Data-driven identification of optimal substitutions in soccer

Deg ee in Ma hema ics Deg ee in Enginee ing Physics Bachelo ’s Deg ee Thesis Da a-d i en iden i ica ion o op imal subs i u ions in socce Joan He nanz i Ib´ a˜ nez Di ec o (KU Leu en): Jesse Da is Tu o (UPC): En ique Rome o Me ino May 2023 Abs ac Da a-d i en oo ball analy ics is a ising ield, and clubs a e spending mo e and mo e esou ces on gaining a compe i i e edge h ough da a-d i en echniques. Subs i u- ions a e he main ool a coach has o in e ene in he cou se o he game, and hei limi a ion and ele ance ha e a ac ed in e es o hei s udy. In his hesis, we seek a da a-in o med app oach o iden i ying and p edic ing op imal subs i u ions. Wi h he ecen change in oo ball legisla ion, now each eam is pe mi ed up o i e subs i u ions pe ma ch. We compa e he new pa adigm wi h he p io one and look o op imal subs i u ions wi h he use o da a-based models. Wi h machine lea ning classi ie s and a subs i u ion sensible in-game win p obabili y model, playe changes ha e been assessed. The same models ha e been used o simula ing al e na i e ypes o subs i u ion and iming, in o de o make da a-d i en app oaches ha inc ease he chances o success. The addi ion o ex a subs i u ions has esul ed in an inc ease in ac ical in e en ions by coaches, bu he ma ch dynamics ha e emained he same. Machine lea ning models ob ain good esul s in he p edic ion o subs i u ion assessmen . The win p obabili y model is sensible o subs i u ions, which ha e gene ally a be e e ec on he subs i u ing eam. O ensi e subs i u ions gene ally inc ease he winning p obabili ies, especially o losing eams. No iming is obse ed o be signi ican ly be e o doing subs i u ions, bu in pa icula cases can be e y ele an . Keywo ds: Foo ball Analy ics, Foo ball, Big Da a, Subs i u ions, Machine Lea ning, Win P obabili y Model, Bayesian Model, Da a-D i en Decisions. Ame ican Ma hema ical Socie y 2020 Ma hema ics Subjec Classi ica ion: 62-07, 60E99 1 Resum L’an`alisi basada en dades del u bol ´es un camp en auge, i els clubs dediquen cada egada m´es ecu sos a ob eni un a an a ge compe i iu mi jan¸can `ecniques basades en dades. Les subs i ucions s´on la p incipal eina de la qual disposa un en enado pe a in e eni en el anscu s del joc, i la se a limi aci´o i elle `ancia han despe a l’in e `es pel seu es udi. En aques a esi busquem un en ocamen anal´ı ic pe a la iden i icaci´o i p edicci´o de subs i ucions `op imes. Amb el ecen can i en la legislaci´o u bol´ıs ica, a a es pe me en a cada equip ins a cinc subs i ucions pe pa i . Compa em el nou pa adigma amb l’an e io i busquem les subs i ucions `op imes amb l’´us de models basa s en dades. Amb classi icado s d’ap enen a ge au om`a ic i un model de p obabili a de ic `o ia en di ec e sensible a les subs i ucions, s’han a alua els can is de jugado s. Els ma eixos models s’han u ili za pe a simula subs i ucions de di e en ipus i en momen s al e na ius, amb la inali a de eali za plan ejamen s basa s en dades que augmen in les p obabili a s d’`exi . L’addici´o de can is addicionals ha p o oca un augmen de les in e encions `ac iques dels en enado s, pe `o la din`amica del pa i es man ´e igual. Els models d’ap enen a ge au om`a ic ob enen bons esul a s en la p edicci´o de la alo aci´o de les subs i ucions. El model de p obabili a de ic `o ia ´es sensible a les subs i ucions, que solen eni un millo e ec e pe a l’equip que les eali za. Les subs i ucions o ensi es solen augmen a les p obabili a s de ic `o ia, especialmen pe als equips que an pe den . No s’obse a que cap momen sigui signi ica i amen millo pe a e el can i, pe `o en casos pa icula s po se elle an . Pa aules clau: Anal´ı ica de Fu bol, Fu bol, Big Da a, Ap ene a ge Au om`a ic, Model de P obabili i a de Vic o ia, Model Bayesi`a, Decisions Basades en Dades. 2 Resumen El an´alisis basado en da os del ´u bol es un campo en auge, y los clubes dedican cada ez m´as ecu sos a ob ene una en aja compe i i a median e ´ecnicas basadas en da os. Las sus i uciones son la p incipal he amien a de la que dispone un en enado pa a in e eni en el anscu so del juego, y su limi aci´on y ele ancia han despe ado el in e ´es po su es udio. En es a esis buscamos un en oque anal´ı ico pa a la iden i icaci´on y p edicci´on de sus i uciones ´op imas. Con el ecien e cambio en la legislaci´on u bol´ıs ica, aho a se pe mi en a cada equipo has a cinco sus i uciones po pa ido. Compa amos el nue o pa adigma con el an e io y buscamos las sus i uciones ´op imas con el uso de modelos basados en da os. Con clasi icado es de ap endizaje au om´a ico y un modelo de p obabilidad de ic o ia en di ec o sensible a las sus i uciones, se han e aluado los cambios de jugado es. Los mismos modelos se han u ilizado pa a simula sus i uciones de dis in o ipo y en momen os al e na i os, con el in de ealiza plan eamien os basados en da os que aumen en las p obabilidades de ´exi o. La adici´on de cambios adicionales ha p o ocado un aumen o de las in e enciones ´ac icas de los en enado es, pe o la din´amica del pa ido se man iene igual. Los modelos de ap endizaje au om´a ico ob ienen buenos esul ados en la p edicci´on de la alo aci´on de las sus i uciones. El modelo de p obabilidad de ic o ia es sensible a las sus i uciones, que suelen ene un mejo e ec o pa a el equipo que las ealiza. Las sus i uciones o ensi as suelen aumen a las p obabilidades de ic o ia, especialmen e pa a los equipos que an pe diendo. No se obse a que ning´un momen o sea signi ica i amen e mejo pa a hace el cambio, pe o en casos pa icula es puede se ele an e. Palab as cla e: Anal´ı ica de F´u bol, F´u bol, Big Da a, A endizaje Au om´a ico, Modelo de P obabilidad de Vic o ia, Modelo Bayesiano, Decisiones Basadas en Da os. 3 Acknowledgemen s Fi s o all, I would like o hank Jesse Da is o his suppo , o always inding a momen o alk and guide me wi h his ideas and o e all clea pic u e, and Maaike Van Roy, o he weekly eedback and always encou aging my wo k. I ha e e y much enjoyed wo king in he oo ball analy ics opic wi h such a wo ld-leading g oup. Thanks o KU Leu en o o e ing all o he se ices o one o he mos p es igious uni e si ies in mainland Eu ope. G acias a En ique Rome o, mi u o po co espondencia elec ´onica, po oda la ayuda, co eciones y ´animo a dis ancia. G `acies en especial al CFIS, pe la con ian¸ca i l’ajuda econ`omica du an o a la ca e a, i pe o e i -me l’opo uni a de passa un any on olia in es igan el que m’ag ada. G `acies als que heu e que la me a es `ancia a B`elgica algui mol la pena: Leu en Team, la Resis encia i els Ea ls o Leu en. I als que m’heu acompanya o a la ca e a: els ma is, i la gen de la FME i EF i la gen del Thau. Una ab a¸cada mol o a pel Ma c, que s´e que es as mol o gull´os de o s nosal es. G `acies a l’And eu pels memes, i pa e, ma e i amilia en gene al pe les isi es i ideo ucades absu des. G `acies, en especial, a la Ma ina, que hi cap en o es les ca ego ies i que es me eix mol m´es que aques a ase. Joan, Leu en, May 2023 4 Con en s 1 In oduc ion 7 1.1 P oblem Desc ip ion .......................... 7 1.2 Resea ch Ques ions ........................... 8 2 Backg ound 10 2.1 In oduc ion o Foo ball Analy ics ................... 10 2.2 The Subs i u ion P oblem ....................... 11 2.3 Rela ed Wo k .............................. 13 3 Da a O e iew and Analysis 14 3.1 Da a Desc ip ion ............................ 14 3.1.1 Da a p o ide s ......................... 14 3.1.2 T aining and es da a ..................... 16 3.2 Da a Analysis .............................. 17 3.2.1 Subs i u ions and goal sco ing ................. 18 3.2.2 Playe s’ pe o mance h ough ime .............. 19 3.3 Compa ison Be ween 3-subs and 5-subs E as ............. 21 4 Subs i u ion E alua ion 24 4.1 P e-S udy Analysis ........................... 24 4.1.1 Assessmen ules de ini ion ................... 24 4.1.2 Rules co ela ion ........................ 26 4.2 Classi ica ion Models .......................... 27 4.2.1 O ensi e and de ensi e subs i u ions ............. 28 4.2.2 Sco eline condi ioned e alua ion ................ 28 4.2.3 Ad anced me ics e alua ion .................. 29 4.3 Win P obabili y Model ......................... 30 4.3.1 Model de ini ion and alida ion ................ 30 4.3.2 E ec o subs i u ions ..................... 32 4.4 Resul s and Discussion ......................... 35 5 Subs i u e Selec ion 36 5.1 Bench Analysis ............................. 36 5.2 Type o Subs i u ion .......................... 39 5.2.1 Win p obabili y simula ions .................. 39 5.3 Playe Selec o . Theo e ical F amewo k ................ 42 5 6 Op imal Timing 44 6.1 Timing Analysis o Posi i e Subs i u ions ............... 44 6.1.1 Subs i u ion classi ica ion ................... 44 6.1.2 Win p obabili y model ..................... 46 6.2 Simula ion o Al e na i e Timing ................... 47 6.2.1 Subs i u ion classi ica ion models ............... 47 6.2.2 Win p obabili y model ..................... 48 6.3 Resul s and Discussion ......................... 