Analysis and Parameterization of Sports Performance: A Case Study of Soccer

Abstract

The importance of Big Data and the analysis of this data in recent years is indisputable, and this boom has spread to all areas of life, including professional sports and, within this, soccer. The significant amounts of money involved in this sport have led to the need for the top clubs to employ these techniques to gain a competitive advantage over their competitors. Despite this, there is very little information on how these tools are used or what parameters they consider. Similarly, there are a multitude of amateur analyses that offer very few conclusions. They simply focus on collecting and presenting the data in the form of a comparison without any analysis or pre-processing. This work describes the implementation of an expert system based on fuzzy logic used to evaluate the talent of a soccer player at all levels, his/her aptitude and attitude, to face his/her individual and collective professional development. For this purpose, the above aspects will be evaluated specifically in the different aspects of the game, which will allow us to evaluate the performance of a soccer team and thus determine the probability of victory, draw, and defeat in a confrontation.

1. Introduction

Today, the impact of professional sports, particularly soccer, on every facet of society is substantial. Economically speaking, the capital tied to sports is significant, resulting in clubs and top-tier athletes deploying vast resources to remain competitively relevant and achieve the best performance, often surpassing established financial capacities [1]. It is unequivocal that data science is proliferating across all life domains. As one would expect, its application has also penetrated the sports industry, especially soccer, with elite teams leveraging it to gain a competitive advantage [2]. Given that data analysis inherently involves handling numerous variables, adopting these tools in professional clubs yields scarce information about their modus operandi or the parameters utilized because any disclosure could jeopardize the team. Analogously, numerous amateur-level analyses and academic studies focus merely on raw data compilation and present them as is or employ them for comparisons without any initial statistical treatments [3,4,5]. In view of the foregoing, there is a growing necessity to harness the potential of Big Data in conducting analysis, designing, and implementing an expert system (ES) that will enable assessment of a player’s or team’s prowess based on specific sensitive, analyzable, and parameterizable factors. In this article, a fuzzy system is proposed as an effective solution for rating a player’s aptitude in various game aspects or determining the probabilities of winning, drawing or losing for their team in a clash. In summary, this study proposes the following objectives:

• Analyzing, designing, and implementing an ES that enables the performance of a soccer player to be parameterized based on raw data collected and statistics from every match.
• Comparing teams based on the parameterization of player characteristics to estimate probabilities of a match’s victory, draw, or defeat.
• Evaluating a player’s talent per parameter established using fuzzy logic functions to categorize players.
• Designing study findings’ visualization that portrays results in a simple, precise, and uncluttered fashion.

Given these objectives, this research represents the work carried out on the design and implementation of a fuzzy logic-based ES that allows the performance of soccer players and teams to be parameterized, facilitating the estimation of an optimal team composition and prediction of a clash’s outcome based on these parameters.

The remainder of the article is organized as follows. Section 2 presents a review of the work related to the proposal put forward in the article. Section 3 introduces the theoretical framework in reference to ESs and fuzzy logic. Then, Section 4 describes the data used for experimentation and the process followed for their treatment. Section 5 describes the application of fuzzy logic functions to our problem. Section 6 shows the system of visualization and presentation of results. Section 7 shows the discussion and a comparison with other forecast tools, and finally, Section 8 shows the conclusions obtained in this work.

2. Related Work

Once defined the research work aim and scope it is necessary to explore other similar works in order to see the degree of innovation of our approach and also to compare the results obtained. There are several initiatives exploring similar issues, such as those listed below:

• Predictive systems about issues of soccer matches, such as a simple framework based on scoring models [6], capable of obtaining accurate forecasts for binary outcomes in soccer matches, are proposed. In this sense and in order to analyze the usefulness of the proposed model, experiments were conducted with the English Premier League and the Italian Serie A, where certain events such as red cards in the match, the score under/over, and the goal/no goal outcome are predicted. Other work related to this kind of prediction is described in [7]. It mainly highlights the effectiveness of the black-box modeling capability of neural networks in the case of soccer match outcome prediction. As the authors indicate, input parameter selection is a serious problem in soccer match prediction systems based on neural networks or statistical methods. In order to do so, they tested two different input vector parameters by learning vector quantization networks in order to emphasize the importance of input parameter selection.
• The use of mathematical methods, including betting odds, for performance analysis, such as the work proposed by [8], defines an approach that uses the information included in betting odds in a soccer forecasting model based on the well-known ELO (proposed by Elo, Arpad E.) ranking system. In this way, the authors show how the proposed model can help to obtain valuable information about the quality of soccer teams and their evolution over time, which is of practical benefit in performance analysis.
• The application of statistical modeling of soccer match results [9] is quite popular among researchers in both academia and industry, not only because of the potential for financial returns but also because of the challenges presented by such modeling. However, the modeling of scores and results, as well as other match outcomes, both simple (e.g., the first player to score) and unusual (e.g., number of player cautions), motivated by the search for inefficiencies in the betting market, has been an important motivating factor in the development of state of the art. The authors of [9] describe the state of the art in the statistical modeling of match outcomes and scoreboards in particular.
• Use of a fuzzy knowledge base relying on the outcome of previous matches [10]. The conclusions presented in this research indicate that it is possible to predict the match outcome based on previous results. Other research related to this topic is presented in [11], which describes an improved system that uses fuzzy logic and ANN (Artificial Neural Networks) to assist managers in the team selection operation. The research results show that the proposed decision support system improves the accuracy in determining the player selection decision.
• Expert support in the formation of a soccer team. The authors of [12] proposed a model to select the best formation of a soccer team through two phases. In the first phase, the players are chosen through fuzzy ranking, selecting the best players, while in the second phase, the best formation is chosen by evaluating the different combinations of the players. Although the judgments of the coaches are essential in evaluating players, this model can help coaches make decisions.

