Keywords

5.1 Stretch Index

5.1.1 Basic Concepts

The Stretch Index of a team is calculated as the average distance of a team’s players and the Geometrical Center, giving a notion of the compactness of the team.

Definition 5.1

[1] The Stretch Index, considering both axes, at a given instant t can be calculated by:

$$\begin{aligned} SI(t) = \frac{\sum _k^N \sqrt{(p_{xk}(t)-C_x(t))^2 + p_{yk}(t)-C_y(t))^2}}{N}, \end{aligned}$$
(5.1)

where C(t) represents the Geometrical Center of the team, N the number of players in the team, \(P_{xk}(t)\) the position along the longitudinal axis for player k at instant t, and \(P_{yk}(t)\) the position along the vertical axis for player k in instant t.

Definition 5.2

[1] The Stretch Index on a single axis, for a given instant t, is given by the following expression:

$$\begin{aligned} SI_x (t) = \sum _k^N |(p_{xk}(t)-C_x(t))^2)|, \end{aligned}$$
(5.2)

where \(C_x(t)\) represents the x coordinate for the Geometrical Center at instant t and \(P_{xk}(t)\) the position along the longitudinal axis for player k at instant t.

Remark 5.1

The same formula is applicable for the calculation along the vertical axis, only replacing \(P_{xk}(t)\) for \(P_{yk}(t)\) and \(C_x(t)\) for \(C_y(t)\).

Definition 5.3

[1] The average values of the Stretch Index both for the coordinates, along the two axes and for each axis separately, is given by the following equation:

$$\begin{aligned} \overline{SI}=\frac{\sum _k^N SI(t)}{N_t}, \end{aligned}$$
(5.3)

where \(N_t\) represents the total number of time instants measured.

Remark 5.2

This same formula is applicable for \(\overline{SI_x}\) and \(\overline{SI_y}\), by replacing SI(t) for \(SI_x(t)\) or \(SI_y(t)\), respectively.

5.1.2 Real Life Examples

The results obtained by a player from Team A in the SSG are presented in Table 5.1 with intervals of 30 s and for the entire 3 min in Table 5.2.

Table 5.1 Values obtained for the Stretch Index of Team A in an SSG, for periods of 30 s
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Table 5.2 Values obtained for the Stretch Index of Team A in an SSG, in the entire period of time of 3 min
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The results obtained by a player from Team B in the SSG are presented in Table 5.3 with intervals of 30 s and for the entire 3 min in Table 5.4.

Table 5.3 Values obtained for the Stretch Index of Team B in an SSG, for periods of 30 s
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Table 5.4 Values obtained for the Stretch Index of Team B in an SSG, in the entire period of time of 3 min
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A screenshot of a representation of the Stretch Index captured from the uPATO software is displayed in Fig. 5.1.

Fig. 5.1
figure 1

Screenshot of the uPATO showing an example game animation with the representation and values of the Stretch Index for both teams

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5.1.3 General Interpretation

Stretch Index was introduced in basketball to measure the expansion and contraction of space, in both axes (longitudinal and lateral), demonstrated by a team during a match [2]. This measure represents the mean deviation of each teammate to the geometrical center [2, 3]. Stretch Index is also known as radius [4].

Dispersion of the players depends on contextual variables and mostly on the moment of the game (with or without possession of the ball) [5]. Dispersion is greater in attacking moments (with possession of the ball) and smaller in defensive pressure (without possession of the ball), in the case of soccer [6]. This follows the main idea that in attacking moments it is necessary to spread the players to attract the opponents towards the outside of the middle and in defensive moments it is necessary to keep the teammates closer to guarantee fewer spaces for opponent’s penetration [7].

This measure works in longitudinal, lateral and/or global, thus different information can be used. In the case of the Stretch Index for the longitudinal axis it can be computed to measure specific situations of counter attacks in which a greater dispersion in goal-to-goal direction can be observed. In the other hand, greater dispersions are found in side-to-side during positional attack (ball circulation). Considering the defensive moments, both axes will drastically decrease in comparison to attacking moments. However, longitudinal dispersion can be used to classify the defensive pressure against positional attacks in which large values of dispersion may suggest that forward teammates are too far away from the defensive colleagues.

5.2 Surface Area

5.2.1 Basic Concepts

The Surface Area of a team is calculated as the area of the polygon defined as the convex polygon with the least number of vertices that can encompass all of the teams’s players, and where the potential vertices are defined as the positions of the players.

Definition 5.4

[8, 9] Given a set of points, the following algorithm is applied to define the Convex Hull:

figure a

Definition 5.5

[9, 10] Given the coordinates of the n vertices that compose the Convex Hull of the team, the Surface Area is given by the following equation:

$$\begin{aligned} SA = \frac{|(x_1 y_2 - y_1 x_2) + (x_2 y_3 - x_3 y_2) + \dots + (x_n y_1 - x_1 y_n) | }{2}, \end{aligned}$$
(5.4)

where \((x_i, y_i)\) are the coordinates of the ith vertex of the Convex Hull.

5.2.2 Real Life Examples

The results obtained by both teams in the SSG and another team in the match are presented in Table 5.5 with intervals of 30 s and for the entire 3 min in Table 5.6.

Table 5.5 Values obtained for the Surface Area of both teams in an SSG and obtained from a team in a match, for periods of 30 s
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Table 5.6 Values obtained for the Surface Area of both teams in an SSG and obtained from a team in a match, in the entire period of time of 3 min
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A screenshot of a representation of the Surface Area captured from the uPATO software is displayed in Fig. 5.2.