50 7 Conclusions 52 Re e ences 57 6 Chap e 1 In oduc ion 1.1 P oblem Desc ip ion Foo ball is he mos impo an spo in he wo ld, wi h ans all o e he globe uning in e e y day o ollow games, and shows discussing e e y hing ela ed o hose games. Wi h he inc easing amoun o da a being gene a ed in oo ball, he indus y has s a ed p omo ing da a-d i en and da a-in o med [13] echniques o op imize each and e e y ask. In ecen yea s, oo ball analy ics has eme ged as a powe ul ool o unde s anding he game and gaining a compe i i e ad an age [30]. While he use o analy ics in spo s is no new, i has been mos p ominen ly associa ed wi h isola ed e en s and highe -sco ing games (such as baseball o baske ball). Da a analysis has been used o decades in hese spo s o e alua e playe s and eams, in o m s a egy, and make in o med decisions [20]. Howe e , he adi ional mindse o oo ball coaches and ans, combined wi h he na u ally andom spi i o oo ball due o i s low-sco ing na u e, has o en made oo ball esis an o he use o da a and s a is ics in he spo . In con as , many oo ball analys s and expe s belie e ha he use o analy ics is essen ial o unde s anding he game and imp o ing pe o mance. The success o eams ha ha e emb aced analy ics in all a eas p o ides compelling e idence o he alue o his app oach. Subs i u ions, in socce , a e impo an esou ces, because hey al e he abili y o change ac ics [23], which can o en di ec ly in luence he inal ou come o he game. Th ough subs i u ions, he coach explici ly de ines wha his in en ion is in ela ion o he game, and can show hei abili y o maneu e mid-ma ch. Nowadays, coaches can pe o m up o i e subs i u ions pe ma ch, in a o al o h ee windows (and possibly a hal - ime) [19]. Coaches ha e o use hese subs i u ions o possible inju ies, a igued playe s, p o ec ion o booked playe s and, po en ially, change he cou se o a ma ch h ough a ac ical o playe change. In his hesis, we will ocus on he la e . Changes in o ma ion, playing s yle and e en a eam’s men ali y can be accomplished wi hou subs i u ing a playe . We a e in e es ed 7 in iden i ying op imal subs i u ions and knowing he e ec i eness o subs i u ions, acco ding o he p og ess o he ma ch. The ype o subs i u ion, whe he i mus change he eam o ma ion o s yle, picking he co ec playe and being spo on wi h iming a e e y impo an o a coach o accomplish hei objec i e. Accomplishing his can esul in al e ing he ou come i hey a e losing, o main aining he ic o y i hey a e al eady winning. Mul iple s a egies can be ollowed, since ac ical decisions a e simul aneously ac o s o he eam s abiliza ion and he opposi ion’s des abiliza ion ([12], [4], [28], [14]). 1.2 Resea ch Ques ions Th ee subs i u ions pe eam and pe ma ch we e he ule un il a ew yea s ago. Wi h he COVID b eak, many leagues accep ed ha eams could do up o i e subs i u ions pe ma ch, due o schedule o e load. E en hough some majo leagues wen back o he h ee subs i u ions pe ma ch, om his season (2022/23) all majo leagues in Eu ope and con inen al compe i ions ha e op ed o allow i e changes pe ma ch [21]. Since he e is no appa en ex a ma ch load, we pose he ollowing ques ion. Resea ch Ques ion 1 Has he new 5-subs i u ions pe ma ch ule made ha mo e subs i u ions become ele an ? Do coaches ha e mo e ways o maneu e ? We will ocus on his ques ion om an obse a ional poin o iew, which will be add essed in sec ion 3.3, by compa ing o he las comple e season p e ious o COVID wi h he seasons ha ha e ollowed ha allowed i e subs i u ions pe ma ch. Nex , we will ocus on assessing subs i u ions. This key pa o he in es iga ion is he base o he whole Thesis, as in es iga ing op imal subs i u ions depends on how you de ine hem. We pose he ollowing esea ch ques ion. Resea ch Ques ion 2 How can we de e mine i a subs i u ion is use ul? Wha is he pe o mance o he subbed playe in compa ison o he s a ing one? Do subs i u ions ha e an e ec on a eam’s win p obabili y? The ques ion will be add essed in chap e 4. Di e en e alua ion me hods will be discussed and analyzed, including changes in esul , ma ch momen um o in subs i u e pe o mance. Machine lea ning models will be ained o he p edic ion o such op imal subs i u ions. A model based on [29] will be de eloped, using subs i u ion in o ma ion o calcula e he win p obabili y du ing he ma ch and discuss how sensible he model is o such ea u es. The models de eloped in his chap e will be used du ing he cou se o his epo . The nex opic ha will be add essed is he selec ion o he adequa e playe o come in. Gi en he momen a which a coach decides o in oduce some kind o 8 Bu SCA a e impo an because hey p o ide in o ma ion on he game p e ious o a goal chance which is jus a sho . Ma ch Shoo ing shee xG, SCA pe each sho Lineup shee S a ing and bench playe s Ma ch epo shee Goals, subs i u- ions and bookings Figu e 3.1: Diag am o in o ma ion ob ained om a ma ch The shoo ing shee is o he u mos impo ance since i gi es xG da a o each sho . This ecen ea u e, ha b e .com has included due o a ecen change in da a-p o ide s 1 , enables us o do a much mo e in-dep h analysis. The low-sco ing na u e o oo ball means ha goals a e a a i y, so xG is a weigh ed way o s udy he eams’ pe o mance. Since he shee p o ides he SCA, we can analyze he in luence o playe s a he sho le el, no only in sho s bu also in he ac ions leading o hem, opening he doo s o many analyses. Following he de ini ion o xG, e en hough he alue assigned o a sho migh di e depending on he p o ide o he da a, a se ies o me ics [15] and e minology is de i ed om i , which a e di ec ly p o ided by he shee s o easily compu ed. De ini ion 3.3 Expec ed Goals Agains (xGA) is he amoun o xG amassed by he opposi ion du ing a pe iod o ime (pe iod o a ma ch, comple e ma ch o season). De ini ion 3.4 Expec ed Goals Di e ence (xGD) is he di e ence in he xG amassed by a eam and he opposi ion du ing a pe iod o ime (pe iod o a ma ch, comple e ma ch o season). De ini ion 3.5 To he playe ha does he pass p e ious o he sho , we assign an Expec ed Assis s (xA) alue equal o he xG assigned o he sho . Sho -by-sho da a gi es his epo a much mo e p o ound scope. This alls sho o he e en -by-e en and acking da a ha clubs and indus y leade s a e cu - en ly wo king on, which gi es he oppo uni y o a mo e in-dep h analysis, which ul ima ely can lead o a compe i i e ad an age [27]. Nex on, we ha e access o he lineups o bo h eams. Fu he analysis o his da a will be conduc ed in sec ion 5.1. Lineups include he posi ion he playe has played du ing he ma ch, including mul iple posi ions i he coach has conside ed ac ical changes h oughou he ma ch. Fo unused bench playe s, posi ion da a is oid, 1 h ps://www.spo s- e e ence.com/blog/2022/11/ b e -sho -le el-xg-now-on-ma ch- epo s- o -20-compe i ions/ 15 since hey did occupy any posi ion. None heless, we can s udy hei posi ions du ing he season o ge access o he playe s’ ans e ma k .com page, whe e hei usual posi ion is a ailable. Finally, om he ma ch epo we ge he commonly known ma ch ea u es: goals, yellow and ed ca ds, and subs i u ions. This in o ma ion gi es us an o e iew o he ma ch s a e. Such da a has been a ailable o much mo e ime han he new in-dep h spa io empo al da a, bu his epo shows ha impo an analysis can be done wi h such simple, and widely a ailable, da a poin s. One missing ea u e om b e .com ’s da a is he inishing imes o bo h i s -hal es and ma ches. Such in o ma ion can be ele an , in pa icula when la ge inju y ime is in oduced, bu since we ha e he minu e o all sho s, bookings and subs i u ions, i will be possible o assess he inal minu e o playing ime wi h su icien accu acy. clubelo.com Elo a ings Elo a ings a e ob ained om clubelo.com page, a e e ence page o Eu opean Foo ball Club Rankings based on he Elo sys em. We can access he comple e ankings on any gi en day, enabling us o ob ain he Elo a ings o he eams on e e y ma ch day. These ankings a e compa able h ough di e en leagues since hey a e weigh ed based on in e na ional ma ches. We do no ha e he pa icula Home Field Ad an age (HFA) o each compe i ion, so we mus add a gene al HFA on all compu a ions. 3.1.2 T aining and es da a As aining da a, we ha e selec ed o s udy he op leagues in Eu opean men’s oo ball. We will di ide be ween da a o he s udy o playe pe o mance and he da a o he s udy o subs i u ions. To s udy playe s’ p oduc i i y, we ha e s udied he sho -by-sho xG o ou comple e seasons, om 2018/19 up o 2021/22, in he ollowing leagues: he English P emie League, he Spanish LaLiga, he I alian Se ie A, he Ge man Bundesliga and he F ench Ligue 1. These i e compe i ions a e known as he 5 big leagues and will be e e ed o as so du ing his Thesis. This sums up o a o al o 7203 ma ches and 177980 sho s aken in o accoun o he aining da a. Un o una ely, we canno use all his da a when aining models ha e alua e subs i u ions, since he 5-changes pe ma ch ule was implemen ed a e he COVID s op. This was h ough he 2019/20 season. The las ma ches o such had 5 subs i u ions, bu he change was jus implemen ed, so coaches migh no ha e i used o ully p o i om such a ule. Thus, we a e coun ing only he 2020/21 and 2021/22 seasons o subs i u ion model aining. The English P emie League decided o go back o he 3-subs pe ma ch uling, so we canno include i in he aining. To minimize he e ec o he educ ion on ma ches, we ha e added hese seasons o he Du ch E edi isie and he Po uguese P imei a Liga, he nex leagues 16 acco ding o he UEFA coe icien 2 . Thus we a e e alua ing 4116 ma ches and 34368 subs i u ions. The numbe o ma ches pe league is 380 o a 20- eam compe i ion, which a e he op- ie leagues in England, Spain, I aly and F ance, and 306 ma ches o he 18- eams leagues, such as he op- ie Ge man, Du ch and Po uguese leagues. The F ench i s di ision, du ing he COVID b eak, had only 279 ma ches played, which explains he odd numbe o ma ches in he co esponding da ase . The es da a a e all ma ches played his season (2022/23) a he male