Two clusterings to capture basketball players’ shooting tendencies using tracking data: clustering of shooting styles and the shots themselves

First, it was necessary to calculate the dissimilarity between players for clustering. The dissimilarity between players is the dissimilarity in the propensity to shoot, i.e., the dissimilarity between sets of shots. However, inter-set similarity, such as the Jaccard coefficient, assumes that the same elements are present and could not be used in this case where no two shots are identical. Therefore, by considering the set of shots for each player as a discrete uniform distribution, we calculated the Wasserstein distance, which is the distance between distributions; the Wasserstein distance is a dissimilarity measure calculated based on the optimal transport problem, recently known as the loss function of the Wasserstein GAN (Arjovsky et al. 2017). Note that Kullback-Leibler divergence is a more well-known measure of dissimilarity of probability distributions, but it cannot be used when the supports of two discrete probability distributions do not completely overlap.

For example, for two random variables X, Y ∈ ℝ^q, given a probability mass function with m supports: μ(x) for X = {x₁, ..., x_m} and a probability mass function with n supports: v(y) for Y = {y₁,...,y_n}, and distance matrix D ∈ ℝ^m×n whose elements are the distances between x_i and y_j with parameter p≥1:xi−yj2p p\left( { \ge 1} \right):\left\| {{{\boldsymbol{x}}_i} - {{\boldsymbol{y}}_j}} \right\|_2^p , the p-Wasserstein distance W_p between μ and ν is defined as follows: (1) Wpμ,ν:=min∑i=1m∑j=1nDijPij1p, {W_p}\left( {\mu ,\nu } \right): = {\left( {min \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{D_{ij}}{P_{ij}}} } } \right)^{{1 \over p}}}, where P ∈ ℝ^m×n is a matrix representing the transportation plan, and P_ij is the quantity (probability mass) to be transported from x_i to y_j. In other words, equation (1) represents the total transportation cost in the most efficient case of carrying the probability mass of one distribution to match that of the other. In particular, when p = 1, it is also called Earth Mover’s Distance (EMD). In this study, Earth Mover’s Distance was calculated for simplicity by considering each shot set for each player as an empirical discrete uniform distribution. In other words, we calculated the dissimilarity of the distributions μ and ν of the shots of the two players when q = 9 (using principal components of the shot) and p = 1 (using distance matrix of Euclidean distance) in the above formula. Note that the data was limited to players for whom data for 30 or more shots existed, and 325 players were included in the analysis.

Next, the players were grouped using Wards method (Murtagh & Legendre 2011), which is based on the calculated distance. Wards method forms clusters so that the sum of squares of deviations within a cluster is minimized when merging two clusters. This method was used because it provides the most evenly distributed number of players in each cluster. Then, we calculated silhouette coefficients to determine the appropriate number of clusters. The silhouette coefficient s_t is defined for data point x_i using the following degree of cohesion a_i within the same cluster and the degree of deviation b_i from the nearest separate cluster: (2) ai=1Cin−1∑j∈Cindxi,xj, {a_i} = {1 \over {\left| {{C_{in}}} \right| - 1}}\sum\limits_{j \in {C_{in}}} {d\left( {{{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}} \right),} (3) bi=1Cnear∑j∈Cneardxi,xj, {b_i} = {1 \over {\left| {{C_{near}}} \right|}}\sum\limits_{j \in {C_{near}}} {d\left( {{{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}} \right),} (4) si=bi−aimaxai,bi, {s_i} = {{{b_i} - {a_i}} \over {max \left( {{a_i},{b_i}} \right)}}, where C_in is the cluster to which x_i belongs, C_near is the nearest cluster from x_t other than C_in, and d is an arbitrary distance function. The silhouette coefficient takes values in [−1, 1] from the definition above and is closer to 1 the more agglomerated the data points are in the same cluster and the further apart the data points are in different clusters. In other words, the closer to 1 can be interpreted as better clustering. In this method, the information on the Earth Mover’s Distance between data points (i.e., players) was used to calculate the mean silhouette coefficients when the number of clusters was between 2 and 20. From the mean silhouette coefficients in Figure 3, a small number of clusters was considered relatively good. However, since a small number of clusters does not provide meaningful information, 13 was adopted as the number of clusters just before the silhouette coefficient decreased, while also taking interpretability into account.

Figure 3:

Mean Silhouette Coefficients in Clustering of Shooting Styles

The characteristics of each cluster were interpreted by looking at histograms of the features of the shots (before dimensionality reduction) contained in each cluster. This allowed us to confirm that the clustering results fit our intuition and to name the clusters.

As another verification, the shot efficiency of players between and within the clusters extracted was compared in True Shooting percentage (TS%) for the 2015–16 season. TS% is an advanced statistics that represents the efficiency of the shot, calculated by the following formula: (5) TS%=PTS2×FGA+0.44×FTA×100, TS\% = {{PTS} \over {2 \times \left( {FGA + 0.44 \times FTA} \right)}} \times 100, where PTS is total points scored, FGA is Field Goal Attempts, and FTA is Free Throw Attempts. Note that the possible range of TS% is from 0 to 150. We examined the top five players in TS% within each cluster and see if they fit our intuition. This analysis is limited to players with 200 or more FGA during the 2015–16 season to ensure the reliability of the TS% value.

