremartialartistforbearers,themixturesofindividualfightersdiffer.SomefighterswouldbereferredtoasstrikersandothersasgrapplerswhospecializeinmartialartssuchasBrazilianJiu-Jitsu(BJJ)[4].Thesestylisticcharacterizationsareimportantwhenestablishingmatchups;buttodate,thereisnoquantitativeprocedureforcharacterizingthestyleofMMAfighters.Furthermore,becauseoftheabsenceofsuchanapproach,theimpactofstyleonsuccessinMMAhasneverbeenquantitativelyinvestigated.

Every sport is enriched by the behaviors that distinguish its athletes and teams.Whether one isconcernedwithapitchersarsenalofpitches,theplaysafootballteamemploysorthelocationsfromwhich a basketball player likes to shoot, style is an important (albeit often nebulous) concept insports.Tomakestylemoreaccessible,data-drivenmodelingapproacheswillbeinvaluable.Thesemodels have enormous potential; they can be used to identify prospects who are similar to

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superstars,buildabalancedteamandguidevisualizationsthatinformcasualfansofeachathletesstrengths.Todate,approachesthathaveaimedtodefineathletes'styleshavegenerallyattemptedtoidentifylatentfactorsthatcouldserveasarepresentationofstyle[5,6,7].Theseapproacheshavebeenrestrictedtospatialsummariesofathletes(e.g.,wheretheyscorepoints).Consequently,theirlatentfactorsreflectcommonspatialpatterns,likethree-pointshootingordunkinginbasketball[6].

Whilelatentvariablebasedapproacheshavebeensuccessfullyappliedtosportsforwhichalargeamountofathlete-specificperformancedataisavailable,suchmethodstendtooverfitpatternswhenlittledataisavailableorwhencomplexpatternsaresought(forexample, ifstylewereallowedtovary between seasons or between games). Most sports are not immune to such overfitting; forinstance,thecapacityforoverfittingobserveddatahasbeenwelldemonstratedwiththedata-richapplicationofbaseballbattingaverages[8,9].Wewouldintuitivelyexpectabatterwith300hitsoutof1000at-bats toperformbetter in the futurethanabatterwith5hitsoutof10at-bats,yet thesecondbatterwouldhaveahigherbattingaverageusingtheMaximumLikelihoodEstimate(MLE)ofbattingaverages.Indeed,itwasdemonstratedthattheMLEwaslessaccuratethantheJames-Steinestimate of batting averages, which shrinks observed batting averages towards the mean of allplayersbasedontheamountofindividual-specificdata[8].Bayesianapproachesofthissorthavethecapacitytobalanceathlete-specificdatawithoverallbehaviorinordertoimproveprediction.

CharacterizingthestylesofMMAfightersisanunprecedentedchallengebecauseofboththeamountandtypeoffighter-leveldataavailable.Mostprofessionalathletescompeteinupwardsof100gamesinasingleseason.Incontrast,asuccessfulMMAfightermayretireafteronly20bouts.Withsuchlimiteddata,afighter'sto-datebehavioraloneisinsufficienttopredicthis/herfutureperformance.Additionally,whilethemannerbywhichafighterfinishesafight(e.g.,finishessuchaspunches,kicksorchokes)isthepurestdepictionofhis/herstyle[3],thesemetricsarecategorical.Consequently,methods for defining spatially-oriented styles from other sports are not directly applicable. WeaddresstheabovetwochallengesbyapplyinglatentDirichletallocation(LDA)[10,11],aBayesianmixedmembershipmodel that naturally accommodates categorical data and shares informationacross fighters, thereby balancing fighter-specific behaviorwith broader trends. The goal of thismodel is to estimate twomixtures that define a plausible underlying structure of fighters' styles(Figure1):(i)Distinctmartialarts(prototypes)havecharacteristicfinishesand(ii)Fightersareamixtureofprototypes(withindividualizedweightssummingtoone).

Figure 1: Identifyingfighter styles based onperformance.WeassumethatanMMAfighter'swinsgenerally reflect his/hertraining, either as a puremartial artist or as apractitioner of multipleprototypicalstyles.Eachofthese fighting stylesutilizesasubsetofpossiblefinishes at characteristicfrequencies.

