Academic Edi o : Ca son K. Leung
Recei ed: 29 Janua y 2025
Re ised: 3 Ma ch 2025
Accep ed: 5 Ma ch 2025
Published: 10 Ma ch 2025
Ci a ion: De-la-C uz-To es, B.;
Na a o-Cas o, M.;
Ruiz-de-Ala cón-Quin e o, A. An
Expec ed Goals On Ta ge (xGOT)
Model: Accoun ing o Goalkeepe
Pe o mance in Foo ball. Big Da a
Cogn. Compu . 2025,9, 64.
h ps://doi.o g/10.3390/
bdcc9030064
Copy igh : © 2025 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license
(h ps://c ea i ecommons.o g/
licenses/by/4.0/).
A icle
An Expec ed Goals On Ta ge (xGOT) Model: Accoun ing o
Goalkeepe Pe o mance in Foo ball
Blanca De-la-C uz-To es 1,* , Miguel Na a o-Cas o 2,* and Anselmo Ruiz-de-Ala cón-Quin e o 3
1Depa men o Physio he apy, Uni e si y o Se ille, c/A icena s/n, 41009 Se ille, Spain
2Depa men o Applied Ma hema ics I, Highe Technical School o A chi ec u e, Uni e si y o Se ille,
A d. Reina Me cedes s/n, 41012 Se ille, Spain
3Foo ball and Handball Academy, S ee nº 12B, O ice 6, 41960 Se ille, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed] (B.D.-l.-C.-T.); [email p o ec ed] (M.N.-C.)
Abs ac : A key challenge in u ilizing he expec ed goals on a ge (xGOT) me ic is
he limi ed public access o de ailed oo ball e en and posi ional da a, alongside o he
ad anced me ics. This s udy aims o de elop an xGOT model o e alua e goalkeepe
(GK) pe o mance based on he p obabili y o success ul ac ions, conside ing no only
he ou comes (sa es o goals conceded) bu also he di icul y o each sho aced. Fo mal
de ini ions we e es ablished o he ollowing: (i) he ini ial dis ance be ween he ball and
he GK a he momen o he sho , (ii) he dis ance be ween he ball and he GK o e ime
pos -sho , and (iii) he dis ance be ween he GK’s ini ial posi ion and he goal, wi h espec
o he y-coo dina e. An xGOT model inco po a ing geome ic pa ame e s was designed o
op imize pe o mance based on he ball posi ion, ajec o y, and GK posi ioning. The model
was es ed using sho s on a ge om he 2022 FIFA Wo ld Cup. S a is ical e alua ion using
k- old c oss- alida ion yielded an AUC-ROC sco e o 0.67 and an 85% accu acy, con i ming
he model’s abili y o di e en ia e success ul GK pe o mances. This app oach enables a
mo e p ecise e alua ion o GK decision-making by analyzing a ep esen a i e da ase o
sho s o es ima e he p obabili y o success.
Keywo ds: gene a i e model; sho on a ge ajec o y; goalkeepe e alua ion; ball posi ion;
da a analysis
1. In oduc ion
In p o essional oo ball (socce ), he p opo ion o a acking plays ha culmina e in
a goal is exceedingly low, wi h only 1% o such sequences and app oxima ely 10% o all
sho s esul ing in a success ul ou come [
1
–
3
]. Despi e his, goals emain he undamen-
al de e minan o ma ch ou comes and a e widely ega ded as he p incipal me ic o
e alua ing he pe o mance o bo h eams and indi idual playe s.
Relying exclusi ely on his me ic, howe e , ails o cap u e he complexi y o playe
and eam con ibu ions. This app oach emphasizes ou comes while neglec ing he p ocesses
ha unde pin hem. To add ess his limi a ion, he ield o oo ball pe o mance analysis
has inc easingly adop ed mo e complex, p ocess-o ien ed me ics. These ad anced me ics
p o ide a comp ehensi e amewo k o assessing pe o mance beyond he e alua ion o
goals, enabling a deepe and mo e nuanced unde s anding o he game. The e o e, he use
o sho cha ac e is ics as a p oxy o success has been inc easingly alida ed in nume ous
oo ball s udies [
4
]. Among hese me ics, he expec ed goals on a ge (xGOT) s ands ou
as pa icula ly no ewo hy [
5
]. The xGOT is an ad anced me ic in oo ball analy ics ha
e ines he s anda d expec ed goals (xG) model by inco po a ing addi ional ac o s ela ed
Big Da a Cogn. Compu . 2025,9, 64 h ps://doi.o g/10.3390/bdcc9030064
Big Da a Cogn. Compu . 2025,9, 64 2 o 13
o he quali y and placemen o sho s on a ge [
6
–
9
]. While xG es ima es he likelihood o
a sho esul ing in a goal by e alua ing he sho ’s quali y be o e i is aken based on ac o s
such as sho loca ion, sho ype, and build-up play, xGOT adjus s his p obabili y a e
he sho is execu ed. I akes in o accoun he ac ual sho placemen and he di icul y o
he goalkeepe (GK) o sa e i . Key elemen s conside ed in xGOT include sho placemen
( he p ecise loca ion wi hin he goal ame whe e he sho is aimed, such as he op co ne
e sus he cen e ), GK in luence ( he posi ioning and eac ion o he GK a he ime o he
sho ), and sho cha ac e is ics like powe and ajec o y (which a ec he sho ’s speed and
mo emen and, consequen ly, i s sa e di icul y) [
10
,
11
]. By in eg a ing hese ac o s, xGOT
p o ides a mo e accu a e assessmen o bo h sho quali y and GK pe o mance, making i a
aluable ool o unde s anding ma ch ou comes and e alua ing playe con ibu ions.
