Theory and data analysis of player and team ball possession time in football.

In this study, the stochastic properties of player and team ball possession times in professional football matches are examined. Data analysis shows that player possession time follows a gamma distribution and the player count of a team possession event follows a mixture of two geometric distributions. We propose a formula for expressing team possession time in terms of player possession time and player count in a team's possession, verifying its validity through data analysis. Furthermore, we calculate an approximate form of the distribution of team possession time and study its asymptotic property.

Validation of a motion model for soccer players' sprint by means of tracking data.

In soccer game analysis, the widespread availability of play-by-play and tracking data has made it possible to test mathematical models that have been discussed mainly theoretically. One of the essential models in soccer game analysis is a motion model that predicts the arrival point of a player in t s. Although many space evaluation and pass prediction methods rely on motion models, the validity of each has not been fully clarified. This study focuses on the motion model proposed by Fujimura and Sugihara (Fujimura-Sugihara model) under sprint conditions based on the equation of motion. A previous study indicated that the Fujimura-Sugihara model is ineffective for soccer games because it generates a circular arrival region. This study aims to examine the validity of the Fujimura-Sugihara model using soccer tracking data. Specifically, we quantitatively compare the arrival regions of players between the model and real data. We show that the boundary of the player's arrival region is circular rather than elliptical, which is consistent with the model. We also show that the initial speed dependence of the arrival region satisfies the solution of the model. Furthermore, we propose a method for estimating valid kinetic parameters in the model directly from tracking data and discuss the limitations of the model for soccer games based on the estimated parameters.

Preferential model for the evolution of pass networks in ball sports.

We propose a theoretical model to evaluate the temporally evolving ball-passing networks whose number of edges increases with time. The model incorporates a preferential selection of edges that chooses an edge based on its frequency of selection. The results are in good agreement with the corresponding ball-passing networks of association football, basketball, and rugby matches, and they enable a quantitative comparison of the passing activity among different teams or ball sports.

Space evaluation in football games via field weighting based on tracking data.

In football game analysis, space evaluation is an important issue because it is directly related to the quality of ball passing or player formations. Previous studies have primarily focused on a field division approach wherein a field is divided into dominant regions in which a certain player can arrive prior to any other players. However, the field division approach is oversimplified because all locations within a region are regarded as uniform herein. The objective of the current study is to propose a fundamental framework for space evaluation based on field weighting. In particular, we employed the motion model and calculated a minimum arrival time [Formula: see text] for each player to all locations on the football field. Our main contribution is that two variables [Formula: see text] and [Formula: see text] corresponding to the minimum arrival time for offense and defense teams are considered; using [Formula: see text] and [Formula: see text], new orthogonal variables [Formula: see text] and [Formula: see text] are defined. In particular, based on real datasets comprising of data from 45 football games of the J1 League in 2018, we provide a detailed characterization of [Formula: see text] and [Formula: see text] in terms of ball passing. By using our method, we found that [Formula: see text] and [Formula: see text] represent the degree of safety for a pass made to [Formula: see text] at t and degree of sparsity of [Formula: see text] at t, respectively; the success probability of passes could be well-fitted using a sigmoid function. Moreover, a new type of field division approach and evaluation of ball passing just before shots using real game data are discussed.

Lifetime distributions for adjacency relationships in a Vicsek model.

We investigate the statistical properties of adjacency relationships in a two-dimensional Vicsek model. We define adjacent edges for all particles at every time step by (a) Delaunay triangulation and (b) Euclidean distance, and obtain cumulative distributions P(τ) of lifetime τ of the edges. We find that the shape of P(τ) changes from an exponential to a power law depending on the interaction radius, which is a parameter of the Vicsek model. We discuss the emergence of the power-law distribution from the viewpoint of first passage time problem for a fractional Brownian motion.

Clustering algorithm for formations in football games.

In competitive team sports, players maintain a certain formation during a game to achieve effective attacks and defenses. For the quantitative game analysis and assessment of team styles, we need a general framework that can characterize such formation structures dynamically. This paper develops a clustering algorithm for formations of multiple football (soccer) games based on the Delaunay method, which defines the formation of a team as an adjacency matrix of Delaunay triangulation. We first show that heat maps of entire football games can be clustered into several average formations: "442", "4141", "433", "541", and "343". Then, using hierarchical clustering, each average formation is further divided into more specific patterns (clusters) in which the configurations of players are different. Our method enables the visualization, quantitative comparison, and time-series analysis for formations in different time scales by focusing on transitions between clusters at each hierarchy. In particular, we can extract team styles from multiple games regarding the positional exchange of players within the formations. Applying our algorithm to the datasets comprising football games, we extract typical transition patterns of the formation for a particular team.