On the other hand, sports performance optimization software tools are engineered to assist athletes, coaches, and sports teams in enhancing their performance and attaining their objectives across various sporting disciplines [13,14]. These applications bear certain common characteristics [15]:

• They incorporate data collection tools suitable for different sporting activities, either individual or group-based. Such tools gather game statistics, physical performance metrics, and biometric data, among others, depending on the nature of the sport.
• Upon data collection, these software systems perform detailed analyses. These analyses allow coaches and athletes to enhance various individual or team parameters with features like visualization of game patterns, potential tactical adjustments, and temporal assessment of athletes’ progress.
• Additionally, this software type enables coaches to construct specialized training programs for each athlete, aiming to boost their individual and team performance.
• Another shared feature beneficial to coaches and sports managers is the live monitoring of athletes. This allows for real-time observation of athletic performance during training and competitions through different sensors and via artificial vision [16].
• A crucial feature associated with sports performance in any discipline is injury management. As such, these tools must document athletes’ injuries, their medical history, and the progression of these injuries. This allows coaches and physicians to make well-informed decisions on the athletes’ availability and, consequently, their performance.
• It is essential to note that depending upon the tool used, these features can vary significantly. Some tools include functionality such as nutrition and rest control, collaborative work, and image analysis, among others. This could potentially increase the cost and complexity of these tools.

It is universally acknowledged that unique business models and analytics are employed by each sports management entity based on their individual requirements. Nonetheless, certain commercially available tools have been extensively utilized in research studies, including the following:

• Catapult Sports [17]: This platform in the lastest versions (Vector Live 2.8; OpenField Console 3.9; OpenField Cloud 4.7) offers sports performance tracking through wearable technology and data analysis software. Sports like soccer, rugby, and basketball exploit it to keep a tab on athletic performance parameters.
• Hudl (latest version 1.16.1) [18]: This software, primarily employed in football, basketball, and soccer, facilitates the analysis of sports videos to evaluate game strategies and plays.
• PlayerTek (latest version 1.2.23) [19]: This system, which offers GPS-based tracking, is extensively utilized in soccer and similar sports to accumulate real-time physical performance data such as speed, distance covered, and acceleration. Based on these data, custom training can be organized for athletes, aiding in their tactical and biomechanical betterment.
• Coach’s Eye (latest version 6.6.0.0) [20]: This video analysis application is favored among athletes and coaches, aiding them in scrutinizing and refining their techniques and strategies across various sporting disciplines.
• Fusion Sport SMARTSPEED (1.12.7) [21]: This software is employed predominantly in track and field sports and soccer and measures the speed and reaction time of athletes, assisting in their speed and mobility training.
• InStat (latest version 2.1.44) [22]: The InStat Scout serves as a digital analyzer of team and player performances, as well as referees in worldwide popular sports such as soccer, basketball, ice hockey, and handball. The platform hosts a myriad of team and individual statistics like passing accuracy, among others, and is unique in its ability to correlate these stats to specific video instances.

Current technologies in the market involve monitoring athletes’ biometrics and performance metrics for data analysis, which can often be intricate to acquire due to sensor requirements. This work centers on establishing an ES capable of processing raw data amassed from each game to parameterize a footballer’s performance and predict potential outcomes of matches, such as a victory, draw, or defeat. This system also has the capability of visualizing both the individual’s performance and all evaluated parameters at the player and team level.

3. Theoretical Framework

The work elucidated herein is predicated on methodologies that are extensively employed in decision-making processes, encompassing ESs and fuzzy logic.

3.1. Expert Systems

Expert systems are a significant tool when it comes to providing decision-making support. Their design allows them to facilitate tasks across diverse application fields, as they yield results equivalent to those provided by a specialist. They emulate human decision-making skills according to contextual conditions by artificially imitating the performance of an individual considered an expert in a particular knowledge domain or activity field [23,24,25]. These systems employ either an inductive or deductive approach to arrive at a conclusion, following the analysis of a series of facts or circumstances. These are problems that, if solved by a human, would necessitate the intervention of an expert with specific knowledge of the subject or discipline from which the problem originates. Their goal is to assist in finding the optimal solution to a specific problem without requiring the input of a subject matter expert. Such a procedure can be performed by the ES even in the presence of incomplete data, as these systems often rely on qualitative rather than quantitative data. They employ what is referred to as ‘fuzzy logic’, an “approximate” reasoning system that ultimately leads to highly probable results [26]. Speaking from a structural perspective, an ES comprises several fundamental components, as outlined in Figure 1 and elaborated on below [25]:

Knowledge acquisition system: Refers to the accumulation, transfer, and transformation of experience from a knowledge source to a software model, with the purpose of developing or enriching the knowledge base. Currently, the most advanced practice involves the collaboration of a knowledge engineer working together with one or more human experts to build this knowledge base.

