# Crediting football players for creating dangerous actions in an unbiased
way: the generation of threat (GoT) indices.

Ali Baouan Centre de Mathématiques Appliquées, Ecole Polytechnique. Email:
<EMAIL_ADDRESS>Sebastien Coustou Parma Calcio 1913 Performance
and Analytics. Email<EMAIL_ADDRESS>Mathieu Lacome Parma
Calcio 1913 Performance and Analytics. Email<EMAIL_ADDRESS>Sergio Pulido Université Paris–Saclay, CNRS, ENSIIE, Univ Évry, LaMME. Email:
<EMAIL_ADDRESS>Mathieu Rosenbaum Centre de Mathématiques
Appliquées, Ecole Polytechnique. Email<EMAIL_ADDRESS>
###### Abstract

We introduce an innovative methodology to identify football players at the
origin of threatening actions in a team. In our framework, a threat is defined
as entering the opposing team’s danger area. We investigate the timing of
threat events and ball touches of players, and capture their correlation using
Hawkes processes. Our model-based approach allows us to evaluate a player’s
ability to create danger both directly and through interactions with
teammates. We define a new index, called Generation of Threat (GoT), that
measures in an unbiased way the contribution of a player to threat generation.
For illustration, we present a detailed analysis of Chelsea’s 2016-2017
season, with a standout performance from Eden Hazard. We are able to credit
each player for his involvement in danger creation and determine the main
circuits leading to threat. In the same spirit, we investigate the danger
generation process of Stade Rennais in the 2021-2022 season. Furthermore, we
establish a comprehensive ranking of Ligue 1 players based on their generated
threat in the 2021-2022 season. Our analysis reveals surprising results, with
players such as Jason Berthomier, Moses Simon and Frederic Guilbert among the
top performers in the GoT rankings. We also present a ranking of Ligue 1
central defenders in terms of generation of threat and confirm the great
performance of some center-back pairs, such as Nayef Aguerd and Warmed Omari.

##  1 Introduction

Which player should be credited for a successful action or sequence in a
football match? In the case of a goal, the striker obviously plays an
important role. However, we all have in mind goals where the striker just
needs to push the ball after a great assist. In that case, the passer is
certainly the most important player involved. Some argue that the second-to-
last pass is actually the most crucial component as it is often this pass that
creates disequilibrium. Sometimes, we even see a clearance by a goalkeeper
being at the origin of a dangerous situation.

In this work, our goal is to build a quantitative and unbiased methodology
enabling us to assess the importance of a player in the generation of
dangerous actions. By a threat, we simply mean a situation where a player of
the team of interest gets the ball in the danger area of the opposing team.
The danger area is defined as a rectangular region around the opponent’s goal
where the likelihood of scoring from a shot is high. To achieve our objective,
we need to model interactions between players, taking into account past events
in the game accurately. This is because we want, for example, to be able to
credit a defender for a great pass that leads to a dangerous situation after
several ball touches following the initial pass. Therefore, at the timestamp
where the action is considered dangerous (in our case when the ball reaches
the danger area), we must "remember" the original pass of the defender.

Thus, at a given time $t$, we want to draw links between past events in the
game and its future. With this objective in mind, simply relying on the
current state of the game (players and ball’s positions) as the information
set is not enough for modeling the game accurately. It is important to
consider the dynamics that occurred prior to time $t$. This is in contrast to
the so-called Markovian approach where one summarizes information obtained
from the beginning of the game until time $t$ by the state of the game at time
$t$. The Markovian setting is in fact underlying some very relevant and
successful metrics introduced recently such as the expected goals Green,
(2012) and expected assists Whitmore, (2021). For example, the expected goal
estimates the probability that a shot results in a goal based on factors such
as the distance to the goal and the angle of the shot, both attributes of the
game state at time $t$. The Markov assumption is in that case natural as these
features give a reasonable estimate of the quality of the chance. Similarly,
the expected assists aim at measuring the probability that a pass leads to a
goal, by looking at a different subset of game state features, such as the
type of the pass and the coordinates of the target. What these two approaches
have in common is that given time $t$ they define a value for an action (pass
or shot), that is determined by the game state at time $t$ only and does not
look at the past patterns of play. In the same spirit, the expected threat
introduced in Singh, (2018) assigns a value to each game state depending only
on the position of the ball. This value combines the possibilities of a direct
shot or a pass to another position in quantifying the expected number of
goals.

To account for the effect of past events in the future dynamics of a game, we
introduce Hawkes processes Hawkes, 1971a ; Hawkes, 1971b  to reproduce
interactions between players. Hawkes processes are stochastic models used to
model sequences of random events. They are widely used in various fields such
as earthquake modeling Adamopoulos, (1976); Ogata, (1988), neuroscience
Lambert et al., (2018); Bonnet et al., 2022a  and finance Jaisson and
Rosenbaum, (2015). In our case, the events are the times when players touch
the ball. Specifically, we implement a Hawkes process with 11 components
(number of players in the team), with component $i$ corresponding to player
$i$ of the team of interest. The value of this component at time $t$ is simply
the number of times player $i$ has touched the ball from the beginning of the
game to time $t$. At each time the player touches the ball, his corresponding
component increases by one. The innovation here is that we collect information
from these timestamps and their correlations from one player to the other
teammates.

The specificity of Hawkes processes is that at time $t$, the probability that
player $i$ gets the ball shortly after $t$ depends on which players had
possession of the ball before $t$ and how long ago they had it. The impact on
this probability of a player touching the ball a long time before $t$ is
negligible compared to a player who had possession right before $t$. The
ability to reproduce the decaying impact of events with time is a particularly
useful property of Hawkes processes in our context. For instance, let us
consider a central defender. At time $t$, the probability that he gets the
ball in the near future should be high if, in the recent past (last few
seconds), he already touched the ball and/or another central defender did. On
the contrary, if the forward players have held the ball for the past minute,
this probability should be low.

Then, we add a twelfth component to our Hawkes process that we call threat.
The value of the threat component at time $t$ is simply the number of times
the ball has reached the danger area of the opposing team between the
beginning of the game and time $t$. Treating this component as part of our
Hawkes process, we are able to model the influence of each player in the
generation of threat.

Calibrating our model allows us to assess the contribution of each player of a
team to the creation of dangerous situations. We are therefore able to
investigate carefully the subtle dynamics and connections leading to ominous
situations. In particular, we can emphasize the crucial role of certain
players that are not spotted by other statistics. Note that our calibration
requires the analysis of a data set of at least ten games. So we are not
evaluating each action occurring in a game but rather the global performance
of players in terms of threat generation over a sequence of games.

More precisely, the structure of Hawkes processes allows us to define the
Generation of Threat (GoT) indices to objectively evaluate a player’s
involvement in the creation of threats over a considered series of games.
These metrics quantify the expected number of dangerous situations for which a
player can be credited. The direct generation of threat indices GoTd and
GoT${}^{d}_{90}$ measure the number of threats the player is directly
responsible for generating per touch of the ball and per 90 minutes,
respectively. Directly generating a threat can be viewed as being the last
link in the chain of events leading to it. On the other hand, the indirect
generation of threat indices GoTi and GoT${}^{i}_{90}$ measure the indirect
contribution per touch and per 90 minutes, respectively, adding the danger
created via the interactions with other players too. In this case, we count
all the instances where the player participates in the chain of events leading
to the dangerous situation. As an application, we use the GoT indices to rank
the Ligue 1 players in the 2021-2022 season. Not surprisingly, the top
positions are dominated by established offensive players. However, we also
identify some surprising picks, including Jason Berthomier, Moses Simon and
Frederic Guilbert, who rank among the top twenty players. We also compare the
performance of the Ligue 1 central defenders in terms of GoT${}^{i}_{90}$.
Naturally, defenders from Paris Saint-Germain stand out and benefit from the
offensive performance of their forwards. However, we also identify other
excellent center-back pairs such as Nayef Aguerd and Warmed Omari from Stade
Rennais, and Facundo Medina and Jonathan Gradit from Lens. Moreover, our
approach allows us to rate these players based on their performance in
specific positions in a formation, providing a tool to identify the optimal
position for each player.

Our approach has the property of being easily interpretable using the
immigration-birth representation of linear Hawkes processes, see Hawkes and
Oakes, (1974). This representation induces a notion of causality between
events and allows us to visualize the interactions between different event
types in a graph. All player touches can be viewed as individuals in a
population, and each individual independently generates offsprings, that are
threat events or ball touches of the same player or other players. In
particular, this enables us to effectively interpret the estimated GoT metrics
as a measure of the causal relationship between the player’s touch and
subsequent threat events. Furthermore, we can construct interaction networks
of football teams and graphically analyze a team’s in-game dynamics and danger
creation circuits. We apply this approach to investigate games from Chelsea in
the 2016-2017 season and Stade Rennais in the 2021-2022 season. We are able to
effectively capture the main threat creation circuits that the opponent should
try to control. Identifying specific patterns and evaluating the ability of
players to create threat with our methodology paves the way to more informed
decisions about tactics.

The article is organized as follows. In Section 2, we provide an overview of
Hawkes processes and recall the results that are useful for our football
application. Section 3 describes the event-based data we have in hand and how
it is processed. Furthermore, we present the interpretation of the estimated
parameters in the context of football and define the Generation of Threat
(GoT) metrics. In Section 4, we briefly describe the maximum likelihood
estimation methodology. We also conduct a study on simulated data to measure
estimation accuracy that can be expected on real datasets depending on the
amount of available data. We find that reliable estimation can be obtained
from 600 minutes of football data. Section 5 presents the results of our
analysis on a collection of Chelsea games in the 2016-2017 season. In Section
6, we establish a ranking of Ligue 1 players in the 2021-2022 season based on
their GoT indices. Finally, in the appendix, we present the analysis of the
Stade Rennais games in the 2021-2022 Ligue 1 season.