Eu opean op 5 leagues up o he FIFA ma ch day o inals o Ma ch since he English league has used 5 subs i u ions pe ma ch his yea [21]. We conside coaches a e used o his uling due o being in he majo i y o Eu ope and in e na ional compe i ions. Thus, ou es da ase s include n = 1305 ma ches played in he i s di isions o England, Spain, I aly, Ge many and F ance since he s a o he season up o he 20 h o Ma ch. Table 3.1 p o ides a summa y o he da ase s used in his Thesis, and he ele an numbe o da a poin s each o hem o e s. Seasons Da a Ma ches Da a poin s Top5 18/22 T aining Da a-Playe s 7203 177980 sho s ES, IT, DE, FR, NL, PT 20/22 T aining Da a-Subs 4116 34368 subs Top5 22/23 (up o 20 h Ma ch) Tes Da a 1305 11377 subs 32461 sho s Table 3.1: Summa y o he da a se s used o aining and es ing. We ha e disca ded in e na ional compe i ions such as he Champions League since i s knockou s age and sho g oup phase can p oduce si ua ions whe e he objec i e o he eam is no o win (depending on esul s om p e ious ma ches) o ha ese e playe s a e used ( eams al eady classi ied o elimina ed). In a simila way, we ha e no aken in o accoun any o he elega ion o i le playo s in Ge many, F ance, he Ne he lands o Po ugal. Thus, we a e conside ing only egula league games, whe e hese si ua ions can happen in a much mo e spo adic way. 3.2 Da a Analysis In a no mal ma ch, du ing he i s -hal , lineups a e wha coaches decided p e ious o he encoun e , and mos o he ac ics a e p e iously decided and gi en o he playe s. in he second-hal , playe s migh ecei e de ailed ins uc ions a hal - ime on how o a ack he opposi ion’s weaknesses, and new playe s a e in oduced o po en ially modi y he ac ics o a eam. Thus, we can dis inguish be ween i s -hal es, whe e he ini ial app oach plays a majo ole, and second-hal es whe e adap abili y becomes a main asse . 2h ps://www.ue a.com/na ionalassocia ions/ue a ankings/coun y/ 17 3.2.1 Subs i u ions and goal sco ing Compa ing he i s and second-hal es can p o ide us wi h an insigh in o he e ec o subs i u ions in a game. Al hough i is ue ha second-hal es a e usually longe , i is assumed ha he inju y ime is in ended o co ec o los ime, so we can assume ha he e ec i e playing ime is simila be ween he wo hal es. Bu he ac s a e ha 55.6% o he xG and 56.1% o he goals happened in second-hal es, which is a signi ican inc ease. When he e we e h ee subs i u ions, goal-sco ing in ensi y inc eased a e he i s and second subs i u ions, bu was educed a e he opposi ion’s hi d [1]. When compa ing his o las season, 21/22, wi h i e subs i u ions h ough 3 subs i u ion windows we ob ain simila esul s. In Figu e 3.2 we see how he xG gene a ed e ol es a e each subs i u ion window, no malized by minu es. xG in ensi y, and consequen ly goal p obabili y, inc eases a e he i s and second coach in e en ions, due o he esh legs and new ac ical in o ma ion b ough in o he pi ch. I also dec eases a e he hi d, which is mos p obably due o he o e lapping wi h he opposi ion’s subs i u ions, which dec eases goal-sco ing p obabili y [1]. Figu e 3.2: A e age xG gene a ed a e each subs i u ion window, g ouped by leagues. Ma ches do end o be mo e open in he second-hal , as mos o he goals and goal oppo uni ies happen he e. Coaches can in oduce ac ical a ia ions and he e o e inc ease hei a ack p oduc ion. Bu a he end o he ma ch, ac ical decisions oppose hemsel es and he goal-sco ing in ensi y dec eases a li le bi , e en hough he numbe o minu es a e he las window is smalle and migh be less signi ican . 18 3.2.2 Playe s’ pe o mance h ough ime The main me ic o assessing he ac ions o playe s will be he newly de ined expec ed Value. De ini ion 3.6 We de ine a new me ic Expec ed Value (xV) as he amoun o xG amassed by a playe in all o hei sho s and SCA. We a e awa e o he sho comings o xV. Me ics such as expec ed h ea [32] o VAEP [11] gi e alue o all ac ions, and a e mo e sensi i e o he amoun o eal haza d he ac ion posed on he de ending eam. We a e no aluing all he ac ions, jus sho s and he wo p e ious eamma es’ ac ions; and we a e gi ing ac ions he alue o he sho hey helped p oduce, which may no be di ec ly ela ed o he quali y o he ac ions. Ex eme examples can also happen, such as a playe doing a d ibble, ge ing ouled and aking he penal y. Since he penal y has an xG o 0.76, and he playe p oduced he wo SCA, hey would amass 2.18 xV, which is much highe han he alue o he playe in ha ci cums ance, bu such cases a e only heo e ical and in p ac ice a e ea ed as ou lie s. Knowing he aul s his me ic can ha e, i gi es much mo e in o ma ion han jus xG o e en xG+xA, since i akes in o accoun many mo e ypes o ac ions, and since we do no wo k wi h e en -by-e en da a, i gi es us he b oade iew possible. Expec ed Value will be used, h ough he Thesis, when alking abou playe s, while expec ed Goals e e s o he eam since o he wise we would be double, o e en iple, coun ing each goal-sco ing chance. Subs i u ions can ha e mul iple easons, and a igue is usually indica ed as he cause o many o he subs i u es. By ha ing he xG da a o e e y sho , we can compu e he xV ha playe s gene a e h ough ime. On Figu e 3.3 we can compa e he P edic ed xV o an a e age op 5 leagues playe , depending on whe he hey s a on he pi ch o he bench. De ini ion 3.7 P edic ed xV is he amoun o xV expec ed o gene a e in he numbe o minu es played, by mul iplying he a e o xV pe minu e by he numbe o minu es. 19 Figu e 3.3: xV gene a ed h ough a ma ch a he a e age playe a e o xV pe minu e. Fo bench playe s, in ed, minu es a e coun ed om he momen hey en e he pi ch The e a e wo main obse a ions o be made. On he one hand, we can see ha he a e o xV gene a ed is almos cons an h ough ime. This happens o bo h s a ing playe s and bench playe s. This is, he a e o a ack gene a ed by a s a ing playe does no decay, which would indica e ha a igue does no eally apply, o is compensa ed by he inc ease in goal oppo uni ies o e all. On he o he hand, subs i u e playe s p oduce a highe amoun o xV. This is a bi o a coun e -in ui i e ac since one would expec ha he bes a acking playe s gene a e mos o he occasions, and ha he bes playe s a e s a e s o hei eams. This happens due o he o e lapping o wo phenomenons: an inc ease in o e all gene a ed xG and a igue o s a ing playe s. Ma ches end o become mo e open and ha e mo e xG when he ma ch is ending, and s a ing playe s p oduce ewe xV in ela ion o he o al gene a ed. Thus, subs i u e playe s usually play in mo e open ma ches and can p oduce mo e a acking oppo uni ies, independen ly o hei quali y in compa ison o s a e s. 20 3.3 Compa ison Be ween 3-subs and 5-subs E as In Resea ch ques ion 1 we wonde ed abou he e ec o he new ule ha allows i e subs i u ions pe eam pe ma ch. This ule, adop ed as an eme gency a he es a a e COVID lockdown, seems o es ablish he no m o he u u e. Subs i u ions a e he main ool coaches ha e o change he cou se o he ma ch. Since hey usually happen in he second-hal , we can ocus on sign changes as a measu e o coaches’ in luence. We ound ha o e he las compe e season be o e COVID, 2018/19, a 42.6% o he ma ches changed hei esul om hal - ime o he end o he ma ch, and also ha 61.7% o he ma ches d awn a e he i s -hal ended up being a ic o y o ei he eam. When we check he wo ollowing comple e seasons wi h 5 subs i u ions, we obse e ha he pe cen age o ma ches ha changed hei esul we e 39.3% in 20/21 and 41.4% in 21/22, e y simila numbe s and i some hing, smalle . Simila ly, a 60.9% o hal - ime d awn ma ches ended up no being a d aw in 20/21 and 61.5% in 21/22, which a e he same pe cen ages wi h a bi o a ia ion. Playe s dis ibu e hemsel es h ough he pi ch, wi h di e en oles depending on hei posi ion [23]. Fo each posi ion, he da a p o ide s de ine, we ha e assigned a alue, he o ensi eness, ela ed o hei p oximi y o he opposi ion’s goal. This way, we can assign an o ensi e alue o he playe depending on he posi ions hey ha e occupied du ing a ma ch. I hey ha e occupied mul iple posi ions, we compu e he a e age o he o ensi eness o all he posi ions illed. In Table 3.2 a ela ion be ween posi ions and hei o ensi eness alue can be ound. Posi ion Abb e ia ion O ensi eness Goalkeepe GK 0 Cen e, Le o Righ back, De ende CB, LB, RB, DF 1 De ensi e Mid ielde , Wingback DM, WB 2 Cen al, Le o Righ Mid ielde CM, LM, RM, MF 3 A acking Mid ielde , Le o Righ Winge AM, LW, RW 4 Fo wa d, Second S ike FW, SS 5 Table 3.2: Di e en posi ions, wi h he abb e ia ions used by he da a p o ide s and he ela i e o ensi eness assigned. De ini ion 3.8 We de ine a playe change as an o ensi e subs i u ion i he o ensi eness o he playe coming in is mo e han 0.5 uni s highe han he playe going ou . Simila ly, a de ensi e subs i u ion is whe e he posi ions occupied by he new playe a e less o ensi e by a leas 0.5 uni s. All o he subs i u ions a e conside ed neu al. The e a e mul iple easons o subs i u e a playe : p o ec ing a booked playe , inju ies, a igue, o ac ical subs i u ions. Acco ding o he ypes o subs i u ion, all o ensi e 21 and de ensi e subs i u ions ca y ac ical adjus men s. A neu al subs i u ion can also be ac ical since di e en playe s ha e di e en p o iles, bu hey migh also be jus eplacemen s due o physical condi ions o he con ex o he ma ch. Due o ou usage o widely a ailable da a, we canno dis inguish be ween hese wo cases. Thus, we will compa e neu al and non-neu al subs i u ions. The pe cen age o non-neu al subs i u ions was 24.6% du ing season 18/19, and con inued o be 24.3% in 19/20 up o he COVID