# Weighted Stretch Index

The stretch index measures the space expansion or contraction of the team on the longitudinal and lateral directions (Bourbousson, Seve, et al., 2010).

FIGURE 4.1 Spatial referential of the field (Clemente et ah, 2013).

Similarly to the team’s centroid, the weighted team’s stretch index may be calculated as follows:

where (/, [f] is the Euclidean distance between player i and the team’s centroid, which is expressed as follows:

Put it differently, the stretch index can be obtained by computing the mean of the distances between each player and the centroid of the team. Hence, this metric represents the mean deviation of each player on a team from its centroid.

# Effective Surface Area

Computing an effective surface area, also known as team coverage area, and as opposed to the majority of the works presented in the literature (e.g., Moura, Martins, Anido, De Barros, & Cunha, 2012), is more complex than the previous tactical metrics. To create a polygon on the planar dimension, i.e., a triangle, at least three points are necessary. Therefore, three players need to be considered to build triangles as the combinations of N players (in which, as stated earlier, N is the total number of players within a team). On the football case, a maximum of 11 players for each team may be in the field at the same time. Consequently, the combination of three out of eleven players results in a total of 165 triangles that may be cumulatively formed (Algorithm 4.1). * 1

Algorithm 4.1. Calculate the surface area of the team.

1 = 0// counter of the combinations of N players taken three at a time

P = Д, // initialize the polygon as the first triangle defined by players 1, 2, and 3 For i = 2 :1

P = PuA,, where P = (pi, ...,pa) ла < N //build the polygon by accumulatively uniting itself to the remaining triangles

a-l

APoi =,iP2.i+l ~ Pi.i+iPl.i)’ with (X < N // calculate the area of the polygon

i=l

The concept of effective play area comes from Grehaigne (1993), defining it as the instant peripheral position of players. That is to say that the surface area should contemplate the effective available space a team can play. Therefore, the effective area of a team should be modelled as the real area that a team covers without intercepting the effective area of the opposing team. This requires a more thorough geometrical analysis of the tactical football strategy to further understand how teams behave over time.

Triangles formed between players are known to be the geometric figures leading to the most successful play along the field (Lucchesi, 2001). The ability of the team to ‘draw up’ such triangles on the field allows developing a good offensive (Clemente et al., 2013). Also, defensive triangles are always being formed in an attempt to create a ‘defensive shadow’, i.e., space through which the opponent cannot pass or dibble due to the triangular-shaped positioning of players (Dooley & Titz, 2010). With this in mind, Algorithm 4.2 calculates all

Algorithm 4.2. Calculate the surface area of team 8 with non-overlapping triangles.

Is =0 // counter of the combinations of N players of team 8 taken three at a time

• ( N5
• s = (p) e Rlxfl, where p = (p,, ..., р„)лД =

V3 /

/,3 = Д? // initialize the polygon as the triangle with the smallest perimeter Af = Д* // initialize the non-overlapping triangles of team 5 TS = 1 // counter of the non-overlapping triangles of team 5 Fori = 2: Is

Г = PS П Af, where Г = (/,,...,уа)л(Х< NS // analyse intersections between triangles

a-1

Л/w = — ,1+1 “ Y;+ Yi.i)with a < NS // calculate the area of the

i=l

intersection

If APl,i = 0 // condition is verified when there is no intersection between triangles

TS = TS+

PS = Ps (J Af // build the polygon by accumulatively uniting the nonoverlapping triangles

Ad\$ = Af // non-overlapping TS triangle of team 6

the non-overlapping triangles formed by players of the same team, generating, at first, the triangles with smaller perimeters.

Trapattoni (2000) claims that, when players are pressed and cannot turn around and dribble, the ball must travel along with triangles until a solution is found, i.e., the offensive triangles are annulled by the defensive triangles. Put it differently, as the number of formed triangles within a team increases, the less effective space is left for the opposing team. Hence, after generating all triangles of each team, Algorithm 4.3 computes the triangles of each team that do not suffer from the intersection of the opposing team, subsequently allowing to calculate the area of each team without an interception.

In Algorithm 4.3, both teams are simultaneously considered, in which 6 and £ are the team superscript, such that 8 = {1,2} and £ = {1,2}, with However, in the presence of interceptions between opposing triangles, and based on the supposition that effective defensive triangles can overlap the offensive triangles (Trapattoni, 2000), the effective area to be considered is one of the defensive triangles (Figure 4.2a), thus reducing the effective area of the offensive team.

Algorithm 4.3. Effective Area - Triangles of team 5 that do not intersect the surface area of the opposing team £.

e6 = 0 // counter of the effective triangles of team 8 AS =0 // effective area of team 8

ES = [ ] // polygon of the effective area of team 8 is initialized as an empty array For i = 1: TS

Г = Af П pt, where Г = (y(, ...,/<*) л a < 6 // analyse intersections between triangles

a-1

APot = |X(yi,,72,,+i -У1,,+|У2,г) with a < 6 // calculate the area of the /=1

intersection

If APl,i =0 // condition is verified when there is no intersection between the triangle from team 8 and the surface area of team £ з

APoi = — ^^(л'|У,+| — Xi+iy,) // calculate the area of the triangle i=l

AS = AS + Apot // cumulative effective area of team 8

£S = £S +1 // counter of the effective triangles of team 8

ES = ES U Af // build the polygon of the effective area of team 8 by accumulatively uniting its effective triangles

FIGURE 4.2 Example of triangles interception (Clemente et al., 2013).