##  2 Hawkes processes

This section provides a short overview of Hawkes processes. It includes
necessary definitions and theoretical results for a better understanding of
the subsequent analysis of football dynamics.

As mentioned in the introduction, Hawkes processes are a class of multivariate
point processes introduced in Hawkes, 1971a . If we consider a vector
$N(t)=(N_{i}(t))_{i\in\\{1,\dots,d\\}}$, where $N_{i}(t)$ denotes the number
of events for the $i$-th component between $0$ and $t$, the associated
intensity process can essentially be defined as:

$\lambda_{i}\left(t\right):=\lim_{h\rightarrow
0^{+}}\frac{\mathbb{P}(N_{i}(t+h)-N_{i}(t)=1|\mathcal{F}_{t})}{h}.$

Here, $\mathcal{F}_{t}$ is the filtration generated by $\\{N_{s},\ s<t\\}$,
that is the information set available at time $t$. The intensity of a counting
process determines the rate at which new jumps occur based on past events, see
Brémaud, (1981) for a more rigorous definition. In the case of Hawkes
processes, the intensity is a linear combination of past jump times.

######  Definition 2.1 (Hawkes process).

A d-variate Hawkes process is a counting-process $N(t)\in\mathbb{R}^{d}$ whose
$i$-th component is determined by its intensity of the form:

$\lambda_{i}(t)=\mu_{i}+\sum_{j=1}^{d}\sum_{t_{k}^{(j)}<t}\phi_{i,j}(t-t_{k}^{(j)}),$

where the $\left(t_{k}^{(j)}\right)_{k\geq 1}$ are the times of events for
dimension $j$ for $j=1,\dots,d$. $\mu_{i}\in\mathbb{R}^{+}$ is a constant
baseline intensity and $\phi_{i,j}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is
a non-negative kernel. We can write the expression for the intensity in the
vectorial form:

$\lambda(t)=\mu+\int_{0}^{t}\phi(t-s)dN(s),$

with $\mu\in\mathbb{R}^{+d}$ and $\phi=\left\\{\phi_{i,j}\right\\}_{0\leq
i,j\leq d}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{d\times d}$ a non-negative
matrix-valued kernel.

The underlying idea behind Hawkes processes is that a constant intensity $\mu$
generates the initial batch of jumps across all dimensions. These jumps are
random but the rate of their occurrence remains constant over time. Then, each
jump increases the intensity in the near future; therefore, exciting new
jumps, that in turn trigger other jumps. This leads to a chain reaction called
the self-excitation property of Hawkes processes.

We need to impose conditions for this system to be stable. These conditions
can be stated in terms of the branching matrix defined below:

######  Definition 2.2 (Branching matrix, stability).

The branching matrix of a Hawkes process is defined as,

$K=\int_{0}^{\infty}\phi(t)dt=\left\\{\int_{0}^{\infty}\phi_{i,j}(t)dt\right\\}_{1\leq
i,j\leq d}.$

Moreover, a Hawkes process is said to be stable if
$\int_{0}^{\infty}\phi_{i,j}(t)dt<\infty$ for all i,j and if the spectral
radius $\rho(K)$ of the branching matrix satisfies:

$\rho(K)<1.$

See Jaisson and Rosenbaum, (2015) for more details.

#### Immigration-birth representation:

Introduced in Hawkes and Oakes, (1974), the immigration-birth representation
provides an intuitive way to understand linear Hawkes processes. Let us
consider a stable $d$-dimensional Hawkes process $N(t)$ with a baseline
intensity $\mu$ and a kernel $\phi$. The law of such point process can be
described through a population approach. Essentially, we consider a population
where immigrants of $d$ types arrive at random times. Each of them gives birth
to children of all types. Then the children, grand-children, grand-grand-
children etc. also give birth to children of all types. More precisely, the
dynamic is constructed as follows:

  * •

For $j=1,2,\dots,d$ , we consider an instance of a Poisson process with rate
$\mu_{j}$, with its elements called immigrants of type $j$. Generation $0$
consists of the immigrants;

  * •

Recursively, given generations $0,1,\dots,n$, each individual born at time $s$
of type $j$ in generation $n$ generates its offspring of type $i$ as an
independent instance of a non-homogeneous Poisson process with rate
$\lambda_{t}^{s,n}:=\phi_{i,j}(t-s)$ for $t\geq s$. The union of these
offspring of all types constitutes generation $n+1$.

  * •

The point process is then defined as the union of all generations.

The resulting process has the law of a Hawkes process. In this representation,
stability means each individual has less than one child on average in the case
$d=1$, which ensures some good mathematical properties for the process. From
now on, we assume that all considered Hawkes processes are stable.
Additionally, under this construction,
$K_{i,j}=\int_{0}^{\infty}\phi_{i,j}(t)dt$ can be interpreted as the expected
number of direct children of type $i$ of an individual of type $j$. The
following proposition provides a closed-form formula for the expected number
of descendants of a single individual. It includes both immediate descendants
and those from later generations. This result is derived similarly to the one-
dimensional case in Jaisson and Rosenbaum, (2015), and allows us to quantify
the average number of events originating from each jump from each dimension.

######  Proposition 2.1.

The entry $i,j$ of the matrix $K(I-K)^{-1}$ gives the expected number of
descendants of type $i$ generated by an individual of type $j$.

In this work, we estimate a branching matrix from football event-based data.
We use the parametric class of exponential kernels in our estimation
methodology.

######  Definition 2.3 (Exponential kernels).

The exponential kernel is defined as

$\phi_{i,j}(t)=\alpha_{i,j}e^{-\beta_{i,j}t}1_{t\geq 0},$

where $\alpha_{i,j}$, $\beta_{i,j}$ are nonnegative real numbers.

Exponential kernels are particularly nice from a computational viewpoint in
estimation. Additionally, their parameters are easy to interpret. In fact, the
branching matrix in this case is simply given by
$K=(\frac{\alpha_{i,j}}{\beta_{i,j}})_{i,j}$ and the decay parameter
$\beta_{i,j}$ indicates the speed at which cross excitation decreases.

##  3 Event-based football data

###  3.1 Description of the data

We use the F24 files provided by Stats-
perform111https://www.statsperform.com/. Each file gives comprehensive
information about a football match. Information includes the formation of each
team and the position of each player on the pitch. Additionally, it lists all
events occurring with the ball within the game specifying the player involved,
the event type, the coordinates on the pitch, and the timestamp for each
action.

In the Stats-perform classification system, each position on the pitch is
assigned a number $p$ in $\\{1,\dots,11\\}$ for each formation. The
distribution of these positions for various formations is shown in Figure 1.
Our study aims at understanding the impact of ball touches in each position
$p$ in $\\{1,\dots,11\\}$ on a team’s offensive performance. To ensure
homogeneity, the analysis is conducted only on games where each position has
the same role. For this purpose, we group formations in clusters of similar
shapes as those presented below and only use matches from the most commonly
used cluster for each team:

  * •

Cluster 1: 433, 4141, 4231, 4321.

  * •

Cluster 2: 442, 41212, 451, 4411, 4222.

  * •

Cluster 3: 532, 352, 31312, 3511, 3412.

  * •

Cluster 4: 343, 541, 3421.

(a) 433 Formation.

(b) 442 Formation.

(c) 532 Formation.

(d) 343 Formation.

Figure 1: The number associated with each position for each group of
formations.

###  3.2 Processing of the data for Hawkes inference

We study our event-based data using Hawkes processes. Doing so, we can gain
insights from timestamps of events and information about the spatial
coordinates of the ball. For a given team and a list of its games in the same
formation cluster, we build a 12-dimensional point process for each game. Each
dimension $p\in\\{1,\dots,11\\}$ records the timestamps of ball touches by the
player occupying position $p$, regardless of his identity. The twelfth
dimension represents the threat state and is triggered every time there is a
ball touch by a player from the considered team in the danger area of the
opponent. The danger area is defined as a box around the opposing goal
covering 50% of the width of the pitch and 25% of its length, as illustrated
in Figure 2. When a player has possession of the ball in this region, the
probability of a shot occurring is high, see Singh, (2018) for an estimate of
the shot probability at each location on the pitch. Compared to the penalty
surface, the danger area is slightly closer to the midfielders and defenders,
enabling us to capture more threat events generated by these positions.

Figure 2: Representation of the danger area.

The following rules are applied when constructing the process:

  1. 1.

Every time a player in the considered team touches the ball, there is a jump
in the dimension $p\in\\{1,\dots,11\\}$ associated with his position.

  2. 2.

Every time a player in the considered team touches the ball inside the
opposing threat area, there is a jump in the twelfth dimension at the
corresponding timestamp. In this case, no jump is recorded in the component
associated with the player.

  3. 3.

Once a threat state is triggered, no jumps or time are recorded until the ball
exits the danger area. We resume counting the jumps when the ball is outside
the danger area by at least two meters.

  4. 4.

When the ball is lost (when there is an event where the opposing team has the
ball), the time and events are not recorded until the ball is won again. Upon
regaining possession, we resume recording the events in our point process by
adding a random duration, with an average of twelve seconds, generated from
the sum of two exponential distributions of parameter six.

  5. 5.

We exclude crossing events coming from a free kick or a corner.

Rule 3 is considered to avoid consecutive threat states. We are not interested
in the auto-exciting property of the threat events. Therefore, we stop
recording once a threat state is achieved and only resume when the team is
outside the opposing surface by at least two meters. In Rule 4, we want to
avoid having large durations where no event occurs. This is the case every
time the considered team loses the ball to the opposition. Thus, the
possession times of the opponent are compressed into an average of twelve
seconds. The choice of the twelve seconds threshold is based on the average
duration between events to which we add another exponential random variable as
a penalization for losing the ball. The constructed point process considers
possession stretches of the team to be uninterrupted. Rule 5 is implemented
because the crossing events are highly correlated with threat events. In
particular, the designated set piece taker of each team is naturally
responsible for more threats. Therefore, we choose to discard these events to
remove bias from our measure of danger creation and ensure fair player
comparisons.