Knowledge base: It incorporates the essential knowledge to understand, pose, and solve problems, consisting of two fundamental components: a special heuristic and rules that guide the application of knowledge in problem-solving within a specific domain.

Factual base: This is an active memory that stores data and facts related to a specific problem, storing the necessary information to address the problems to be solved.

Inference engine: This component, often referred to as the “brain” of the ES, also known as the control structure or the rule interpreter, plays a key role in its performance. In essence, it is composed of algorithms that offer methods for reasoning about the information contained in the knowledge base, providing guidelines on how to use the system’s knowledge to establish the calendar that organizes and supervises the necessary steps to solve a problem when a query is performed. It consists of three main elements:

• Interpreter: executes the selected calendar.
• Scheduler: maintains control over the calendar.
• Consistency control: attempts to maintain a consistent representation of the solutions found.

Justification subsystem: Its main function is to describe how the ES works when finding a solution, making it easier for users to ask the system questions to understand the reasoning patterns it has followed. This is particularly useful for non-expert users who wish to learn how to perform a specific task.

Considering these subsystems, there are multiple types of ES, which differ based on the reasoning method they employ.

Rule-based expert systems: These ESs apply these rules and compare results. They implement a logical inference that begins with established evidence in a specific situation and moves toward finding a cure or solution. Alternatively, they can begin with hypotheses about potential solutions and work backward to identify or deduce existing evidence that supports a particular hypothesis.

Case-based expert systems: Solve new problems based on previous solutions to similar problems. For instance, a physician might treat an ailment by recalling a case where a patient demonstrated similar symptoms. Lawyers also use this method when they cite legal precedents to make arguments. Moreover, biomedical engineers can implement case-based reasoning by using nature as a database of potential solutions. This approach is very powerful for both computational and routine problem-solving, and it has even been suggested that all reasoning is based on prior experiences.

Expert systems underpinned by Bayesian networks: They use a probabilistic graphical model that showcases a set of random variables and their dependent relationships via a directed acyclic graph. It can be used, for example, to model probabilistic relationships between diseases and symptoms and allows for calculating the probabilities of the occurrence of several diseases given a set of symptoms.

Fuzzy Expert Systems: They are built on fuzzy logic, a technique that deals with uncertainty. The system is based on the mathematical modeling of fuzzy sets, and it mimics the human reasoning process, enabling computers to make decisions that may not be precisely accurate but are more logically sound compared with other methods. Fuzzy Expert Systems are commonly used in situations where decision-making is not binary but consists of a wide array of possibilities.

3.2. Fuzzy Logic

Today, traditional logic is struggling to navigate numerous challenges in the sphere of Artificial Intelligence (AI). The human cognitive process is inherently ambiguous, lacking clear-cut distinctions of true or false. However, computing systems historically employ this form of conventional logic. The impediment of classical logic underscores the relevance of fuzzy logic [27,28]. Fuzzy logic is a component of AI that equips computers with the facility to examine real-world data on a graduated scale between false and true, manipulate nebulous concepts and provide engineers with the means to create devices that evaluate hard-to-define information. Fuzzy logic, within the context of AI, helps resolve a wide range of issues, predominantly those tied to intricate industrial process control, decision-making systems, and data resolution and compression. Systems built on fuzzy logic can mirror human decision-making processes with the added benefit of superior speed. They are typically robust and capable of handling inaccuracies and noise in the input data. The time factor plays a significant role because these control systems may require feedback within specified timelines, making previous data essential for an average evaluation of prior circumstances. Fuzzy logic endeavors to craft mathematical approximations for resolving issues by generating precise outcomes from imprecise data, making it particularly beneficial in electronics and computation. The term “fuzzy” is a nod to the inherent incertitude associated with the non-deterministic truth values employed. The concept of fuzzy sets was first introduced by Lofti A. Zadeh in his 1965 research paper titled “Fuzzy Sets” [29]. This publication establishes the principles of fuzzy logic extrapolated from the definition and properties of fuzzy sets.

“Let X be a classical set. A fuzzy set, A, in X is characterized by the membership function fA(x), which associates to each point x ∈ X a real number from the interval [0, 1], where the values of fA(x) represent the degree of membership of x in A, such that, the closer the value of fA(x) is to 1, the greater the degree of membership of x to A”.