The p ima y challenge in u ilizing he xGOT me ic lies in he limi ed public a ailabil-
i y o de ailed oo ball e en and posi ional da a, as well as o he ad anced me ics. To
he bes o he au ho s’ knowledge, such da ase s emain la gely inaccessible [
12
,
13
]. Com-
panies like Op a and STATSBOMB collec hese da a independen ly and dissemina e he
esul s di ec ly ia hei pla o ms. This lack o anspa ency and he opaci y su ounding
he algo i hms used o calcula e hese me ics—commonly e e ed o as “black-boxing”—
hinde s he unde s anding and in e p e a ion o how speci ic me ic alues a e de i ed.
This unde sco es he necessi y o de eloping an xGOT model ha explici ly iden i ies
he in luencing a iables and quan i ies hei espec i e impac s, he eby enhancing in e -
p e abili y and analy ical u ili y [12,13].
In he exis ing li e a u e, e o s o c ea e in e p e able xGOT models ha e p ima -
ily ocused on le e aging publicly a ailable da ase s, such as hose om pla o ms like
Wyscou [
14
] o open-sou ce ini ia i es (e.g., STATSBOMB’s ee da ase s) [
15
]. Indeed,
Go ini GA [
16
] pe o med a hesis in which hey emphasized xGOT as a key me ic o
e alua ing goalkeepe pe o mance wi hin speci ic models, ye wi hou o e ing he echni-
cal o ma hema ical ounda ions behind i s compu a ion (i.e., a “black box”). The au ho s
e e enced s udies and o ganiza ions such as STATSBOMB ha employ xGOT, bu ea ed
his me ic as an ex e nal esou ce a he han one ha was de eloped o eplica ed wi hin
he scope o hei s udy. Despi e hese e o s, exis ing s udies ha e ye o de elop an
xGOT model ha bo h iden i ies he in luencing a iables and quan i ies hei espec i e
impac s [
12
,
13
,
16
]. This aspec is c ucial o enhancing coaches’ unde s anding and o he
de elopmen o aining s a egies aimed a imp o ing ma ch pe o mance.
The e o e, his s udy aims o b idge his gap by p esen ing a sho p edic ion model
de eloped using e en and posi ional da a. The accu acy o he model is e alua ed h ough
s a is ical analysis and alida ed h ough consul a ions wi h p o essional ma ch analys s,
each wi h o e 25 yea s o expe ience ac oss all le els o compe i ion. Expe inpu is
also inco po a ed in o he ea u e selec ion p ocess and he in e p e a ion o hei impac
on p edic ions. Fu he mo e, p ac ical applica ions a e demons a ed using da a om
he 2022 FIFA Wo ld Cup ma ches, highligh ing use cases ele an o coaching s a and
decision-make s in oo ball analy ics.
The s uc u e o his pape is o ganized as ollows: Sec ion 2p o ides an o e iew
o he da a and key de ini ions used in he s udy. Sec ion 3ou lines he me hodology
o cons uc ing he xGOT model. In Sec ion 4, we p esen he p ac ical applica ions o
he model. In Sec ion 5, we compa e he xGOT alues be ween ou own model and he
ou pu om he STATSBOMB model. Finally, Sec ion 6discusses he implemen a ion o
ou app oach using da a om he 2022 FIFA Wo ld CUP, accompanied by a c i ical analysis
o he esul s.
Big Da a Cogn. Compu . 2025,9, 64 3 o 13
2. Da a and De ini ions
In his s udy, we analyzed 1536 sho s om he 2022 FIFA Wo ld Cup. F om he o al
numbe o sho s, we selec ed only hose aken om he g ound o ou model, esul ing in
a inal da ase o 344 sho s. This in e na ional ou namen ea u ed 32 men’s eams om
i e con ede a ions, compe ing ac oss 64 ma ches. The e en da a we e sou ced om an
openly accessible websi e [
17
]. Below, Table 1p o ides a de ailed o e iew o he di e en
alues obse ed o each sho .
Table 1. Fea u es ex ac ed om synch onized posi ional and e en da a ha we used o ain
ou model.
Fea u es Value De ini ions
Foo ball ield dimension Nume ical A 3-elemen uple (x, y, z)whe e
x∈[0, 68 m],y∈[0, 105 m],z∈[0, ∞m).
Ball diame e Nume ical 0.225 m
Ball sho loca ion Nume ical The x-, y-, and he z-coo dina e o he ball a he
ins an ime o he sho .
Goalkeepe loca ion Nume ical The x-, y-, and he z-coo dina e o he goalkeepe
a he ins an ime o he sho .
Speed o ball Nume ical The speed o he ball. We will assume i o
be cons an .
Type o sho Ca ego ical Two ypes: s aigh line sho and pa abolic sho .
3. Expec ed Goals On Ta ge (xGOT) Modeling
Fo he de elopmen o ou model, we ha e analyzed and conside ed a ious ypes
o da a ha in luence goal sco ing. A he ou se , i is impo an o cla i y ha we ha e
assumed all playe s and GKs o be iden ical. The e o e, ou a iable assumes ha any
indi idual aking a sho unde he same ini ial condi ions will achie e he same ou come.
Table 1p o ides he de ini ions o he a iables equi ed o he de elopmen o he model.
All he a iables men ioned abo e can be calcula ed using a ious exis ing ools.