Given a collection of games of a team, the point processes built from each
game are assembled into one process. An example of the resulting point process
is shown in Figure 3. We use information on the timestamps and spatial
coordinates on the field to define the threat state. The aim is to extract the
causal relationship between player touches. We are interested in identifying
the positions where a ball touch is directly correlated to a future jump in
the twelfth dimension, which represents a threat. We also want to measure the
indirect contribution of a player to the generation of threat through his
interaction with other players.

Figure 3: Example of constructed point process.

######  Remark 3.1.

In the following, we aim at evaluating a player’s performance when he plays in
a specific position. To achieve this, we only consider sequences of games
where the player in question is playing in that position. We record ball
touches in the other positions regardless of the identity of the player
occupying them. In Section 5 and Appendix A where we analyze the interactions
between the starting eleven players in given teams, we only record sequences
of games where the same eleven players play in their respective positions. The
way we deal with substitutions is detailed for each case in Sections 5 and 6.

######  Remark 3.2 (A different twelfth dimension).

In this work, we have incorporated a twelfth dimension that tracks the
instances of entering the opposing danger area. This is done because we want
to identify the players who are responsible for creating the threat events.
Our approach can be extended for various analyses by selecting an alternative
twelfth state. For example, we can choose to record the timestamps of ball
losses in the twelfth dimension instead of threats. This would enable us to
identify the players who are most accountable for losing possession and
measure the correlation between their touches and subsequent turnovers.

###  3.3 Generation of Threat (GoT) indices

The immigration-birth representation of Hawkes processes explained in Section
2 allows us to establish connections between the events in a football match.
Essentially, each ball touch or threat event can be seen as an individual in a
population, that generates first-generation children of various types - ball
touches from other players and threat situations. These offspring, in turn,
generate additional ball touches or threat events etc. When we say that an
event generates a ball touch or a dangerous situation, we mean that it is
responsible for its occurrence. This is a subtle definition because being
responsible for an action does not necessarily mean providing the pass that
leads to it. In some instances, the second-to-last pass is the most crucial
step in creating the dangerous situation. There may even be several events
between the generating ball touch and the dangerous action. Our approach
eliminates these "noisy" in-between events and associates events through
parent-child connections. Hawkes processes impute the responsibility of
generating a threat to the most likely parent event, even if it occurred prior
to other ball touches. In particular, they allow us to quantify the average
number of dangerous actions that can be attributed to a given player.

Using this population representation, we define the following GoT indices to
assess the ability of a player to generate threat when he plays in a given
position. The first two indices evaluate the impact of one touch of the player
whereas the latter two measure the impact of the player’s touches over 90
minutes.

#### Direct GoT per touch (GoTd):

A ball touch from the player in position $p$ generates first-generation
children of type threat. We refer to these instances as the direct threat
events generated by the player touch. We define GoTd as the average number of
these threat events that occur because of one touch from player $p$. This
metric describes the intrinsic ability of the player to create dangerous
situations. It can be calculated through the estimated branching matrix:

$\textup{GoT}^{d}(p)=K_{12,p}.$

#### Indirect GoT per touch (GoTi):

A ball touch from a given player can be directly responsible for a threat
event, but can also generate other ball touches that then generate danger. To
quantify the total impact of a single player touch on the danger creation
process, we use Proposition 2.1 and consider the matrix

$M=K(I-K)^{-1}.$

The coefficient $M_{12,p}$ represents the expected number of threat events
where the ball touch from the player $p$ originates the chain of events
leading to it. This includes the threat directly generated but also the one
resulting from a sequence of other player touches. The difference with the
GoTd index is that we credit the player touch for being at the root of the
generation process and not for the crucial creative step.

$\textup{GoT}^{i}(p)=M_{12,p}.$

#### Direct GoT per 90 minutes (GoT${}^{d}_{90}$):

We may want to account for the involvement in the game of a given player by
normalizing GoTd by his expected number of touches. We define the direct GoT
per 90 minutes as the expected number of dangerous actions over 90
minutes222Note that here 90 minutes corresponds to 90 minutes of data after
processing which does not translate to 90 minutes in a football match. This is
notably because of the concatenation of sequences of possession explained in
Section 3.2. for which we credit the player:

$\textup{GoT}_{90}^{d}=\mathbf{E}\left(N_{p}(T)\right)\times\textup{GoT}^{d}(p),$

where $T=90\textup{ minutes}$. The expected number of touches vector can be
approximated thanks to the law of large numbers:

$\mathbf{E}\left(N(T)\right)\approx(I-K)^{-1}\mu T.$

#### Indirect GoT per 90 minutes (GoT${}^{i}_{90}$):

This index measures the expected number of threats over 90 minutes where a
given player is involved in the building circuit. We define the indirect GoT
per 90 minutes as the average number of threat events subtracted by the
average number of threat events if the considered player is removed from the
pitch. The GoT${}^{i}_{90}$ index is therefore calculated as follows:

$\textup{GoT}_{90}^{i}=\mathbf{E}\left(N_{12}(T,K,\mu)-N_{12}(T,K^{(-p)},\mu^{(-p)})\right),$

where $K^{(-p)}$ is defined as the matrix K where the $p^{th}$ row and
$p^{th}$ column are set to zero. Likewise, $\mu^{(-p)}$ is defined as the
vector $\mu$ where the $p^{th}$ coordinate is set to 0. The expected number of
threats can be approximated using the branching matrix and the baseline
intensity $\mu$:

$\mathbf{E}\left(N_{12}(T,K,\mu)\right)\approx\left((I-K)^{-1}\mu
T\right)_{12}.$

######  Remark 3.3.

Calculating the GoT${}^{i}_{90}$ by multiplying the GoTi index by the average
number of ball touches of the player would overestimate the player’s
involvement in danger creation. In fact, we would count multiple times the
circuits leading to threat where the player touches the ball more than once.

Additionally, a ball touch from a player can also be responsible for
generating ball touches from other players or himself. In this case as well,
this is not necessarily achieved through a direct pass. Hawkes processes allow
us to estimate the expected number of these generated ball touches. Similar to
the GoTd index definition, the branching coefficient $K_{p_{1},p_{2}}$
indicates the expected number of touches of player $p_{1}$ that happen because
a given ball touch from player $p_{2}$ occurred before. The graphical
representation of these interaction indices through a graph helps us gain a
better understanding of the danger creation process. In particular, it allows
us to identify the patterns of play that end in a threat.

##  4 Maximum Likelihood estimation

###  4.1 Likelihood of Hawkes process

This section describes briefly parameters estimation for multivariate Hawkes
processes, see Ogata et al., (1978); Bonnet et al., 2022b . Consider a
$d$-dimensional point process $(N(t))$ on $[0,T]$ with intensity of the form

$\lambda_{i}(t,\theta^{*})=\mu_{i}^{*}+\sum_{j=1}^{d}\sum_{t_{k}^{(j)}<t}\alpha_{i,j}^{*}\exp\left(-\beta_{i,j}^{*}(t-t_{k}^{(j)})\right),$

where $\theta^{*}=(\mu^{*},\alpha^{*},\beta^{*})$ are some unknown parameters.
Given fixed parameters $\theta=(\mu,\alpha,\beta)$ and a realization of the
Hawkes process, the log-likelihood is calculated as follows:

$\ell(\theta)=\sum_{i=1}^{d}\left(-\int_{0}^{T}\lambda_{i}(s,\theta)ds+\sum_{t_{k}^{(j)}<T}\log\left(\lambda_{i}(t_{k}^{(i)},\theta)\right)\right).$
(1)

The maximum likelihood estimator is the parameter that maximizes the above
function. It can be observed from Equation (1) that the likelihood can be
separated into $d$ distinct subfunctions, each dependent on the parameters
$\mu_{i}$ and $(\alpha_{i,j},\beta_{i,j})_{j=1,\dots,d}$ for $i$ in
$\\{1,\dots,d\\}$. As a result, the optimization can be performed separately
$d$ different times to estimate each subset of parameters. It is shown in
Ogata et al., (1978) that this estimator is consistent. Additionally, the log-
likelihood can be simplified in the case of exponential kernels and computed
in time complexity of $\mathcal{O}\left(d^{2}N(T)\right)$, see Ogata, (1981).
For example, for $d=1$ and $T=t_{n}$, the likelihood is given by :

$\ell(\theta)=\sum\limits_{i=1}^{n}\log\left(\mu+\alpha R(i)\right)-\mu
t_{n}+\frac{\alpha}{\beta}\sum_{i=1}^{n}\left(e^{-\beta(t_{n}-t_{i})}-1\right),$

where $R(i)=\sum\limits_{j=1}^{i-1}e^{-\beta(t_{i}-t_{j})}$ can be computed
recursively for i in $\\{2,\dots,n\\}$ :

$R(i)=e^{-\beta(t_{i}-t_{i-1})}\left(1+R(i-1)\right).$

######  Remark 4.1.

The likelihood function is not concave with respect to
$(\beta_{k,l})_{k,l=1,\dots,d}$ in the exponential case. This means that
convergence to the global maximum is not guaranteed, especially in large
dimensions. Fixing $\beta_{k,l}=\beta_{k}$ for all $l=1,\dots,d$ as proposed
by Bonnet et al., 2022b  produces very good results for $d=12$. In this case,
each of the objective functions is not concave in only one parameter instead
of $d$.

######  Remark 4.2.

In the context of football, the effect of a ball touch on the intensity of the
process should last no longer than a few seconds. When $n$ realizations of
football matches are concatenated and treated as one long game, the likelihood
function should not be altered by much. In fact, the rapid decay of the
exponential kernel compared to the duration of games makes the induced error
negligible.

