*6.2. Results and Discussion*

forwards (players 9–11).

Table 2 depicts the overall values of the distance covered, average values, standard deviation and entropy of the fractional coefficient, and entropy of the heat maps.


**Table 2.** Descriptive statistics of the overall results for each player.

Players 7 and 8 covered the largest overall distance (!8CW8 and !8CWWY km, respectively). On the other hand, the goalkeeper (player 1: VCW\<4) and central defenders (player 3: XCZW km; player 4: XCVWW km) covered smaller overall distances. Both cases are in line with the literature [30,42,43]. Generally, the largest distances are covered by midfielders since they act as links between defence and attack [36,42]. Bangsbo [44] reported that elite defenders and forwards cover approximately the same average distance, which is significantly less than the distance covered by midfield players. This study shows that central defenders, excluding goalkeepers since they are more constrained than other players, cover (with a large difference) a smaller distance than any other tactical position.

In terms of heat maps entropy, the results are in line with the overall distance. All the midfielders have a larger entropy than the remaining teammates (player 7: E  8CPZ; player 8: E  8CPPX; player 6: E  8CVZ8). On the other hand, the goalkeeper (player 1) presents the lower entropy value (E  CW!W), followed by the right forward (player 10) with E  8C8P, and the central defender (player 3) with E  8C\V. These results can be easily explained by the tactical roles of each position. In football, midfielders act as a link between the defenders and the forwards [42]. Therefore, they present a higher level of participation in the periods of time with or without ball possession. Also, as the goalkeeper and the central defenders have different roles in specific confined *TR*, they present a smaller spatial distribution (lower heat maps entropy). Conversely, lateral positions (defenders and forwards) have a larger *TR*. In some cases, the lateral defenders participate in offensive attempts. The inverse is observed in the lateral forwards players, because they regularly help in the defensive moments. Hence, the low values of entropy from player 10 can be explained by his reduced participation in the defensive phase.

The fractional coefficients show that the midfielders are the most unpredictable players. Players 6 and 7 are characterised by values close to H :) ,  CY\. These values are in line with the combination linear-random trajectories. This tendency makes sense since midfielders cover more distances. On the other hand the goalkeeper's trajectory is defined by larger fractional coefficient values (H :) ,  C\Y). This result is in line with the combination constant-chaotic. In point of fact, this also makes sense since the goalkeeper stays most of the time around the same tactical region. The remaining players are somewhere between the combination constant-chaotic and linearrandom, with more tendency for the linear-random.

Going further on this analysis, the attraction domain previously defined was considered so as to study the number of times that each player remained within their stability region. In a quantitative point-of-view, if a player's trajectory is classified as stable (based on : ") , *and* H:) ,), then the stability is defined as 1. Otherwise, the stability is defined as -1. Putting differently, a player that is as often within the stable region and the unstable one, will have an overall stability level of 0. From this analysis it is possible to obtain the stability values *per* player on the 5-axis and I-axis coordinates (see Figure 13).

From the results shown, it is possible to observe that the goalkeeper (player 1) is the more unstable elements in both axes. On the other hand, defenders (players 2–5) are the elements with higher stability values. These values can be supported by the specific tactical missions of each player. Defenders should keep a large defensive stability by remaining in their tactical position, giving some equilibrium to the team. As a point of interest, a considerable number of goals suffered results from the defensive instability. Therefore, defenders should maintain their trajectories within their specific regions so as to ensure the possibility of recovering the ball in the offensive attempts by the opponent team. In contrast, the goalkeeper's *TR* is evidently smaller than all his teammates. As such, at many moments of the match (mainly in the offensive situations) the goalkeeper moves outside his *TR*, towards his remaining teammates. Such movements decrease the goalkeeper's stability.

**Figure 13.** Players' stability levels at 5-axis and I-axis coordinates.

Using both concepts (predictability and stability) it is possible to observe that one player can be highly predictable (in terms of trajectory) while unstable (going outside his *TR*). On the other hand, a highly unpredictable player can be very stable if he stays most of the time inside his *TR*. Therefore, those two concepts are different and they provide an interesting set of information for coaches and their staff. The predictability level can be used to classify the oscillations during the football match while the stability level can be used to identify player's responsiveness to his *TR*. To illustrate the relationship between predictability and stability, let us present Figure 14. The 3D chart on Figure 14 depicts how the level of stability on the *x*-axis, : , is related to the level of stability on the *y*-axis, : <sup>7</sup>, and the level of predictability represented by the fractional coefficient H:.

As one may observe, the relationship between these measures is represented by a plane. Moreover, as already concluded from Figure 13, although players are more stable in the *x*-axis, there is a clear dependency between the stability on both axes. On the other hand, the level of predictability seems to vary in a significant manner depending on the positional main role of players. For instance, it is possible to divide the points into four clusters, wherein the goalkeeper

(red circle) represents the first cluster, the defenders represent the second cluster (blue triangles), the central players represent the third cluster (green lozenges) and the forwards represent the forth cluster (purple squares).

The information retrieved from the fractional coefficient cannot be compared neither with the outcome provided by the total distance covered nor with the heat maps entropy. The distance covered can be the same for all players, without providing a specific characteristic about the behaviour of the player. The heat maps entropy only provides information about the spatial distribution of players on the field. This distribution may also be similar for two players without describing their trajectory over time. Only understanding the specific properties of each football player's trajectory can improve the performance analysis and, likewise, improve the quality of the football training.