Following he egula ions and he a e age dimensions o oo ball ields, we ha e
ex ac ed he measu emen s. The coo dina e o igin (0, 0, 0) will be se a one o he co ne s,
depending on he websi e om which he da a we e sou ced. The x-coo dina e ep esen s
he ho izon al dis ance ac oss he wid h o he ield ( on al axis); he y-coo dina e indica es
he dep h o dis ance owa d he goal (sagi al axis); and he z-coo dina e ep esen s he
heigh o he playe s and he ball ela i e o he g ound ( e ical axis). Consequen ly, he
z- alue is gene ally 0, excep o when a sho is aken o a playe jumps (Figu e 1).
Big Da a Cogn. Compu . 2025, 9, x FOR PEER REVIEW 3 o 14
2. Da a and De ini ions
In his s udy, we analyzed 1536 sho s om he 2022 FIFA Wo ld Cup. F om he o al
numbe o sho s, we selec ed only hose aken om he g ound o ou model, esul ing
in a inal da ase o 344 sho s. This in e na ional ou namen ea u ed 32 men’s eams om
i e con ede a ions, compe ing ac oss 64 ma ches. The e en da a we e sou ced om an
openly accessible websi e [17]. Below, Table 1 p o ides a de ailed o e iew o he diffe -
en alues obse ed o each sho .
Table 1. Fea u es ex ac ed om synch onized posi ional and e en da a ha we used o ain ou
model.
Fea u es Value De ini ions
Foo ball ield dimension Nume ical A 3-elemen uple 󰇛x,
y
,z󰇜 whe e 𝑥∈
󰇟0,68 m󰇠,𝑦∈󰇟0,105 m󰇠,𝑧∈󰇟0,∞ m󰇜.
Ball diame e Nume ical 0.225 𝑚
Ball sho loca ion Nume ical The 𝑥-, 𝑦-, and he 𝑧-coo dina e o he ball
a he ins an ime o he sho .
Goalkeepe loca ion Nume ical The 𝑥-, 𝑦-, and he 𝑧-coo dina e o he
goalkeepe a he ins an ime o he sho .
Speed o ball Nume ical The speed o he ball. We will assume i o
be cons an .
Type o sho Ca ego ical Two ypes: s aigh line sho and pa abolic
sho .
3. Expec ed Goals On Ta ge (xGOT) Modeling
Fo he de elopmen o ou model, we ha e analyzed and conside ed a ious ypes
o da a ha in luence goal sco ing. A he ou se , i is impo an o cla i y ha we ha e
assumed all playe s and GKs o be iden ical. The e o e, ou a iable assumes ha any
indi idual aking a sho unde he same ini ial condi ions will achie e he same ou come.
Table 1 p o ides he de ini ions o he a iables equi ed o he de elopmen o he
model. All he a iables men ioned abo e can be calcula ed using a ious exis ing ools.
Following he egula ions and he a e age dimensions o oo ball ields, we ha e ex-
ac ed he measu emen s. The coo dina e o igin (0, 0, 0) will be se a one o he co ne s,
depending on he websi e om which he da a we e sou ced. The x-coo dina e ep esen s
he ho izon al dis ance ac oss he wid h o he ield
( on al axis); he y-coo dina e i
ndica es he
dep h o dis ance owa d he goal
(sagi al axis); and he z-coo dina e ep esen s he heigh
o he playe s and he ball ela i e o he g ound ( e ical axis). Consequen ly, he z- alue
is gene ally 0, excep o when a sho is aken o a playe jumps (Figu e 1).
Figu e 1. The coo dina e sys em used o ep esen he posi ion o he ball o a playe consis s o
h ee axes: he x-axis ( on al axis), which ep esen s he ho izon al dis ance ac oss he wid h o he
ield; he y-axis (sagi al axis), which indica es he dep h o dis ance owa d he goal; and he z-axis
Figu e 1. The coo dina e sys em used o ep esen he posi ion o he ball o a playe consis s o
h ee axes: he x-axis ( on al axis), which ep esen s he ho izon al dis ance ac oss he wid h o he
ield; he y-axis (sagi al axis), which indica es he dep h o dis ance owa d he goal; and he z-axis
( e ical axis), which ep esen s heigh ela i e o he g ound, wi h a alue o 0 when he ball o
playe is on he g ound and which inc eases when he ball is ai bo ne o a playe jumps.
Based on he a iables p esen ed in he able, we can now pose he ollowing ques ion:
Can we model he mo emen o he ball and he GK? The answe is yes. To add ess his,
Big Da a Cogn. Compu . 2025,9, 64 4 o 13
we will use di e en ial equa ions. Fo simplici y and o ensu e accessibili y o spo s
p o essionals, we ha e chosen o employ simpli ied equa ions in his s udy.
As indica ed in he able o a iables and gi en ha oo ball in ol es wo dis inc ypes
o sho s on goal, each ype will be analyzed and discussed in de ail in sepa a e subsec ions.
3.1. S aigh -Line Sho Equa ions
We ha e made use o he di e en ial equa ions o calcula e he mo emen in each o
he coo dina es, i.e., ou sho will appea as ollows (Figu e 2):
Big Da a Cogn. Compu . 2025, 9, x FOR PEER REVIEW 4 o 14
( e ical axis), which ep esen s heigh ela i e o he g ound, wi h a alue o 0 when he ball o
playe is on he g ound and which inc eases when he ball is ai bo ne o a playe jumps.
Based on he a iables p esen ed in he able, we can now pose he ollowing ques-
ion: Can we model he mo emen o he ball and he GK? The answe is yes. To add ess
his, we will use diffe en ial equa ions. Fo simplici y and o ensu e accessibili y o spo s
p o essionals, we ha e chosen o employ simpli ied equa ions in his s udy.