In view of the above, the membership functions utilized will be assigned based on the particular features of the reality to which they correspond. Nonetheless, some classical functions, as suggested in [30], are often employed. From a pragmatic perspective, the requirement is to convert vague linguistic ideas into fuzzy numerical variables. Following this, they need to be amalgamated utilizing the fuzzy logic operators, derive every rule’s deduction in a numerical format, compile the different reactions of the distinct rules, and finally return from the gathered numerical response to the linguistic one. This procedure complies with a model scheme that facilitates the management of inference rules on fuzzy sets, which is succinctly illustrated in Figure 2:

• Fuzzifier: this component modifies clear, definitive numerical inputs into indeterminate variables using a process that we call fuzzification.
• Knowledge base (fuzzy rules): this is essentially a repository where all the expert-derived principles are saved.
• Inference engine: this module emulates human cognitive processes by making logical deductions based on received inputs and rule sets.
• Defuzzifier: this final section takes the ambiguously determined variable created by the inference engine and converts it back to a clear numerical value for future use.

4. Dataset and Parameterization

For the establishment of the dataset, we have conducted an examination of various European football leagues, including the Premier League, Serie A, Ligue 1, Bundesliga, and La Liga Santander [31,32,33]. We chose to focus on data from the 2021–2022 La Liga Santander season, as our prior knowledge would be beneficial in modeling different aspects of the ES. We collected data from a group of 617 players who participated for at least one minute with their clubs during that season, including the 20 teams that took part in the tournament. Notably, there were several instances of players moving between two different clubs during the season. In these cases, their performance at each club was evaluated independently. Information is detailed below regarding the data collected, their distribution, and the importance given to each raw data in its specific context, determined as the weight. These were determined based on our previously gained knowledge, models already developed, and adjustments made considering the test results [34,35,36,37].

Table 1. Identification Data.

Parameter	Description	Abbreviation
Id	Player identificator	id
Name	Player’s name	name
Nick	Nickname	nick
Date of bird	Date of bird	date_bird
Country	Nationality	country
Demarcation	Demarcation	demarcation
Club	Club	club
Height	Height	height

identifies the players’ general data. From this data, we know the origin, club, and name of the player, among others.

Table 2. Defense parameters and sub-parameters.

Defense
Tackle
Sub-parameter	Weight
Tackles won × 90 min	20
% tackles success	40
Tackles off dribble won × 90 min	10
% tackles off dribble success	30
Pressure
Sub-parameter	Weight
Pressures gained × 90 min	20
% success in pressure	80
Blocking
Sub-parameter	Weight
Blocks × 90 min	25
Blocked shots × 90 min	40
Shots between the three posts × 90 min	15
Blocked passes × 90 min	20
Interception
Sub-parameter	Weight
Interceptions × 90 min	100
Clearances
Sub-parameter	Weight
Clearances × 90 min	100
Recovery
Sub-parameter	Weight
Recoveries × 90 min	100

specifies the defensive parameter and the sub-parameters that define it with their respective weights so that every player has a score on his defensive performance. The percentage on a sub-parameter indicates how accurate the player is in that particular action.

In

Table 3. Psychological Aspects.

Psychological Aspects
Cards and fouls committed
Sub-Parameter	Weight
Yellow cards × 90 min	30
Second yellow cards × 90 min	15
Red cards × 90 min	15
Fouls committed × 90 min	40
Offside
Sub-parameter	Weight
Offside × 90 min	100
Mistakes
Sub-parameter	Weight
Errors leading to kicks × 90 min	100
Penalties committed
Sub-parameter	Weight
Penalties committed × 90 min	100
Own goals
Sub-parameter	Weight
Clearances × 90 min	100
Experience
Sub-parameter	Weight
Own metric	100

, psychological data pertaining to the player on the field is specified. Factors considered include errors made, fouls committed, cards received, and any other events that could potentially impair the player’s performance due to psychological reasons.

When assessing a player’s experience, a range of metrics derived from [38] have been utilized. It is apparent that there is a decline in physical performance in athletes over the age of 30, becoming even more pronounced when they exceed 35 years. However, prior to these age milestones, no significant disparities in performance were observed other than potential strength enhancements. Conversely, older players appear to excel in technical and tactical proficiency compared to their junior counterparts. A 3–5% increase in successful pass rates is observed in players over 30 as opposed to those aged between 16 and 29. The hypothesis is that the physical decline in older players could be offset by enhancements in other skills, such as decision-making or game intelligence. Regarding the apex of a soccer player’s career, considering both physical and mental aspects, it is noted that it coincides mostly with ages 27–28. Therefore, it could be deduced that players reach their prime at those ages, as illustrated in Figure 3.

Table 4. Distribution aspects.