lockdown. When leagues esumed, allowing i e subs i u ions, only 18.0% o subs i u ions we e non-neu al. The same a e con inued in 20/21 wi h a 17.2%. The numbe o non-neu al subs i u ions pe eam pe ma ch emained he same, close o 0.7. This means ha coaches made a simila numbe o ac ical changes, keeping hei usual numbe o ac ical al e a ions h ough a ma ch, bu inc eased non- ac ical ones, p obably due o a igue, COVID sequels and a mo e compac schedule. A summa y o hese a es is in Table 3.3. These pe cen ages may a y depending on he de ini ion o o ensi e and de ensi e subs i u ions [7]. On season 21/22, wi h coaches ha ing ime o adap , he pe cen age o non-neu al subs aised back o 24.4% o he o al ones, and coaches did 1.1 non-neu al subs i u ions pe eam pe ma ch. This end seems o be con adic ed by he esul s o 22/23 season so a , going back o 17.8% o o als. This migh be due o he Wo ld Cup being held in Decembe , which has made eams play mo e equen ly and playe s play mo e ma ches, hus inc easing he need o physical subs i u ions. Season Non-neu al subs pe cen age Non-neu al subs pe ma ch Subs pe ma ch 18/19 24.6% 0.72 2.93 19/20 p e-COVID 24.3% 0.70 2.87 19/20 pos -COVID 18.0% 0.77 4.29 20/21 17.2% 0.72 4.17 21/22 24.4% 1.1 4.33 22/23 17.8% 0.77 4.35 Table 3.3: Summa y o he non-neu al subs i u ions pe season, bo h as pe cen age o o al subs i u ions and pe ma ch, and also he numbe o subs i u ions pe eam pe game. Finally, we would like o see i eams a e adap ing o he new ule, so we look a he numbe o o e all subs i u ions made by he eams o e ime. A e pe o ming a Welch Two Sample - es on he subs i u ions pe eam pe ma ch h ough seasons 20/21 and 21/22 we ob ain ha he mean o he la e , 4.33 subs pe eam wi h a s anda d de ia ion o 0.88, is s a is ically signi ican (p- alue 4 . 328 e− 15) bigge han he 4.17 o he i s season, wi h 0.92 de ia ion. The compa ison wi h he ongoing 22/23 season shows ha he alue emains s able a a ound 4.35 subs pe ma ch, and wi h a simila s anda d de ia ion. 22 Including a new subs i u ion is a e y ele an change in he ules [34]. We ha e ob- se ed ha e en hough coaches ha e mo e maneu e abili y, he change in dynamics in second-hal es emains e y simila . A possible explana ion is a Red Queen e ec [33], whe e bo h eams e ol e so ha he coaches’ e ec cancels i sel . When he new ule was in oduced, he ac ical subs i u ions pe ma ch emains he same, bu he pe cen age dec eased. Wi h he adap abili y o he eams’ s a , which can be seen in he inc ease o o e all subs i u ions a eam uses, he amoun o non-neu al subs i u ions has inc eased. 23 Chap e 4 Subs i u ion E alua ion This sec ion ocus on Resea ch ques ion 2 om di e en pe spec i es. Fi s , we will y o de ine wha is a use ul o posi i e subs i u ion acco ding o di e en c i e ia. We will also y o compu e he winning p obabili y and hus de i e he impac subs i u ions ha e. Foo ball’s low-sco ing na u e gene a es mul iple si ua ions whe e be e -playing eams do no necessa ily con e in o a posi i e esul . Thus, i a subs i u ion has an impac on he game ha is no e lec ed in he da a we ha e a ailable, we migh no be able o e alua e i p ope ly. In his chap e , we use he a ailable in o ma ion o y and assess subs i u ions in he bes possible way. 4.1 P e-S udy Analysis E alua ing a subs i u ion can be a di icul ask, and mo e so no ha ing e en -by- e en da a, so we canno ge alua ions on he e ec o a pa icula playe ’s ac ions, such as xT [32] o VAEP [11]. Wha we do ha e is sho -by-sho da a, and he e o e we can compu e he expec ed Value gene a ed by a playe wi h hei sho s and Sho C ea ing Ac ions. We a e going o pack subs i u ions in o windows, as he e ec o wo simul aneous changes can no be disce ned, so in his sec ion he e m subs i u ion will be used also o e e ei he o a single sub o a window o mul iple playe swaps, acco ding o con ex . The analysis in his sec ion is new, wi h ules de eloped especially o his hesis. E en hough some o he s udies use simila assessmen ules ([26], [31], [7]), we in oduce a wide ange o ways o s udy and e alua e subs i u ions. 4.1.1 Assessmen ules de ini ion Fo hose windows including an o ensi e o de ensi e subs i u ion, we a e going o de ine simple ules: an o ensi e change is success ul i a goal is sco ed a e wa ds; a de ensi e one is success ul i no goal is conceded om ha momen o he end o he 24 gi∼B ( , θ ,i ). By adding ha o he al eady sco ed goals, G ,i , we can simula e he inal esul . Di e en app oaches we e aken in he way o p edic θ ,i . The in ui i e idea is ha he goal-sco ing p obabili y a he nex minu e inc eases wi h ime, bu di e en ea u es can a ec i in di e en ways. I has been shown ha he ela i e impo ance o a iables e ol es non-linea ly o ime [29], bu also in ui i ely close ime ames should ha e a close ela ionship be ween he a iables. The ou algo i hms we e: •Mul iple Logis ic Reg essions (mLR) A logis ic eg ession is calcula ed o each ime emaining . We use his app oach and no a gene al logis ic eg esso because he impo ance o ce ain ea u es a ies wi h ime. θ ,i =ew x ,i 1 + ew x ,i (4.1) •Random Fo es (RF) A Random Fo es eg esso . Random o es is able o deal wi h non-linea in e ac ions be ween he ea u es. •Mul iple Random Fo es s (mRF) A di e en Random Fo es o each ime ame, bu using also close- ime alues also as aining o p e en a ce ain ime ame o be an ou lie . This app oach is basically he same as a simple andom o es bu wi h a e y much inc eased numbe o ees. •Gaussian Walk (GW) This Bayesian app oach also uses an in e se logis ic unc ion wi h a se ies o weigh s α ,i which ake as a p io he weigh s o he p eceding ime ame, when he minu es emaining a e . θ ,i = in logi (α ·x ,i +β)α ∼N(α +1,2) β∼N(0,10) (4.2) Compa ing his pa icula model o he one de eloped by Robbe ech s, Van Haa en and Da is [29], we see a ew impo an di e ences. Since we wan o look a subs i u ions, we only s udy he p obabili ies o he second-hal o ma ches. We wo k wi h he numbe o minu es emaining , while Robbe ech s e al. di ide he ma ch in o 100 ime ames. Ou app oach gi es each minu e he same impo ance, e en i he ma ch coun ed wi h a big amoun o s oppage ime. Bu he model aces he same p oblem i i was o be implemen ed in eal li e since inju y ime is unknown. Fo he es da a, we assume he 96 h minu e o be he las , since i is he mode, and we belie e his model should be e alua ed wi hou knowing he eal las minu e. The second main di e ence is we p edic θ ,i and hen simula e he 31 ma ch, while Robbe ech s e al. implemen i in hei model and di ec ly p edic he p obabili ies o each esul . In o de o alida e he pe o mance o a win p obabili y model, we need o look a he whole pic u e. I he model gi es a eam an 8% o win p obabili y on a pa icula ma ch, ha esul is ei he going o happen o no , because he ma ch only happens one ime. Thus, we g oup all he p edic ions o an 8% and we expec ha such a pe cen age o he cases ac ually happen. We call p edic ed p obabili y, o he p obabili y he model p edic s o a ce ain esul , and ac ual p obabili y he pe cen age o imes such esul comes ue. In Figu e 4.2 we see he g aphs o he ou algo i hms and hei pe o mances on he h ee di e en esul s: home win, away win and d aw. The size o he poin ep esen s he numbe o such p edic ions ha we e made. This is a di e ence wi h espec o he model o Robbe ech s e al. Since we calcula e p obabili ies as a esul o he simula ion, gi en he α ,i , and no as a di ec p edic ion o he model, we ge a much mo e e en dis ibu ion o p edic ions. This is, o he Random Fo es o mul iple Logis ic Reg esso , hey go a non-uni o m dis ibu ion o p edic ions, ha ing ew p edic ions wi h high p obabili ies. When implemen ing he Bayesian app oach, ha p oblem was pa ially sol ed. In ou case, o all models we ha e a much mo e uni o m dis ibu ion, e en hough p edic ions o 0-2% a e a bi mo e ypical. 4.3.2 E ec o subs i u ions A eam migh ake mul iple app oaches o a ma ch, such as maximizing hei win p obabili y o minimizing he losing one [3]. These decisions migh depend on many ac o s: league posi ions, ela i e s eng h, psychological momen um, e c... In his hesis, we will ocus on egula league games, and based on ha 3-poin pe win, 1-poin pe d aw ule, we will measu e expec ed poin s as a uni ied app oach. De ini ion 4.7 Expec ed Poin s (EP) is he expec ed alue o poin s o a eam gi en hei winning, d awing and losing p obabili ies in a game. I is logical ha a eam would like o maximize he poin s hey ob ained, since i may gi e hem be e long- e m esul s. Thus, we will use EP as a min-max-s yle app oach, and we will assess possible s a egies as a esul o hei e ec on EP. This me ic weigh s he d aw and win p obabili ies in he same way he league sco ing sys em does, and he e o e we belie e i can be accep ed as a gene al s a egy o all eams. We compa e he winning p obabili y a e he subs i u ion o he minu e be o e i happened. Consequen ly, when a goal and a change happen a he same minu e, so i epo s b e .com . Bu we canno disce n wha happened be o e. This p oduces a massi e change in win p obabili y, and a e y high ∆ EP , which is due o he goal and no he subs i u ion, so we disca d hose subs i u ions om he analysis. To ensu e ha he subs i u ion ea u es we e ele an we ained he same model bu 32 (a) Mul iple Logis ic Reg esso s (b) Random Fo es (c) Mul iple Random Fo es (d) Gaussian Walk Figu e 4.2: E alua ion o di e en echniques o calcula e θ ,i and he inal pe o mance o he in-game win p obabili y model, compa ing p edic ed p obabili y o ac ual p obabili y o each ype o p edic ion. Size indica es he numbe o p edic ions. wi hou hose ea u es. This is, p edic θ ,i om x ,i = ( , i, τ ,i ). We shall call his model simple as opposed o he comple e model which we will e e o as complex. In Figu e 4.3 we see he dis ibu ion o he a ia ion o expec ed poin s when a subs i u ion is made o he wo models. The simple model is obli ious o he ac ha he subs i u ions ha e been made, so he dis ibu ion should be as i we picked andom momen s in ma ches’ second-hal es and calcula ed hei ∆EP . 