###  4.2 Simulation study

The goal of this section is to evaluate the maximum likelihood estimation
using a simulated dataset that reproduces similar dynamics as those in a
football game. We want to determine the amount of data required for an
accurate estimation of the branching matrix. We also want to assess the
model’s ability to detect a null kernel between two dimensions. A null kernel
$\phi_{i,j}$ means a jump in dimension $j$ has no exciting effect on dimension
$i$. In the context of football, it is particularly informative to detect such
an absence of connection between players.

We perform simulations over different horizons. The parameters are sampled as
follows:

  * •

$\mu$ is chosen from a uniform random variable over $[0.006,0.01]$.

  * •

$\beta$ is chosen to be constant for all $i,j$ in $\\{1,\dots,12\\}$ sampled
from a uniform random variable over $[0.5,1]$ .

  * •

The $\alpha_{i,j}$ are chosen independently from a geometric distribution of
parameter $p=0.4$ scaled by $40$ for all $i,j$ so that $40\%$ of the values
are equal to $0$.

Horizon (minutes) | False positive | Error on false negative | Relative error
---|---|---|---
300 | 1.1% | 0.0054 | 25.7%
600 | 0.0% | 0.0045 | 19.3%
1200 | 0.0% | 0.0030 | 12.4%
2400 | 0.0% | 0.0020 | 11.4%
Table 1: Accuracy results of the maximum likelihood estimation of Hawkes
parameters on the simulated dataset.

Then we fit a 12-dimensional Hawkes process to this data using the algorithm
from Bonnet et al., 2022b . We analyze the resulting accuracy as a function of
the simulation horizon. Table 1 presents the results through three different
metrics:

  * •

False positive: Percentage of branching matrix coefficients
$\hat{\alpha}_{i,j}$ wrongly estimated as null when $\alpha_{i,j}>0$. Our
estimation correctly detects existing links even for small horizons.

  * •

Error on false negative: Our estimation detects accurately 60% of null links
$\alpha_{i,j}=0$. The estimation on the remaining 40% is generally very low as
can be seen in Table 1.

  * •

Relative error: The weighted mean absolute percentage error when
$\alpha_{i,j}>0$. This metric is defined as the mean absolute error divided by
the average value of $\alpha_{i,j}$:

$\textup{wMAPE}=\frac{\sum\limits_{i,j}\left|\hat{\alpha}_{i,j}-\alpha_{i,j}\right|\mathbb{1}_{\alpha_{i,j}>0}}{\sum\limits_{i,j}\alpha_{i,j}\mathbb{1}_{\alpha_{i,j}>0}}.$

The maximum likelihood estimate is good enough for our purposes given the high
dimensionality. Figure 4 shows the estimated branching matrix from a
simulation of horizon 600 minutes. We observe that the estimated branching
matrix appears to correctly approximate the true branching matrix in Figure 5.

Figure 4: Estimated branching matrix.

Figure 5: True branching matrix.

######  Remark 4.3 (Confidence intervals).

Given regularity assumptions on the kernel of the Hawkes process, we can
retrieve the rate of convergence of the maximum likelihood estimator and build
asymptotic confidence intervals. We do not include confidence interval values
here to ease reading but our choice of minimal number of games is dictated by
them and the analysis in this section.

##  5 Analysis of Chelsea FC in the 2016-2017 season

As a first example, we perform our analysis on a selection of Chelsea FC
matches from the 2016-2017 season. The team had a stable formation and a
constant starting eleven over thirteen games in the Premier League. This is
quite convenient because we retrieve a large amount of data where each
position $p$ in $\\{1,\dots,11\\}$ is associated with one player. Similar
analysis for Stade Rennais in the 2021-2022 season is provided in Appendix A.

###  5.1 Selected games

In Table 2, we give the list of selected games for Chelsea FC. In each of
these games, the flat 343 formation is used for at least sixty minutes and the
starting eleven remains the same:

  * •

Thibaut Courtois.

  * •

Gary Cahill - David Luiz - Cesar Azpilicueta.

  * •

Marcos Alonso - Nemanja Matic - N’Golo Kante - Victor Moses.

  * •

Eden Hazard - Diego Costa - Pedro Rodriguez.

Therefore, we use the data before the first substitution from Chelsea FC in
each game to build the counting process.

Date | Opponent | Home or Away | Competition
---|---|---|---
Oct 15, 2016 | Leicester City | Home | English Premier League
Oct 23, 2016 | Manchester United | Home | English Premier League
Oct 30, 2016 | Southampton | Away | English Premier League
Nov 5, 2016 | Everton | Home | English Premier League
Nov 20, 2016 | Middlesbrough | Away | English Premier League
Nov 26, 2016 | Tottenham Hotspur | Home | English Premier League
Dec 11, 2016 | West Bromwich Albion | Home | English Premier League
Jan 4, 2017 | Tottenham Hotspur | Away | English Premier League
Jan 22, 2017 | Hull City | Home | English Premier League
Feb 4, 2017 | Arsenal | Home | English Premier League
Feb 12, 2017 | Burnley | Away | English Premier League
Apr 8, 2017 | Bournemouth | Away | English Premier League
Apr 30, 2017 | Everton | Away | English Premier League

Table 2: List of selected games with the same starting eleven for Chelsea FC.

###  5.2 Results and discussion

In Table 3, we display the different GoT indices for the Chelsea players.
Figure 6 graphically represents the direct interactions between players as
well as their GoTi indices and Figure 7 shows the estimated branching matrix.
We can identify two buildup schemes along the wings with two triangles:
Cahill-Alonso-Matic and Kante-Azpilicueta-Moses. The main channel of
communication between both sides is based on the Matic-Kante link.

Below is a list of observations on players:

Figure 6: Graph summarizing the interactions between Chelsea players. The
width of an arrow from player $p_{1}$ to player $p_{2}$ is proportional to the
expected number of touches of player $p_{2}$ generated by one touch from
player $p_{1}$. The size of the circle of player $p$ is proportional to the
sum of the arrow sizes received, indicating the involvement of the player in
the considered games. The color of the circle represents the GoTi index for
each player.

Player name | GoTd | GoTi | GoT${}^{d}_{90}$ | GoT${}^{i}_{90}$
---|---|---|---|---
Eden Hazard | 0.16 | 0.21 | 14.2 | 15.0
Victor Moses | 0.07 | 0.11 | 5.7 | 7.5
Pedro Rodriguez | 0.08 | 0.12 | 5.5 | 6.7
N’Golo Kante | 0.02 | 0.07 | 2.7 | 6.2
Nemanja Matic | 0.01 | 0.06 | 1.5 | 5.2
Marcos Alonso | 0.02 | 0.06 | 1.9 | 5.1
Diego Costa | 0.07 | 0.10 | 3.6 | 4.8
Cesar Azpilicueta | 0.00 | 0.04 | 0.0 | 4.1
Gary Cahill | 0.00 | 0.04 | 0.0 | 3.0
David Luiz | 0.00 | 0.01 | 0.0 | 1.0
Thibaut Courtois | 0.00 | 0.01 | 0.0 | 0.6

Table 3: Generated threat metrics for the players of Chelsea FC. The table is
sorted by GoT${}^{i}_{90}$.

#### Eden Hazard:

Unsurprisingly, the offensive player, ranked second in the PFA Players’ Player
of the Year 2017 award, leads all GoT metrics. In particular, there is no
significant difference between his GoT${}^{d}_{90}$ and GoT${}^{i}_{90}$
indices, indicating that his primary way of creating danger is through direct
threat. Hazard was well known for his aggressive and direct play as well as
for his dribbling.

#### N’Golo Kante:

Ranking fourth in GoT${}^{i}_{90}$ is evidence to Kante’s important role in
Chelsea’s success in the 2016-2017 season. The winner of the PFA Players’
Player of the Year 2017 award is definitely not limited to defense as the
numbers show that he is largely involved in danger creation. This is explained
by the fact that Kante is a box to box midfielder and that he is at the center
of multiple circuits that end in a threat:

  * •

Kante $\rightarrow$ Pedro $\rightarrow$ Threat.

  * •

Kante $\rightarrow$ Moses $\rightarrow$ Pedro $\rightarrow$ Threat.

  * •

Kante $\rightarrow$ Matic $\rightarrow$ Hazard $\rightarrow$ Threat.

#### David Luiz:

The contribution of the central defender David Luiz in the generation of
threat is minimal. This is not surprising as the flat 3-4-3 system relies
heavily on the wings. David Luiz naturally passes the ball to either Gary
Cahill or Azpilicueta in the build-up to spread the play.

#### Diego Costa:

Costa generates a small number of threats despite being a striker. This is
expected as he is responsible for transforming the goalscoring chances rather
than being at the origin of the danger. Moreover, his GoT${}^{i}_{90}$
statistic is particularly low since he has a low number of touches per time
unit and many of his touches in the danger zone are not recorded in the
constructed counting process.

We can clearly see that considering indirect contribution to threat generation
is important for defenders and midfielders. These positions are generally at
the base of the danger creation process. They have small GoTd indices.
However, indirect generated threat combined with the consideration of the
number of touches allows us to effectively compare players playing in deeper
positions.

From the graphical representation in Figure 6, we can identify some patterns
that lead to a dangerous situation. When facing a team like Chelsea in the
2016-2017 season, some strategies can be derived from this analysis:

  * •

As illustrated in Figure 6, the right side of Chelsea combines a lot for
threat generation and should be disrupted at the root. Azpilicueta should be
stopped from feeding the ball to the midfielders or directly to Pedro.

  * •

The left side relies much more on the direct offensive output of Eden Hazard.
In fact, all of Gary Cahill, Matic and Marcos Alonso mostly aim at delivering
the ball to the left winger. To neutralize the threat of the left side, it is
essential to prevent the ball from reaching Hazard. This can be achieved by
marking him closely or by constantly closing the passing lanes to him.