As indica ed in he able o a iables and gi en ha oo ball in ol es wo dis inc
ypes o sho s on goal, each ype will be analyzed and discussed in de ail in sepa a e sub-
sec ions.
3.1. S aigh -Line Sho Equa ions
We ha e made use o he diffe en ial equa ions o calcula e he mo emen in each o
he coo dina es, i.e., ou sho will appea as ollows (Figu e 2):
Figu e 2. G aphic o s aigh -line sho on a ge .
Thanks o ou p e ious conside a ions, he mo ion o he ball will be desc ibed by
he equa ions:
𝑥󰇛𝑡󰇜𝑥𝑣𝑡𝑐𝑡
𝑦󰇛𝑡󰇜𝑦𝑣𝑡𝑐𝑡
𝑧󰇛𝑡󰇜0.225
whe e he a iable ep esen s ime; he alues x
0
and y
0
deno e he ini ial posi ion o he
ball on he ield;
0x
and
0y
co espond o he ini ial eloci y componen s o he ball a he
momen i is kicked; and 𝑐 is he coefficien o ic ion, which, being he ic ion be ween
ubbe and d y ea h, will be be ween 󰇟0.4,0.6󰇠. To see his coefficien , we ha e accessed
whe e hey gi e a lis o ic ion coefficien s [18], among which is he coefficien be ween
g ass and plas ic.
Addi ionally, he unc ion z( ) is a cons an unc ion, independen o ime. In his ype
o sho , he ball emains on he g ound. This alue ep esen s he heigh o he ball ela i e
o he g ound; howe e , some da ase s assign alues wi hin he ange [0, 0.4], despi e he
ball s aying on he g ound in all cases [19]. None heless, z( ) emains cons an .
3.2. Pa abolic Sho Equa ions
We used he diffe en ial equa ions o calcula e he mo emen in each o he coo di-
na es, i.e., ou sho will appea as ollows:
Thanks o ou p e ious conside a ions, he mo ion o he ball will be desc ibed by
he equa ions (Figu e 3): 𝑥󰇛𝑡󰇜𝑥𝑣𝑡󰇛sinψcosα󰇜
𝑦󰇛𝑡󰇜𝑦𝑣𝑡󰇛sinψsinα󰇜
𝑧󰇛𝑡󰇜𝑧𝑣𝑡cosψ𝑔𝑡
2
Figu e 2. G aphic o s aigh -line sho on a ge .
Thanks o ou p e ious conside a ions, he mo ion o he ball will be desc ibed by
he equa ions:
x( )=x0+ 0x −c 2
y( )=y0+ 0y −c 2
z( )=0.225
whe e he a iable ep esen s ime; he alues x0and y0deno e he ini ial posi ion o he
ball on he ield;
0x
and
0y
co espond o he ini ial eloci y componen s o he ball a he
momen i is kicked; and
c
is he coe icien o ic ion, which, being he ic ion be ween
ubbe and d y ea h, will be be ween
[0.4, 0.6]
. To see his coe icien , we ha e accessed
whe e hey gi e a lis o ic ion coe icien s [
18
], among which is he coe icien be ween
g ass and plas ic.
Addi ionally, he unc ion z( ) is a cons an unc ion, independen o ime. In his ype
o sho , he ball emains on he g ound. This alue ep esen s he heigh o he ball ela i e
o he g ound; howe e , some da ase s assign alues wi hin he ange [0, 0.4], despi e he
ball s aying on he g ound in all cases [19]. None heless, z( ) emains cons an .
3.2. Pa abolic Sho Equa ions
We used he di e en ial equa ions o calcula e he mo emen in each o he coo dina es,
i.e., ou sho will appea as ollows:
Thanks o ou p e ious conside a ions, he mo ion o he ball will be desc ibed by he
equa ions (Figu e 3):
x( )=x0+ 0x (sin ψcos α)
y( )=y0+ 0y (sin ψsin α)
z( )=z0+ 0z cos ψ−g 2
2
whe e
ψ∈0, π
2
ep esen s he sho angle wi h espec o he
XZ
plane, and
α∈[0, 2π]
ep esen s he sho angle wi h espec o he
XZ
plane. I
α
= 0 o 2
π
, he sho does no
a ec he y-coo dina e, and he shadow o he ball will desc ibe a linea ajec o y. Simila ly,
when ψ=π
2 o any alue o α, he ball will ollow a pu ely e ical mo ion.
Big Da a Cogn. Compu . 2025,9, 64 5 o 13
Big Da a Cogn. Compu . 2025, 9, x FOR PEER REVIEW 5 o 14
whe e ψ∈󰇣0,
󰇤 ep esen s he sho angle wi h espec o he 𝑋𝑍 plane, and α∈󰇟0,2π󰇠
ep esen s he sho angle wi h espec o he 𝑋𝑍 plane. I α = 0 o 2π, he sho does no
affec he y-coo dina e, and he shadow o he ball will desc ibe a linea ajec o y. Simi-
la ly, when 𝜓
 o any alue o α, he ball will ollow a pu ely e ical mo ion.
Figu e 3. G aphic o pa abolic sho on a ge .
Con e sely, when 𝜓0, he sho desc ibes a pa abolic ajec o y along he g ound.