33 Figu e 4.3: Dis ibu ion o he a ia ion o EP a e a subs i u ion. Compa ison be ween he simple and he complex model. Ins ead, in he dis ibu ion o he complex model we see how he numbe o subs i- u ions o none o e y li le impac in EP is much lowe , and ha he subs i u ions wi h an inc ease o dec ease in EP g ow, as i can be seen in bo h ails o he complex dis ibu ion cu e. I we sepa a e he subs i u ions in posi i e and nega i es, as in Figu e 4.4, we can ocus on he di e ence be ween he posi i e and nega i e ail. In o de o be e isualize hese ails in smalle numbe s, we apply he ans o ma ion o he x-axis and expose √∆EP . Figu e 4.4: Dis ibu ion o √∆EP o posi i e and nega i e subs i u ions in he complex model. 34 He e we can see how he wo cu es a e no symme ic, which means subs i u ions a e no jus andom momen s acco ding o he model. We can also see how he densi y is bigge o he subs i u ions wi h a posi i e impac on he ma ch. One o he conclusions is ha he majo i y o subs i u ions ha e a posi i e e ec on he eam’s goal-sco ing p obabili y, while no diminishing he de ense so ma ch, so he win and d aw p obabili y inc ease and so do he expec ed poin s. 4.4 Resul s and Discussion In his chap e , we wan ed o s udy he p oblem o p ope ly e alua ing he e ec i e- ness o a subs i u ion. This is a di icul hing o do wi h he a ailable da a since many o he playe ’s ac ions go unno iced. Di e en app oaches ha e been aken o he assessmen , and e en hough hey a e co ela ed, in many cases a single playe ’s pe o mance, o e en he whole eam, migh no ansla e in o a palpable esul in he o m o goals. We ained many models o p edic he success o he subs i u ions. In a simila s udy on he B azilian League [7], hey ob ained model pe o mances o 75-85%. Ou esul s oscilla e be ween he high six ies and high se en ies. As we desc ibed, me hods in bo h s udies di e , and his migh explain why a mo e de ailed da ase gi es wo se p edic ions. I is ue, hough, ha B u i e al. ocus much on he eam’s s eng h, while we only coun ed o Elo a ings. A mo e eam-based app oach migh be necessa y o imp o e p edic ions on subs i u ions. We obse ed ha playe -speci ic in o ma ion did be e he esul s o he models p edic ing xV Change Rule. The second app oach consis ed in calcula ing he impo ance o subs i u ions as a byp oduc o he a ia ion in win p obabili y. Ou in-game win p obabili y model, based on he one de eloped by Robbe ech s, Van Haa en and Da is [29], p o ed o be sensi i e o subs i u ions, as seen in Figu e 4.3. Ou change in he model, ying o p edic u u e goals and hen simula ing he ma ch, ins ead o p edic ing esul pe cen ages, u ned ou o be e ec i e in ha ing p edic ions o all possible pe cen ages o all esul s. Simila ly, as in Robbe ech s e al., he Bayesian app oach wi h he Gaussian Walk has esul ed as he mos e ec i e, ha ing a high ela ion be ween p edic ed and ac ual p obabili y. One aspec o discuss abou he win p obabili y model is he e ec ha is shown ha subs i u ions ha e. Mos o he subs i u ions ha e a small impac , bo h o posi i e and nega i e ou comes. In ac ual ma ches, a subs i u ion can ha e a much bigge e ec , since he playe ha comes in migh be especially inspi ed and ha e an abo e-a e age inishing, o on he con a y ha e a bad day. The model a e ages he e ec o he subs i u ions gi en a game s a e. I does no accoun o o he con ex ual ea u es ha migh ha e an impac on he de elopmen o he game, such as he psychological aspec s o a subs i u ion. In gene al, hough, he impac o a playe eplacemen is usually bene icial, as seen om Figu e 4.4. 35 Chap e 5 Subs i u e Selec ion A na u al ques ion a coach may come up wi h is which playe should be in oduced. Resea ch ques ion 3 is he o mula ion o such doub and his chap e aims o p o ide an answe . Wi h he new egula ions, in many o he leagues, he numbe o playe s on he bench has now inc eased, hough i is di e en o each coun y. Deciding which playe comes in o he ma ch depends mos ly on he a ailable esou ces. Two playe s a e in ol ed in a subs i u ion: he playe on he ield ha is emo ed, and a bench playe who is in oduced. Deciding on which playe should lea e he pi ch is a e y in e es ing opic. Un o una ely, we canno add ess i since we a e wo king wi h public and widely a ailable da a, ha does no include e en -by-e en , he e o e, a de ailed analysis o he playe ’s pe o mance h ough he game canno be made, making i di icul o p ope ly assess which playe is ha ing a below pa ma ch. Mo eo e , many o he ac o s, such as physical o m o he ac ical needs o he eam need o be aken in o accoun . O cou se, some o hese p oblems a ise as well when selec ing he bench playe ha will pa icipa e in he ma ch. Physical p oblems a e no a conce n o subs i u es, p o essionals can play a ma ch as bench playe s, and ha e been p o en o co e mo e spaces and do mo e sp in s han s a ing playe s [4]. We gi e a da a-based app oach o subs i u e selec ion, e en hough ac ical knowledge and speci ic on- ma ch si ua ions canno be cap u ed in o da a a e s ill impo an . 5.1 Bench Analysis Fo he pu pose o simula ions and p ope subs i u e selec ion, we need o know which playe s a e on he bench. On a gene al basis, i we wan o decide be ween a ious possible ypes o subs i u ions, knowing a eam’s o ma ion and he a ailable playe s is e y impo an . In his sec ion, we aim o p o ide some g ound ules o simula ion. In he 2020/21 and 2021/22 seasons, eams could ha e up o 12 playe s on he bench o he Spanish LaLiga, he I alian Se ie A and he Du ch E edi isie; bu only up o 36 (a) Goalkeepe s His og am (b) De ende s His og am (c) Mid ielde s His og am (d) Fo wa ds His og am Figu e 5.1: His og ams o he dis ibu ion o each posi ion, sepa a ed by maximum bench size. 9 in he Ge man Bundesliga, he F ench Ligue 1 and he Po uguese P imei a Liga. Thus, ou analysis will ha e o be dual. We i s s a by seeing which is he ypical dis ibu ion o a bench o see which asse s a coach has a hei disposal. Figu e 5.1 shows he dis ibu ion o he main posi ions a playe can be: goalkeepe , de ende , mid ielde o o wa d. A ew obse a ions abou he dis ibu ion o playe s. In Figu e 5.1a we see ha , while in smalle benches eams mainly ha e jus one bench goalkeepe , when wel e spo s a e a ailable, usually wo goalkeepe s s and a he bench. Subs i u e goalkeepe s a ely come in o play ( oughly 0.3%), and needing wo ex a goalkeepe s is unhea d- o , bu eams a he ha e hem bo h on he bench, possibly due o no needing 11 subs i u e ield playe s. In Figu e 5.1c we obse e ha some benches do no ha e mid ielde s a all. While his sounds odd, some eams migh play hei mid ielde s in o he posi ions, since hey a e he mos e sa ile, o plan on playing in mid ield a playe which he da a p o ide s ag as ei he de ende o o wa d. Addi ionally, h ough he season squads ge a ec ed by inju ies and he numbe o playe s ini ially planned could no be a ailable in ce ain ma ches 37 Selec ing adequa e bench playe s is key o he coach’s success because i de ines hei ools du ing he ma ch, he pieces hey can i in o he playing XI o change he game dynamics. Beal e al. [3], de elop a o mal model o a p e-Ma ch Bayesian game. Teams ha e di e en ac ics, which a e he o ma ions ha hey use, and a mul i-class classi ica ion deep neu al ne wo k is ained o lea n he payo s o he di e en s a egies. In such a way, hey y o p edic he opposi ion’s s a egy and sugges ed an op imal ac ic. The esul s showed eams wi h ac ics simila o he sugges ed ones achie ed be e esul s. The p e-ma ch modeling o he game is an a ea whe e bench managemen could be an impo an asse . Al hough we we e unable o explo e he idea in his epo , we belie e i is an impo an a ea o u u e esea ch. Including he in o ma ion on he opposi ion’s bench could imp o e he model by gi ing mo e aluable in o ma ion. In his analysis, we aim o gi e an app oxima e idea o he bench composi ion o accu a e and eal simula ions. 12-Playe Bench 9-Playe Bench Fo ma ion Gk De Mid Fo Gk De Mid Fo 4-5-1 2 3 3 4 1 3 2 3 4-4-2 2 4 2 4 1 3 2 3 4-3-3 2 3 3 4 1 3 2 3 3-5-2 2 4 3 3 1 2 2 4 3-4-3 2 4 2 4 1 2 2 4 5-3-2 2 3 3 4 1 2 2 4 Table 5.1: Numbe o playe s pe posi ion in he bench, sepa a ed by he wo bench sizes. In Table 5.1 we ha e summed up he bench ha we will be wo king on du ing he simula ions, based on he a e ages o playe s om such posi ion and such ha hey a e a ull-sized bench. The ypes o o ma ions a e o de ed by appea ances on ou da ase , and all ypes o o ma ions ha appea mo e han 50 imes can be included in one such o ma ion. Fo ma ions a e de ined by he numbe o de ende s, mid ielde s and o wa ds sepa a ed by a dash. This is a simpli ica ion o he p e iously de ined posi ions in Table 3.2, so mul iple ac ical app oaches a e los . Fo example, bo h he 4-2-3-1 and 4-1-4-1, wo widely used o ma ions, a e packed unde 4-5-1, e en hough hey migh ha e mo e playe s de ined as o wa ds. Following he example o [2], we will assume ha on he pi ch we can ha e a minimum o 3 and a maximum o 5 de ende s, same wi h mid ielde s, and 1 up o 3 a acke s. E en