Tactical analysis of football games by vector calculus of last-pass performance

The traditional approach to analyzing football (soccer) gameplay is to observe the movements of the ball and players closely to gain insight into the tactics and interactions between players. This study introduces a more advanced mathematical approach based on observational data to elucidate typical game patterns and tactical characteristics. Specifically, we applied vector analysis to the direction and length of the last-passes observed in numerous games and derived potential fields from the last-pass vector fields. Our approach allows for the visualization of natural pass flows along the gradient of the potential and the tactical characteristics of the last-passes that do not follow the gradient. Vector analysis also revealed the spontaneous formation of low-potential areas where passes were concentrated in front of the goal area, visualizing the typicality of crosses near the penalty area. Additionally, a detailed analysis of vector components not aligned with the gradient revealed the tactical characteristics of attacks or responses to defenders in the central and side areas. The outcomes of this study provide useful insights into tactical analysis and strategy optimization in matches, offering new perspectives in sports science.