This section presents an overview of the established methods used to study the spatial formation of football teams, which is referred to as “team formation analysis” in the remainder of this paper. In football, teams are reported to play in specific spatial configurations, called team formations (Kaminka et al., 2003; Kuhlmann et al., 2005). A team formation can be defined as “A specific structure defining the distribution of players based on their positions within the field of play” (Ayanegui-Santiago, 2009). While several methods for analyzing team formations have been described in various team sports (e.g., American football: Atmosukarto et al., 2013; basketball: Lucey et al., 2014; and volleyball: Jäger and Schöllhorn, 2012), this analysis can be regarded as particularly challenging in football because of the fluent and dynamic character of the game (Atmosukarto et al., 2013).

For a coach, it is important to correctly estimate the strengths and weaknesses of his or her own team. In this way, he/she can choose the most suitable team formation for a game (Carling, 2011). The selected team formation will thereby be dependent on the expected formation of the adversary team as well as the characteristics of the theoretical formation and the players that are fit to play. Aquino et al. (2019), for example, studied the effects of the applied team formation on metrics such as ball possession. Low et al. (2021) studied the impact of team formation on defensive passing behavior and collective team movements. Moreover, the manipulation of the players’ positions on the field has been described as one of the key elements of how team tactics can be applied during football matches (Rein and Memmert, 2016). The analysis of team formations in football can be performed in various ways, but often starts with a visual exploration of the data, followed by the actual detection of the formations (Stein et al., 2016, 2018). Müller-Budack et al. (2019) thereby emphasize the importance of visualizing individual situations to understand the application of team formations in football, which further illustrates the complexity of this task. Some other methods make use of various Key Performance Indicators (KPIs) to determine and analyze team formations in football (Memmert et al., 2017). These KPIs are calculated based on the positions of the players on the field. For example, Sampaio and Macãs (2012) suggested the team centroid, team entropy, a team stretch index and the surface area of the team as KPIs for team formation analysis. Going further on this, Frencken et al. (2012) added the inter-team distance, i.e., the distance between the centroids of both teams, as a key performance indicator to observe goals or shots on target during a football match.

Bialkowski et al. (2014) introduced a method for the automated detection of the type of team formation performed in a match, based on the average positions of the players on the field. They argue that, because the players are swapping positions during the game, team formation needs to be analyzed using a dynamic instead of a static ordering of players. Using data from an entire Premier League season, Lucey et al. (2013) and Bialkowski et al. (2014) found no significant difference between formations of different teams, but could detect that English Premier League teams used more offensive team formations during home games. Narizuka and Yamazaki (2019) proposed a clustering technique to compare team formations in football based on the positions of the players summarized in so-called heatmaps.

Various new methods use principles of (artificial) neural networks (McCulloch and Pitts, 1943) for sports analytics in football. Visser et al. (2001) used artificial neural networks to recognize the team formation of the opponent team by using a set of predefined reference team formations. Expanding on this, Ayanegui-Santiago (2009) suggested the inclusion of multiple relations between players for the recognition of team formations in football. For this purpose, players were divided into three groups (defenders, midfielders, and attackers) based on their role in the game and their averaged positions. Next, notions from graph theory were applied to study the team formation, thereby only including links between nodes of adjacent groups.

While most methods use quantitative metrics, Perin et al. (2013) argue that quantitative analysis alone might not be sufficient to understand the team formation of a game or an entire season. Perin et al. thereby focus on the qualitative fragmentation of a football match to obtain meaningful temporal fragments (e.g., similar defending or attacking situations) that can then be compared. Unfortunately, qualitative team formation analysis in football is at present mostly performed by human experts and is therefore rather labor-intensive (Bialkowski et al., 2014). Moreover, few studies have analyzed team formations on real size football fields, or on real football matches. Gonçalves et al. (2017); however, demonstrated that there are important differences between team tactics (and thus team formations) of matches played on a football field with a reduced size and games played on a full field, suggesting that research should be conducted on official football matches. As in other sports disciplines, mixed methods—which combine both quantitative and qualitative principles—have been gaining importance in sports analytics for football in recent years (Anguera et al., 2017). There are, however, only a limited number of mixed methods that are specifically aimed at analyzing and defining team formation in football (Frencken et al., 2012; Sampaio and Macãs, 2012; Aquino et al., 2019). These methods primarily combine quantitative performance metrics (e.g., ball possession, running speeds or number of shots, and goals) with qualitative observations of the performed team formations in order to evaluate the effectiveness of the played team formation(s). In the remainder of the paper, however, we propose a method that primarily aims at detecting the performed team formation rather than evaluating its effectiveness or impact on quantitative performance metrics.

