A Potential Function Approach to the Flow of Play in Soccer

There is a growing literature on the statistical analysis of data from association-football/soccer games, seasons or groups of seasons. In contrast this paper is concerned with a single play, that is a sequence of successful passes. The play studied contained 25 passes and ended in a goal for Argentina in World Cup 2006. One question addressed is how to describe analytically the spatial-temporal movement of such a particular sequence of passes. The basic data are points in the plane, successively joined by straight lines. The resulting figure represents the trajectory of the moving soccer ball. The approach of this study is to develop a useful potential function, a concept arising from physics and engineering. In particular the potential function leads to a regression model that may be fit directly by linear least squares. The resulting potential function may be used for simple description, summary, comparison, simulation, prediction, model appraisal, bootstrapping, and employed for estimating quantities of interest. The purpose illustrated here is to simulate play in a game where the ball goes back and forth between two teams each having their own potential function.