# Analysis and Parameterization of Sports Performance: A Case Study of Soccer
## 4 Dataset and Parameterization
\begin{array}{ll}\dfrac{x-40}{20},&if\ 40\leq x<60\\ \dfrac{80-x}{20},&if\ 60\leq x<80\end{array} \right. \tag{5}\]

\[f(x)_{good}=\left\{ \begin{array}{ll}\dfrac{x-60}{20},&if\ 60\leq x<80\\ \dfrac{100-x}{20},&if\ 80\leq x<100\end{array} \right. \tag{6}\]

\[f(x)_{elite}=\left\{\dfrac{x-80}{20},&if\ 80\leq x<100\right. \tag{7}\]

Thus, a player is classified as (very bad, bad, average, good, or elite) if the degree of participation in that function exceeds a reference value of 0.5, with \(x\) representing the score achieved in each criterion, as shown in Figure 4.

### Fuzzy Logic for Games

Equations (8)-(10) show the membership functions defined, and using the difference between the scores of two teams as input to the fuzzy logic functions, the probabilities of victory, tie, and loss in a confrontation will be determined by the following:

\[f(x)_{victory}=\dfrac{1}{1+e^{-(x-\ln{(2)})}} \tag{8}\]

\[f(x)_{lost}=\dfrac{1}{1+e^{-(-x-\ln{(2)})}} \tag{9}\]

\[f(x)_{tie}=1-\left(f(x)_{victory}+f(x)_{lost}\right) \tag{10}\]

Assuming that:

\[x=\dfrac{average_{home}-average_{visitor}}{10} \tag{11}\]

As shown in Figure 5, the win and loss functions are two mirror sigmoids that have a value of 0.3. for \(x=0\). Therefore, the tie function at that point has the same value of 0.3. This is important because at \(x=0\), the two teams are evenly matched, and the probability of victory, tie, and loss is the same. Figure 4: Membership functions for players. Similarly, when the difference is negative (the home team is worse than the away team), the chances of a draw are greater than those of a home win. In the opposite case, when the difference is positive (the home team is better than the away team), the chances of a draw are greater than those of a home defeat. ### Membership of the Team Skeleton

The membership function (12) is defined to apply fuzzy logic to the minutes played by each player and thus determine if he belongs to the skeleton of the team or, in other words, how important his presence is in the team. The variable \(x\) in Equation (12) represents the minutes played. \[f(x)_{skeleton}=\frac{x}{3420} \tag{12}\]

A player is then considered to belong to the skeleton of the team if the degree of belonging to that function is greater than 0.5. ## 6 Display System

A visualization is designed to represent the parameterization of each player, as showcased in Figure 6. These visualizations present the player's score in each studied parameter, in addition to the team, position, overall average, area-specific average, birthdate, nationality, and height. Moreover, the visualization's color varies based on their position (goalkeeper, defender, midfielder, or forward). Regarding the parameters, they are shown in red if considered very poor, yellow if poor or average, and green if considered good or elite according to fuzzy logic functions. Another visualization that has been designed, as seen in Figure 7, represents the face-off between two teams. In this view, the possibilities of victory, tie, or defeat of the home team concerning the visiting team are presented along with each squad's lineup. Within the lineups section, positions, nicknames, and each player's overall averages are depicted together with a symbol indicating whether each player belongs to the team's main lineup or not. Table 10 shows the ranking at the end of the league by the ES compared to the actual ranking that took place in the 2021-2022 season. Figure 5: Fuzzy logic functions for games. Figure 4.0 Player Display. Figure 4.1 Face-off between two teams. ## 7 Discussion

In this section, a comparison is made with different tools that have been used for the prediction of the results in the 2021-2022 season. In [39], a prediction is made after each day. In this regard, the prediction made on 29 September 2021 on the final ranking of the teams has been taken. This date was chosen because the model used is fed back each day, so the comparison has to be made without prior knowledge, as it is done in the model proposed in this work.