Aydın Tiryaki (May 18, 2026)
This model is a 3-layer and multi-dimensional analysis engine developed to mathematically detect radical squad operations, unethical rotations in head coach preferences, and anomalies that undermine competitive fairness in the industrial football ecosystem, while objectively measuring player stability.
Layer 1: Player Data Function
Mathematical Notation: P(t, n, w, i)
This atomic function officially records the physical data produced on the pitch and the legitimacy or eligibility status of each player in a given team for each matchweek played in the league as a digital footprint.
Dimensions and Set Boundaries:
- t (Team ID): Unique codes of the teams in the league from 1 to 18.
- n (Jersey Number): Squad numbers of the players in the club from 1 to 99. When combined with the team code, it functions as a unique player identity (Player ID).
- w (Matchweek): The relevant matchweek of the season from 1 to 34.
- i (Status Index): Status and action codes of the player for that week from 1 to 6.
i Status Index Values and Sharp Precision Standards:
- i = 1 [Played / Appearance]: Can only take the value of 0 or 1. If the player took part in that match, it is loaded as 1. For those who were in the official match squad but did not get playing time, or those who were not in the squad at all, it is 0.
- i = 2 [Starting XI / Line-up]: Can only take the value of 0 or 1. If the player started the match in the starting XI, it is 1; if they were on the substitute bench or not in the squad, it is 0.
- i = 3 [Minutes Played Ratio]: It is a decimal number ranging between 0.0 and 1.0. To eliminate stop-time gaps between matches and external noises in extra time, the active duration the player spent on the pitch is proportioned to the entirety of the match finalized by the referee’s whistle.
- Normalization Formula: P(t, n, w, 3) = Active Duration the Player Spent on the Pitch / Total Duration of the Match Finalized by the Referee’s Whistle
- i = 4 [Suspended]: Can only take the value of 0 or 1. If the player cannot officially play due to a card suspension or federation/disciplinary committee ban, it is 1; if there is no suspension, it is 0.
- i = 5 [Out of Squad]: Can only take the value of 0 or 1. If the player is not injured or suspended but is excluded from the match squad by the technical staff due to club-internal administrative or disciplinary decisions, it is 1; if there is no obstacle, it is 0.
- i = 6 [Injured]: Can only take the value of 0 or 1. If the player cannot play because they are injured or ill according to the official report of the medical staff, it is 1; if they are healthy, it is 0.
Sparse Matrix Feature:
Squad numbers that are vacant or not worn by any athlete during the season remain as empty (Null) records in the database. These cells are not included in the calculation, do not occupy space in memory, and do not slow down the software system.
Concrete Model Examples for Layer 1:
- Example A — P(1, 10, 34, 2) = 1
- Explanation and Meaning: The player with jersey number 10 (Jersey Number = 10) of team number 1 (Team ID = 1) started the match in the starting XI (Status Index = 2) in the 34th matchweek of the league (Matchweek = 34). The value being 1 confirms that this situation occurred.
- Example B — P(3, 7, 20, 3) = 0.85
- Explanation and Meaning: The player with jersey number 7 of team number 3 produced a value of 0.85 in the minutes played ratio breakdown (Status Index = 3) in the match played in the 20th week of the league. If that match lasted a total of 100 minutes including stoppages, it means this player remained actively on the pitch for exactly 85 minutes. Match durations are normalized in this manner.
- Example C — P(2, 99, 15, 6) = 1
- Explanation and Meaning: The value of the player with jersey number 99 of team number 2 in the injury/health breakdown (Status Index = 6) in the 15th week of the league is 1. This condition documents that the player was officially injured according to the medical staff report and that their absence from the squad is based on a medical justification.
Layer 2: Player Cumulative Average Profile
Mathematical Notation: C(t, n, j, i)
It is the structure that produces the weekly average (Cumulative Average) of a football player’s cumulative data in designated past time-series blocks (Windows), independent of individual league weeks. Normalizing all indicators to the 0.0 – 1.0 range allows for a direct comparison between the instantaneous raw match data (P) of that week and the past cumulative data (C) in perfect dimensional harmony.
Dimensions and Set Boundaries:
- t (Team ID) and n (Jersey Number): Fixed club and player identifiers fully synchronized with Layer 1.
- j (Evaluation Series / Time Windows): Time blocks forming the mathematical series from 1 to 3.
- j = 1 [Short-Term]: Cumulative average data of the last 5 weeks preceding the target match.
- j = 2 [Medium-Term]: Cumulative average data of the last 10 weeks preceding the target match.
- j = 3 [Long-Term / Post-Transfer]: Cumulative average data of the entire second half of the season from the closing of the winter transfer window to the target match.
