Last week I participated in the open division of ESPN’s 3rd annual hackathon, part of the MIT Sloan Sports Analytics Conference. It was my first time at the conference and first time participating in the hackathon as well. It was a great experience, getting to talk with people who know very well both math and sports and get to play with the holly grail of basketball, i.e., spatial tracking data!

The topic of the hackathon was to measure the immeasurable, the latter being things like team chemistry, hustle, heart etc. Some of these notions are hard to even define in plain English, so it is even harder to measure them. However, 47 participants of the event came up with some great ideas on how to quantify various intangibles of basketball. Here I will briefly present my approach in quantifying team chemistry with fractal theory.

My approach to defining team chemistry is through the expected observable from a team that exhibits good chemistry. In particular, I would expect that a team with good chemistry will have good ball movement. However, ball movement itself seems to be hard to define. Luckily, one of the data that the league is collecting is detailed tracking data for both the players as well as the ball itself. Every 4msec there is location information sample for each player and the ball and data from 5 randomly selected games were provided to us as part of the hackathon.

However, **how can you use the ball trajectory during a possession to quantify ball movement? **The approach I took was to describe the spatial distribution of the trajectory for every possession through the fractal dimension of the underlying point-set. Fractal dimension is a real number (compare to topological dimension that is an integer number) and hence, allows for fine-grained comparisons between different spatial *objects*. Nevertheless, one additional problem that we have here is that we do not know what is a good value for fractal dimension in order to say that a team (possession) had good ball movement? I answered this by seeing which possessions (i.e., of what fractal dimension) ended up in a made FG as compared to *failed *possessions (e.g., TO). More specifically I ran a regression model where the dependent variable is the outcome of the possession (1 for a successful one and 0 for a failed one) and the only independent variable is the fractal dimension of the ball trajectory. The results are presented in the following table.

As we can see the fractal dimension of the ball trajectory is significantly correlated with the outcome of the possession. The lower the fractal dimension, the more probable the possession will end up in a made field goal. The following box plot is very illustrative in the fact that successfully possessions exhibit lower ball fractal dimension. We also visualize two representative examples of a successful and a not successfull possession and the corresponding ball movement.

Obviously the above considers missed FG possessions as possessions with bad ball movement. Of course, this not necessarily true as a good ball movement possession can just end up with an open missed FG. So there is clearly more research that needs to be done on this topic but it seems that using complex systems theory is a direction worth exploring for quantifying intangibles. For the record, my approach was a finalist in the open division.

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