Fig. 5.2
figure 2

Screenshot of the uPATO showing an example game animation with the representation and values of the surface area for both teams

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5.2.3 General Interpretation

Surface Area represents the area of a polygon constituted by all teammates (dots) [11]. The measure can be defined as the total space covered by a team considering the area within the convex hull [9]. The Surface Area measures the contraction and expansion of teams across the soccer match, as does the Stretch Index [3]. However, this measure represents all the area covered by the team, while the Stretch Index only measures the mean deviation to the Geometrical Center of the team.

Typically, the area of the teams is significantly bigger in possession of the ball than without possession [3, 5]. Moreover, the variability of Surface Area decreases across the game [4, 12], thus suggesting a stabilization of the team in attacking and defensive moments.

In a study conducted in elite Spanish soccer teams values between 800 and 2800 \(\mathrm {m^{2}}\) in attacking and [1000; 2000] \(\mathrm {m^{2}}\) in defensive moments were found [13]. Values in attacking varied between 1638 and 1831 \(\mathrm {m^{2}}\) in possession of the ball and [1277; 1369] \(\mathrm {m^{2}}\) without possession in an elite Portuguese team [14].

The visualization of Surface Area can help coaches identify the space between sectors (defensive, middle and forward) and the dispersion of the team in specific moments. The generated triangulations can provide an immediate analysis of the collective behavior, particularly in moments of positional attack and defensive ‘block’.

5.3 Team Length and Team Width

5.3.1 Basic Concepts

The Team Length and Width is defined by its most advanced and rear players for the length, and its rightmost and leftmost players for the length.

Definition 5.6

[4] Given a set of points for team player positioning along a time-series of length N, P, and where \(P_x\) and \(P_y\) represent the set of longitudinal and lateral coordinates for every player of the team on every measured time instant, the Team Length and Team Width on a given time instant i can be calculated as follows:

$$\begin{aligned} t_l (i) = max(P_x(i)) - min(P_x(i)) \end{aligned}$$
(5.5)
$$\begin{aligned} t_w (i) = max(P_y(i)) - min(P_y(i)), \end{aligned}$$
(5.6)

where \(t_l\) represents the Team Length, \(t_w\) represents the Team Width and \((P_x(i), P_y(i))\) represents the set of coordinates of the team’s players in instant i.

5.3.2 Real Life Examples

The results obtained by both teams in the SSG are presented in Table 5.7 with intervals of 30 s and for the entire 3 min in Table 5.8.

Table 5.7 Values obtained for the team width and team length of both teams in an SSG, for periods of 30 s
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Table 5.8 Values obtained for the team width and team length of both teams in an SSG, in the entire period of time of 3 min
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The results obtained by the team in the match are presented in Table 5.9 with intervals of 30 s and for the entire 3 min in Table 5.10.

Table 5.9 Values obtained for the team width and team length of a team in a match, for periods of 30 s
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Table 5.10 Values obtained for the team width and team length of a team in a match, in the entire period of time of 3 min
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5.3.3 General Interpretation

The Team Length represents the maximum length of a team considering the minimum and maximum position of a player in the longitudinal (goal-to-goal) direction [4]. The same application is applied in the case of Team Width (side-to-side) [4]. Therefore, the length and width measures provide information about how stretched are the two farthest players of a team in longitudinal and lateral directions.

Coaches can use this information to understand the optimal distances to readjust some tactical tasks in training sessions based on usual length and width found in official matches. An interesting approach used the width and length to suggest specific sizes to work positional attack and counter-attack in small-sided games [15]. Moreover, coaches can compare the length and width in different moments of the match and identify in which moments the extreme size can be associated with critical moments (e.g., shots, goals).

5.4 Length per Width Ratio

5.4.1 Basic Concepts

The ratio between a team’s length and width is the Length per Width Ratio (lpwratio) of that team.

Definition 5.7

[16] Given \(t_l\) and \(t_w\) as the Team Length and Team Width of a team on a given instant, the lpwratio of a team is given by the following equation:

$$\begin{aligned} lpwratio = \frac{t_l}{t_w}. \end{aligned}$$
(5.7)

5.4.2 Real Life Examples

The results obtained by both teams in the SSG and the other team in the match are presented in Table 5.11 with intervals of 30 s and for the entire 3 min in Table 5.12.

Table 5.11 Values obtained for the lpwratio of both teams in an SSG and a team in a match, for periods of 30 s
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Table 5.12 Values obtained for the lpwratio of both teams in an SSG and a team in a match, in the entire period of time of 3 min
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A screenshot of a representation of the lpwratio captured from the uPATO software is displayed in Fig. 5.3.

Fig. 5.3
figure 3

Screenshot of the uPATO showing an example game animation with the representation and values of the width, length and lpwratio for both teams

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5.4.3 General Interpretation

The lpwratio quantifies the relationship between the length (maximum distance between the two farthest players in longitudinal direction) and width (maximum distance between the two farthest players in lateral direction) during the match [16]. The authors of this measure argue that small variations of this measure may suggest a team’s higher adherence to width and concentration on “principles of play” [16]. In the other hand, large variations of this ratio may suggest a more individual and less collectively coordinated approach to the soccer game [16].

Coaches can use this measure to classify the teams in different moments of the match. Some teams will tend to play in counter-attack, thus increasing the length and reducing the width. In the other hand, teams that opt to attack with circulation of the ball will have increased width and reduced the length. This relationship will help to understand some patterns of play. The same case can be applied for defensive moments in which teams that opt to defend in ‘block’ closer to the goal will present decreased length and teams that opt to extend the defensive ‘block’ for middle or forward zones of the pitch will have increased length.