September 19, 2018

Analysis and Sports Technology
Team Sports

Degrees of freedom or degrees of slavery?

What is the appropriate approach to understanding soccer scientifically? Here, it is proposed that a first step toward developing dynamical models that are more sensitive to the context of the game is to solve the problem of choosing “the right degrees of freedom” (DoFs) (Krieger, 1992, p. 36). According to Martin Krieger DoFs are parts in which one decomposes a system to get a “handle” on them, that is, variables that one can change and manipulate expecting the induced changes in overall behavior to be systematic and illuminating as to the way that system works. Interestingly, with the use of Dynamical Systems Theory (DST) as a modeling framework, one can, in principle, use any DoFs of our choice to describe a system, but only a subset of all the possible sets of DoFs will properly correspond to variables that allow us to predict and modify the behavior (Krieger, 1992). This is no minor issue. To some extent, the task of discovering the right DoFs can be as difficult as that of designing models, and in soccer this seems to be the case. In the past, when studying intentional movements, say Bernstein’s famous hammering task, measurements of muscle forces or even joint angles, have been clearly identified as purely anatomical units that provide “poor” measures (Turvey, 1986). Indeed, the numerous units at the structural level makes it extremely challenging to account for an explanation of such action, and therefore, motor control theories have advocated for a more functional approach to motor movement (Shaw & Turvey, 1981; Turvey, et al., 1981). A good example of motor theorists’ perspective at the time are Ghiselin’s (1981) words: “the structures that support action should not be confused with action” (p. 200).

Nowadays, I encounter the same problem in soccer. When measuring, say for example, distances of players relative to the centroid of a team, rather than functional units that capture the lawful relations of the system, these metrics are an example of the anatomical (i.e. structural) level of the system. That is, borrowing Ghiselin’s (1981) conception, the structure of soccer players being distributed at certain distances from a statistical point (i.e., centroid) should not be confused with the higher order behavior that emerges in a lawfully structured way to function in particular contexts of the game. To be sure, it is true that soccer generates emergent processes, that behavior of individuals is swarm-like under some aspect, that some statistical figures can shed light on current and future states of the game, etc. Nevertheless, what’s needed in soccer and team sports in general, first and foremost, is a principled and structured guide to the fundamental questions a theory of soccer must answer, and these questions will in turn hint the relevant DoFs from which adequate modeling can be derived.

Figure 1. It represents the similar challenges that scientists and coaches have to face to work on the system of interest. In both cases, the incorrect choice of the degrees of freedom will radically change the output (i.e., numerical or behavioral) forcing (slaving) the scientist or coach to comprehend the output based on their choice of DoFs rather than serving as evidence for testable predictions of a principled theory/playing style.

When a person plays soccer or trains other players, she is confronted to the same problem of finding the right DoFs to alter and improve her performance and that of her team. And from this rather practice-oriented approach, a few initial observations can be derived to steer the more theory-driven scientific approach towards appropriate questions. Playing soccer is a skill that evolves during one’s lifetime. This variation in skill over time, is where one can detect what remains invariant in terms of the important parameters to which soccer players become better and better attuned. I contend that one such parameter is immediately obvious and indeed the most important one, that is, the position of the ball. During the first stages when children play together, the position of the ball seems to drive their behavior directly and almost exclusively. Little do they know about teamwork, tactical movements, triangulations or even dribbling. Novice children will simply follow the ball, all at the same time, regardless of what other players are doing. They show no sensitivity to team structure nor coordination with other team mates and the opposite team. When in possession of the ball, a child will often simply try to run with it towards the goal and when not in possession, a child will simply try to get it back directly by going after it. The ball’s position and the player in possession of it thus constitute the fundamental predictors for the behavior of the players. In other words, the first questions that need to be asked are “where is the ball?”  and “who is in possession of it?”.

Consider, now, the case of more experienced soccer players. Certainly, their behavioral patterns are significantly more complex and seem to depend on quite a few parameters of the game. Are the ball’s “where” and “who” still the primary parameters driving their behavior?

Figure 2. It shows the location of certain players of the team in recovery phase during the unfolding of a play. As suggested in López-Felip & Turvey (2017), the existence of three functional zones emerge as a function of players’ position relative to the ball. The blue curvy line indicates players in an immediate zone of intervention due to their short distance relative to the ball; orange line indicates a zone of mutual help with players in the intervention zones; and a green line indicates a zone of cooperation with the other zones due to their considerable distance from the ball.

Although my FCB soccer masters would confirm this, their scientists’ peers would suggest that whether those parameters can help model the behavior of soccer players is precisely what the model is supposed to demonstrate empirically. Thus, any model of soccer dynamics, no matter which aspect of the game or the level of analysis it targets, insofar as it is a model of soccer, will have to contain parameters describing the position and the possessor of the ball. Soccer is a kind of collective phenomena, to be sure, but unlike, say, the flocking behavior of birds, it is one centered around the position and the player in possession of a very special element of the collective, namely, the ball.

Figure 3. a) It represents the overall synchronization of the team in recovery phase for a play (~ 20 seconds) represented graphically in Figure 1 (see details of the analysis in López-Felip et al., 2018).  b) It shows the behavioral patterns of intra-team coordination as a function of players’ distance to the ball during the play of Figure 1. Results from analysis show that there is a decay of functional coordination of the players as they get further away from the ball.

To wrap up, this post hinted that the ball and the player in possession of it seem to be relevant DoFs to take into consideration when trying to understand (scientists) and train (coaches) the system under study (i.e., soccer). The take-home message, thus, is to warn coaches and scientists that their choice of DoFs, if not appropriate, can slave their theory of the game (scientists) or their playing style (coaches). In other words, when a scientist’s choice of DoFs does not seem to be right, then, their theories become slaves of the output’s interpretation from mathematical modeling. In the same way, if a coach does not manipulate the right variables in each training drill to constraint the team behavior to the desired team’s playing style, then, their playing style becomes slave of coaches’ misinterpretations of the variables of the game (DoFs) to achieve a desired playing style.

Next month’s post will address the ontology of soccer to answer questions such as: What relationships among DoFs hold the systemic nature in soccer? What is the system in first place? and others of similar nature.


Maurici A. López-Felip. Center for the Ecological Study of Perception and Action, USA. Team Sports Department at Futbol Club Barcelona, Barça Innovation Hub, Spain.




Ghiselin, M. T. (1981). Categories, life and thinking. The Behavioral and Brain Sciences, 4, 269-313.

López-Felip, M. A., & Turvey, M. T. (2017). Desideratum for GUT_ A functional semantics for sport. Human Movement Science, 1–0.

López-Felip, M.A., Davis, T.J., Frank, T.D. & Dixon, J.A. (2018). A Cluster Phase Analysis for Collective Behavior in Team Sports. Manuscript submitted for publication. Human Movement Science, 59, 96-111 .

Shaw, R.& Turvey, M. T. (1981). Coalitions as models for Ecosystems: A Realist Perspective on Perceptual Organization. In M. Kubovy & J. Pomerantz, Perceptual Organization. Pages 343 – 415. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Turvey, M.T. (1986). Beyond anatomical specificity. Comment on Berkinblit, M. B., Feldman, A. G., & Fukson, O. I. Adaptability of innate motor patterns and motor control mechanisms. The Behavioral and Brain Sciences, 9, 624-625.

Turvey, M.T., Shaw, R.E., Reed, E.S., & Mace, W.M. (1981). Ecological laws of perceiving and acting: In reply to Fodor and Pylyshyn. Cognition, 9, 237-304.





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