  * •

Goalkeeper Courtois is successful in targetting Marcos Alonso directly. This
passing pattern should be considered when pressing Chelsea.

Figure 7: Estimated branching matrix for Chelsea FC.

##  6 Ligue 1 2021-22 season analysis

In this section, we provide a ranking of players and teams from Ligue 1 in the
2021-2022 season based on their generation of threat. To maintain homogeneity,
we only consider for each team the games where they use their main formation
cluster, see Table 12 in Appendix B for the list of formation clusters of each
team.

###  6.1 Generated threat to rank players in a position

Each position on the pitch imposes a different role on the player who occupies
it. In particular, we cannot expect the same player to produce the same GoT
metrics at two different positions. Therefore, we choose to evaluate players
when they play in a particular position. This approach will also allow us to
determine the optimal position for a player to maximize a GoT metric of
interest. Additionally, we apply a filter to consider only players who play at
least 600 minutes at a given position, with playing time calculated based on
games in which the player features for at least 45 minutes.

Given a player and a position, we record the games in which the player
occupies the position. The remaining positions may feature different players
at each game. Whenever a player from his team is substituted, we do not
consider the rest of the game in the construction of the counting process. We
fit a Hawkes process and assign to the player the generated threat indices of
his position. Tables 4 and 5 present the top twenty players in Ligue 1 in
terms of GoTd and GoT${}^{i}_{90}$, respectively (see Tables 10 and 11 in
Appendix B for the Top 100). We display these two indices because they
quantify the two extremes of the danger generation process. GoTd isolates the
direct impact of players while GoT${}^{i}_{90}$ measures their participation
in the chain of events leading to threats.

In reference to the results in Section 4.2, one should keep in mind that the
fewer minutes a player plays in a position, the less accurate the estimate of
his generated threat is. Moreover, our estimation relies on selected games
only. When a player has a limited number of minutes in a position, a good GoT
metric should be interpreted as a measure of performance across the considered
games only. For example, Moses Simon ranking third in GoTd should not be
surprising as he provided seven assists in the 1200 minutes but only gave one
more assist in the remaining games when the team plays in a different
formation or when he plays in a different position.

Below are some observations based on the results:

Rank | Name | Position | Team | Minutes | GoTd
---|---|---|---|---|---
1 | Lionel Messi | 10 | Paris Saint-Germain | 630 | 0.130
2 | Ángel Di María | 10 | Paris Saint-Germain | 1171 | 0.128
3 | Moses Simon | 11 | Nantes | 1222 | 0.120
4 | Kylian Mbappé | 9 | Paris Saint-Germain | 1338 | 0.110
5 | Lionel Messi | 9 | Paris Saint-Germain | 675 | 0.109
6 | Martin Terrier | 11 | Rennes | 1386 | 0.108
7 | Kylian Mbappé | 11 | Paris Saint-Germain | 1066 | 0.107
8 | Romain Faivre | 7 | Brest | 630 | 0.106
8 | Houssem Aouar | 7 | Lyon | 810 | 0.106
10 | Sofiane Boufal | 9 | Angers | 771 | 0.100
11 | Jonathan Ikoné | 7 | Lille | 767 | 0.096
12 | Wissam Ben Yedder | 9 | Monaco | 1625 | 0.094
13 | Franck Honorat | 11 | Brest | 838 | 0.093
13 | Karl Toko-Ekambi | 11 | Lyon | 1855 | 0.093
15 | Benjamin Bourigeaud | 10 | Rennes | 1719 | 0.092
16 | Sofiane Boufal | 11 | Angers | 665 | 0.091
17 | Justin Kluivert | 11 | Nice | 1207 | 0.090
19 | Dimitri Payet | 9 | Marseille | 617 | 0.088
19 | Kevin Gameiro | 10 | Strasbourg | 673 | 0.088
19 | Neymar | 11 | Paris Saint-Germain | 1258 | 0.088

Table 4: Ranking of Ligue 1 players in terms of GoTd.

#### GoTd vs GoT${}^{i}_{90}$:

GoTd captures the intrinsic ability of a player to advance the ball to the
opponent’s danger area while GoT${}^{i}_{90}$ incorporates possible
combinations with teammates. Therefore, the style of play and the ability of
teammates can have an impact on the value of GoT${}^{i}_{90}$. These two
indices describe different ways to contribute to threat generation and allow
us to select different profiles of players. For example, the Paris Saint-
Germain midfielder Verratti produces high values of GoT${}^{i}_{90}$ while
Moses Simon from FC Nantes features in the top positions in terms of GoTd.

#### Jason Berthomier as a surprising pick:

In his only season in Ligue 1, Jason Berthomier delivered excellent values of
GoT${}^{i}_{90}$. The Clermont Foot midfielder ranks 43rd in terms of GoTd and
climbs up to the tenth position in the GoT${}^{i}_{90}$ ranking. This proves
that he is consistently involved in the generation of dangerous situations for
his team and is successful in feeding the forward players.

#### Téji Savanier excels in midfield:

Téji Savanier stands out as an interior midfielder in the 433 formation of
Montpellier. With eight goals and seven assists, it is no surprise that he is
central to the process of threat generation of his team. He ranks eighth in
GoT${}^{i}_{90}$ and outperforms many offensive players in the league. This
confirms the quality of Téji Savanier and his good performance during the
2021-2022 season.

#### A defender in the Top 20:

Frederic Guilbert of Strasbourg is a defender who excels at creating threats,
ranking 18th in GoT${}^{i}_{90}$. In fact, his team deploys a 532 formation
that provides enough cover for the fullbacks to play offensively. The same
holds for Jonathan Clauss who acts almost as a right midfielder in the Lens
formation and ranks 33rd in GoT${}^{i}_{90}$. This is also not surprising as
Clauss ranks third in the league in the number of passes that lead to a shot,
another proof of his creative play.

#### A good season from Messi in generated threat:

Despite underperforming in terms of scoring goals, Lionel Messi delivers
outstanding values of generated threat both directly given his dribbling and
passing quality, and indirectly given his involvement in ball possession.
Additionally, we observe that his performance increases slightly when playing
in his natural position as a 10 in the 433 formation. The right wing is
Messi’s best position as he poses more of a threat cutting inside from the
right.

#### Optimal position for some players:

Romain Faivre stands out in both GoTd and GoT${}^{i}_{90}$, ranking among the
top twenty players. This is in fact expected because, when playing as a right
midfielder in the 442 formation of Brest, the player performed well and was
involved in six goals in just 660 minutes. Similarly, Houssem Aouar was
successful as an interior midfielder in the 433 formation. He scored three and
assisted three more in the considered period, earning him a top spot on our
list.

#### A metric that does not value center forwards:

Very few strikers make the Top 20 in the two metrics. This is because the role
of some center forwards is to receive the ball in the danger area and not
necessarily to be at the origin of the threat. This is even more pronounced
when looking at GoT${}^{i}_{90}$. For example, Mbappé, the top scorer in the
league, barely makes it to the Top 20. Mbappe is not known for participating
in possession and touching the ball a lot but as an aggressive transition
player. In contrast, midfielders such as Verratti and Guimarães, that are
involved in the build-up of a lot of dangerous situations, feature in the top
positions in terms of GoT${}^{i}_{90}$.

Rank | Name | Position | Team | Minutes | GoT${}^{i}_{90}$
---|---|---|---|---|---
1 | Lionel Messi | 10 | Paris Saint-Germain | 630 | 14.911
2 | Ángel Di María | 10 | Paris Saint-Germain | 1171 | 13.218
3 | Neymar | 11 | Paris Saint-Germain | 1258 | 12.724
4 | Marco Verratti | 4 | Paris Saint-Germain | 602 | 12.581
5 | Lionel Messi | 9 | Paris Saint-Germain | 675 | 12.353
6 | Romain Faivre | 7 | Brest | 630 | 10.402
7 | Houssem Aouar | 7 | Lyon | 810 | 10.077
8 | Téji Savanier | 7 | Montpellier | 2209 | 9.608
9 | Marco Verratti | 8 | Paris Saint-Germain | 1069 | 9.446
10 | Jason Berthomier | 7 | Clermont | 1244 | 9.340
11 | Benjamin Bourigeaud | 10 | Rennes | 1719 | 9.211
12 | Sofiane Boufal | 9 | Angers | 771 | 9.100
13 | Bruno Guimarães | 4 | Lyon | 900 | 8.817
14 | Dimitri Payet | 9 | Marseille | 617 | 8.815
15 | Moses Simon | 11 | Nantes | 1222 | 8.790
16 | Martin Terrier | 11 | Rennes | 1386 | 8.639
17 | Kylian Mbappé | 11 | Paris Saint-Germain | 1066 | 8.577
18 | Frédéric Guilbert | 2 | Strasbourg | 2428 | 8.421
19 | Ruben Aguilar | 2 | Monaco | 1205 | 8.019
20 | Lovro Majer | 7 | Rennes | 1302 | 7.927

Table 5: Ranking of Ligue 1 players in terms of GoT${}^{i}_{90}$.

###  6.2 Ranking the central defenders’ involvement in terms of GoT

To quantify the involvement of central defenders in danger creation, we use
the indirect generation of threat per 90 minutes. This is because the direct
generation of threat (GoTd) values are particularly low for defenders and
therefore cannot be used to compare players. While GoT${}^{i}_{90}$ is
influenced by the quality of the offensive players and team style of play, it
also provides valuable information on the role of defenders in the team’s
build-up scheme. For instance, a center-back who is technically proficient but
avoids taking risks and does not contribute much to ball progression will have
a low value of GoT${}^{i}_{90}$. This metric strikes a balance in measuring a
player’s intrinsic ability as well as their involvement within the team. Table
6 displays the Top 10 best central defenders with the highest values of
GoT${}^{i}_{90}$.

