A Mathematical Model to Study Defensive Metrics in Football: Individual, Collective and Game Pressures

Performance analysis, utilizing video technology and recent technological advancements in soccer stadiums, provides a wealth of data, including player trajectories and real-time game statistics, which are crucial for tactical evaluation and decision-making by coaches and players. This data allows for the definition of metrics that not only enriches the experience for soccer fans through enhanced visual displays, but also empowers coaching staff and managers to make informed, real-time decisions that directly impact match outcomes. Ultimately, it serves as a pivotal tool for improving team strategy based on comprehensive post-match data analysis. In this article, we present a mathematical model to study the concept of pressure between players and, subsequently, between teams. We first explore the concept in a fixed frame of a match, determining what we call {\sl influence areas} between players. We introduce the unit pressure function and analyze the total number of pressure interactions. Then, we apply these concepts to football matches, considering various factors such as players and the radius of the area of influence, examining pressure efficiency through mean unitary pressure. Lastly, a real case study is presented, showcasing visualizations like a heatmap matrix displaying individual and collective pressure, as well as team pressure balance.