Clustering shooting styles, or typing players based on their shooting tendencies, provides an intuitive understanding of shooting tendencies. However, it is not very specific and does not quantitatively tell us what kind of shots they take more often. Therefore, by clustering the shots themselves, it is possible to calculate the percentage of how many shots a player is taking in each cluster, leading to a quantitative understanding.

We first attempted clustering using the 9 principal component dimensions of the shot features extracted in the Preprocessing Section. However, some clusters had a mixture of 2-point and 3-point shots, which made interpretation difficult. To resolve this, categorical features with information on whether they were 2- or 3-point features were added. As a dissimilarity measure for clustering this quantity-quality mixed data, we used Gower distance, a dissimilarity measure that incorporates the Manhattan and Dice distances; Gower distance D_gower between data points x_i and x_j is expressed by the following equation: (6) Dgowerxi,xj=1−1K∑k=1KSkxik,xjk, D_{gower}\left( {{{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}} \right) = 1 - \left\{ {{1 \over K}\sum\limits_{{\boldsymbol{k}} {\boldsymbol {=}} {\boldsymbol{1}}}^{\boldsymbol{K}} {{S_k}\left( {{x_{ik}},{x_{jk}}} \right)} } \right\}, where K is the number of feature dimensions, x_ik and x_jk refer to the k-th feature dimension of the data points x_i and x_j respectively, and (7) skxik,xjk=1−xik−xjkRk if k−th feature is a quantitative variable,1 if k−th feature is a qualitative variable and xik=xjk,0 if k−th feature is a qualitative variable and xik≠xjk, {S_k}\left( {{x_{ik}},{x_{jk}}} \right) = \left\{ {\matrix{{1 - {{\left| {{x_{ik}} - {x_{jk}}} \right|} \over {{R_k}}}\;{\rm{if}}\;k - {\rm{th}}\;{\rm{feature}}\;{\rm{is}}\;{\rm{a}}\;{\rm{quantitative}}\;{\rm{variable}},} \hfill \cr {1\;\;\;if\; k - {\rm{th}}\;{\rm{feature}}\;{\rm{is}}\;{\rm{a}}\;{\rm{qualitative}}\;{\rm{variable}}\;{\rm{and}}\;{x_{ik}} = {x_{jk}},} \hfill \cr {0\;\;\;if\; k - {\rm{th}}\;{\rm{feature}}\;{\rm{is}}\;{\rm{a}}\;{\rm{qualitative}}\;{\rm{variable}}\;{\rm{and}}\;{x_{ik}} \ne {x_{jk}},} \hfill \cr } } \right. where R_k is the range (difference between maximum and minimum) of the k-th feature.

Based on the calculated Gower distance, hierarchical clustering was performed using the Ward method. The Ward method was again employed here because, as before, it tended to produce clusters of uniform size and the clusters were easy to interpret. The number of clusters was determined from the change in the average silhouette coefficient as well as the clustering of shooting styles. From Figure 4, we decided to adopt the number of clusters 10 just before the mean of silhouette coefficient decreases.

Figure 4:

Mean Silhouette Coefficients in Clustering of the Shots Themselves

Interpretation of the characteristics of each cluster was done by viewing images representation of the trajectory of the shots in that cluster.

As an example of how players’ shooting tendencies can be captured, we will also present an actual case study comparing how the percentage of each shot cluster differs among players with the same shooting style.

In the clustering of shooting styles, the individual clusters were interpreted using the features of the shots in each cluster. For example, regarding the distance between the shooter and the rim 3 seconds before the shot in cluster 1 and 2 (Figure 5), the players in these clusters were often about 4 meters from the rim in common. Compared to the other clusters, the players in these clusters are more likely to be near the rim before the shot, indicating that the players in these clusters is a Big Men (common name for large players). Regarding the location of the shot (Figure 6), both clusters prefer to shoot near the rim, but compared to cluster 1, players in cluster 2 tend to shoot mid-range shots as well. From the above, we named cluster 1 “Close-Range Big” and cluster 2 “Mid-Range Big”. For all clusters, we also confirmed that the clusters to which the players belonged were not counterintuitive.

Table 1 describes the name of each cluster, the players’ mean height, examples of players, and a brief description. The third cluster was named “Mid-Range All-Rounder” due to their tendency to attempt mid-range shots, especially near the high post. The fourth cluster was named “Mid-Range Slasher” owing to the tendency to hold the ball slightly longer than clusters 1 and 2, although it prefers shots near the rim. The fifth cluster was named “Driving All-rounder” because it tended to aim for shots from any location and was interpreted as being aggressive in driving while attempting triples. The sixth cluster was named “Pull-up Ball-Hander” due to the tendency to hold the ball longer and has more threes from outside the corners and shots just inside its arc. The seventh cluster was named “Driving Ball-Handler” because, like cluster 6, it had a longer ball-holding time and more shots were made near the rim. The eighth cluster was named the “Stretch Four” due to its higher frequency of 3-pointers from the corners and top and a slightly taller average height. The ninth cluster was named “Corner Shooter because of its preference for shooting threes, especially from the corner. The tenth cluster preferred the 3-point, but unlike the ninth cluster, it tended to aim from any position, hence the name “Pure Shooter”. The eleventh cluster was named “Slashing Finisher” because players tended to hold the ball longer and preferred to shoot near the rim but not so much for 3-pointers. The twelfth cluster was similar to the ninth cluster but was named “Driving Shooter” due to a slight preference for shots near the rim. The 13th cluster prefers close-range to mid-range shots, but can also attempt a reasonable number of 3-pointers, thus we named it “Stretch All-Rounder”. Based on the distance matrix, the players were plotted in a 2-dimensional coordinate plane with cluster labels using t-SNE (Figure 7). In this figure, players with similar shot tendencies are placed closer together, while players with dissimilar shot tendencies are placed farther apart.