#### *6.3. Practical Remarks*

The information retrieved from the fractional coefficient cannot be compared with either the outcome provided by the total distance covered, or with the heat maps entropy. The distance covered can be the same for all players, without providing any specific characteristic about the behaviour of the player. The heat maps entropy only provides information about the spatial distribution of players on the field. This distribution may also be similar for two players without describing their trajectory over time. Only understanding the specific properties of each football player's trajectory can improve the performance analysis and, likewise, to improve the quality of the football training.

We should note that we are not redefining the concept of variability. Instead, this work proposes to analyse such variability by studying the regularity of players in returning to their own *TR* (stability), and by studying how predictable their trajectory may be (predictability). Stability, in science, is defined as its resistance to perturbations. In fact, this is a typical property shared by many dynamical systems, in which we could state that the stability is not only the resistance to a perturbation, but also the ability to return to the equilibrium point (or initial position) [45]. The existence of a stable equilibrium point implies the existence of a "restoring force" which is directed towards the equilibrium point. For instance, in the simple pendulum case study, this is a combination of the tension in the string and the force of gravity. Nevertheless, as opposed to the simple pendulum, the results presented here classify football players as non-linear dynamical systems, thus presenting chaotic or even stochastic trajectories. Although we assume that there is a steady-state point from which players converge, the truth is that they converge to an equilibrium point which is defined by their *TR*. In other words, a player's orbit spirals in towards the equilibrium. The same can be said about the predictability of players. By definition, predictability is the degree to which a correct estimation of a system's state can be made. This is in line with the concept of predictability in mathematics, wherein a process is classified as predictable if it is possible to know the "next" state at the present time.

In many situations, the choice regarding the players from the first team is based on each player's specific properties so as to adjust the team against their opponents. As such, one may choose between more stable players to focus on the defence, or more unpredictable ones to focus on the attack. The fractional coefficient can also be a useful method to improve the understanding about

decision-making in sports. The main techniques used so far for this specific issue have been the approximate entropy and the Lyapunov exponent. Nevertheless, the applicability of such methods depends on the variable that better explains the level of predictability. For a more specific tactical analysis, one should go further into understanding the fractional coefficient variability by resorting to stability theory. This is very important in understanding the player's regularity on returning to his own *TR*. The stability confined to on attraction region has a great potential for use by coaches to classify the tactical oscillations of players, thus adjusting or readjusting the desired tactical behaviours. Also, the opponent coach can use this information to identify some unstable points and exploit them during the match. Nevertheless, it should be highlighted that neither the fractional coefficient, nor the stability analysis *per se*, are the ultimate answer to one's needs in the context of football. Such team sport, as a complex and dynamic game, should be analysed using collective nonlinear methods. The classical perspective of the performance analysis has been overtaken using new technologies to improve the understanding of the individual and tactical parameters, mainly trying to explain the process variables. For the collective analysis, some metrics have been proposed based on the position of players over time [3,46,47]. Nevertheless, for the individual performance, the researchers have been emphasizing on the notational information (*i.e.*, product variables) and kinematical information [43]. This paper provides a new take-home message on the individual performance of a football match, with the main purpose being the understanding of the specific properties of each player and their dynamical behaviour during the match.

It is noteworthy that the herein proposed methodology proposed here was evaluated using one match. Its usefulness for coaches and sports analysts needs to be further assessed over multiple matches, with and without professional players. Note, however, that this requires the use of automatic tracking systems, such as *AMISCO Pro* and *ProZone* [48]. These systems provide online information to coaches and their staff about players' movements (e.g., energy spent by a player). Nevertheless, despite of their efficiency and autonomous properties, player-to-player occlusion, similar player appearance, number of players changing over time, variability of players' motion and noises or video blur present themselves as open problems [49]. Therefore, although generally autonomous, these tracking systems still require some human input as well as continual online verification by an operator to make sure that players are correctly tracked by the computer program [48]. Hence, beyond their expensive devices (e.g., many high-definition video cameras), those systems may benefit from the outcome provided by the fractional calculus methodology provided in this paper and previously presented in Couceiro *et al.* [17], to accurately and autonomously estimate a given player's position over time.

#### **7. Conclusions**

New technological devices and mathematical methods have been used recently to analyse the performance of football players. Despite these developments, a gap still remains on understanding a player's dynamical behaviour during the match. Some of the most important variables one may look at are inherent in a player's variability, which one may classify based on the predictability and stability of his trajectory. This study proposed an approach to measure the predictability and stability levels of player's trajectories based on the concepts inherent in Fractional Calculus. Furthermore, the variability of each player was measured using the well-known Shannon's entropy and the approximate entropy. The fractional coefficient, explaining a given player's trajectory, was used in order to estimate his predictability. The addition of a new parameter, herein denoted here as the stability vector, gave rise to an attraction domain defining the player's stability. The results showed that the goalkeeper was the most predictable and unstable player. The most unpredictable players were the midfielders while the most stable players were the defenders. All this information can be used by coaches to adjust and readjust the team's strategy, as well as the tactical behaviour of players.

## **Acknowledgments**

This work was supported by a PhD scholarship (SFRH/BD /73382/2010) by the Portuguese Foundation for Science and Technology (FCT). Also, this paper reports research work carried out within the project "Towards a technological approach of the match analysis: Using tactical metrics to evaluate football teams" from the Instituto de Telecomunicações, granted by the Portuguese Foundation for Science and Technology (FCT) with the ref. PEst-OE/EEI/LA0008/2011.

## **Conflicts of Interest**

The authors declare no conflict of interest.

### **References**


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