It is no surprise that Marquinhos and Kimpembe take the first two spots, given
that they are part of Paris Saint-Germain, the most dominant team in Ligue 1.
This is of course due to their technical ability, but there is also a factor
due to the high possession values and danger creation ability of their team.
The same holds for Nayef Aguerd and Warmed Omari that contribute significantly
to ball progression, primarily through accurate long balls. The third-placed
is Facundo Medina. The Lens defender is well known for his range of passing
and for his ability to switch play from one side to the other. In particular,
he ranks tenth in the league in terms of accurate passes per 90 minutes.
William Saliba naturally completes the Top 5. The Marseille player excels with
the ball at his feet and ranks third in accurate passing in Ligue 1. The
player has now moved to Arsenal, a team that likes to play from the back, and
continues to deliver in that aspect of the game.

Rank | Name | Position | Team | Minutes | GoT${}^{i}_{90}$
---|---|---|---|---|---
1 | Marquinhos | 5 | Paris Saint-Germain | 2340 | 5.625
2 | Presnel Kimpembe | 6 | Paris Saint-Germain | 1840 | 5.230
3 | Facundo Medina | 4 | Lens | 1329 | 4.953
4 | Nayef Aguerd | 6 | Rennes | 1698 | 4.908
5 | William Saliba | 5 | Marseille | 1800 | 4.652
6 | Jason Denayer | 6 | Lyon | 630 | 4.591
7 | Jonathan Gradit | 6 | Lens | 1710 | 4.535
8 | Warmed Omari | 5 | Rennes | 1710 | 4.407
9 | Damien Da Silva | 5 | Lyon | 612 | 4.182
10 | Dante | 6 | Nice | 2880 | 3.707
11 | Duje Caleta-Car | 6 | Marseille | 1397 | 3.462
12 | Lucas Perrin | 6 | Strasbourg | 2329 | 3.185
13 | Kevin Danso | 5 | Lens | 1620 | 3.105
14 | Benoît Badiashile | 6 | Monaco | 975 | 3.075
15 | Castello Lukeba | 6 | Lyon | 1375 | 3.074
16 | Mamadou Sakho | 6 | Montpellier | 1962 | 2.998
17 | Florent Ogier | 6 | Clermont | 2329 | 2.949
18 | Guillermo Maripán | 6 | Monaco | 810 | 2.916
19 | Guillermo Maripán | 5 | Monaco | 605 | 2.915
20 | Jean-Clair Todibo | 5 | Nice | 3123 | 2.864

Table 6: Ranking of Ligue 1 central defenders in terms of GoT${}^{i}_{90}$.

###  6.3 GoTd to rank teams

To verify the consistency of our metrics, we rank Ligue 1 teams based on their
aggregate values of GoTd. This metric can be considered as an indicator of
squad quality. For each club, we fit a 12-dimensional Hawkes process to all
matches in which they use their primary formation cluster, regardless of the
players occupying each position. We then sum the estimated direct threat per
touch GoTd for all the positions.

Table 7 shows the resulting Top 10 based on generated threat. Our metric
describes an important part of the offensive performance but obviously does
not cover all aspects of the game. Nevertheless, it remains a very good
measure of the quality of the team. Our ranking shows a significant $62\%$
Kendall correlation with the realized ranking of Ligue 1. This is achieved
while only looking at ball touch and threat event timestamps to infer player
abilities. Below are some observations from the ranking:

  * •

Rennes climbs to the second position in our ranking. This is because the team
was very attack-minded in the 2021-2022 season and managed to score 82 goals,
one of the highest totals in Europe. Their expected threat is proof of their
offensive output.

  * •

Olympique Lyonnais, ranked eighth in Ligue 1, still had a very prolific season
offensively. They have the third-highest total of goals and the second-highest
total of expected goals. It is therefore natural they are fourth with respect
to our offensive metric.

Team | GoTd | Ligue 1 ranking | Goals scored
---|---|---|---
Paris Saint-Germain | 0.42 | 1 | 90
Rennes | 0.41 | 4 | 82
Monaco | 0.41 | 3 | 65
Lyon | 0.36 | 8 | 66
Marseille | 0.36 | 2 | 63
Lens | 0.30 | 7 | 62
Nice | 0.28 | 5 | 52
Strasbourg | 0.26 | 6 | 60
Lille | 0.26 | 10 | 48
Reims | 0.26 | 12 | 43

Table 7: Top 10 Ligue 1 teams with respect to aggregated GoTd of starting
eleven.

##  7 Conclusion and future work

In order to measure a player’s ability to create threat in football, we
develop model-based metrics that rely on Hawkes processes. These processes
provide an easy to interpret way to capture causation between event times.
Thanks to this modeling, we are able to identify the players whose touches are
most consistently correlated with subsequent threats. We derive four different
metrics each describing different ways to create danger. On the one hand, the
direct generation of threat metrics GoTd and GoT${}^{d}_{90}$ allow us to
isolate the intrinsic ability of players. On the other hand, GoTi and
GoT${}^{i}_{90}$ indicate the indirect contribution to the generation of
threat through interactions with other positions. Beyond crediting players for
danger generation, our approach can also be used to quantify and visualize the
synergies between players on the pitch and identify the patterns that lead to
dangerous situations.

We demonstrate our methodology can successfully detect and rank the key
players in the 2021-2022 Ligue 1 season, who contribute to their team’s
offensive output. The results we find are consistent with the observed
performances of the retrieved players, but also reveal some surprising
choices. Through the example of Chelsea in the 2016-2017 season, we show that
our model-based approach can help teams make data-driven decisions about their
tactics. By primarily looking at timestamps of ball touches, we gain a deeper
understanding of the threat generation process of a team.

Future work will include exploring the application of our model-based metrics
for optimal team selection. In fact, if we are capable of inferring the
branching matrix parameters linking players from different teams, we can
measure the impact of a potential transfer on the danger creation process. In
addition, we can use this framework to capture interactions of players with
other game states different from threats. In particular, by replacing the
threat events with ball losses, we can effectively analyze the defensive
aspect of the game and determine players whose touches are most correlated
with a turnover.

#### Acknowledgment:

The authors thank Anna Bonnet for her help with the estimation of Hawkes
processes in large dimensions. They are also grateful to Charlotte Dion and
Céline Duval. The authors gratefully acknowledge financial support from the
chairs “Machine Learning & Systematic Methods in Finance” and “Deep Finance
and Statistics”.

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Appendix

##  Appendix A Stade Rennais

In this appendix, we present a detailed analysis of one of the Ligue 1 teams
in the 2021-2022 season.

###  A.1 Selected games

In the same spirit as Section 5, we choose a collection of games where the
formation is the same and starting lineup is as stable as possible. Table 8
shows the selected matches for Stade Rennais. The team plays in a 433
formation in all of these games but the starting eleven is not always exactly
the same. In fact, some players are sometimes rotated for a game or two, but
we assume that the substitute behaves approximately the same as the starting
player. Stade Rennais line up as follows in the selected games, where the main
player in each position is in bold:

  * •

Gomis/Alemdar

  * •

Traore \- Omari/Bade - Aguerd/Bade/Santamaria - Truffert/Meling

  * •

Majer \- Santamaria/Martin - Tait

  * •

Bourigeaud \- Laborde/Guirassy - Terrier

We construct a 12-dimensional counting process from the selected Stade Rennais
games regardless of the players starting. We use the data from each game as
long as the eleven players on the pitch correspond to the scheme provided
above. We then fit a 12-dimensional Hawkes process and associate the estimated
metrics of each position with the main player occupying it.

Date | Opponent | Home or Away | Competition
---|---|---|---
May 11, 2022 | Nantes | Away | French Ligue 1
Apr 2, 2022 | Nice | Away | French Ligue 1
May 14, 2022 | Marseille | Home | French Ligue 1
Dec 22, 2021 | Monaco | Away | French Ligue 1
Mar 20, 2022 | Metz | Home | French Ligue 1
Apr 15, 2022 | Monaco | Home | French Ligue 1
Apr 30, 2022 | St Etienne | Home | French Ligue 1
Apr 24, 2022 | Lorient | Home | French Ligue 1
Nov 20, 2021 | Montpellier | Home | French Ligue 1
May 21, 2022 | Lille | Away | French Ligue 1
Nov 7, 2021 | Lyon | Home | French Ligue 1

Table 8: List of selected games for Stade Rennais F.C.

###  A.2 Results and discussion

In Table 9, we rank the Stade Rennais players with respect to generated threat
metrics. Figure 8 graphically represents the direct interactions between them
and Figure 9 displays the estimated branching matrix.

Figure 8: Graph summarizing the interactions between Stade Rennais players.
The width of an arrow from player $p_{1}$ to player $p_{2}$ is proportional to
the expected number of touches of player $p_{2}$ generated by one touch from
player $p_{1}$. The size of the circle of player $p$ is proportional to the
sum of the arrow sizes received, indicating the involvement of the player in
the considered games. The color of the circle represents the GoTi index for
each player.

We can see that the team adopts a 433 shape that progresses mainly through the
wings. The danger creation is asymmetric with more combinations occurring on
the right side, where Majer is the most creative midfielder. Interestingly,
despite being a central midfielder, Flavien Tait delivers a large value of
GoT${}^{i}_{90}$, indicating that he is a significant contributor to the
team’s offensive efforts. In contrast, although Santamaria has more
possession, he has limited involvement in creating threats. This difference in
their threat generation can be attributed to their distinct roles on the
field. On one hand, Tait is a more box-to-box midfielder who frequently
projects forward and has a considerable direct threat metric. On the other
hand, Santamaria belongs to a class of defensive midfielders who act as anchor
points. They participate in the buildup close to the center backs and have
limited interactions with the forward positions.