This ype o sho is a e in oo ball. A s udy by Howa d Masu [20] de i ed he equa ions
o such a ajec o y in he con ex o billia ds. When bo h angles a e ze o, he sho ollows
a linea ajec o y in i s g ound p ojec ion. Mo eo e , since ψ = 0, he ball emains on he
g ound, meaning i s ac ual mo ion coincides wi h i s p ojec ion. In his case, we apply he
equa ions om he p e ious sec ion. Consequen ly, we assume ha bo h angles a e non-
ze o in ou analysis. In his sec ion, we employ a simpli ied model, p e iously published
in an ea lie s udy [21], whe e angula eloci y is no conside ed. The alues x
0
and y
0
deno e he ini ial posi ion o he ball on he ield, while
0x
and
0y
ep esen he ini ial
eloci y componen s a he momen o he kick. The pa ame e g deno es he g a i a ional
cons an .
3.3. Goalkeepe Equa ions
Based on he da a analys specializing in oo ball wi h o e 25 yea s o expe ience
ac oss all le els o compe i ion, and because we aim o de elop a simple algo i hm, we
ha e decided ha he GK’s mo emen will ollow a ec ilinea ajec o y. Thus, he equa-
ions a e as ollows: 𝑥󰇛𝑡󰇜𝑥
𝑣
𝑡
𝑦󰇛𝑡󰇜𝑦
𝑣
𝑡
𝑧󰇛𝑡󰇜𝑧
𝑣

𝑡
The alues x
, y
, and z
 ep esen he GK’s ini ial posi ion on he ield. Simila ly,
𝑣
, 𝑣
, and 𝑣
 deno e he GK’s ini ial eloci y a he momen he playe kicks he ball.
3.4. Build o xGOT
Th ough he cons uc ion o he diffe en ial equa ions and he way we modeled he
ield, we a e able o calcula e he ime ins ances o bo h he ball and he GK. Howe e , o
achie e ou objec i e, we need o de e mine when bo h coincide, which only occu s unde
speci ic condi ions. To simpli y his, we will conside only he ins ances when hey lie
wi hin he same 𝑋𝑍 plane a a gi en alue o 𝑦. This allows us o ocus on sol ing he
ollowing p oblem:
𝑦
𝑣
𝑡𝑦𝑣𝑡󰇛sinψsinα󰇜
This leads o he ollowing:
𝑡 𝑦
𝑦
𝑣󰇛sinψsinα󰇜𝑣

Figu e 3. G aphic o pa abolic sho on a ge .
Con e sely, when
ψ=
0, he sho desc ibes a pa abolic ajec o y along he g ound.
This ype o sho is a e in oo ball. A s udy by Howa d Masu [
20
] de i ed he equa ions
o such a ajec o y in he con ex o billia ds. When bo h angles a e ze o, he sho ollows
a linea ajec o y in i s g ound p ojec ion. Mo eo e , since
ψ
= 0, he ball emains on he
g ound, meaning i s ac ual mo ion coincides wi h i s p ojec ion. In his case, we apply he
equa ions om he p e ious sec ion. Consequen ly, we assume ha bo h angles a e nonze o
in ou analysis. In his sec ion, we employ a simpli ied model, p e iously published in an
ea lie s udy [
21
], whe e angula eloci y is no conside ed. The alues x
0
and y
0
deno e
he ini ial posi ion o he ball on he ield, while
0x
and
0y
ep esen he ini ial eloci y
componen s a he momen o he kick. The pa ame e gdeno es he g a i a ional cons an .
3.3. Goalkeepe Equa ions
Based on he da a analys specializing in oo ball wi h o e 25 yea s o expe ience
ac oss all le els o compe i ion, and because we aim o de elop a simple algo i hm, we ha e
decided ha he GK’s mo emen will ollow a ec ilinea ajec o y. Thus, he equa ions a e
as ollows:
ˆ
x( )=ˆ
x0+ˆ
0x
ˆ
y( )=ˆ
y0+ˆ
0y
ˆ
z( )=ˆ
z0+ˆ
0z
The alues
ˆ
x0
,
ˆy0
, and
ˆz0
ep esen he GK’s ini ial posi ion on he ield. Simila ly,
ˆ
0x
,
ˆ
0y, and ˆ
0zdeno e he GK’s ini ial eloci y a he momen he playe kicks he ball.
3.4. Build o xGOT
Th ough he cons uc ion o he di e en ial equa ions and he way we modeled he
ield, we a e able o calcula e he ime ins ances o bo h he ball and he GK. Howe e ,
o achie e ou objec i e, we need o de e mine when bo h coincide, which only occu s
unde speci ic condi ions. To simpli y his, we will conside only he ins ances when hey
lie wi hin he same
XZ
plane a a gi en alue o
y
. This allows us o ocus on sol ing he
ollowing p oblem:
ˆ
y0+ˆ
0 =y0+ 0y (sin ψsin α)
This leads o he ollowing:
=ˆ
y0−y0
0y(sin ψsin α)−ˆ
0y
In his manne , we can calcula e:
- The ini ial dis ance be ween he ball and he GK, deno ed as d0.
-
The dis ance be ween he ball and he GK when bo h sha e he same y-coo dina e
alue, deno ed as dgb.
-
The dis ance be ween he ini ial posi ion o he GK and he goal, wi h espec o he
y-coo dina e, deno ed as dg.

Big Da a Cogn. Compu . 2025,9, 64 6 o 13
Wi h hese de ini ions, we will de ine ou alue o xGOT as:
xGOT =min1, 
dg+dgb
d0
The alue ob ained om he xGOT me ic ep esen s he p obabili y o sco ing a goal
a e he sho has been aken and is on a ge . This alue is exp essed on a scale om 0 o 1,
whe e a highe xGOT alue indica es a g ea e p obabili y o sco ing, while a lowe xGOT
alue e lec s a educed likelihood o sco ing.