According to Dooley and Titz (2010), effective defensive triangles in football can only be established if defensive players are able to ensure a maximum distance of 12 m between them. Generalizing this rationale, an offensive triangle will overlap a defensive triangle with a perimeter above 36 m since there are no guarantees that the defensive players will be able to intercept the ball (Figure 4.2b). It is noteworthy that this rationale takes into account the specific football dynamics and that effective defensive formations from other team sports will inevitably lead to changes in the algorithm. However, to keep it as generalized as possible, the maximum perimeter of the triangle is defined by pe which, for this particular case, is set to pe = 36. With this in mind, Algorithm 4.4 considers that all triangles of the defensive team with perimeter inferior to 36 m overlap the interceptive offensive triangles (Algorithm 4.4).

Algorithm 4.4. Effective Area - Defensive

triangles of team ё that intersect the surface area of the opposing team £.

I f hcilt(£) = // condition is verified when team C has the possession of the ball Fori = 1 : ZS

Г = Д fnP , where Г = (yj, ..., ) л O' < 6 // analyse intersections between

triangles

a-l

Ap„l = — ,1+1 “ 7i,;+ 72,) with a < 6 // calculate the area of the

i=i

intersection

3

ppol =J'J(xi-xi> }'. - Yj ) • with 1 * j A i < j i=i

If Ap<,i > 0лрм < pf // condition is verified when there is an intersection between the defensive triangle from team о and the surface area of team C and the perimeter of the defensive triangle is smaller than pe

APoi = — (*,)',+i *,+i)',) // calculate the area of the triangle

i=i

AS = AS + Ap,,/ // cumulative effective area of team 8

£S = £S +1 // counter of the effective triangles of team 8

s s s

P = P U Д(- // build the polygon of the effective area of team 8 by accumulatively uniting its effective triangles

At last, Algorithm 4.5 takes into account all offensive triangles that are not intercepted by the defensive triangles with perimeter inferior to pE = 36 m, leading to the effective areas of both teams at every instant.

Algorithm 4.5. Effective Area - Offensive

triangles of team <5 that are not intersected by the defensive triangles of the opposing team £.

Ifbiil/jwsscssUm (<5) = 1 // condition is verified when team a has the possession of the ball For i = 1: TS

Г = ДfП’ where Г = (ya..., 7«)лСГ<6 /7 analyse

intersections between offensive triangles and the effective area of both teams

a-1

A Pot = ,/72,/+i - fu+iTb)with a < 6 // calculate the area of the

/=1

intersection

If APl,i =0 // condition is verified when there is an intersection between the defensive triangle from team 8 and the surface area of team C and the perimeter of the defensive triangle is smaller than pe

з

APl,i = — ^^(.Y,y,_l_i - x,+1y,) // calculate the area of the triangle

i=i

AS = AS + APoi // cumulative effective area of team 8

£S = £S +1 // counter of the effective triangles of team 8

ps = p& u A? // build the polygon of the effective area of team 8 by accumulatively uniting its effective triangles

The effective surface area of the polygon P^ [f] formed by team <5 is computed at each time t, As[t], with the same being done for team £. This is illustrated in Figure 4.3, in which the offensive formation from team 8 is being intercepted by a set of efficient defensive triangles. This allows analysing if a team, in the defensive phase, acts as a defensive ‘block’, i.e., the union (U) of the defensive triangles forms a defensive polygon that constrains the opponents to lose the ball. It also allows analysing if the midfielders’ triangles are large enough to allow offensive triangle moving forward without effective opposition. Additionally, to the team’s effective area, the number of effective triangles of each team may characterize the efficiency of the team’s tactical organization.

FIGURE 4.3 Example of effective area with defensive and offensive effective triangles.

Put together, the effective area and the number of triangles can give different and complementary information, since a team with the same number of effective triangles may have a different effective area in different situations.

Just like all other computational metrics presented in this chapter, the computation of the effective area needs to be undertaken in real time (while the football match is running), which, considering its complexity, calls upon high- performing computers and an efficient selection of certain subroutines. Before computing the team’s effective area, one needs to compute the surface area with non-overlapping triangles (Algorithm 4.2). To form a triangle between the players from team 8, a three combination of the players may be considered

c

as a subset of three distinct players of N . The time complexity to compute the team’s surface area is further increased as one also needs to sort the perimeters of all triangles (cf., Algorithm 4.2). For a list of a computationally efficient sorting algorithm, refer to Bhalchandra, Deshmukh, Lokhande, and Phulari (2009).

The team’s effective area is then divided into three algorithms. Algorithms 4.3 and 4.4 linearly depend on the number of non-overlapping triangles from team 8, i.e., TS. However, Algorithm 4.5 depends on the number of non-overlapping triangles from the opposing team, i.e., T’. Moreover, the computation from the effective area from both teams depends on one another. At last, integrating all metrics, the computing requirements, for both teams, can be mathematically described as follows: