2.1 Dynamic time warping

DTW is based on Levenshtein distance and finds an optimal coupling pathing between two sequences in a precomputed distance matrix [10]. In a general sense, the Levenshtein distance between two sequences is the minimum number of single-site edits (insertions, deletions, substitutions) required to change one sequence into another. Viewing two trajectories as a stream of data couplets, the distance matrix required for the Levenshtein distance is composed of distance values of the previous, current and next alignment of data pairs. As such, phase shifts between trajectories may be detected with the DTW algorithm which are not generally possible using Euclidean distance. This may be visualised in Fig 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Visualising DTW.

Two sample trajectories are analysed with the distance matrix provided. The optimal pathing may be found, represented by the red dots, which are alignment of the two trajectories (Anderberg, 1973).

https://doi.org/10.1371/journal.pone.0272848.g001

Consider two trajectories s₁ and s₂. DTW achieves the optimal coupling by following a path of the smallest leading edge along a DTW distance matrix (also known as local cost matrix), M:

1. Initialise matrix of size (1: length(s₁)×1: length(s₂))
2. For each Row, iterate through the columns and find the Euclidean distance of item i on the row element to item j on the column element. This is computed as a n dimensional Euclidean distance.
3. Find the minimum Euclidean distance with neighbouring elements including itself; add the Euclidean distance calculated on the previous step and store in the respective location in the initialised matrix M.
4. Return matrix M.

The full DTW Algorithm is given in S3 –Appendices, Dynamic Time Warping Algorithm.

For each time point on a trajectory, the algorithm compares the distance to the same point on the second trajectory, and at the previous and following points (steps 2 and 3 above). The distance with the minimum distance is selected, where:

• M[i−1][j]: Known as a deletion, denotes s₂ is accelerated, where s₁ aligns with previous s₂ values.
• M[i][j−1]: Known as an insertion, denotes s₂ is decelerated, where s₂ aligns with previous s₁ values.
• M[i−1][j−1]: Known as a match, denotes s₁ aligns with s₂.

Hence the DTW algorithm allows for “warping” of two trajectories to find a measure of minimum distance between them. We propose the application of DTW to three dimensional trajectories (x,y,t) to capture not just the pathway taken by the ball (dimensions in x and y), but also the speed at which it moved along the trajectory (represented as time t). This is critical as the speed of execution of a play is just as important as the play (i.e. ball movements) itself in affecting scoring outcomes [17]. The full trajectory of a play was used to identify the variability and impact of temporal-based elements and was noted by industry partner the first 8 seconds of a play to be crucial in scoring [18].

In the case study below, we standardised each dimension prior to the computation steps of DTW. This differences in the range and variability amongst the x,y, and t values; t is bounded by the shot clock, and x,y values bounded by the court dimension. To allow for greater stability and consistency when comparing across teams, the global standard deviation was used during the standardisation process.

The complexity of optimising the alignment of two ball movement trajectories increases exponentially with the number of time steps [18]. DTW provides one way to address this computational complexity by comparing discrete time points immediately before and after each point and by imposing the following conditions:

• Boundary condition: The path initialises from bottom left of matrix to top right. Mathematically: the i and j index initialises at (1,1) to (X,Y) (leading diagonal of matrix). This ensures that the analysis of trajectories is initialised correctly.
• Continuity condition: The path can only advance one step at a time. Mathematically: the i and j index can only increase by increments of 1. This ensures that each point within trajectories is matched in a sequential manner.
• Monotonic condition: The path may not trace back on itself. Mathematically: the i and j index cannot be decreased. The monotonicity condition mitigates potential infinite loops and mismatching in the temporal aspect. It also constrains the shape of trajectories, ensuring that the alignment follows through the leading diagonal of the distance matrix.
• Warping window condition: An optimal path is near the leading diagonal of the matrix.

2.2 Clustering methodology

Although DTW helps to find a measure of the closest match between every pair of trajectories, clustering is still a challenge. This arises due to the sheer diversity of ball movement trajectories with differing starting points (out of bounds, under the basket and so on), and unequal effect of parts of the trajectory on scoring outcomes. Furthermore, as temporal factors were included, trajectories may have a similar temporal path, but have very different spatial features. It was found that the endpoint of the trajectory from where the shot was taken was critically important from a scoring effectiveness and strategic perspective [19]. To address these challenges, we adopted a two-step approach. In the first step, we clustered trajectory endpoints using unsupervised K-means clustering. We refer to clusters of endpoints as areas in this paper. In the second step, for each area, we clustered using both spatial and temporal features from the data through hierarchical clustering of trajectories based on DTW distances to learn about the plays leading to endpoints in those areas.

