A Validation Procedure for the Estimation of Reachable Regions in Football

TL;DR

This paper introduces a validation routine for football player motion models, assessing their ability to predict reachable positions accurately and demonstrating that motion along two segments of constant speed best approximates real player movement.

Contribution

It proposes a validation method for trajectorial motion models in football and compares four models, highlighting the effectiveness of a two-segment constant speed approach.

Findings

Constant speed and acceleration models are less accurate.

Two-segment constant speed motion provides the highest accuracy.

The validation routine helps optimize and select better motion models.

Abstract

Modelling the trajectorial motion of humans along the ground is a foundational task in the quantitative analysis of sports like association football. Most existing models of football player motion have not been validated yet with respect to actual data. One of the reasons for this lack is that performing such a validation is not straightforward, because models of player motion are usually phrased in a way that emphasises possibly reachable positions rather than expected positions. Since positional data of football players typically contains outliers, this data may misrepresent the range of actually reachable positions. This paper proposes a validation routine for trajectorial motion models that measures and optimises the ability of a motion model to accurately predict all possibly reachable positions by favoring the smallest predicted area of reachable positions that encompasses all observed reached positions up to a manually defined threshold. We demonstrate validation and optimisation on four different motion models, assuming (a) motion with constant speed, (b) motion with constant acceleration, (c) motion with constant acceleration with a speed limit, and (d) motion along two segments with constant speed. Our results show that assuming motion with constant speed or constant acceleration without a limit on the achievable speed is particularly inappropriate for an accurate distinction between reachable and unreachable locations. Motion along two segments of constant speed provides by far the highest accuracy among the tested models and serves as an efficient and accurate approximation of real-world player motion.