# Spatial-Temporal Metrics

Online spatial-temporal metrics may provide relevant information about how teams, as a collective system, behave over time throughout the match (Clemente et al., 2013). In fact, such metrics can be used as an important tool to improve the coaching opportunities to make changes on the team’s strategy, allowing detecting and acting upon its weaknesses during the match. For this, the weighted centroid, weighted stretch index, and surface area are hereby described.

## Weighted Centroid

In football and all other team sports, the centroid is often calculated through the geometric mean position of all players of a given team. As before, the position of player i in the field is decomposed in its dimensions as x, [f] = (x,- [f], y, [f]).

Then, the centroid (x[f],y[f]j can be calculated based on the geometric

position of all N players (x, [f], y,[/]) for each team. According to Frencken and Lemmink (2008), the centroid provides three relevant measurements: (i) the лг-distance, representing forward-backward displacement (i.e., length of the field); (ii) the у-distance, representing lateral displacement (i.e., width of the field); and (iii) the radial distance, comprising both forward-backward and lateral displacements. These are obtained based on the centroid position relative to the origin O, i.e., (0,0), defined at the centre of the field, as follows:

wherein the position of the ith player is defined as (x, [f], y, (?)) and w, [f] is the weighting factor of such player. Many studies do not consider the positions of the goalkeeper and the ball (Bourbousson, Seve, & McGarry, 2010; Frencken, Lemmink, Delleman, & Visscher, 2011). Nevertheless, both should be considered for different reasons: the goalkeeper due to his/her preponderance in the defensive phase, and the ball since the player’s influence shall decrease with the distance to it. In other words, the relevance of the ith player to the team’s centroid, i.e., ip, [f] [/], is based on the Euclidean distance from the ith player to the ball as follows:

where (лу [f], yi, [t]) corresponds to the position of the ball and rfmax [t] is the Euclidean distance of the farthest player to the ball at each time t (Figure 4.1).