Approaches such as K-means and mixture modelling are ideally suited to clustering endpoints as clusters are formed around a centroid which has a practical spatial interpretation on the court. Here we chose K-means for its simplicity and efficiency for single point classification [20–22]. For step (ii), we chose hierarchical clustering to minimise the variance within the clusters derived from pairwise DTW distance measures. The “ward.d2” method [23] in the “hclust” package in R (Anderberg, 1973) was used, which begins with every trajectory in its own cluster, and pairs of clusters are incrementally combined in order of similarity (i.e. least distance). This agglomerative process produces a tree where in which the distance (i.e. height) to the leaves represents a threshold for similarity within clusters; the tree can be pruned at arbitrary thresholds ranging from every trajectory in its own cluster to all trajectories in one cluster. As the clustering task is unsupervised, i.e. there is no ground truth of how trajectories should be assigned to clusters, we used validation indices and specifically the Silhouette index.

Note that this dataset contained a number of trajectories composed of only a single point. These were found to be an artefact of the trajectory encoding process in situations such as an offensive rebound, or a violation occurred prior to the ball moving. These encoding artefacts were filtered out and removed prior to analysis.

2.3 Validation indices

A key challenge of unsupervised clustering is the selection of the number of clusters, K. Given the volume and diversity of plays over multiple games, it is infeasible to manually annotate trajectories to form a ground truth for clustering. We adopted the Silhouette Index (SI) to find an appropriate value of K because it is a normalised metric balancing similarity within a cluster and dissimilarity between clusters [24]. Other indices are available and were also evaluated (See S3 –Appendices, Validation indices and Results) [25]. The SI, defined in Eq 2, is based on cohesion (distance of data relative to own cluster centre) and separation (distance of data relative to other cluster centres); maximal cohesion to separation occurs when the index is 1, suggesting a clustering configuration where the trajectories are well matched to their respective clusters.

1

The number of clusters can be chosen by plotting the SI index against the number of clusters and finding the ‘elbow’ [22] as this helps to achieve a near-maximal SI index. This approach was applied to both the choice of the number of clusters for endpoint clustering and hierarchical clustering of trajectories.

2.4 Dynamic time warping barycentre averaging

To represent each cluster, a representative trajectory was computed using the DTW Barycentre Averaging DBA algorithm [10]. This algorithm aims to minimise the sum of DTW distances computed between the average trajectory to trajectory in the cluster. Assume that N trajectories are grouped in K clusters. The set of clusters is denoted by C_k = {c₁, c₂,…c_K}. The k^th cluster is represented as a list of trajectories , where U_k is the number of trajectories in the cluster. The centroid of cluster c_k is calculated as the Barycentre average of the component trajectories (1) with the following algorithm:

1. Initialise an arbitrary trajectory as the centroid .
2. Iterate through the trajectories inside the chosen cluster c_k.
   1. Compute the DTW aligned trajectories between the centroid and each trajectory i in a cluster.
   2. Update the centroid based on the barycentre of the coordinates iteratively, as defined in Eq 1.

(2)

2.5 Play outcomes

We used a number of metrics to characterise the outcomes of a play (i.e. possession). These included the mean and standard deviation in: play duration, distance travelled, heading angle, and change in heading angle. Note that the heading angle is measured relative to the horizontal line joining the two baskets, and is computed for each segment (pair of two consecutive sample points j and j+1) of a trajectory.

(3)

In addition, the scoring rate of trajectory clusters was used as a measure of scoring outcome. This is the ratio of scored shots to total number of shots attempted (3).

4

2.6 Case study

Existing data from 72 women’s international basketball games ranging from the 2014 FIBA World Championships to the 2016 Rio Olympic Games were analysed [26]. The data consist of manually tracked x and y coordinates with a frequency between 1~6 Hz, and metadata of event sequences manually encoded whilst the coordinates were tracked. Metadata includes events such as turnovers, fouls, violations, shots and shot outcomes.

In this paper, we focused on a case study on three teams with the greatest number of games: Australia (11 games), USA (12 games), and Japan (12 games).

4.1 Scoring rate of clusters

One approach for analysing successful plays is to compare trajectory clusters with high scoring rates and which terminate in the same area of the court. For instance, Table 1 shows that both Area 3 Cluster 3 and Area 3 Cluster 16 for Australia have high scoring rates of 60.0% and 72.7%, respectively, but that arises from very different types of play. The mean duration for cluster 3 is 3.8 seconds, compared to 15.2 seconds for cluster 16. The former corresponds to fast breaks, which are known to typically result in a higher scoring rate [27,28].

However, our trajectory clustering also reveals that cluster 3 has lower mean heading changes compared to cluster 16, 74 degrees compared to 304 degrees, respectively (Table 1). Potentially, changes in direction, either more frequent and/or larger in magnitude, could be contributing towards the higher scoring rate. We then sorted Table 1 by mean heading change to identify the top ten and bottom ten clusters in terms of heading change. The scoring rate for the top ten was 29%, on average, compared to just 14%, on average, for the bottom ten. This builds upon findings from previous work that found entropy was a contributing factor to play effectiveness [26]. Based on the plays in cluster 3 and 16, end users could review video footage to support game preparation in terms of plays and their execution.

4.2 Spatial tendencies

High level summary analysis may help to understand the overall playstyle and score rate for a country. Here we look at patterns from a spatial context.

