Energy-Variation Analysis and Orbit-Complexity Quantification

In state-recurrence analysis, recurrences are considered either in the space that constitutes of the observed quantities, or in an embedding space that is manufactured by time-delayed variables. An alternative approach is proposed here. In the so-called energy-variation analysis, the complexity of an orbit is quantified in terms of the statistics of the constant-speed geodesic condition. Energy-variation analysis requires significantly less operations than state-recurrence analysis, in particular, for multivariate time-series, while numerical experiments demonstrate that the resulting energy-recurrence matrix encodes information that is sufficient for quantifying the complexity of orbits.

Football Match Dynamics Explored by Recurrence Analysis

A widely accepted notion of football matches in performance analysis (PA) is to consider them as dynamic interaction processes with emerging behaviors. The description and analysis of these processes requires specific methods. Recurrence analysis is a technique for analyzing complex systems in many domains like astrophysics, earth sciences, engineering, biology, cardiology, and neuroscience. Its general concept is to examine the recurrence behavior of a system, as in when, how often and how close its trajectory in a phase space returns to a previous state. The aim of the study is to apply recurrence analysis to football matches. Positional data from 21 football matches of a German Bundesliga team were examined. The phase space was made up of the field players' x,y-positions at each second of the match. For each pair of seconds, the average distance of all the players between their positions at these two time points was calculated. Recurrence plots (RPs) were obtained by color-coding these distances. With a recurrence threshold of rt = 9 m and a minimum line length of lmin = 3 s, general recurrence parameters were calculated to characterize the individual recurrence behaviors of each match. Three football-specific recurrence parameters were defined to represent recurrence properties of open play. RPs showed commonalities (typical features indicating set plays and continuous gameplay) as well as unique structures during each match (number, distribution, and sequence of typical features). The recurrence parameters showed several significant correlations with traditional performance indicators like number of goals and passes completed, e.g., the correlation between number of goals and recurrence rate is r = −0.622 (p = 0.003). By extending the sample and design of recurrence studies, there is great potential for recurrence analysis to improve both the practical and theoretical potential of performance analysis.