Space evaluation in football games via field weighting based on tracking data

The game of football, which is also commonly referred to as soccer, is a complex system wherein 22 players in two different teams interact with each other in order for their team to win. In the process of carrying a ball to their respective goal, a vast number of movement options are available to players, leading to the emergence of various behaviors from individual to team levels, such as ball passing, dribbling, marking an opponent player, and organizing into formations. The emergence of such behaviors in football, and recent developments in data acquisition methodologies and tools [1] have promoted a wide range of analyses for football games from the perspective of statistics as well as physics [2, 3]. Examples of such analyses include the universality of goal distribution [4], complex network analysis for ball passing [5, 6], dynamics-based analysis of ball motion [7, 8], and characterization of formations [9, 10].

Characterization of space in a field is a key issue in football game analysis because it is directly related to the quality of ball passing or player formations. However, the definition of space in the context of football is not well-defined and neither are approaches for its evaluation. In general, there are two approaches for space evaluation, namely field division and field weighting. A well-known approach for field division is determining a “dominant region”—as introduced by Taki et al. [11, 12]—wherein a certain player can arrive in a region prior to any other players. A typical example of field division is defining a Voronoi region for each player, which corresponds to their dominant region as defined by the Euclidean distance between their position and each location in the field [13]. The Voronoi region provides a first approximation of the territory of any player on the field, and its basic properties in football games have been previously investigated[14, 15, 16].

However, because these Voronoi regions lack information about the velocity and acceleration of players, Fujimura and Sugihara proposed a more realistic definition of the dominant region based on a “motion model” [17]. In the motion model, each player is assumed to move according to an equation of motion with acceleration and resistance terms. Thus, given the initial position and velocity of a player, their arrival times at any location on the field can be calculated based on the equation of motion. Accordingly, using the equations of motion, the Voronoi regions of players can be suitably modified as per their velocity and acceleration. Another approach for extension of the definition of a dominant region is a data-driven one wherein individual dominant regions on the field could be estimated using machine learning [18, 19].

Though field division based on dominant regions is one approach for space evaluation when positions and motions of players are known, it is an oversimplified approach, because all locations within a dominant region yield the same space evaluation in that only one specific player can reach a location first. For further detailed characterization of space, the field division approach could be replaced by a field weighting approach using appropriate variables. Several recent studies have addressed the problem of field weighting based on variables such as pass probability [20], scoring opportunity [21], and space occupation and generation gain [22].

In a similar vein, in this study, we proposed yet another framework for space evaluation based on field weighting, which provides a definition of space from a new perspective. In particular, our space evaluation approach is based on a “minimum arrival time” $\tau$ to all locations in the field, which can be calculated based on the physics-based motion model. Herein, we considered two variables $\tau_{\textrm{of}}$ and $\tau_{\textrm{df}}$ corresponding to the minimum arrival time for offense and defense teams, respectively. Using $\tau_{\textrm{of}}$ and $\tau_{\textrm{df}}$, we introduced new orthogonal variables $z_{1}$ and $z_{2}$ that represent degrees of safety and sparsity of each location, respectively. To elucidate the quantitative meaning of $z_{1}$ and $z_{2}$, we carried out ball-passing analyses based on real datasets. As applications of our proposed space evaluation method, a new field division approach is discussed and an evaluation of ball passing just before shoots for a dataset of football games is presented.

The properties and significance of our proposed framework for space evaluation are described below. First, instead of a field division approach, our framework is based on a field weighting approach derived from a previous motion model. Second, the variables for field weighting have an explicit physical meaning, i.e., minimal arrival times of players when they move to all locations by sprinting. Third, each location is evaluated using two variables; $\tau_{\textrm{df}}$ and $\tau_{\textrm{of}}$, and the influence of both the defense and offense teams at each location can be simultaneously evaluated using our approach. Fourth, most importantly, two orthogonal variables $z_{1}$ and $z_{2}$, which correspond to the degrees of safety and sparsity of a location, were introduced. It is important to note that our framework does not depend on the definition of a motion model; in addition, $z_{1}$ and $z_{2}$ provide a quantitative definition of space on the football field. Therefore, our framework essentially yields a starting point to answer the question, ”what is space in football games?”

