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Dynamical Stability and Predictability of Football Players: The Study of One Match

Couceiro, M. Couceiro ; Clemente, F.M.C. ; Martins, F. ; Machado, J. A.

Entropy Vol. 16, Nº 2, pp. 645 - 674, January, 2014.

ISSN (print): 1099-4300
ISSN (online):

Scimago Journal Ranking: 0,50 (in 2014)

Digital Object Identifier: 10.3390/e16020645

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Abstract: The game of football seeks for new computational approaches to measure the individual and collective performance. Understanding about these phenomena involved in the game may foster the identification of strengths and weaknesses not only of each player, but also of the whole team. The development of assertive quantitative methodologies constitutes a key element in sports training. In football, the predictability and stability inherent in the motion of a given player may be seen as one of the most important concepts to fully characterise the variability of the whole team. This paper characterises the predictability and stability levels of players during an official football match. A Fractional Calculus (FC) approach to define a player’s trajectory is considered. By applying FC, one can benefit from a new modelling perspective, such as the fractional coefficient, to estimate a player’s predictability and stability. This paper also formulates the concept of attraction domain, related to the tactical region of each player, inspired by stability theory principles. To compare the variability inherent in the player’s process variables (e.g., distance covered) and to assess his predictability and stability, entropy measures are considered. Experimental results suggest that the most predictable player is the goalkeeper while, conversely, the most unpredictable players are the midfielders. We also conclude that, despite his predictability, the goalkeeper is the most unstable player, while lateral defenders are the most stable during the match.

Keywords: Fractional Calculus; Entropy; Stability; Predictability; Dynamical Systems; Football; Performance Analysis; Variability.

PACS Codes: 37Fxx; 37Mxx; 01.80.+b; 05.45.Tp.