- i (Average Metrics): Indicators giving the cumulative average volume of the player in the selected time block from 1 to 3.
i Breakdown Calculation Logic:
- i = 1 [Average Appearance]: Found by dividing the number of matches the player played in that period by the total number of weeks in that period (Value range: 0.0 – 1.0).
- i = 2 [Average Starting XI]: Found by dividing the number of matches the player started in the starting XI in that period by the total number of weeks in that period (Value range: 0.0 – 1.0).
- i = 3 [Average Minutes Ratio]: Found by dividing the total of the proportional normalized minutes the player received in that period by the total number of weeks in that period (Value range: 0.0 – 1.0).
Concrete Model Examples for Layer 2:
- Example A — C(1, 10, 1, 2) = 0.80
- Explanation and Meaning: The rate of starting in the starting XI (Average Metric i = 2) of the player with jersey number 10 of team number 1 in the short-term time window of the last 5 weeks (Time Window j = 1) is 0.80. This numerical data expresses that the player started in the starting XI in exactly 4 out of the last 5 matches, meaning they are part of the short-term core framework of the team.
- Example B — C(4, 5, 2, 3) = 0.95
- Explanation and Meaning: The average on-pitch duration ratio per match (Average Metric i = 3) of the player with jersey number 5 of team number 4 in the medium-term period of the last 10 weeks (Time Window j = 2) is 0.95. This data demonstrates that the player was almost never substituted out in the last 10 weeks, remaining on the pitch for 95% of the matches’ duration and serving as the physical leader of the team.
Layer 3: Match Integrity Function
Mathematical Notation: M(t, w, j, i)
This is the highest level of decision-making, auditing, and macro analysis organ of the model. In a designated matchweek w, it individually scores the extent to which the squad fielded by a team t (up to 16 players receiving playing time in the match) remains loyal to its own past cumulative average stability curve on a team scale. All output values in this layer are now completely scaled between 0.0 and 1.0.
Macro Formulation:
M(t, w, j, i) = [Sum of C(t, n, j, i) values of all players receiving playing time in week w] / Total number of players receiving playing time in that match
3 Strategic Macro Reports Produced by i Breakdowns:
- M(t, w, j, 1) — Squad Experience and Familiarity Score (0.0 – 1.0): Shows how much league rhythm and mutual playing practice the fielded players possess at a macro level.
- M(t, w, j, 2) — Core Squad Loyalty Score (0.0 – 1.0): Documents to what extent the team fielded its ideal core starting squad in that week. Values close to 0.0 are mathematical proof that a radical substitute operation was executed.
- M(t, w, j, 3) — Physical Readiness and Conditioning Score (0.0 – 1.0): Shows the team-wide average of the past average on-pitch duration ratios of the fielded players. It measures how physically prepared and rhythmic a squad quality the team possessed on the pitch.
Legitimacy Filter and Caught-in-the-Act Mechanism:
The system checks the excuse indexes of core players who are suddenly dropped while performing the analysis. If at least one of the values of i = 4 (Suspended), i = 5 (Out of squad), or i = 6 (Injured) in Layer 1 for a non-playing core player is 1, this absence is accepted as legitimate and the team is not subjected to a score penalty. However, if all of these excuse indexes are 0 but the player is suddenly dropped in the final week, and a player whose past C value is close to zero is fielded instead, the value of the M function rapidly drops, and the system directly generates a “Squad Deformation Anomaly” violation report.
Concrete Model Examples for Layer 3:
- Example A — M(1, 34, 1, 2) = 0.45
- Explanation and Meaning: The Core Squad Loyalty Score (Average Metric i = 2) calculated according to the past 5-week reference (Time Window j = 1) for team number 1 in the 34th week (Matchweek w = 34), which is the final week of the league, came out as 0.45. This data is a caught-in-the-act document; because the average of entering the starting XI in the last 5 weeks for the players who played in that match is 45%. In other words, the team suddenly benched its core players who carried the team recently without any legitimate excuse (injury/suspension) and fielded a radical substitute squad.
Pre-Match Live Audit and Early Warning Mechanism
This model’s most radical innovation is not merely performing a retrospective forensic analysis after the match is over, but acting as a “Deterrent and Preventive Security Radar” before the match begins.
Working Principle and Mathematical Projection:
According to official rules, the starting XI lineups and substitute lists of both teams are entered into the federation system and announced to the public a certain period before the match starts (usually 60 minutes before the opening whistle). The software is automatically triggered as soon as the lineups are announced. The system immediately runs the “Pre-Projectional M Score” simulation based on the assumption that the match will be completed with this squad structure, encompassing all players announced to start in the starting XI and the substitutes on the list.