Based on the ball-passing analysis discussed above, our framework also provides a new approach for field division based on the degrees of safety and sparsity. In particular, using the two axes, $z_{1}=0$ and $z_{2}=2$, the football field can be divided into the following four spaces: (A) safe dense space ($z_{1}>0$ and $z_{2}<2$), (B) safe sparse space ($z_{1}>0$ and $z_{2}>2$), (C) risky sparse space ($z_{1}<0$ and $z_{2}>2$), and (D) risky dense space ($z_{1}<0$ and $z_{2}<2$). We show a typical example of field division using our approach into spaces (A)–(D) in Fig. 5. From the figure, it can be confirmed that the dense spaces (A) and (D) defined as $z_{2}<2$ are almost located within the formation, i.e., $\tilde{R}<1$.

For example, in the case shown in Fig. 5, the blue leftward triangle overlapping the blue open circle indicates that a player has the ball. In order for this player to execute a shoot from this location, the ball has to be sent to a space in front of the opponent’s goal; however, this space is risky because it corresponds to (C) or (D). We show that passes just before shoots have a tendency to become risky generally based on our framework. For passes just before shoots, Figs. 6(a) and (b) present the scatter plots for $z_{1}(\vec{x}_{\mathrm{e}},t_{\mathrm{o}})$ and $z_{2}(\vec{x}_{\mathrm{e}},t_{\mathrm{o}})$, and the probability distribution of $z_{1}$ for successful passes, respectively. From Fig. 6(b), we can observe that the peak value $z_{1}\simeq 0$ is smaller than that for the case of all passes ($z_{1}=0.69$ in Fig. 3(b)). This observation indicates that passes just before a shoot tend to become risky because the success probability of a pass is a monotonically increasing function of $z_{1}$, as shown in Fig. 3(c).

As shown in Fig. 3(c), the success probability $P(q=1|z_{1})$ of passes being well-fitted by the sigmoid function (5) could be a consequence of the normal distributions for $P(z_{1}|q=1)$ and $P(z_{1}|q=0)$ (see Fig. 3(b)). In particular, it is known that $P(q=1|z_{1})$ can be expressed as follows:

which transforms to Eq. (5) when $P(z_{1}|q=1)$ and $P(z_{1}|q=0)$ follow a normal distribution with the same standard deviation [23]. In our case, the standard deviations obtained from fitting our data are $\sigma_{1}=0.54$ and $\sigma_{0}=0.38$, and $\sigma_{1}\neq\sigma_{0}$. The deviation of the success probability curve shown in Fig. 3(c) from the sigmoid function curve where $z_{1}<0$ can be attributed to this difference in standard deviation values.

Finally, here, we present directions for future research related to the present study by comparing our framework with other relevant studies. First, Fernández and Bornn proposed a field weighting approach for space evaluation in football games [22]. In their method, a player’s influence on each location is defined by a multivariate normal distribution and is transformed into a single value to evaluate which team is dominant at a location. In contrast, our method evaluates a location in the field based on two variables, $z_{1}$ and $z_{2}$, which are defined based on the motion model. It is noteworthy that each of $z_{1}(\vec{x},t)$ and $z_{2}(\vec{x},t)$ has an explicit physical meaning, i.e., the degree of safety for a pass made to $\vec{x}$ at $t$ and degree of sparsity of $\vec{x}$ at $t$, respectively. Several extensions to our proposed framework are required before it can be applied to real game analyses. For example, the information of the distances from the locations of the ball and goal should be incorporated into variables for field weighting as in the case of Fernández and Bornn’s approach, because the outcome of passes or shoots typically depends on these distances. In regard to the motion of the ball, Spearman et al. proposed a model based on the equation of motion for the ball [20, 21]. Our proposed framework could also be similarly extended to the ball, which would enable us to evaluate the accuracy of passes. Because our framework is independent of the definition of the motion model, we could employ any extended motion model based on the equations of motion or machine learning technique [18, 19] for calculating $z_{1}$ and $z_{2}$. A player-specific motion model might also allow us to assess the playing ability of players.