Distribution
Short passes
Sub-parameter	Weight
Passes completed × 90 min	60
% passes completed	40
Medium passes
Sub-parameter	Weight
Passes completed × 90 min	60
% passes completed	40
Long passes
Sub-parameter	Weight
Passes completed × 90 min	60
% passes completed	40
Progressive passes
Sub-parameter	Weight
Progressive passes × 90 min	75
(progressive distance passes/distance passes)	25
Changes of play
Sub-parameter	Weight
Changes of play × 90 min	100
Centers
Sub-parameter	Weight
Cross passes × 90 min	40
Crosses into the opponent’s box × 90 min	60
Assist (expected vs. actual)
Sub-parameter	Weight
Assists × 90 min	80
(assists-fail_assists)	20
Key passes
Sub-parameter	Weight
Passes that generate shots × 90 min	30
Passes in last 1/3 × 90 min	10
Passes in opponent’s area × 90 min	30
Passes with the ball in play that generate a shot × 90 min	10
Set-piece passes that generate a shot × 90 min	5
Passes with the ball in play generating goal × 90 min	10
Set-piece passes that generate goal × 90 min	5
Passes under pressure
Sub-parameter	Weight
Passes under pressure × 90 min	75
(passes under pressure/total passes attempted)	25

delineates the various sub-parameters affecting each player’s field distribution during play. It offers statistical information on aspects such as passing, turnovers, and assists, all of which contribute to the fluidity of a team’s defensive and offensive maneuvers. The percentage associated with each sub-parameter reflects the precision demonstrated by a player in executing the specified action.

Table 5. Skill aspects.

Skill
Dribbles
Sub-parameter	Weight
% successful dribbles	15
Successful dribbles × 90 min	40
Players dribbled × successful dribbles	15
Dribbles that result in a shot × 90 min	15
Dribbles leading to goal × 90 min	15
Drives
Sub-parameter	Weight
Driving × 90 min	15
Progressive driving × 90 min	10
Dribbling that enters the last 1/3 × 90 min	15
Drives into the opponent’s area × 90 min	15
(driving distance/No. of drives)	15
(progressive driving distance/No. of drives)	10
(progressive driving distance/driving distance)	20
Losses
Sub-parameter	Weight
Control errors × 90 min	80
Errors by rival tackle × 90 min	20
Receptions
Sub-parameter	Weight
% passes successfully received	30
Passes received × 90 min	40
Progressive passes received × 90 min	30
Touches
Sub-parameter	Weight
Touches × 90 min	70
% touches with ball in play	30
Left-handed/Right-handed ability
Sub-parameter	Weight
(|left-handed passes − right-handed passes|)/(left-handed passes + right-handed passes)	100
Fouls received
Sub-parameter	Weight
Assists × 90 min	80
(assists-fail_assists)	20
Key passes
Sub-parameter	Weight
Fouls received × 90 min	70
Fouls conceded, leading to a shot × 90 min	20
Fouls conceded, leading to goals × 90 min	10

outlines a set of sub-parameters focused on the technical aptitude of each individual player. These specific sub-parameters facilitate the assessment of a player’s skill level. The calculated percentage linked to each sub-parameter serves as an indicator of a player’s accuracy in executing a specific action.

Table 6. Physical condition aspects.

Physical Condition Aspects
Aerial duels
Sub-parameter	Weight
% aerial duels won	60
Aerial duels won × 90 min	40
Height
Sub-parameter	Weight
Height percentile	100
Minutes played
Sub-parameter	Weight
Minutes played	70
Minutes played per game	15
Games completed	15
Age/Injuries/Recovery
Sub-parameter	Weight
Own metric according to age (0–100)	100
Speed
Sub-parameter	Weight
Pace	100
Strength
Sub-parameter	Weight
Strength	100

presents the findings associated with the player’s physical performance. These sub-parameters hold significant importance in terms of physical endurance and power, factors that directly influence a player’s performance, particularly towards the final stages of a match or during any potential extended play due to unexpected occurrences during the game.

Table 7. Attack aspects.

Attack
Shooting opportunities
Sub-parameter	Weight
Shooting opportunities generated × 90 min	60
Shots that generate rebound and subsequent shot × 90 min	25
Defensive actions that generate a shot × 90 min	15
Goals generated
Sub-parameter	Weight
Goal creation actions × 90 min	75
Defensive actions that generate goals × 90 min	15
Shots that generate a rebound and goal × 90 min	10
Penalties
Sub-parameter	Weight
% effectiveness in penalties	100
Goals
Sub-parameter	Weight
Goals × 90 min	80
(goals-failed_goals)	20
Kicks
Sub-parameter	Weight
Shots × 90 min	30
% shots on goal	50
Average distance shots	20

illustrates varying parameters pertaining to a patient’s performance in attack. In consideration of this, it is pertinent to account for variables that qualify all forms of interactions with the opponent’s goal. The percentage on a sub-parameter signifies the level of precision a patient exhibits in that specific maneuver.

In conclusion,

Table 8. Aspects of goalkeeper.