4. P ac ical Applica ions o he Model
This model e alua es he decision-making p ocesses o GKs by analyzing hei posi-
ioning and mo emen du ing c i ical game si ua ions. The amewo k combines spa ial,
empo al, and e en -based da a o quan i a i ely assess a GK’s pe o mance. Key a iables
include sho loca ion, ajec o y, ball speed, and he GK posi ioning a bo h he ini ia ion
and conclusion o he play. To u he cla i y hese ideas, he au ho s p esen wo examples
o sho s on goal o illus a e he inal xGOT alues ob ained (Table 2, Figu e 4). The i s
example (A) is a sho on goal ha is sa ed by he goalkeepe , esul ing in an xGOT alue
o 0.07. This indica es ha he sho has a 7% chance o becoming a goal, which may be
a ibu ed o he low quali y o he sho o he GK’s good pe o mance. The second example
(B) is a sho on goal ha is no sa ed by he goalkeepe , esul ing in an xGOT alue o 0.82.
This indica es ha he sho has an 82% chance o becoming a goal, which may be due o he
high quali y o he sho o he GK’s poo pe o mance.
Table 2. An example o calcula ing he xGOT alue o wo ypes o sho s on goal.
Example Ball’s Ini ial
Posi ion
Ball’s Final
Posi ion
GK’s Ini ial
Posi ion
GK’s Final
Posi ion
A. Sho on goal
s opped
x = 89.0 m
y = 34.7 m
z=0m
x = 104.5 m
y = 38.0 m
z = 0.7 m
x = 103.9 m
y = 38.9 m
z=0m
x = 104.5 m
y = 38.0 m
z = 0.7 m
d0: 15.48 m; dgb: 0 m; dg: 1.10 m
xGOT alue = 0.07 (7%)
B. Sho on goal
sco ed
x = 95.2 m
y = 39.1 m
z=0m
x = 105.0 m
y = 32.5 m
z = 1.1 m
x = 102.6 m
y = 37.0 m
z=0m
x = 102.6 m
y = 35.4 m
z=0m
d0: 7.69 m; dgb: 3.92 m; dg: 2.4 m
xGOT alue = 0.82 (82%)
Abb e ia ions: d
0
, he ini ial dis ance be ween he ball and he GK; d
gb
, he dis ance be ween he ball and he GK
a a gi en ins an ime; and d
g
, he dis ance be ween he ini ial posi ion o he GK and he goal, wi h espec o he
y-coo dina e.
Big Da a Cogn. Compu . 2025, 9, x FOR PEER REVIEW 6 o 14
In his manne , we can calcula e:
- The ini ial dis ance be ween he ball and he GK, deno ed as 𝑑.
- The dis ance be ween he ball and he GK when bo h sha e he same y-coo dina e
alue, deno ed as 𝑑.
- The dis ance be ween he ini ial posi ion o he GK and he goal, wi h espec o he
y-coo dina e, deno ed as 𝑑.
Wi h hese de ini ions, we will de ine ou alue o xGOT as:
𝑥𝐺𝑂𝑇min󰇧1,𝑑𝑑
𝑑󰇨
The alue ob ained om he xGOT me ic ep esen s he p obabili y o sco ing a goal
a e he sho has been aken and is on a ge . This alue is exp essed on a scale om 0 o
1, whe e a highe xGOT alue indica es a g ea e p obabili y o sco ing, while a lowe
xGOT alue e lec s a educed likelihood o sco ing.
4. P ac ical Applica ions o he Model
This model e alua es he decision-making p ocesses o GKs by analyzing hei posi-
ioning and mo emen du ing c i ical game si ua ions. The amewo k combines spa ial,
empo al, and e en -based da a o quan i a i ely assess a GK’s pe o mance. Key a ia-
bles include sho loca ion, ajec o y, ball speed, and he GK posi ioning a bo h he ini i-
a ion and conclusion o he play. To u he cla i y hese ideas, he au ho s p esen wo
examples o sho s on goal o illus a e he inal xGOT alues ob ained (Table 2, Figu e 4).
The i s example (A) is a sho on goal ha is sa ed by he goalkeepe , esul ing in an
xGOT alue o 0.07. This indica es ha he sho has a 7% chance o becoming a goal, which
may be a ibu ed o he low quali y o he sho o he GK’s good pe o mance. The second
example (B) is a sho on goal ha is no sa ed by he goalkeepe , esul ing in an xGOT
alue o 0.82. This indica es ha he sho has an 82% chance o becoming a goal, which
may be due o he high quali y o he sho o he GK’s poo pe o mance.
Figu e 4. Expec ed goal on a ge (xGOT) model: d
0
, he ini ial dis ance be ween he ball and he
GK; d
gb
, he dis ance be ween he ball and he GK a a gi en ins an ime; and d
g
, he dis ance be-
ween he ini ial posi ion o he GK and he goal, wi h espec o he y-axis.
Figu e 4. Expec ed goal on a ge (xGOT) model: d
0
, he ini ial dis ance be ween he ball and he GK;
d
gb
, he dis ance be ween he ball and he GK a a gi en ins an ime; and d
g
, he dis ance be ween
he ini ial posi ion o he GK and he goal, wi h espec o he y-axis.
Big Da a Cogn. Compu . 2025,9, 64 7 o 13
The model compu es a ‘decision e iciency’ sco e. Using his sco e, a hea map (Figu e 5)
can be gene a ed o assis coaches in analyzing op imal decisions ac oss a ious sho ypes.
Figu e 5illus a es he p obabili y o sco ing a goal as a unc ion o he ball’s inal posi ion
o he sho , based on he xGOT me ic, o bo h s aigh -line and pa abolic sho s on a ge .
