Statistical properties for directional alignment and chasing of players in football games

TL;DR

This paper investigates the statistical properties of player interactions in football, revealing how alignment and chasing behaviors depend on interpersonal distance, supported by empirical data and a simple predictive model.

Contribution

It introduces a novel analysis of directional alignment and chasing in football players, identifying characteristic distances and modeling player interactions.

Findings

Strong velocity vector alignment at ~500 cm distance

Chasing behavior initiates below ~1000 cm distance

Distribution of velocity angle shifts with alignment strength

Abstract

Focusing on motion of two interacting players in football games, two velocity vectors for the pair of one player and the nearest opponent player exhibit strong alignment. Especially, we find that there exists a characteristic interpersonal distance $ r\simeq 500 $ cm below which the circular variance for their alignment decreases rapidly. By introducing the order parameter $ \phi(t) $ in order to measure degree of alignment of players' velocity vectors, we also find that the angle distribution between the nearest players' velocity vectors becomes wrapped Cauchy ($ \phi \lesssim 0.7 $) and the mixture of von Mises and wrapped Cauchy distributions ($ \phi \gtrsim 0.7 $), respectively. To understand these findings, we construct a simple model for the motion of the two interacting players with the following rules: chasing between the players and the reset of the chasing. We numerically show that our model successfully reproduce the results obtained from the actual data. Moreover, from the numerical study, we find that there is another characteristic distance $ r\simeq 1000 $ cm below which player's chasing starts.