Figure 5:

Distance between the shooter and the rim 3 seconds before shot in two clusters as examples. Normalized frequencies are displayed. On the left is Cluster 1, which tends to have the shortest distance from the rim 3 seconds before the shot. On the right is Cluster 2, which also tends to have a shorter distance from the rim 3 seconds before the shot.

Figure 6:

2D histogram of shot location, in two clusters as examples. Normalized frequencies are displayed. On the left is Cluster 1, with shot locations concentrated near the rim. On the right is Cluster 2, which exhibits shots well distributed from the rim to mid-range.

Figure 7:

Visualization of shooting style clusters by t-SNE

Tables 3 shows the top five TS% players and their respective field goal attempts for several clusters in the 2015–16 season (the higher the value, the more reliable the value of the TS%). The remaining clusters are listed in the Supplementary Materials. From the tables, we can see that many of the players are from teams that were strong at the time, such as Golden State Warriors (GSW), San Antonio Spurs (SAS), and Oklahoma City Thunder (OKC). This suggests that players from stronger teams tended to have better shooting efficiency (when viewed in conjunction with remaining clusters in the Supplementary Materials, many players from other strong teams of the time, the Los Angeles Clippers (LAC) and the Toronto Raptors (TOR), are also represented), which fits our intuition. The results indicate that this typology of shooting styles is highly valid, as well as its usefulness in comparing players' shooting efficiency.

The results of the clustering of the shots themselves were determined primarily by looking at the visualized image of the trajectory of each cluster just prior to the shot, as described in Materials and Methods.

Cluster 1 was named ‘3-pointer from the right corner or right wing’ because the cluster consisted of 3-point shots from around the right corner or right wing. Cluster 2 is consisted of a three-point shot from the left corner, hence the name ‘3-pointer from the left corner’. Cluster 3 consisted of three-point shots near the top of the key or left wing, thus the name ‘3-pointer from the top of the key or left wing’. Cluster 4 consisted of shots near the high-post, hence the name ‘Mid-range shot near the high-post’. Cluster 5 was named ‘Shot from a mid-post move’ because it consisted of shots from post-play near the mid-post. Cluster 6 was named “Close-range shot from a post move” because it consisted of close-range shots from a post move. Cluster 7 was named “Cutting layup/dunk” because it consisted of shots after receiving the ball from the cutting. Cluster 8 was named “Mid-range shot from the right side” because it consisted of midrange shots, including pull-up jumper, on the right side. Cluster 8 was named “Mid-range shot from the left side” because it consists of mid-range shots from the left side, including pull-up jumpers. Cluster 10 was named “Driving layup/dunk” because it consisted of shots from the drive.

Here, the detailed players' shooting tendencies are interpreted by comparing two players belonging to the same shooting style. As an example, the percentage of each shot cluster for Stephen Curry and Kevin Durant in the Driving All-Rounder and Anthony Davis and Dirk Nowitzki in the Mid-Range All-Rounder is shown in a side-by-side bar graph (Figure 8). Both Curry and Durant have driving layups/dunks that account for about 20% of their total shots, but there is a difference in their 3-point shot tendencies. Curry hits about 20% of his threes from the right corner and right wing and 10% from the left corner, while Durant hits about 10% and almost 0% of those respectively. There is also a difference in the middle shot near the high post, with Curry hitting this shot only about 5% of the time, while Durant seems to hit it nearly 20% of the time. As for the other pair, Davis and Novitsky, there are also differences in tendencies. First, Davis hits only about 5% of his 3-pointers near the top of the key or left wing, while Novitsky hits about 15%. There is also a difference in mid-range tendencies, with both players strongly favoring shots near the high post at around 25%, but Novitski's percentage is higher than Davis' for shots from the mid-post and mid-shots from the right side. Furthermore, Nowitzki's close-range shot from a post move, cutting layup/dunk, and driving layup/dunk all have smaller percentages, suggesting that Nowitzki did not take many close-range shots. In summary, these are considered to be consistent with the characteristics of each player's actual shot tendencies, and reflect each player's signature moves and preferences of shots.

Figure 8:

Percentage of each shot cluster for Stephen Curry, Kevin Durant, Anthony Davis, and Dirk Nowitzki. Note that the total number of shot data is 216, 248, 280, and 262, respectively.