The axis limi s co espond o he ac ual dimensions o a p o essional oo ball goal. The
colo ba on he igh indica es he p obabili y alues associa ed wi h di e en colo s,
whe e a goal p obabili y o 1.0 signi ies ha he sho on a ge always esul ed in a goal.
No eco ded sho s had a goal p obabili y o ze o. A eas o he goal ep esen ed by he
co esponding blue colo (goal p obabili y o 0.0) indica e egions whe e no sho s on a ge
en e ed. Figu e 5clea ly demons a es a posi i e ela ionship be ween he posi ion o he
sho on a ge , ega dless o he sho ype, and he p obabili y o sco ing. In his way,
he model p oduces ou pu s such as maps o he op imal posi ioning zones, mo emen
success p obabili ies, and compa a i e benchma ks agains pee s. This app oach enables
coaches o pinpoin a eas o imp o emen and suppo s GKs in e ining hei an icipa o y
decision-making. Ul ima ely, i enhances hei abili y o p e en goals h ough e ec i e
posi ioning. To u he cla i y hese concep s, he au ho s p esen ou examples o sho s on
goal wi h a ying inal posi ions o he ball o illus a e he xGOT alue spec um (Figu e 6).
Hea maps we e gene a ed om hese examples o isualize high- isk goal zones ( ed a eas),
emphasizing he egions whe e GKs should posi ion hemsel es o minimize he xGOT
(blue a eas). (A) is a hea map o a oo sho wi h he GK posi ioned a he cen e o he goal
(ball posi ion: x = 34 m, y = 11 m, z = 0 m); (B) is a hea map o a oo sho wi h he GK
mo ing o he le (ball posi ion: x = 34 m, y = 11 m, z = 0 m); (C) is a hea map o a oo sho
on a ge wi h he GK jumping o he igh (ball posi ion: x = 34 m, y = 11 m, z = 0 m); and
(D) is a hea map o a heade sho om a he le side o he goal (ball posi ion: x = 23 m,
y = 5.50 m, z = 1.80 m). To simpli y he examples, all sho s we e conside ed o ha e been
conceded by he GK.
Big Da a Cogn. Compu . 2025, 9, x FOR PEER REVIEW 8 o 14
Figu e 5. This hea map illus a es he x- and y-coo dina es o he goal and he p obabili y o sco ing
a goal a e he sho has been aken and is on a ge . The x-axis ep esen s he wid h o he goal (7.32
m), while he y-axis ep esen s he heigh o he goal (2.44 m), co esponding o he dimensions o a
eal oo ball goal. The GK is assumed o be posi ioned a he cen e o he goal. The colo spec um
e lec s he xGOT alues o he sho s on a ge ha esul ed in a goal, anging om ed o blue.
A eas whe e he ball is mo e likely o a i e a e shown in ed (indica ing high xGOT alues),
whe eas a eas whe e he ball is less likely o a i e a e depic ed in blue (indica ing low xGOT al-
ues).
Taking hese calcula ions in o accoun , he main p ac ical applica ion o his model is
ha GK coaches and analys s could use i o assess he s eng hs and weaknesses o bo h
hei own eam’s GKs and opposing GKs, allowing o a mo e a ge ed and pe sonalized
app oach o aining and ac ics. By unde s anding speci ic GKs’ endencies, such as p e-
e ed posi ioning, eac ion imes, and a eas o ulne abili y, coaches can ailo hei ain-
ing sessions o add ess weaknesses and build on s eng hs. Addi ionally, his insigh helps
in de eloping ac ical plans, such as de e mining which a eas o he goal o a ge o when
o adjus shoo ing echniques o exploi an opponen ’s GK beha io .
Mo eo e , his model goes beyond jus ac ical p epa a ion. I can be ex ended o
enhance playe decision-making du ing c i ical momen s in a ma ch. Fo example, playe s
can be ained o ecognize when a GK is mo e likely o make a sa e based on he GK’s
habi s and posi ioning endencies, allowing hem o make be e decisions abou sho
placemen in eal- ime. This in eg a ion o GK analysis in o playe decision-making os-
e s a mo e s a egic app oach o bo h a acking and de ending, ul ima ely imp o ing he
o e all eam pe o mance. By le e aging hese da a, eams gain a compe i i e edge no
only in game p epa a ion bu also in adap ing o in-ma ch dynamics o mo e effec i e
play.
Figu e 5. This hea map illus a es he x- and y-coo dina es o he goal and he p obabili y o sco ing a
goal a e he sho has been aken and is on a ge . The x-axis ep esen s he wid h o he goal (7.32 m),
while he y-axis ep esen s he heigh o he goal (2.44 m), co esponding o he dimensions o a
eal oo ball goal. The GK is assumed o be posi ioned a he cen e o he goal. The colo spec um
e lec s he xGOT alues o he sho s on a ge ha esul ed in a goal, anging om ed o blue. A eas
whe e he ball is mo e likely o a i e a e shown in ed (indica ing high xGOT alues), whe eas a eas
whe e he ball is less likely o a i e a e depic ed in blue (indica ing low xGOT alues).
Big Da a Cogn. Compu . 2025,9, 64 8 o 13
Big Da a Cogn. Compu . 2025, 9, x FOR PEER REVIEW 9 o 14
Figu e 6. Hea map o xGOT based on he ball’s inal posi ion. Hea map illus a ing diffe en exam-
ples o sho s on goal: (A) A oo sho on a ge wi h he GK posi ioned a he cen e o he goal (ball
posi ion: x = 34 m, y = 11 m, z = 0 m); (B) a oo sho on a ge wi h he GK mo ing o he le (ball
posi ion: x = 34 m, y = 11 m, z = 0 m); (C) a oo sho on a ge wi h he GK jumping o he igh (ball
posi ion: x = 34 m, y = 11 m, z = 0 m); and (D) a heade sho om he le side o he goal (ball posi ion:
x = 23 m, y = 5.50 m, z = 1.80 m). To simpli y he examples, all sho s we e conside ed o ha e been
conceded by he GK.