In Table 1, we found that there were 534 plays by Australia that had a right-hand bias where much of the play occurred with the ball in the right half of the court, vis a vis the mean heading angle was positive. In contrast, 400 plays had left-hand bias (negative mean heading angle). However, clusters with right-hand court bias had, on average, a 30% scoring rate compared to 25% for left-hand court biased clusters. This type of analysis could be helpful for training of the home team, or for preparing against an opponent team.

4.3 Temporal context

By sorting the summary table by possession time, coaches and players can view clusters that perform well during fast breaks compared to long possession time. The fastest ten clusters have an average scoring rate of 30% whilst the slowest ten clusters have an average of 19%, which may indicate fast plays have better scoring rates. However, upon closer inspection, the cluster size in these extremities (both slowest and fastest trajectories) have low occurrence rates (proportion of trajectories). Thus, it is important to further analyse these trajectories to support decision making. For instance, trajectory plots, such as those used in Figs 5 and 6 may provide high level insight on both spatial temporal aspects of a play (i.e. court location and speed of movement). However, review of video footage is likely also necessary; the proposed model helps to guide end users to identify which plays to review in light of time pressure and/or limited time.

There can be considerable variation in the scoring rate of fast break plays terminating in the same area. Consider for instance cluster 20 of Area 3 with a scoring rate of 33% versus cluster 3 of Area 3 with 60% (Table 1). The former accounts for a greater proportion of plays (3.2%) compared to the latter (0.96%). Although the possession time is quite similar (around 3 seconds of play) for both, further investigation, such as of spatial patterns of the trajectories and review of video footage could help identify the reason for this disparity. An example of a high-level spatial pattern is that out of the top ten fastest plays, the scoring rate was 41% on average for plays with a right-hand court bias (i.e. positive mean heading angle), compared to 13% for those with left-hand court bias.

4.4 Comparisons between teams

Comparisons between teams can also offer insight to support preparation for competition. The equivalent trajectory cluster for Australia’s Area 3 Cluster 16 was Area 5 Cluster 27 for team USA based on endpoint area and trajectory plot. Factors such as the frequency of execution, possession time, mean heading and scoring rate can be used identify potential differences in the play and its execution between the two teams. In this example, cluster 27 for team USA was run relatively rarely (0.31% of the time) and with a significantly lower scoring rate (0%) than cluster 16 for Australia (1.18% proportion of plays with 72.73% scoring rate). A key difference appears to be the speed of execution, which is slower for team USA (19s) compared to Australia (15.81s).

4.5 Conclusion

Our DTW based framework enables analysis and interpretation of complex trajectories over many games and teams, and the generation of key summary metrics of play outcomes. Comparison of trajectory clusters within and between countries can assist in identifying key patterns to support coaching and preparation for games. A key component of this is the ability to highlight key plays for targeted video review from a large database of game footage, assisting end users with limited time resources. A common methodology used in current and previous research utilises Euclidean distances [3–5] which does not consider data alignment and will miss out similar plays when analysed under unsupervised conditions. Existing works that used DTW in basketball tended to focus on player movements, and did not explicitly relate trajectories to play outcomes with a focus on both spatial and temporal characteristic of the trajectories [15,16]. Our framework contributes a practical means to cluster spatio-temporal trajectories and relate them to play outcomes, using ball movement data. However, the use of k-means to cluster trajectory endpoints is limited in that it cannot take into account boundary conditions associated with court boundaries. In addition, there is immense diversity in trajectories and determining how many clusters to use is somewhat subjective; potentially, a combination of using a vocabulary of discrete actions [16] and a Diricihlet Process Mixture Model [29] could better cluster trajectories and inherently finds the optimal number of clusters.

This paper illustrated some insights that could be obtained through DTW clustering, including support for conventional wisdom of higher scoring rates with fast breaks, and clusters where more changes in direction (higher mean heading change) in the play was also associated with high scoring rates. Interestingly, the top ten fast break clusters for team Australia revealed a significant disparity between those that had a bias towards the right-hand side of the court, which were more effective than those with a left-hand side bias. In addition to these summary metrics, trajectory plots could also be used to capture spatial layout of the trajectory and its speed of execution. This provides an opportunity for coaches and players to understand the effects of spatio-temporal factors into plays on a macro level, in addition to the micro level of video review.

Potential future work includes optimisation of the clustering process to better account for large numbers of clusters with a small number of trajectories. The current framework utilises a basic implementation of both DTW and k-means clustering; more sophisticated approaches, such as PrunedDTW, SparseDTW, FastDTW, and MultiscaleDTW, could potentially enable faster computation and better clustering results [30]. Alternative techniques for clustering and even mixture modelling could be investigated, especially if integrating clustering with trajectory alignment. A higher resolution dataset with better sampling consistency may also assist in the DTW and clustering process; further investigation could include the breakdown of trajectories into discrete actions as part of a vocabulary of basketball actions [16]. Finally, extension of the work to include both ball and player trajectories could provide further detail and insight to support coaches and athletes.