Fractional Dynamics in Soccer Leagues

Abstract

This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees and a multitude of other effects. The scale of observation focuses the teams’ behavior at each round. Two approaches are considered, namely the evaluation of the team progress along the league by a variety of heuristic models fitting real-world data, and the analysis of statistical information by means of entropy. The best models are also adopted for predicting the future results and their performance compared with the real outcome. The computational and mathematical modeling lead to results that are analyzed and interpreted in the light of fractional dynamics. The emergence of patterns both with the heuristic modeling and the entropy analysis highlight similarities in different national leagues and point towards some underlying complex dynamics.

Theory of Cooperative-Competitive Intelligence: Principles, Research Directions, and Applications

“…In recent research, entropy as a partial measure of performance has been extensively used either in individual or in collective sports. Researchers have investigated a large set of variables in many sports disciplines (e.g., Hristovski et al, 2006 ; Passos et al, 2009 ; Fewell et al, 2012 ; Vilar et al, 2013 ; Couceiro et al, 2014 ; Silva et al, 2016 ; Castellano et al, 2017 ; Gonçalves et al, 2017 ; Neuman et al, 2018 ; Lopes and Tenreiro Machado, 2019 , 2020 ), showing that behavioral entropy, as measured by different entropy measures, has considerable effect on a number of teams and individual performance indexes. These immensely important results seem to point to a possible unifying explanation of how entropy, as a property of these systems, enters and becomes relevant for performance.…”

Uniform Manifold Approximation and Projection Analysis of Soccer Players

“…Presently, soccer has a huge economical and social relevance [ 1 , 2 ], but the study using advanced numerical and computational tools is still limited. We note that distinct levels of competition have been tackled, namely the technical progress of a player during his/her career [ 3 , 4 , 5 ], the time–space trajectories of the players in a match [ 6 , 7 , 8 , 9 , 10 ], or the performance of a number of teams along a league and season [ 11 , 12 , 13 , 14 , 15 ].…”