Goalkeeping
Goal against
Sub-parameter	Weight
Goals against × 90 min	35
(goals against − failed_goals against)	35
(clean sheets/minutes as goalkeeper)	30
Saves
Sub-parameter	Weight
% shots saved	70
Saves × 90 min	30
Penalties saved
Sub-parameter	Weight
% penalties saved	90
% penalties missed	10
Interception of crosses
Sub-parameter	Weight
% crosses intercepted	70
Intercepted crosses × 90 min	30
Defensive actions
Sub-parameter	Weight
Defensive actions × 90 min	80
Average distance of actions	20

outlines the parameters required for evaluating the performance of goalkeepers. These parameters are exclusively considered when a player’s position is that of a goalkeeper. The percentage quoted under each sub-parameter demonstrates the accuracy of the player in executing that specific task.

Therefore, based on the predefined criteria for measuring a player’s experience and physical fitness, we have devised certain metrics intended to assess these factors relative to the athlete’s age. Our aim is for these metrics to fall within the range of [0, 100], and the results we attain are represented in the values found in

Table 9. Values of physical-mental metrics according to age.

Age	Experience	Physical	%
15	10	50	30%
16	20	60	40%
17	30	70	50%
18	40	75	58%
19	45	80	63%
20	50	85	68%
21	55	90	73%
22	65	95	80%
23	75	95	85%
24	80	100	90%
25	85	100	93%
26	90	100	95%
27	95	100	98%
28	95	100	98%
29	95	95	95%
30	95	92.5	94%
31	100	85	93%
32	100	80	90%
33	100	70	85%
34	100	60	80%
35	100	55	78%
36	100	50	75%
37	100	40	70%
38	100	30	65%
39	100	25	63%
40	100	20	60%
41	100	17.5	59%
42	100	15	58%
43	100	12.5	56%
44	100	10	55%
45	100	7.5	54%

.

It should be noted that by the age of 30, a soccer player is considered to have 100 points of experience and that he is at that same point when the valuation of his physique begins to drop drastically, even more so from the age of 35 onward.

With respect to the physicality of the players, it is considered that they reach their fullness between 22 and 29 years of age in correspondence with the data in Table 9; hence, in that age range, values between 95 and 100 are assigned in terms of physicality. From 15 to 21 years of age, they are still in the process of development.

In order to quantify these factors as a function of age, 3rd-degree polynomial trend lines have been generated (Equations (1) and (2)) from which the following formulas are extracted, where x is the age of the athlete:

$$f_{experience}={0.0062x}^3-{0.7504x}^2+29.955x-293.72$$

(1)

$$f_{physical}={0.0155x}^3-{1.6693x}^2+53.424x-434.14$$

(2)

It is pertinent to highlight that these functions, being automatically generated, will not adhere flawlessly to the predetermined range of [0, 100]. Consequently, any values that fall below zero will be rounded up to 0, while those exceeding 100 will be reduced to 100. The concluding phase of the parameterization procedure involves assigning an overall score to each unit. We will utilize this comprehensive mean in subsequent calculations to estimate the likelihood of winning, tying, or losing a game. We will derive this team rating in reference to the Algorithm 1 as delineated below:

Algorithm 1: Calculation of the overall average of a team

INPUT: team_name OUTPUT: average_list INICIO:    average_list <- []    ids <- pids_team(team_name)    score_gk, min_gk <- 0                   //goalkeeper rating    score_def, min_def <- 0                 //defense rating    score_mid, min_mid <- 0              //middle rating    score_att, min_att <- 0                   //attacker rating    score_overall, min_overall <- 0    //overall rating    media <- 0    FOR EACH id IN ids DO:       pos <- retur_data(“overall”, “position”, id)       min_player <- retur_data(“physical”, “min_player”, id)       stats <- calc_stats(id, pos)       avg_player <- stats       IF (pos == “GK”) THEN:          min_gk = min_gk + min_player          score_por = score_por + (avg_player × min_player)       ELSE IF (pos == “DF”) O (pos == “LT”) THEN:          min_def = min_def + min_player          score_def = score_def + (avg_player × min_player)       ELSE IF (pos == “MCD”) O (pos == “MC”) O (pos == “MCO”) THEN:          min_mid = min_mid + min_player          score_mid = score_mid + (avg_player * min_player)       ELSE IF (pos == “EXT”) O (pos == “DC”) THEN:          min_att = min_att + min_player          score_att = score_att + (avg_player × min_player)       min_overall <- min_overall + min_player       score_overall <- score_overall + (avg_player × min_player)    score_gk <- (score_por/min_por)    score_def <- (score_def/min_def)    score_mid <- (score_med/min_med)    score_att <- (score_del/min_del)    score_overall <- (score_grl/min_grl)    average <- ((score_gk × 0.1) + (score_def × 0.15) + (score_mid × 0.2) + (score_att × 0.3) + (score_overall × 0.25))    average_list <- [average, score_gk, score_def, score_mid, score_att, score_overall] RETURN: average_list

As detailed in the algorithm, the overall team average is calculated from the scores by lines (goal, defense, midfield, and forward) and another average score of the team’s players. All this is determined considering the minutes played by each player. Considering the above, each of these scores is given a specific weight for the calculation of this team rating.