Figu e 6. Hea map o xGOT based on he ball’s inal posi ion. Hea map illus a ing di e en examples
o sho s on goal: (A) A oo sho on a ge wi h he GK posi ioned a he cen e o he goal (ball
posi ion: x = 34 m, y = 11 m, z = 0 m); (B) a oo sho on a ge wi h he GK mo ing o he le (ball
posi ion: x = 34 m, y = 11 m, z = 0 m); (C) a oo sho on a ge wi h he GK jumping o he igh
(ball posi ion: x = 34 m, y = 11 m, z = 0 m); and (D) a heade sho om he le side o he goal (ball
posi ion: x = 23 m, y = 5.50 m, z = 1.80 m). To simpli y he examples, all sho s we e conside ed o
ha e been conceded by he GK.
Taking hese calcula ions in o accoun , he main p ac ical applica ion o his model is
ha GK coaches and analys s could use i o assess he s eng hs and weaknesses o bo h
hei own eam’s GKs and opposing GKs, allowing o a mo e a ge ed and pe sonalized
Big Da a Cogn. Compu . 2025,9, 64 9 o 13
app oach o aining and ac ics. By unde s anding speci ic GKs’ endencies, such as
p e e ed posi ioning, eac ion imes, and a eas o ulne abili y, coaches can ailo hei
aining sessions o add ess weaknesses and build on s eng hs. Addi ionally, his insigh
helps in de eloping ac ical plans, such as de e mining which a eas o he goal o a ge o
when o adjus shoo ing echniques o exploi an opponen ’s GK beha io .
Mo eo e , his model goes beyond jus ac ical p epa a ion. I can be ex ended o
enhance playe decision-making du ing c i ical momen s in a ma ch. Fo example, playe s
can be ained o ecognize when a GK is mo e likely o make a sa e based on he GK’s
habi s and posi ioning endencies, allowing hem o make be e decisions abou sho
placemen in eal- ime. This in eg a ion o GK analysis in o playe decision-making os e s
a mo e s a egic app oach o bo h a acking and de ending, ul ima ely imp o ing he
o e all eam pe o mance. By le e aging hese da a, eams gain a compe i i e edge no
only in game p epa a ion bu also in adap ing o in-ma ch dynamics o mo e e ec i e play.
S a is ical E alua ion o he Sho P edic ion Model
The sho p edic ion model es ima es he likelihood o a sho esul ing in a goal based on
spa ial and empo al a iables, including ball ajec o y, ball eloci y, and GK posi ioning.
The p ima y objec i e o his algo i hm was o e alua e GK pe o mance and decision-
making. To assess i s e icacy, we employed k- old c oss- alida ion, achie ing an A ea
Unde he Cu e–Recei e Ope a ing Cha ac e is ic (AUC-ROC) sco e o 0.67 and an
accu acy o 85%. These me ics we e compu ed using he FIFA Wo ld Cup 2022 da abase
desc ibed in Sec ion 2. We used Py hon 3.11.9 code o ead he da a and classi y sho s as
ei he sa es o goals. To e alua e all he p ope ies o ou model, we used he R lib a y
wi hin Py hon. The model alida ion is p esen ed in Table 3below, whe e we compa e he
alues o ou model wi h hose o he STATSBOMB model.
Table 3. Compa ison be ween own model and ou pu om STATSBOMB model.
Own Model
(Mean ±DS)
STATSBOMB Model
(Mean ±DS) pValue d Cohen
xGOT ( o al sho s on
goal) 0.38 ±0.12 0.33 ±0.09 0.05 -
xGOT (goal sho s) 0.73 ±0.08 0.55 ±0.09 <0.001 2.11
xGOT (sho s sa ed) 0.19 ±0.03 0.20 ±0.04 0.48 -
Abb e ia ions: xGOT, expec ed goals on a ge .
Building on his amewo k [
22
], decision-making can be modeled as a game heo y
p oblem, whe e he GK mus e alua e hei posi ioning wi hin he penal y a ea o iden i y
he mos ad an ageous posi ion. As illus a ed in he hea map abo e, by knowing he
ini ial posi ions o he ball and he GK, simula ions can gene a e he op imal a eas o he
GK o posi ion hemsel es p io o he sho . This ensu es ha , ega dless o ball speed, he
GK minimizes he equi ed mo emen dis ance o make a sa e.
5. Compa a ion o xGOT Values Be ween Ou Own Model and Ou pu s
o STATSBOMB Model
Table 3p esen s he esul s o a compa a i e analysis o he xGOT da a gene a ed
using ou p oposed model and he model de eloped by STATSBOMB. Fo his analysis, he
ollowing a iables we e conside ed: he xGOT o o al sho s on goal, he xGOT o sho s
esul ing in goals, and he xGOT o sho s sa ed. A desc ip i e analysis was conduc ed by
calcula ing he mean and s anda d de ia ion, while a compa a i e analysis was pe o med
using an independen S uden ’s - es . S a is ical signi icance was es ablished a p< 0.05.
Addi ionally, e ec sizes (Cohen’s d) we e calcula ed o de e mine he p ac ical signi icance
o he indings, wi h h esholds de ined as small (<0.2), medium (0.5), and la ge (>0.8). All