hough he numbe o o wa ds on he pi ch is smalle , we know ha mos o he subs i u ions in ol e a acking playe s [4] so coaches wan o ha e mo e op ions in hese posi ions. These cons ain s will be applied o he simula ion o ull ma ches. 38 5.2 Type o Subs i u ion We ha e classi ied subs i u ions as o ensi e, de ensi e o neu al, acco ding o De i- ni ion 3.8, depending i he e is a easonable inc ease o dec ease in he o ensi eness on he posi ion o he incoming playe . We can keep he momen a coach decides o make a subs i u ion cons an . Fo his, we will assume ha no subs i u ions a e due o an inju y, and hence equi e a simila playe as a eplacemen . Du ing chap e 4 we s udied di e en ways o e alua e he e ec i eness o a subs i u ion. We a e going o use hose same models o e alua e i ano he ype o subs i u ion could ha e been be e acco ding o he models. Fo each window subs i u ion, we ha e simula ed he expec ed esul i hose subs i u ions had had ano he na u e. Fo o ensi e subs i u ions, we simula e hem being neu al o de ensi e, and e alua e wha he pe o mance would ha e been. Simila ly, we ha e done he same o neu al and de ensi e subs i u ions. The wide spec um o a ge s de ined in subsec ion 4.1.1, mul iplies he oppo uni ies we ha e he e, so we a e jus going o explo e he op ions we ind mo e in e es ing. A eal-li e applica ion o his kind o model could be o coaches, o decide he ype o subs i u ion hey ha e o do o o be da a-in o med as o which is mo e likely o accomplish hei goals. Addi ional models, ha no only di e en ia e be ween o ensi e and de ensi e subs i u ions bu also inco po a e o he ac o s, ha e he po en ial o equip echnical s a wi h a b oade ange o ools o conduc ing simula ions. 5.2.1 Win p obabili y simula ions Simula ions wi h he in-game win p obabili y model, de ailed in sec ion 4.3, no only allow us o see i ce ain simula ed subs i u ions would be success ul bu since we wo k wi h Expec ed Poin s (De ini ion 4.7), we a e p o ided wi h a nume ical weighing o he e ec o changing he ype o playe swap. Fo each subs i u ion, we ha e simula ed ha modi ica ion in an o ensi e, neu al and de ensi e way. The Bayesian win p obabili y model is applied o each gene a ed game s a e. This is, gi en a game s a e x ,i = ( , i, τ ,i, s ,i ) we modi y s ,i , so we ha e s ,i,o , s ,i,neu and s ,i,de . Simila ly, o ha same , we modi y he game s a e o he opposing eam i∗ in o x ,i∗ = ( , i∗, τ ,i∗, s ,i∗,o /neu/de ), depending on he case. Thus, o e e y case, we ge h ee pai s o each case: ( θ ,i,o , θ ,i∗,o ) o he o ensi e subs i u ion s a e, ( θ ,i,neu, θ ,i∗,neu ) o he neu al one and ( θ ,i,de , θ ,i∗,de ) o de ensi e. Wi h he de ini ion o g ,i ∼B ( , θ ,i ) and adding he cu en sco eline ( G ,i, G ,i∗ ), we simula e and ge he home win, d aw and away win p obabili ies. No e ha his analysis is independen o which o he na u e o he subs i u ions. Fo ei he o ensi e, de ensi e o neu al subs i u ion, we simula e he h ee s a es. 39 Figu e 5.2: Va ia ion in he amoun o Expec ed Poin s. x-axis is squa e oo scaled. Sepa a ion be ween he compa ison o o ensi e subs i u ions wi h espec o neu al ones (blue) and nega i e ones wi h espec o neu al ( ed). The g aphical ep esen a ion o he simula ion esul s can be obse ed in Figu e 5.2. F om he p e ious simula ions we ob ain EPo , EPneu and EPde . In blue, we ha e he dis ibu ion o ∆ EPo = EPo −EPneu , and in ed we ha e ∆ EPde = EPde −EPneu . The x-Axis has been escaled by a squa e oo ans o ma ion o be e isualiza ion, bu main aining he sign o he change, so he amoun plo ed is ac ually sgn ( EP ) √∆EP . I we plo simply ∆ EP he cu e is cen e ed and spiked. This is due o 60.3% o he subs i u ions ha ing a ∆ EP o less han 0.03, which co esponds o less han 1% change in EP since we wo k wi h a 3-poin sys em. This is impo an since we see ha almos wo o each i e subs i u ions could ha e a signi ican displacemen in e ms o win p obabili y i he p ope ype o subs i u ion is aken. In Figu e 5.2 we see ha , on he one hand, he blue cu e is displaced o he igh , which means ha , in gene al, an o ensi e subs i u ion gi es you be e odds o winning han a neu al one. On he o he hand, he ed cu e is sligh ly displaced o he le , gi ing you wo se win p obabili y pe cen ages. This does no mean i is always he case since we see a big pa o he ∆ EPo cu e is nega i e, gi ing you wo se chances han a neu al sub, and simila ly wi h he ∆ EPde in he posi i e changes. Looking a he numbe s, we see ha o e 20% o he subs i u ions could ha e e y signi ican , mo e han 0.3 expec ed poin s, swing in expec ed poin s. This means ha , by p ope ly selec ing he subs i u ion ype, one o e e y i e subs i u ions could ha e a eal impac on he ma ch and he winning p obabili y. 40 ob ain a win p obabili y ad an age. We do no ice ha he model is mo e sensi i e o subs i u ions ea ly in he second-hal . Figu e 6.3: Dis ibu ion o he a ia ion o EP a e a subs i u ion. Dis ibu ion a each minu e. 6.2 Simula ion o Al e na i e Timing In his sec ion we will s udy he al e na i e iming o he subs i u ions coaches did du ing he s udied ma ches. This is, o each subs i u ion ha had been made, we a e simula ing mo ing i o wa d and backwa d h ough ime. We ha e espec ed he coaches’ choices in e ms o he ype o subs i u ion and he di e en subs i u ions windows used, so i a eam made he i s wo changes a he same ime and hen wo isola ed ones, we ha e also done so in he simula ions, while also including a i e-minu e minimum space be ween he simula ed subs i u ions, since i is no eal scena io was ing wo windows in wo consecu i e playe changes, excep o ex ao dina y cases such as an inju y, a ed ca d, o ying o lose ime. Simula ions ha e ollowed he same s uc u e o ou wo ypes o models. Knowing how he subs i u ions we e packed in o windows and hei o ensi e o de ensi e na u e, we we e able o gene a e new game s a es x ,i by a ying he subs i u ion in o ma ion. This applied bo h o he classi ica ion models, which had he upda ed s a e inpu , and he win p obabili y model, in a simila way o he simula ions p e iously explained in his hesis. 6.2.1 Subs i u ion classi ica ion models A e gene a ing he adequa e game s a es o simula ion, we applied he ained subs i u ion classi ica ion models. We g ouped subs i u ions acco ding o hei 47 p oximi y o he eal ma ch case, and he esul s a e shown in Table 6.1. Time g oupings a e no homogeneous because we wan o ema k he minu es close o he eal subs i u ion. We show he change in he pe cen age o posi i e o success ul subs i u ions. This is, o ensi e subs i u ions, when simula ed om 6 o 10 minu es p io o hei eal minu e, ob ained be e p edic ions 3.6% o he ime. Change in posi i e/success ul subs i u ions (%) Time o eal sub [-20,-11] [-10,-6] [-5,-1] [1,5] [6,10] [11,20] O ensi e Subs +8.2 +3.6 +0.4 -0.2 -1.3 -6.5 De ensi e Subs -7.6 -3.4 -1.2 +0.9 +2.5 +7.3 2C-SER +3.2 +2.5 +0.4 +0.2 -0.1 -0.4 3C-SER +1.5 +0.7 0.0 -0.2 -0.7 -1.1 xV-Change Rule +10.2 +6.1 +1.4 -2.1 -4.2 -8.3 Table 6.1: Resul s o iming simula ions wi h he classi ica ion models. Resul s g ouped by di e en assessmen echniques and ime wi h espec o he eal subs i u ion. Numbe s ep esen he a ia ion in he pe cen age o posi i e o success ul subs i u ions. Nega i e numbe s ep esen a dec ease in he numbe o success ul ou comes Resul s in Table 6.1 a e in line wi h mos o he o he esul s. O ensi e subs i u ions ha e a highe p obabili y o success i done ea lie , while de ensi e ones beha e he o he way a ound: success inc eases as hey a e done la e in he ma ch. This ag ees wi h he esul s in Figu e 6.2, and he main easons a e he same: ime le o achie e he goal. The Sco eline E alua ion Rule, bo h o he wo and h ee-class p oblems, seems o be a bi be e when subs i u ions a e done ea lie han in eal li e, bu he numbe s a e e y small and no e y signi ican . The bigges esul s a e o he xV Change Rule, which compa ed he incoming playe pe o mance o he ou going one. We see ha ha ing mo e ime as a subs i u e is co ela ed wi h a highe p obabili y o a success ul subs i u ion. 6.2.2 Win p obabili y model F om Figu e 6.3, we obse ed ha he e is no appa en gene al be e momen o do a subs i u ion, bu on a single ma ch we ha e obse ed signi ican changes in he win p obabili ies. Thus we simula e o each subs i u ion an al e na i e momen . The esul s, in gene al, ag ee ha subs i u ions should be done, i any hing, p e iously o he momen when hey happen in eali y. In Figu e 6.4 we ha e plo ed he dis ibu ion o ∆ EP depending on he ime p e ious o which he subs i u ion ook place. The bigges changes a e when subs i u ions a e made 30 minu es p io o hei eal- ime, which p ac ically could ne e happen. 