5. Fuzzy Logic Applied Functions

5.1. Fuzzy Logic for Players

Equations (3)–(7) show the membership functions to apply fuzzy logic to the different parameters and thus determine the player’s talent in each aspect of the game:

$${f\left(x\right)}_{very bad}=\left\{\begin{array}{c}1, if 0\le x<20 \\ \frac{40-x}{20}, if 20\le x<40\end{array}\right.$$

(3)

$${f\left(x\right)}_{bad}=\left\{\begin{array}{c}\frac{x-20}{20}, if 20\le x<40 \\ \frac{60-x}{20}, if 40\le x<60\end{array}\right.$$

(4)

$${f\left(x\right)}_{average}=\left\{\begin{array}{c}\frac{x-40}{20}, if 40\le x<60 \\ \frac{80-x}{20}, if 60\le x<80\end{array}\right.$$

(5)

$${f\left(x\right)}_{good}=\left\{\begin{array}{c}\frac{x-60}{20}, if 60\le x<80 \\ \frac{100-x}{20}, if 80\le x<100\end{array}\right.$$

(6)

$${f\left(x\right)}_{elite}=\left\{\frac{x-80}{20}, if 80\le x<100\right.$$

(7)

Thus, a player is classified as (very bad, bad, average, good, or elite) if the degree of participation in that function exceeds a reference value of 0.5, with x representing the score achieved in each criterion, as shown in Figure 4.

5.2. Fuzzy Logic for Games

Equations (8)–(10) show the membership functions defined, and using the difference between the scores of two teams as input to the fuzzy logic functions, the probabilities of victory, tie, and loss in a confrontation will be determined by the following:

$${f\left(x\right)}_{victory}=\frac{1}{1+{\mathcal{e}}^{-(x-\ln \left(2\right))}}$$

(8)

$${f\left(x\right)}_{lost}=\frac{1}{1+{\mathcal{e}}^{-(-x-\ln \left(2\right))}}$$

(9)

$${f\left(x\right)}_{tie}=1-({f\left(x\right)}_{victory}+{f\left(x\right)}_{lost})$$

(10)

Assuming that:

$$x=\frac{{average}_{home}-{average}_{visitor}}{10}$$

(11)

As shown in Figure 5, the win and loss functions are two mirror sigmoids that have a value of 0.$\widehat{3}.$ for x = 0. Therefore, the tie function at that point has the same value of 0.$\widehat{3}$ This is important because at x = 0, the two teams are evenly matched, and the probability of victory, tie, and loss is the same.

Similarly, when the difference is negative (the home team is worse than the away team), the chances of a draw are greater than those of a home win. In the opposite case, when the difference is positive (the home team is better than the away team), the chances of a draw are greater than those of a home defeat.

5.3. Membership of the Team Skeleton

The membership function (12) is defined to apply fuzzy logic to the minutes played by each player and thus determine if he belongs to the skeleton of the team or, in other words, how important his presence is in the team. The variable x in Equation (12) represents the minutes played.

$${f\left(x\right)}_{skeleton}=\frac{x}{3420}$$

(12)

A player is then considered to belong to the skeleton of the team if the degree of belonging to that function is greater than 0.5.

6. Display System

A visualization is designed to represent the parameterization of each player, as showcased in Figure 6. These visualizations present the player’s score in each studied parameter, in addition to the team, position, overall average, area-specific average, birthdate, nationality, and height. Moreover, the visualization’s color varies based on their position (goalkeeper, defender, midfielder, or forward). Regarding the parameters, they are shown in red if considered very poor, yellow if poor or average, and green if considered good or elite according to fuzzy logic functions.

Another visualization that has been designed, as seen in Figure 7, represents the face-off between two teams. In this view, the possibilities of victory, tie, or defeat of the home team concerning the visiting team are presented along with each squad’s lineup. Within the lineups section, positions, nicknames, and each player’s overall averages are depicted together with a symbol indicating whether each player belongs to the team’s main lineup or not.

Table 10. Raking by ES vs. True Ranking.

ES Ranking			Actual Ranking
Rank	Team	Average	Rank	Team	Score
1	Real Madrid CF	75.40	1	Real Madrid CF	86
2	FC Barcelona	68.97	2	FC Barcelona	73
3	Sevilla FC	68.85	3	Atl. de Madrid	71
4	RC Celta de Vigo	68.26	4	Sevilla FC	70
5	Real Betis	68.05	5	Real Betis	65
6	Villarreal CF	67.64	6	Real Sociedad	62
7	Rayo Vallecano	67.12	7	Villarreal CF	59
8	Atl. de Madrid	66.87	8	Ath. de Bilbao	55
9	Real Sociedad	66.22	9	Valencia CF	48
10	RCD Espanyol	65.18	10	CA Osasuna	47
11	Ath. de Bilbao	64.41	11	RC Celta de Vigo	46
12	CA Osasuna	64.38	12	Rayo Vallecano	42
13	Getafe CF	64.07	13	Elche CF	42
14	Granada CF	64.04	14	RCD Espanyol	42
15	Cádiz CF	63.89	15	Getafe CF	39
16	Elche CF	63.43	16	RCD Mallorca	39
17	Valencia CF	63.30	17	Cádiz CF	39
18	Levante UD	61.88	18	Granada CF	38
19	RCD Mallorca	61.09	19	Levante UD	35
20	Deportivo Alavés	60.92	20	Deportivo Alavés	31

shows the ranking at the end of the league by the ES compared to the actual ranking that took place in the 2021–2022 season.