QTC is a qualitative calculus for describing spatiotemporal relations between two or more Moving Point Objects (MPOs; Van de Weghe and De Maeyer, 2005). While QTC typically describes movement between multiple objects, it was extended to a new variant named the QTC_S by Beernaerts et al. (2018). While multiple variants of the dynamic QTC exist, the QTC_S method presented in this paper makes use of the principles of the most basic dynamic variant QTC_B (Van de Weghe and De Maeyer, 2005). QTC_S describes the static formation of Point Objects (POs) in a qualitative manner and can be used to describe the spatial formation of players in football, which is shown for a basic example in Figure 1. Its use for analyzing team formations of (parts of) football teams was demonstrated by Beernaerts et al. (2018) by means of some basic examples.

By constructing a QTC_S-matrix for each team formation of interest (i.e., timestamp), QTC_S can be used to describe the team formation at different moments during a match, by calculating the distance between the QTC_S-representation at each moment and the QTC_S-representation of a series of reference team formations, thereby assigning the reference team formation with the lowest distance (the highest similarity) as a label for the team formation of that respective timestamp. The distance between two QTC_S-matrices can be calculated by summing up the pairwise distances between the corresponding elements (QTC_S-relations), thereby using the conceptual distance between QTC-relations (Van de Weghe and De Maeyer, 2005). By dividing the total distance by the maximal possible distance (depending on the matrix dimensions), the relative distance is calculated. For ease of understanding, the relative distance is recalculated to a similarity value between 0 and 1. One could argue that QTC_S, a method that originated from the geographical research domain, can therefore be considered a mixed method for analyzing team formations in football, as it combines qualitative spatial reasoning with a quantitative distance calculation.

The experiment aims at defining the team formation of the Belgian national football team during two football matches (Belgium—Japan and Brazil—Belgium). Most studies are limited to the analysis of the impact of different team formations (labeled manually by football experts) on different football metrics. In this experiment, however, we aim at investigating the use of QTC_S as a method for the (automated) detection (thus labeling) of team formations. The data was extracted from the official FIFA match reports (FIFA, 2018), which are publicly available and consist of a theoretical team formation (filed by the coach before the start of the game) and a series of actual positions of the players averaged over every 15, 45, and 90 min of both matches, visualized in Figure 2. Unfortunately, no generally accepted dataset of reference football team formations has been reported in literature (Football Bible, 2014; Wilson, 2014). Such a dataset is required for labeling the theoretical and actual performed team formations. Moreover, every coach or football analyst has his/her own interpretation of each team formation, based on personal experiences or expectations of specific player roles for that formation (Wilson, 2014). For these reasons, the reference team formation dataset was derived from the popular football simulation game FIFA 2016, as it includes a top-down view of 25 standard team formations (Figure 3), which are well-known by millions of people and are documented rigorously (FifaUTeam, 2016).

All 11 players are included in the analysis, where the keeper can be seen as a static point, fixating the rotation of the team formation (Beernaerts et al., 2018). Indeed, the goalkeeper will always be positioned in front of his own goal, thus with an almost identical orientation with respect to the rest of the players of his team. As players tend to switch roles and positions during the game (Bialkowski et al., 2014), it is necessary to include permutations of the QTC_S-matrices when calculating the distance between the team formation at a certain moment and the different reference team formations. Calculating all permutations, however, requires 39.916.800 iterations of the distance calculation. To reduce the complexity of the experiment, four players were chosen as “fixed” based upon common sense, thereby reducing the number of players to permute to 7 and the number of permutations to 5,040. Indeed, assuming that a football team always plays with one goalkeeper and at least three defenders, these players can be omitted. This corresponds with the reference team formations dataset, which does not contain a team formation without goalkeeper or with less than 3 defenders. Furthermore, defensive players are characterized by a high level of positional rigidity (Gonçalves et al., 2014), meaning that position switches are not likely to occur.