48 Figu e 6.4: Dis ibu ion o he a ia ion o EP depending on he ime p e ious o he eal subs i u ion Again, subs i u ions in a ma ch a e ela ed, so conside ing hei ela ions is ele an . I we apply i o he Le an e 3-3 Real Mad id ma ch we discussed ea lie , whose expec ed poin s e olu ion is in Figu e 5.3, we can see ha Mad id could ha e s a ed hei comeback be o e, acco ding o he model. This is due o Mad id ha ing a highe Elo a ing, so he model gi es a highe goal-sco ing p obabili y by doing he subs i u ions, so i sugges s doing i be o e and hus enhancing he winning p obabili ies. On he o he hand, o Le an e, i shows e y li le change when jus changing he iming. Rey e al. [28] showed ha losing eams subs i u e be o e while leading eams end o delay changes, and ou analysis sugges s i is he op imal s a egy. While he p e ious ma ch may ha e been mo e ene ic and high-sco ing, we can ocus also on a quie e and mo e measu ed game, which now p esen s an equally aluable oppo uni y o analysis. In ac , by s udying he s a egies and ac ics used in a mo e low-key se ing, we may be able o unco e new insigh s and app oaches ha could p o e use ul in u u e ma ches. This is he case o Osasuna 0-0 Espanyol, played also a he s a i he 2021/22 LaLiga. Bo h eams had e y simila Elo a ings, and i was a close goal-less ma ch. Espanyol made a hal - ime sub, and hen a double subs i u ion a he 63 d minu e, a e which hey had 1.07 EP. Had hey done he subs i u ion be o e, he EP a e he subs i u ion would ha e been 0.02 EP abo e he alue hey go in he eal ma ch. When a i ed o he 63 d minu e, he simula ion gi es he same alue, since he game s a e x ,i is he same. The ac is, du ing he minu es he subs i u ion had been done, Espanyol had a highe win 49 p obabili y. When we make his double subs i u ion la e in he ma ch, Espanyol’s EP goes lowe while Osasuna’s goes up. In ha ma ch, Osasuna jus made h ee changes. In his pa icula case, he a ia ion wi h ime o hose subs i u ions has almos no e ec , less han 0.01 EP on he winning p obabili ies. Wha simula ions do say is ha , i hose changes had been o ensi e, a eal change could ha e been seen, as we discussed in sec ion 5.2. Had Osasuna’s coach decided o in oduce a ou h and i h bench playe in o he ma ch, hei EP would ha e gone down by close o 0.02 EP, so jus s aying wi h he playe s on he ield is he decision he model would ha e sugges ed. 6.3 Resul s and Discussion Timing o subs i u ions is key. Mos o he subs i u ions occu be ween he 60 h and he 85 h minu e [4], in ou case we ound ha 65.3% o he subs i u ions we e on ha ime pe iod. Ea ly subs i u ions showed a highe endency o u n he esul in o a desi ed one, while la e subs i u ions ha e a highe pe cen age o ailu e. By b inging esh legs and new ac ics on o he ield, coaches ha e he oppo uni y o injec ene gy and c ea i i y in o hei eam’s pe o mance, po en ially shi ing he momen um in hei a o . E en hough gene a ed chances, measu ed by xG, and ac ual goal sco ing a e no di ec ly ela ed, he analysis o bo h cases is e y simila . The simula ions wi h al e na i e iming on he Sco eline E alua ion Rule esul ed in no signi ican change. The dis ibu ion in ime o success o o ensi e and de ensi e subs i u ions is he logical consequence o hei de ini ion. Since we need a goal o conside he o ensi e subs i u ion as accomplished, he p obabili y is highe i he change is done ea lie . Simila ly, he p obabili y o keeping a clean shee a e a de ensi e change inc eases he less ime he opposi ion has o goal-sco ing. Mo eo e , he esul s align wi h he a ia ion o goal-sco ing in ensi y by Amez e al. [1]. Simila ly, he numbe o inconsequen ial subs i u ions, hose ha do no a y he esul , inc eases as he inal whis le ge s close . Simula ions wi h al e na i e iming ga e acco ding o esul s, bo h o o ensi e and de ensi e subs i u ions. O ensi e and de ensi e subs i u ions, which we de ined as a change in he posi ion occupied by he playe s has an e ec on he cen oid o he whole eam shi ing [23]. In subsec ion 3.2.2, especially cap u ed in Figu e 3.3, we discussed how subs i u es end o p oduce mo e expec ed alue, which makes a logical decision o ad ance an o ensi e subs i u ion i a playe is pe o ming poo e . While de ensi e subs i u ions ha e highe chances o being success ul i done la e in he ma ch, i does no mean ha he op imal s a egy is such. Teams ha a e winning, hough, ha e been obse ed o end o delay hei subs i u ions [28]. The in-game win p obabili y model seems o ha e a highe sensibili y owa ds ea ly 50 subs i u ions. This is logical due o he e ec o he subs i u ions being able o ha e epe cussions in he game, bo h o good and bad subs i u ions. This esul , oge he wi h he analysis o he di e en assessmen s, poin s owa ds he same di ec ion: ea ly subs i u ions ha e a highe impac on he game. The highe a e o xV pe minu e o subs i u es is one o he easons we ob ain a endency o be e esul s o ea lie subs i u ions. In he i s s age o he in es iga ion, no cons ain s we e included, and he win p obabili y model sugges ed he winning combina ion was o bu n all subs i u ions as soon as possible. This, wi h he p ope amoun o in o ma ion, does no make sense, since playing o e 30 minu es wi h no subs i u ions le can be ha m ul o inju y and a igue wise. Since eams ne e use his s a egy, and he las subs i u ions a e done e y la e in o he game ([4], [28], [14]), he model has no lea ned ha i could be a p oblem. Adding esh playe s does inc ease you goal-sco ing p obabili ies, so he model sugges s adding e en mo e esh playe s in o he pi ch. When doing ime simula ions wi h he xV Change Rule we ob ained ha doing ea lie he subs i u ions esul ed in mo e success. This means he incoming playe s a e expec ed o ou pe o m hei colleagues i enough ime is gi en o hem because bench playe s ou pe o m s a e s on a e age. Simula ions on comple e ma ches exposed he p oblem o he model sugges ing oo ea ly subs i u ions. We could no ind any e idence o Mye s’ subs i u ion ule [26] as an op imal s a egy. E en hough we ha e now i e subs i u ions and s udy a di e en da ase o ma ches, he p oblems discussed by Sil a and Swa z [31] s ill apply: simila s a egies a e equally use ul and some bad pa e ns can bias he analysis. 51 Chap e 7 Conclusions Subs i u ions a e he main asse a coach has o di ec ly in luence he game. By subs i u ing playe s on he ield wi h eshe bench playe s, clubs’ echnical s a a e able o in oduce new ac ics, make mo e adjus men s and possibly change momen um. The objec i e o his hesis is o s udy op imal subs i u ions h ough he use o he da a. In o de o do so, we s udied he subs i u ion p oblem om di e en poin s o iew. Following he COVID pandemic, he majo i y o leagues in oduced a ou h and i h subs i u ion o each eam, as a empo a y measu e o he big amoun o ma ches played in a small pe iod o ime. Some leagues decided o keep hese ex a subs i u ions, and his now seems o be he new ule o in e na ional op- ie oo ball. Th ough he analysis o he Eu opean male op leagues, we esponded o Resea ch ques ion 1, in which way did he in oduc ion o he addi ional subs i u ions a ec he coaches’ maneu e abili y and he ela i e impo ance o subs i u ions. The esul s show how clubs h oughou Eu ope ha e needed a pe iod o adjus men . The cu en season, 2022/23, is he hi d comple e season whe e he 5 subs i u ions pe ma ch ule is alid. The numbe o ac ual playe changes done by a eam has inc eased signi ican ly speaking since his ule was in oduced and now seems o be s able, wi h 4.35 subs i u ions pe eam pe game in a e age. E en hough subs i u ions ha e inc eased, he pe cen age o ma ches ha changed esul s a e he second-hal , which is he one a ec ed by subs i u ions, has emained he same o p e-COVID e a. The pe cen age o ma ches ha a e d awn a hal - ime and inish wi h a winne has also emained e y simila . This is he esul o eams e ol ing o con inue o be compe i i e wi h each o he , and we canno see an e ec o he inc ease o subs i u ions in ma ch sco elines. We de ined subs i u ions as o ensi e o de ensi e depending on he posi ions he incoming and ou going playe s occupy. I he posi ions a e dis inc , i means ha he subs i u ion has a ac ical componen , and he beha io o he eam is changed [23]. O cou se, same posi ion subs i u ions can be due o ac ical easons o simply 52 a igue, inju ies o bookings, bu due o he da a we a e wo king wi h, we ha e no way o know. When he new 5-subs ule was in oduced, coaches main ained hei numbe o non-neu al subs i u ions pe ma ch in absolu e alue, which by he second season had been aised, equaling he pe cen age o almos 1 o e e y 4 subs i u ions being non-neu al. This is u he p oo o he adap a ion p ocess eams ha e aken since he COVID ou b eak, and how he new 5-subs ule has changed he ac ical landscape, wi h non-neu al subs i u ions inc easing by a 50% in absolu e numbe s. Du ing he whole hesis, we ha e wo ked wi h public da a a ailable o many leagues, no only Eu opean male op- ie compe i ions, so ha his analysis could be assimila ed in o o he leagues. The coun e pa is he lack o da a, which in his case means many subs i u ions ha e a ac ical e ec we canno see h ough he da a. Ha ing a be e desc ip ion o he subs i u ions would be a g ea asse o s udy how he change be ween 3-subs pe ma ch and 5 has changed he coaches’ ac ics. In pa icula , ha ing e en -by-e en da a is he main poin o imp o emen o his hesis and h oughou he conclusions will be men ioned mul iple imes. Mo ing o wa d, we explo ed Resea ch ques ion 2, whe e we ocus on he e alua ion o subs i u ions. The aim o he hesis is o iden i y op imal subs i u ions, so i s we needed o s a e wha we de ine as a success ul one. We based ou assessmen on h ee poin s o iew: he sco eboa d and i s e olu ion, he goal-sco ing chances and he playe s’ pe o mance. All h ee ways o assessing subs i u ions a e co ela ed, bu on a pa icula game o subs i u ion he assessmen s can be dis inc . Also, o all he pe spec i es we looked a he p oblem as a wo-class p oblem, whe e subs i u ions a e ei he posi i e o nega i e, and a h ee-class p oblem whe e we in oduced inconsequen ial ags o he subs i u ion wi hou a su icien ly ele an e ec . Once we had he ules whe e we de ined wha we conside as success ul, we ained di e en machine lea ning models, mainly k Nea es Neighbo s, Suppo Vec o Machines and Random Fo es classi ie s. The esul s we e close, bu ela i ely wo se, han