The main threat sources are Bourigeaud, Majer, and Terrier. These three
players are outstanding going forward. Terrier is the leader of the team in
goalscoring and ranks third in Ligue 1 but seems to be involved in danger
creation as well. Bourigeaud generating the most threat is not surprising
since he is the creative force of the team. In fact, he ranks first in the
league in terms of key passes with 3.2 per game, and first in accurate crosses
with 104 in the season.

As expected, the center backs have zero direct threat contribution. However,
in terms of indirect threat per 90 minutes GoT${}^{i}_{90}$, Aguerd and Omari
rank fourth and sixth in the team respectively. The pair generates danger
through their involvement in team build-up and possession. In particular,
Aguerd and Omari are comfortable with the ball at their feet and rank eighth
and twentieth in the league, respectively, in the number of passes per game
with high success rates.

Player name | GoTd | GoTi | GoT${}^{d}_{90}$ | GoT${}^{i}_{90}$
---|---|---|---|---
Benjamin Bourigeaud | 0.14 | 0.16 | 11.8 | 12.6
Martin Terrier | 0.13 | 0.17 | 8.6 | 9.5
Lovro Majer | 0.08 | 0.11 | 6.9 | 8.1
Flavien Tait | 0.06 | 0.09 | 5.5 | 7.2
Adrien Truffert | 0.02 | 0.06 | 2.4 | 5.2
Hamari Traoré | 0.01 | 0.05 | 1.5 | 4.6
Nayef Aguerd | 0.00 | 0.04 | 0.0 | 4.5
Baptiste Santamaría | 0.00 | 0.05 | 0.4 | 4.3
Warmed Omari | 0.00 | 0.04 | 0.0 | 4.0
Gaëtan Laborde | 0.04 | 0.08 | 2.0 | 3.0
Alfred Gomis | 0.00 | 0.01 | 0.0 | 0.7

Table 9: Generated threat metrics for the players of Stade Rennais. The table
is sorted by GoT${}^{i}_{90}$.

Finally, we can observe from Figure 8 some remarkable circuits that lead to
dangerous situations. These patterns of play should be taken into account by
an opposing team when facing Stade Rennais:

  * •

Aguerd $\rightarrow$ Truffert $\rightarrow$ Terrier $\rightarrow$ Threat.

  * •

Terrier $\rightarrow$ Tait $\rightarrow$ Threat. Terrier is highly effective
in generating direct threats, but he also frequently combines with Flavien
Tait to create danger. Similarly, Bourigeaud often gives the ball to Lovro
Majer to generate indirect threat.

  * •

Omari $\rightarrow$ Traoré $\rightarrow$ Bourigeaud $\rightarrow$ Threat.

  * •

Omari $\rightarrow$ Bourigeaud $\rightarrow$ Threat. This is a straightforward
pattern from defense to attack that should be controlled. Omari is highly
successful in progressing the ball, both through slow build-up play by passing
the ball to the right-back Traoré, as well as through fast transitions with
direct passes to Bourigeaud.

Figure 9: Estimated branching matrix for Stade Rennais.

##  Appendix B Top 100 ranking of Ligue 1 player in terms of GoT

Rank | Name | Position | Team | Minutes | GoTd
---|---|---|---|---|---
1 | Lionel Messi | 10 | Paris Saint-Germain | 630 | 0.130
2 | Ángel Di María | 10 | Paris Saint-Germain | 1171 | 0.128
3 | Moses Simon | 11 | Nantes | 1222 | 0.120
4 | Kylian Mbappé | 9 | Paris Saint-Germain | 1338 | 0.110
5 | Lionel Messi | 9 | Paris Saint-Germain | 675 | 0.109
6 | Martin Terrier | 11 | Rennes | 1386 | 0.108
7 | Kylian Mbappé | 11 | Paris Saint-Germain | 1066 | 0.107
8 | Romain Faivre | 7 | Brest | 630 | 0.106
8 | Houssem Aouar | 7 | Lyon | 810 | 0.106
10 | Sofiane Boufal | 9 | Angers | 771 | 0.100
11 | Jonathan Ikoné | 7 | Lille | 767 | 0.096
12 | Wissam Ben Yedder | 9 | Monaco | 1625 | 0.094
13 | Franck Honorat | 11 | Brest | 838 | 0.093
13 | Karl Toko-Ekambi | 11 | Lyon | 1855 | 0.093
15 | Benjamin Bourigeaud | 10 | Rennes | 1719 | 0.092
16 | Sofiane Boufal | 11 | Angers | 665 | 0.091
17 | Justin Kluivert | 11 | Nice | 1207 | 0.090
19 | Dimitri Payet | 9 | Marseille | 617 | 0.088
19 | Kevin Gameiro | 10 | Strasbourg | 673 | 0.088
19 | Neymar | 11 | Paris Saint-Germain | 1258 | 0.088
21 | Jodel Dossou | 10 | Clermont | 1762 | 0.086
22 | Lucas Da Cunha | 10 | Clermont | 606 | 0.084
22 | Jim Allevinah | 11 | Clermont | 858 | 0.084
24 | Frédéric Guilbert | 2 | Strasbourg | 2428 | 0.082
25 | Armand Laurienté | 9 | Lorient | 842 | 0.080
26 | Ludovic Blas | 7 | Nantes | 810 | 0.078
27 | Gaël Kakuta | 9 | Lens | 976 | 0.077
28 | Arnaud Kalimuendo-Muinga | 11 | Lens | 724 | 0.075
29 | Cengiz Ünder | 10 | Marseille | 1047 | 0.074
29 | Florent Mollet | 10 | Montpellier | 1269 | 0.074
31 | Téji Savanier | 7 | Montpellier | 2209 | 0.071
32 | Ghislain Konan | 3 | Reims | 1007 | 0.070
33 | Lovro Majer | 7 | Rennes | 1302 | 0.068
34 | Kevin Volland | 7 | Monaco | 1131 | 0.067
34 | Jonathan Clauss | 2 | Lens | 1940 | 0.067
36 | Angelo Fulgini | 8 | Angers | 630 | 0.066
36 | Andy Delort | 10 | Nice | 1478 | 0.066
38 | Javairô Dilrosun | 9 | Bordeaux | 675 | 0.064
39 | Vanderson | 10 | Monaco | 619 | 0.061
40 | Lucas Paquetá | 7 | Lyon | 1248 | 0.060
40 | Burak Yilmaz | 9 | Lille | 1900 | 0.060
43 | Jonathan Bamba | 11 | Lille | 1763 | 0.059
43 | Thomas Foket | 2 | Reims | 631 | 0.059
43 | Jason Berthomier | 7 | Clermont | 1244 | 0.059
45 | Amine Gouiri | 9 | Nice | 1749 | 0.058
46 | Gaëtan Laborde | 9 | Rennes | 1305 | 0.057
47 | Ibrahima Sissoko | 7 | Strasbourg | 826 | 0.055
48 | Jonathan David | 10 | Lille | 2072 | 0.054
49 | Dimitri Lienard | 3 | Strasbourg | 1728 | 0.050
51 | Flavien Tait | 8 | Rennes | 1129 | 0.046

Rank | Name | Position | Team | Minutes | GoTd
---|---|---|---|---|---
51 | Florian Sotoca | 10 | Lens | 1119 | 0.046
51 | Jérémy Le Douaron | 10 | Brest | 759 | 0.046
53 | Andy Delort | 9 | Nice | 795 | 0.045
55 | Youcef Atal | 2 | Nice | 1032 | 0.044
55 | Randal Kolo Muani | 9 | Nantes | 876 | 0.044
55 | Stephy Mavididi | 11 | Montpellier | 1585 | 0.044
55 | Aleksandr Golovin | 11 | Monaco | 607 | 0.044
58 | Elbasan Rashani | 11 | Clermont | 1588 | 0.043
59 | Mohamed Bayo | 9 | Clermont | 2331 | 0.042
60 | Abdu Conté | 3 | Troyes | 695 | 0.041
60 | Sanjin Prcic | 11 | Strasbourg | 652 | 0.041
64 | Bruno Guimarães | 4 | Lyon | 900 | 0.040
64 | Anthony Caci | 3 | Strasbourg | 1400 | 0.040
64 | Kevin Gameiro | 9 | Strasbourg | 1458 | 0.040
64 | Hicham Boudaoui | 7 | Nice | 1304 | 0.040
64 | Gerson | 8 | Marseille | 704 | 0.040
67 | Sofiane Diop | 11 | Monaco | 631 | 0.039
68 | Igor Silva | 2 | Lorient | 1197 | 0.038
69 | Renato Sanches | 8 | Lille | 951 | 0.037
69 | Issa Kaboré | 2 | Troyes | 1679 | 0.037
71 | Mohamed-Ali Cho | 10 | Angers | 807 | 0.035
71 | Angelo Fulgini | 9 | Angers | 751 | 0.035
74 | Akim Zedadka | 2 | Clermont | 3330 | 0.034
74 | Pol Lirola | 2 | Marseille | 863 | 0.034
74 | Adrien Thomasson | 7 | Strasbourg | 1853 | 0.034
76 | Habib Diallo | 10 | Strasbourg | 859 | 0.033
76 | Xavier Chavalerin | 11 | Troyes | 949 | 0.033
78 | Jean-Ricner Bellegarde | 11 | Strasbourg | 1454 | 0.032
78 | Maxence Caqueret | 8 | Lyon | 1389 | 0.032
80 | Vital N’Simba | 3 | Clermont | 2731 | 0.031
80 | Ruben Aguilar | 2 | Monaco | 1205 | 0.031
83 | Seko Fofana | 8 | Lens | 1861 | 0.030
83 | Ismail Jakobs | 3 | Monaco | 650 | 0.030
83 | Terem Moffi | 10 | Lorient | 911 | 0.030
87 | Valère Germain | 9 | Montpellier | 1083 | 0.029
87 | Stéphane Bahoken | 10 | Angers | 657 | 0.029
87 | Youssouf Fofana | 8 | Monaco | 1116 | 0.029
87 | Mattéo Guendouzi | 7 | Marseille | 1350 | 0.029
87 | Ludovic Ajorque | 10 | Strasbourg | 1334 | 0.029
90 | Ludovic Ajorque | 9 | Strasbourg | 1243 | 0.028
90 | Marco Verratti | 4 | Paris Saint-Germain | 602 | 0.028
93 | Baptiste Santamaría | 8 | Rennes | 675 | 0.027
93 | Vincent Le Goff | 3 | Lorient | 1440 | 0.027
93 | Ricardo Mangas | 3 | Bordeaux | 613 | 0.027
96 | Caio Henrique | 3 | Monaco | 1454 | 0.026
96 | Mihailo Ristic | 3 | Montpellier | 1150 | 0.026
96 | Junior Sambia | 2 | Montpellier | 732 | 0.026
98 | Souleyman Doumbia | 3 | Angers | 1797 | 0.025
98 | Przemyslaw Frankowski | 3 | Lens | 1191 | 0.025
100 | Florian Tardieu | 8 | Troyes | 1530 | 0.024

Table 10: Ranking of Ligue 1 players in terms of GoTd.