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Usingdata from>250,000 fights,weapplied themixedmembershipmartialarts (3MA)model toidentify10prototypicalmartialartsstyles,eachcomposedofasetofcharacteristicfinishes.Thestyleof each of ~18,000 fighters was defined based on his/her individualized combination of theseprototypes.Thesestylisticmakeupsgreatlyimprovedthepredictionofheldoutresults.Beyondtheirvalueasanimplicitreadoutofafightersabilities,fighterstylesareamajordeterminantofsuccessinthecage.Inparticular,winningfightsintheupperechelonsofMMArequiresdynamicstrikingandtheabilitytogothedistancebywinningfightsthroughjudgesdecision.DefendingUFCchampions,themostsuccessfulathletesinthesport,generallyconformtothesestyles,providinginsightsintohowfightingstrategyaffectssuccessinMMA.2. DatasetUsingtheSherdogFightFinder[12],weobtainedrawsummariesof257,582previousamateurandprofessionalboutsamong151,746fightersthroughOctober29,2016.Eachboutwassummarizedbasedonthepairoffighters,theresultofthebout(win,loss,draw,nocontest)andthewayinwhichthe bout was finished (e.g. by punches or unanimous decision). Rare finishes were manuallycombined with the most similar high-frequency finish to generate 50 distinct high-frequencycategories (shown in Figure 2). Only fights that ended in awin for one fighterwere considered,leaving230,126boutsforthisanalysis.

3. IdentifyingpatternsinfinishusageAttheirmostbasiclevel,martialartsaresetsoftechniquesthatareusedtowinfights.Eachfightiswonbyusingasingletechnique,butifwelookatafighter'scareer,patternsmayemergewhereinsetsoffinishesrecuracrossafighter'svictories.Ifthesepatternsareduetoamartialartthatissharedamongfighters,wewouldexpectsetsoffinishestoco-occurnotonlyinasinglefighterbutalsoinarecurringmanneracrossmany individuals.Forexample,MuayThai fightersusebothelbowsandknees,soweexpectthatfighterswhowinwithelbowstendtowinbyknees.Thus,elbowsandkneesshouldco-occur.

To determine whether any subsets of the 50 finishes are frequently used in conjunction, wequantifiedhowofteneachpairoffinishesco-occurs(i.e.,isusedtowinfightsbythesamefighter),aggregatingacrossall fighters.Toanswerthisquestion,wecomparedobservedco-occurrencesofeachfinishpair(numberofpairsofafinishxandyinafighter'srecord,summedoverfighters)withthe expectation under independence. Then, using permutation testing,wedeterminedhowoftensuch large deviations were expected. Out of 1,225 pairs of finishes, 289 pairs co-occur morefrequentlythanexpected,and292pairsco-occurlessfrequentlyatafalsediscoveryrateof0.05[13].

Pairs of finishes that significantly co-occur are far from random; rather, they are organized intocliques of generally mutually correlated finishes (Figure 2). For example, kicks/knees form onetightlyconnectedcliqueandlegsubmissions,another.Fromthisanalysis,wealsoseethatpatternsaremorestronglyinfluencedbythepositionfromwhichasubmissionisapplied,ratherthanbywhattypeofsubmissionisapplied,perse.Forexample,theanacondachoke,appliedfromafrontheadlockposition,isindependentfromthetrianglechoke,whichisappliedfromguard.Incontrast,trianglechokesandarmbarsarefrequentlyappliedfromguardandthustendtoco-occur.

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Figure 2: Structure ofMMA finishes based onpairwiseco-occurrence.289 pairs of finishessignificantly co-occurredamongfighters.Thethreestrongest connections toeach finish (dark edges)were used to construct alayout; other significantedges are shown as faintbackgroundlines.

4. Determiningfighter-specificstylesByaggregatingdataoverall fighters,wedemonstrated that finishesexist in co-occurringgroups.Directly applying this information at the fighter-level is challenging, however. Although wecollectivelyhavealargeamountofinformationaboutfightersperformances,wehaveamoremodestamountoffighter-specificdata.Nofighterhaswonafightwitheachofthe50possiblefinishes.Infact,thevastmajorityofexperiencedfightersdonotevenhave50winstodate.Becauseweonlyhaveanincompletereadoutofafighterscapabilities,weneedtobalancewhatweknowaboutafighterwithwhatwe do not, hedging the uncertainty in fighter-specific data by using broader patternsacrossfighters.Specifically,weusethebehaviorofsimilarfighterstodefinecommonfinishpatterns.Wethencharacterizeindividualfightersinthecontextofhowfrequentlytheyhavefoughtandtheclassesoffinishestheyhaveemployed.

4.1. Themixedmembershipmartialarts(3MA)modeloffighterstyles

Increatingamodeloffighterstyles,ourultimategoalistodeterminethefinishprob