he ones ob ained by B u i e al. in a simila s udy [7], e en hough me hodologies a e sligh ly di e en and use dis inc da a. The bes p edic ions we e ob ained o he h ee-class p oblems, due o accu acy being highe o he amoun o inconsequen ial subs i u ions. We did no ind any signi ican di e ence be ween he algo i hms used o p edic he classi ica ion o subs i u ions. The second way in which we assessed subs i u ions was as a byp oduc o an in-game win p obabili y model, based on he idea by Robbe ech s e al. [29]. Ou model p edic s, gi en a game s a e, he p obabili y o sco ing a goal and simula es he es o he ma ch. O he di e en echniques ied o he p edic ion o goal sco ing, he one ha esul ed in mo e accu a e p edic ions was he Gaussian Walk, as he Bayesian app oach eally ans e s in o ma ion be ween ime ames. The alida ion cu es shown by he o he me hods, he Mul iple Logis ic Reg esso s, Random 53 Fo es and Mul iple Random Fo es we e wo se han he Bayesian me hod bu s ill achie ed a easonable le el o accu a e p edic ions. F om his model, we s udied how he win, ie and loss p obabili ies changed wi h he subs i u ions. While compa ing o a model wi hou he subs i u ion in o ma ion, we p o ed ha playe modi ica ions a e ele an o p edic ed goal sco ing and, he e o e, winning p obabili y. We also obse ed ha subs i u ions end, in a majo i y, o gi e a boos o he eams’ pe o mance, gi en ha he majo i y o he changes p oduced a posi i e swing in expec ed poin s. These wo main models, he classi ie s and he win p obabili y, a e he base o he hesis. Classi ica ion models could be imp o ed wi h mo e in o ma ion abou eams, such as hei o ensi e and de ensi e pe o mances, o a mo e de ailed game s a e. The win p obabili y model has a main poin o imp o emen , which is ha due o he obse ed highe pe o mance o subs i u es, i usually sugges s mo e subs i u ions o be done ea lie . Teams always wai un il he end o he ma ch o inish hei subs i u ion possibili ies, o e en ual ed ca ds o inju ies. The model has no lea ned he consequences o ha ing he subs i u ions done so ea ly, because he e a e no such cases in he aining se . Tu ning o Resea ch ques ion 3, we in es iga ed he p oblem o knowing he expec ed pe o mance o he bench playe s. Be o e p oceeding o he simula ions, we did an analysis o he dis ibu ions o he bench, depending on he numbe o allowed playe s and he s a ing o ma ion. This allows us o make a gene al analysis o he subs i u ions wi hou depending on each pa icula bench om each ma ch. We ha e wo ked wi h o ensi e, neu al and de ensi e subs i u ions o he ype o subs i u ion in he simula ions, which a e p o en o modi y he eam’s ac ics [23]. Fo e e y subs i u ion ha happened in a ma ch, we simula ed as i i was any o he h ee op ions, using he win p obabili y model o ob ain a nume ic alue o he impac o he subs i u ion. Ha ing access o mo e de ailed da a, such as he e en -by-e en o acking da a, would allow hese same models o in oduce mo e ea u es and di e en kinds o subs i u ions. I a simila model we e o be implemen ed o eal use, coaches would like a wide ange o he ypes o a ailable subs i u ions, wi h mo e ac ical nuances han a simple o ensi e-de ensi e choice. The esul s o he simula ions we e in e es ing. Fi s , we saw ha he change in expec ed poin s o o ensi e subs i u ions wi h espec o neu al was in majo i y posi i e. Ins ead, doing he subs i u ion de ensi ely gi es, on a e age, a wo se EP alue han neu al. This means ha he model ends o be e alue he o ensi e subs i u ions. Bu we saw ha de ensi e changes can ac ually induce a posi i e swing in EP. I was he case o he Le an e - Real Mad id ma ch, whe e a lowe Elo a ed eam, Le an e, wi h a a o able esul , he model sugges ed ha a de ensi e playe swap was he co ec op ion. O e all, he esul s o he simula ions showed 54 ha a easonable amoun o subs i u ions can ha e a signi ican change in hei expec ed poin s alue depending on he ype o subs i u ion. The selec ion o a subs i u e has many imp o emen ways. The expec ed pe o mance o a playe is an analysis o i s own, and in his hesis could only be heo ized and e y small wo ked wi h. Many me ics om ex ended seasons can be aken in o accoun o p ope ly assess he playe , and oo ball clubs could e en include aining da a which is e iden ly ne e public. Mo eo e , s udying ela ionships be ween eamma es on he oo ball pi ch [6] can also be a powe ul asse o iden i y op imal subs i u ions. The c ea ion o a Bayesian model, whe e e e y piece o in o ma ion abou a playe can be inpu ed o a be e pe o mance p edic ion, is a p omising way o s a he in es iga ion on bench playe s’ achie emen o ecas . Finally, we add essed Resea ch ques ion 4 and deal wi h he iming p oblem. We mo ed he subs i u ions h ough he ime, espec ing he windows on which hey we e made, and applying easonable cons ain s o he simula ions. The gene a ed game s a es we e ed in o he models, bo h he classi ie s and he win p obabili y. The esul o o ensi e subs i u ions was ha i was be e o make hem as soon as possible, which is logical since i gi es he eam mo e ime o sco e a goal. In a simila way, when assessing subs i u ions by xV o he playe , ea ly subs i u ions a e much mo e e ec i e ha he end o he ma ch. Re e sely, he de ensi e subs i u ions a e p o en mo e e ec i e in he la es pa o he ma ch, since he opposi ion has less ime o sco e. These indings a e consis en wi h o he analyses o goal-sco ing equencies and hei co ela ion wi h subs i u ions [1]. Ob iously, he esul s a e a consequence o he de ini ion. While bench playe s gene a e mo e xV and is logical o subs i u e a playe pe o ming poo ly, i is no clea ha delaying de ensi e subs i u ions is he op imal s a egy, e en hough is wha eams do [28]. When s udying he subs i u ions wi h espec o he Sco eline E alua ion Rules, we see ha posi i e subs happen be o e, in gene al, han nega i e ones. Bu when doing he simula ions, we did no ind a e y signi ican change in he amoun o posi i e and nega i e subs i u ions, nei he o he wo-class o h ee-class p oblem. Simila ly, he simula ions wi h he win p obabili y model did no show much a ia ion in ime. We did obse e ha he subs i u ions ha ha e a highe e ec on he winning p obabili ies a e he ones made ea ly in he second-hal and a hal - ime. These esul s align wi h he s udy by Sil a and Swa z [31], whe e hey ound no appa en be e momen o make he subs i u ion, con adic ing Mye s’ [26] iming ule. While wi h he ype o subs i u ion we ob ained a high p opo ion o subs i u ions wi h a ele an change in expec ed poin s, wi h he iming he esul s a e much mo e es ained. In some cases, we ob ain ha doing he subs i u ion ea lie can p o ide he eam wi h a highe winning p obabili y du ing a ew minu es, bu he inc emen s in EP a e lowe han wi h he ype o subs i u e. 55 Du ing he hesis we ha e wo ked sepa a ely on wo di e en pa s o he subs i u ion p oblem: he incoming playe and he ime a which hey do. A comple e sys em o da a-d i en o ecas o subs i u ions, as u u e wo k on his p oblem, should include a coo dina ion o hose wo p oblems, and also he decision o he ou going playe . Wi h access o e en -by-e en o acking da a, a mo e p o ound analysis can be done o a playe ’s ac ions ([32], [11]) and hei e ec on he ma ch, while joining wi h physical in o ma ion h ough GPS [36] o mul i-came a acking sys ems ([8], [5], [4]), he sys em would be able o p o ide wi h a mo e clea iew o he unde -pe o me s o he game. Such a comple e model could implemen a ious ideas: ying o p edic he decline in physical pe o mance, iden i ying de ensi e unde -pe o mance om an opposi ion high alue. Such in o ma ion, adequa ely combined wi h a iming analysis and a p ope subs i u e selec ion, could p o ide coaches and hei s a wi h a powe ul da a-in o med ool o gain a compe i i e ad an age and o know he bes da a- based decisions. The key idea is o wea e all he analysis o mo e eal in o ma ion abou he p oblem. In his hesis, we simula ed al e na i e subs i u e selec ion and al e na i e iming sepa a ely. Being able o in oduce simula ions o he ou going playe and combining he h ee ac o s could lead o a well- unc ioning comple e model o o ecas ing and iden i ica ion o op imal subs i u ions in socce . In his hesis we de eloped wo ypes o models: classi ica ion models and a subs i u e sensible win p obabili y model. We ha e used hem o s udy he mos bene icial subs i u ions and simula e al e na i e se ings o be e pe o mance. We saw ha coaches can de ini ely use subs i u ions o change he cou se o ma ches and ha by selec ing mos ly he app op ia e ype o subs i u ion can ob ain a big inc ease in hei win p obabili ies. Timing is less impo an nume ically wise, bu can play a ele an ole in some cases. In conclusion, his hesis has p o ided a comp ehensi e analysis o subs i u ions in socce , explo ing a ious app oaches and me hods o e alua ing hei impac on he game. Th ough ou in es iga ion, we ha e s udied some ac o s ha in luence he e ec i eness o subs i u ions, including he iming o he subs i u ion and he ac ical app oach o he eam om he ype o subs i u ion hey choose. Ou indings p o ide insigh s ha can in o m decision-making p ocesses o coaches and analys s alike. Mo ing o wa d, i is clea ha u he esea ch is needed o ully unde s and he complexi ies o subs i u ions in socce and o de elop mo e sophis ica ed me hods o e alua ing hei impac . None heless, we belie e ha his hesis ep esen s a aluable con ibu ion o he ield o oo ball analy ics, and o e s a solid ounda ion o u u e wo k on he subs i u ion p oblem. 56