7. Discussion

In this section, a comparison is made with different tools that have been used for the prediction of the results in the 2021–2022 season.

In [39], a prediction is made after each day. In this regard, the prediction made on 29 September 2021 on the final ranking of the teams has been taken. This date was chosen because the model used is fed back each day, so the comparison has to be made without prior knowledge, as it is done in the model proposed in this work.

In [40], the CIES Football Observatory was also used for the comparison. It presents forecasts for the 2021/22 season for the five major European leagues. The statistical model used for this season includes player experience, investments in transfers to build the squads, as well as the performance of each team over the last 365 days.

Considering the predictions made by the tools proposed (

Table 11. Predictions based on other tools.

FiveThirtyEight [39]		CIES Football Observatory [40]		Actual Rank
Rank	Team	Rank	Team	Rank	Team
1	Real Madrid CF	1	Real Madrid CF	1	Real Madrid CF
2	FC Barcelona	2	Atl. de Madrid	2	FC Barcelona
3	Atl. de Madrid	3	FC Barcelona	3	Atl. de Madrid
4	Sevilla FC	4	Sevilla FC	4	Sevilla FC
5	Real Sociedad	5	Real Sociedad	5	Real Betis
6	Villarreal CF	6	Valencia CF	6	Real Sociedad
7	Real Betis	7	Villarreal CF	7	Villarreal CF
8	Ath. de Bilbao	8	Ath. de Bilbao	8	Ath. de Bilbao
9	Valencia CF	9	Real Betis	9	Valencia CF
10	Rayo Vallecano	10	CA Osasuna	10	CA Osasuna
11	RC Celta de Vigo	11	RCD Mallorca	11	RC Celta de Vigo
12	CA Osasuna	12	Levante UD	12	Rayo Vallecano
13	RCD Mallorca	13	RC Celta de Vigo	13	Elche CF
14	RCD Espanyol	14	Granada CF	14	RCD Espanyol
15	Levante UD	15	Getafe CF	15	Getafe CF
16	Elche CF	16	Rayo Vallecano	16	RCD Mallorca
17	Getafe CF	17	RCD Espanyol	17	Cádiz CF
18	Deportivo Alavés	18	Elche CF	18	Granada CF
19	Cádiz CF	19	Cádiz CF	19	Levante UD
20	Granada CF	20	Deportivo Alavés	20	Deportivo Alavés

), the prediction of the classification of this type of data cannot be accurate. The models used for this purpose are often fed back based on the performance of the teams; however, physical and psychological parameters such as those established in the present work are not taken into account. In all predictions, including those made with the proposed ES, teams remain in the actual quartiles; however, there are different outliers that either climb or move to the lower part of the ranking.

In the model developed in this work, there are two clear cases such as Atl. de Madrid, which is positioned in a clearly lower place, as well as RC Centa de Vigo, which is attributed to a position much higher than the real one. It should be noted that the proposed ES only has data on individual players, but the collective capabilities of the team and those of the coaching staff have not been used. We consider that this additional information could be very useful to determine the final classification and obtain a better model fit so that these parameters will be taken into account for future research.

8. Conclusions

In the present work, we have carried out the implementation of an ES that allows us to make estimations about the players of a soccer team and their respective teams. For this purpose, a previous parameterization of each player was carried out, considering data about his physical condition, skills, psychological state, and certain characteristics of his game. Based on this data and the application of different fuzzy logic functions, a series of scores were established, both for each player and for the teams, so that a comparison can be established and an estimate of the result of a match can be made.

Table 10 shows a comparison between the result offered by the ES and the real ranking of the season where this work has been applied so that it is able to distinguish the best and worst team of the 2021–2022 LaLiga Santander season, Real Madrid CF, and Deportivo Alavés. In addition, the ranking of the ES agrees 75% with reality if the top four ranked teams and from 5th to 12th positions are considered. Finally, the ranking of the ES agrees 87.5% with reality if the last eight teams of the championship (from 13th to 20th) are taken into account, and the results obtained in the analysis of these data have been satisfactory and promising. It would be necessary to apply it to other data sets and on other types of sports so that its capacity to make these predictions could be adjusted.

Considering the results obtained, we can state that the performance of a team cannot be calculated as the sum of the individual skills of the players because there are other determining factors that need to be taken into account. The relationship among team members, as well as their ability to work together in a match, are crucial aspects that are not being valued in the ES, and therefore, this would give us an idea not of the collective performance of the team but of the individual performance, i.e., the potential of these players. While it is true that the performance of the ES is promising in terms of the predictions made, it is necessary to continue working on adding these factors, in addition to the coach’s strategy and tactics, as they are fundamental elements that can significantly influence the team’s performance.