At present, tactical skill is increasingly being seen as the key property of successful coaches and individuals like Pep Guardiola or Diego Simeoni are praised for the tactical genius. Indeed, research confirms that tactical behavior is a crucial aspect for successful performance in modem association soccer (Kannekens, Elferink-Gemser, & Visscher, 2011; Memmert, Lemmink, & Sampaio, 2016). However, at present tactical analysis in a strict sense has seen much less attention in the literature compared to classical notational approaches investigating event sequences in soccer (Rein & Memmert, 2016). Following the insights by Garganta (2009), the tactics specify how a team manages space, time, and individual actions throughout a game (Fradua et al., 2013; Garganta, 2009). Thus, by changing the tactics a team can modify the way space is available for their team as well as for the opponent limiting attacking and defending opportunities. Consequently, this leads to the question of how space control can be quantified from a scientific point of view. In the literature, various attempts have been proposed to quantify space control during a game (Fonseca, Milho, Travassos, & Araujo, 2012; Kim, 2004; Nakanishi, Murakami, & Naruse, 2008; Taki & Hasegawa, 2000). A common thread across all of these approaches is the use of variations of so-called Voro-noi diagrams. A Voronoi diagram partitions a plane into different cells according to the distances between points distributed across the plane. Each of these points is called a seed and is associated with a single unique cell. The crucial property of a Voronoi cell is that the geometry of the cell is chosen in such a way that all points contained in that cell are closest to the seed compared to all other seeds in the plane yielding an arrangement of convex cells. More formally, Q={p e ^IIp ~ st| < Ip - sf\,j = 1,2, ...i - l,t + l, where Ci is the i-th Voronoi cell, s* is the associated seed point, i e N and p are all seed points in the plane P. Accordingly, the standard Voronoi-cell uses the Euclidean distance to determine the cell geometries and results in a simple approach to partition a plane into disjoint areas. With respect to space control in soccer, by interpreting each player as a seed, this procedure allows therefore to quantify the space control exerted by each player and provides an interesting point of departure to assess space control in association soccer (compare Fonseca et al., 2012; Rein, Raabe, Perl, & Memmert, 2016 for examples). However, when using the Euclidian distance as the measure to determine the geometry of each cell the model introduces a few crucial simplifications. All players are assumed static and can equally fast accelerate in all directions which is obviously not true in reality. Further, as the players are assumed static the current running and heading direction are discarded which also influences how long it takes to reach specific points. For example, when a player sprints into a specific direction all points in front of him will be reached much faster compared to points behind him as he has to stop, turn and reaccelerate into the opposite direction first. Acknowledging this simplification, already the first applications using Voronoi-diagrams addressed these short-comings. For example, Taki and Hasegawa (2000) defined a constant velocity function based on the concept of reachability. The authors introduced a player running-model with an average finite running velocity associated to each player yielding a weighted Voronoi-diagram Q={p e P\f(p(x,y, v),Si) < f{p(x,y,v),Sj),j = 1,2, ...i - l,i + 1, ...,n}. Thus, instead of the simple Euclidean distance a more complex distance function f is used to determine the geometric region associated with the individual cells. The results obtained by Taki and Hasegawa (2000) showed that the attacking team occupies a greater area compared to the defending team (compare also Gudmundsson & Wolle, 2014; Taki & Hasegawa, 2000) although a similar finding has been been obtained using a standard Voronoi-diagram in an experimental Futsal game (Fonseca et al., 2012). The model by Taki and Hasegawa (2000) has been later extended by introducing a resistive force into the distance function which decreases acceleration with increasing running speed (Fujimura & Sugihara, 2005). All these approaches have in common that the same running model is applied to all players. More recently, Gudmundsson and Wolle (2014) however proposed to use past tracking data from each individual player to generate individualized running models. Following this proposal we sought to establish an empirical running function to model reachability with respect to different angle deviations from the current running directions and to calculate the Voronoi-diagram based on this empirical model.
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