Rank | Name | Position | Team | Minutes | GoT${}^{i}_{90}$
---|---|---|---|---|---
1 | Lionel Messi | 10 | Paris Saint-Germain | 630 | 14.911
2 | Ángel Di María | 10 | Paris Saint-Germain | 1171 | 13.218
3 | Neymar | 11 | Paris Saint-Germain | 1258 | 12.724
4 | Marco Verratti | 4 | Paris Saint-Germain | 602 | 12.581
5 | Lionel Messi | 9 | Paris Saint-Germain | 675 | 12.353
6 | Romain Faivre | 7 | Brest | 630 | 10.402
7 | Houssem Aouar | 7 | Lyon | 810 | 10.077
8 | Téji Savanier | 7 | Montpellier | 2209 | 9.608
9 | Marco Verratti | 8 | Paris Saint-Germain | 1069 | 9.446
10 | Jason Berthomier | 7 | Clermont | 1244 | 9.340
11 | Benjamin Bourigeaud | 10 | Rennes | 1719 | 9.211
12 | Sofiane Boufal | 9 | Angers | 771 | 9.100
13 | Bruno Guimarães | 4 | Lyon | 900 | 8.817
14 | Dimitri Payet | 9 | Marseille | 617 | 8.815
15 | Moses Simon | 11 | Nantes | 1222 | 8.790
16 | Martin Terrier | 11 | Rennes | 1386 | 8.639
17 | Kylian Mbappé | 11 | Paris Saint-Germain | 1066 | 8.577
18 | Frédéric Guilbert | 2 | Strasbourg | 2428 | 8.421
19 | Ruben Aguilar | 2 | Monaco | 1205 | 8.019
20 | Lovro Majer | 7 | Rennes | 1302 | 7.927
21 | Lucas Da Cunha | 10 | Clermont | 606 | 7.882
22 | Kylian Mbappé | 9 | Paris Saint-Germain | 1338 | 7.733
23 | Ghislain Konan | 3 | Reims | 1007 | 7.721
24 | Sanjin Prcic | 11 | Strasbourg | 652 | 7.624
25 | Sofiane Boufal | 11 | Angers | 665 | 7.547
26 | Lucas Paquetá | 7 | Lyon | 1248 | 7.450
27 | Karl Toko-Ekambi | 11 | Lyon | 1855 | 7.365
28 | Jonathan Ikoné | 7 | Lille | 767 | 7.285
29 | Franck Honorat | 11 | Brest | 838 | 7.207
30 | Achraf Hakimi | 2 | Paris Saint-Germain | 1781 | 7.125
31 | Idrissa Gueye | 8 | Paris Saint-Germain | 662 | 7.090
32 | Dimitri Lienard | 3 | Strasbourg | 1728 | 7.050
33 | Gerson | 8 | Marseille | 704 | 7.016
34 | Vanderson | 10 | Monaco | 619 | 6.939
35 | Ibrahima Sissoko | 7 | Strasbourg | 826 | 6.892
36 | Ludovic Blas | 7 | Nantes | 810 | 6.816
37 | Gaël Kakuta | 9 | Lens | 976 | 6.802
38 | Jonathan Clauss | 2 | Lens | 1940 | 6.757
39 | Justin Kluivert | 11 | Nice | 1207 | 6.700
40 | Flavien Tait | 8 | Rennes | 1129 | 6.671
41 | Kevin Gameiro | 10 | Strasbourg | 673 | 6.569
42 | Florent Mollet | 10 | Montpellier | 1269 | 6.465
43 | Angelo Fulgini | 8 | Angers | 630 | 6.459
44 | Renato Sanches | 8 | Lille | 951 | 6.427
45 | Henrique | 3 | Lyon | 619 | 6.385
46 | Danilo Pereira | 4 | Paris Saint-Germain | 879 | 6.346
47 | Pol Lirola | 2 | Marseille | 863 | 6.327
48 | Jim Allevinah | 11 | Clermont | 858 | 6.303
49 | Youcef Atal | 2 | Nice | 1032 | 6.258
50 | Maxence Caqueret | 8 | Lyon | 1389 | 6.229

Rank | Name | Position | Team | Minutes | GoT${}^{i}_{90}$
---|---|---|---|---|---
51 | Vital N’Simba | 3 | Clermont | 2731 | 6.066
52 | Anthony Caci | 3 | Strasbourg | 1400 | 6.023
53 | Emerson | 3 | Lyon | 1848 | 6.019
54 | Aleksandr Golovin | 11 | Monaco | 607 | 5.973
55 | Birger Meling | 3 | Rennes | 776 | 5.936
56 | Juan Bernat | 3 | Paris Saint-Germain | 777 | 5.929
57 | Jodel Dossou | 10 | Clermont | 1762 | 5.864
58 | Caio Henrique | 3 | Monaco | 1454 | 5.855
59 | Jonas Martin | 4 | Rennes | 1115 | 5.810
60 | Thomas Foket | 2 | Reims | 631 | 5.805
61 | Jonathan Bamba | 11 | Lille | 1763 | 5.632
62 | Marquinhos | 5 | Paris Saint-Germain | 2340 | 5.625
63 | Florian Sotoca | 10 | Lens | 1119 | 5.598
64 | Jordan Ferri | 8 | Montpellier | 2129 | 5.516
65 | Aurélien Tchouaméni | 4 | Monaco | 1620 | 5.431
66 | Malo Gusto | 2 | Lyon | 1369 | 5.382
67 | Cheick Oumar Doucouré | 7 | Lens | 1350 | 5.372
68 | Armand Laurienté | 9 | Lorient | 842 | 5.345
69 | Akim Zedadka | 2 | Clermont | 3330 | 5.281
70 | Presnel Kimpembe | 6 | Paris Saint-Germain | 1840 | 5.230
71 | Ismail Jakobs | 3 | Monaco | 650 | 5.199
72 | Fábio | 3 | Nantes | 726 | 5.154
73 | Thilo Kehrer | 2 | Paris Saint-Germain | 632 | 5.144
74 | Cengiz Ünder | 10 | Marseille | 1047 | 5.079
75 | Léo Dubois | 2 | Lyon | 1246 | 5.060
76 | Mattéo Guendouzi | 7 | Marseille | 1350 | 4.995
77 | Facundo Medina | 4 | Lens | 1329 | 4.953
78 | Adrien Thomasson | 7 | Strasbourg | 1853 | 4.921
79 | Nayef Aguerd | 6 | Rennes | 1698 | 4.908
80 | Abdu Conté | 3 | Troyes | 695 | 4.891
81 | Javairô Dilrosun | 9 | Bordeaux | 675 | 4.845
82 | Hamari Traoré | 2 | Rennes | 1878 | 4.836
83 | Przemyslaw Frankowski | 3 | Lens | 1191 | 4.778
84 | Wissam Ben Yedder | 9 | Monaco | 1625 | 4.685
85 | Vincent Le Goff | 3 | Lorient | 1440 | 4.669
86 | Valentin Rongier | 2 | Marseille | 899 | 4.668
87 | Angelo Fulgini | 9 | Angers | 751 | 4.661
88 | William Saliba | 5 | Marseille | 1800 | 4.652
89 | Boubacar Kamara | 4 | Marseille | 1497 | 4.636
90 | Nuno Mendes | 3 | Paris Saint-Germain | 1246 | 4.599
91 | Jason Denayer | 6 | Lyon | 630 | 4.591
92 | Baptiste Santamaría | 8 | Rennes | 675 | 4.536
93 | Jonathan Gradit | 6 | Lens | 1710 | 4.535
94 | Youssouf Fofana | 8 | Monaco | 1116 | 4.517
95 | Florian Tardieu | 8 | Troyes | 1530 | 4.496
96 | Jordan Lotomba | 2 | Nice | 1410 | 4.454
97 | Mihailo Ristic | 3 | Montpellier | 1150 | 4.439
98 | Warmed Omari | 5 | Rennes | 1710 | 4.407
99 | Melvin Bard | 3 | Nice | 2470 | 4.350
100 | Seko Fofana | 8 | Lens | 1861 | 4.345

Table 11: Ranking of Ligue 1 players in terms of GoT${}^{i}_{90}$.

Team | Formation cluster
---|---
Angers | 3
Bordeaux | 3
Brest | 2
Clermont | 1
Lens | 3
Lille | 2
Lorient | 3
Lyon | 1
Marseille | 1
Metz | 3
Monaco | 1
Montpellier | 1
Nantes | 1
Nice | 2
Paris Saint-Germain | 1
Reims | 3
Rennes | 1
St Etienne | 4
Strasbourg | 3
Troyes | 4

Table 12: The main formation clusters for each team in Ligue 1 in the
2021-2022 season.