# Automated Discovery of Successful Strategies in Association Football
## 2 Materials and Methods
### Preparing and Representing Corner Kick Event Data
#### 2.1.2 Abstract Representation for Corner Kick Plays  
In this section, we create an abstract representation of corner kick plays. Notably, our representation retains only the play's crucial information for our tactical discovery--the location, name, and sequential order of each event--while other details, such as the player executing the action or the match period, are either disregarded or stored as play metadata. A crucial component of our representation is the use of a unified frame of reference to represent corner kick plays. Note that a corner kick can be taken from any of the four corners of the field. However, the data set we use measures an event's position relative to the offensive team. The X-coordinate value indicates the nearness of the event (in percentage) to the opponent's goal. In contrast, the Y-coordinate value indicates the nearness of the event (in percentage) to the right side of the field. While technically, there are four possible origin locations for our plays, when measured from the offensive team's point of view, there are only two: a left-side corner, starting from \((x=100,y=0)\), and a right-side corner, starting from \((x=100,y=100)\). Under this convention, and due to the different orientations, our analysis needs to be separated into two parts: left-side corners and right-side corners. However, one can consider that left and right-sided corner kicks are mirrored versions of each other, and therefore, they are, to some extent, equivalent. By placing all the corner kicks into a unified frame of reference, we are left with a single type of corner kick, avoiding the need to separate the analysis into two parts and increasing the number of corner kick instances available during the analysis. In our new frame of reference, we establish that all corner kicks start at \((x=100,y=100)\). Figure 3 shows an example of this transformation. Note that after the mirroring operation, it is not possible to determine the initial side of the corner kick. Previous works [10] have suggested that the side of the field from which the corner kick originates could influence its outcome. Therefore, before executing the mirroring operation, we save the side (left or right) from which the corner kick originates as part of the play metadata. Another crucial element of our representation is the spatial abstraction of events. Unlike other approaches [25; 42], which use a grid to divide the field into symmetric regions, our regions are proposed using, to some extent, domain knowledge. Figure 4 shows our representation of the field together with the names of the regions. In this new representation, the field is divided into 12 unique regions. Notice that the size of our regions drives a trade-off between the expressiveness of our plays and our ability to detect similarities between them. If the regions are too large, the events in the plays are likely to fall into the same region, making it easier for us to compare them. However, the plays will have little expressive power, meaning that we cannot tell what happened during the game with much detail. On the other hand, if the regions are too small, we may obtain very expressive plays; however, because of the increased number of spatial locations, it will be harder to detect similarities between them. Figure 3: Example frame of reference transformation for a corner kick play starting at (x = 100, y = 0). (**a**) Play shown in its original frame of reference. (**b**) Play shown in the new frame of reference. By creating a histogram of the final positions (destination regions) of the events in our plays, we can validate our assumption that most events in our plays will be targeted toward the penalty and goal areas (accumulated frequency of 58.37%). Notably, region _Penalty Middle_ (PM) is the most frequent destination (around 25% of the events), which is reasonable considering that this region is immediately in front of the opponent's goal.