Gap and Malicious Intent Detection:
If, as a result of this early calculation, a massive mathematical gap or a structural collapse is detected between the 9 macro values obtained before the match starts and the short-term stability baseline especially for the last 5 weeks (j = 1), the malicious intent, lack of motivation, or tendency to disrupt competitive integrity within the team is clearly deciphered before the opening whistle even blows.
Sanction and Regulation Power:
It is of vital importance that this mechanism is written into regulations as a formal sporting fair play rule in written football regulations. As soon as the lineups are entered, the system sends an automated “Squad Deformation Alarm” and an official warning mechanism to the club: “This squad choice has exceeded the mathematical anomaly threshold according to your club’s recent stability curve; proceeding to the match with this squad structure will trigger the automatic points deduction or financial sanction process.”
In this way, a mathematical sanction threat creates a binding protective shield over clubs without requiring the subjective intent reading or initiative of any administrative board. If the club insistently continues to field that deformed squad despite this warning, the anomaly report generated by the system directly prepares the legitimate ground for legal and sporting punitive sanctions.
Real-Time System Dynamics
The designed model functions as a Live Audit Dashboard within a software integration rather than just being a static database. The operational mechanism follows this automation sequence:
- Instant Data Entry: As soon as the match ends, the identities of up to 16 players who took part on the pitch, their net durations spent on the field, and the medical or administrative excuse codes (i = 4, 5, 6) of the core players who were excluded from the squad are entered into the software.
- Automated Processing: The system takes Layer 1 raw data and collides it with Layer 2 cumulative averages in its memory within seconds.
- 9-Dimensional Matrix Output: Exactly 9 independent audit indexes for that week come to life on the screen for each team.
In-System Comparison Options
The software allows for real-time comparison and analysis of this generated 9-dimensional data across three different scientific axes:
- Vertical Analysis (Comparison with Own History): The team’s values for that week are compared with its past stability curve. A sudden drop in a loyalty score that has been trending around 0.95 throughout the season is flagged as an anomaly.
- Horizontal Analysis (Comparison with League Rivals): The squad integrity scores of rivals who have a direct conflict of interest in the championship or relegation battle within the same week are brought side-by-side to audit competitive equality.
- Curve Analysis (Comparison of Time Windows): The gap between the short-term rhythm of the last 5 weeks (j = 1) and the long-term rhythm of the entire second half of the season (j = 3) is examined to reveal whether the squad change is a temporary tactical rotation or a permanent relaxation.
By-Products
- Macro Dimension (Audit): It audits the suspicious squad choices of clubs in the final weeks, protects the competitive integrity of the league, and presents an objective “Sporting Fair Play” audit mechanism to UEFA/FIFA boards.
- Micro Dimension (Performance Analytics): It functions as an “Indispensability Index” measuring the real contribution of football players to the club, a “Reliability Score”, and a “Hidden Form Drop Detection Motor” that foresees administrative crises between the head coach and the player.
Conclusion: 9-Dimensional Output and Statistical Analysis Phase
Through this established mathematical architecture, a total of 9 “Macro Competition Integrity Scores” (3 time windows × 3 status indicators) are calculated for each match in every remaining league week for the teams.
As soon as the squad lineups are entered into the software, all past stability references of the players who played and did not play are compiled, and these 9 values (all normalized between 0.0 and 1.0) are generated instantaneously. With this phase, the data science and structural modeling task is completed.
The process after this phase is the scientific evaluation stage of this rich 9-dimensional data set and falls entirely within the expertise of statistical science. Examining the internal variances of teams, calculating week-to-week standard deviations, determining scientific thresholds for detecting anomalies (outliers), and proving unethical rotations through statistical models are now at the discretion of statisticians and data analysts who will process this data.
| aydintiryaki.org | YouTube | Aydın Tiryaki’nin Yazıları ve Videoları │Articles and Videos by Aydın Tiryaki | Bilgi Merkezi│Knowledge Hub | ░ Virgülüne Dokunmadan │ Verbatim ░ |░ Futbolda Kadro İstikrarı ve Rekabet Denetimi Teorik Modeli │Squad Stability and Competition Audit Theoretical Model in Football ░ 18.05.2026
A Note on Methods and Tools: All observations, ideas, and solution proposals in this study are the author’s own. AI was utilized as an information source for researching and compiling relevant topics strictly based on the author’s inquiries, requests, and directions; additionally, it provided writing assistance during the drafting process. (The research-based compilation and English writing process of this text were supported by AI as a specialized assistant.)
Aydın Tiryaki 