Modeling Football as Interacting Processes with Dynamical Systems Trees

These are some additional experiments to accompany:

T. Jebara and A. Howard. Dynamical Systems Trees. Submitted. [.ps, .pdf]

We used tracked trajectories of real American football plays to model the interaction of players at different levels. Players are treated as simple (leaf) processes modeled with Linear Dynamical Systems (LDSs) or Switched Linear Dynamical Systems (SLDSs). Higher level interactions are modeled via discrete intra-team variables influencing each member of a team (community), and higher level inter-team discrete variables mediating the interactions between the teams. We explore various different graph topologies within the framework of Dynamical Systems Trees (DSTs).

For a discussion of the various models used, take a look at the models page.

We start with a toy problem. Sample from a trivial DST then learn the distribution and correctly infer the hidden switching states.

The first and second plots are the sampled outputs from two leaf processes. The third plot is the aggregating state mediating the switching in the leaf SLDSs. The fourth plot is the inferred probabilities from the learned model of the aggregating state. The fifth and seventh plots are the switching states of each SLDS leaf process. The sixth and eighth are the corresponding inferred state probabilities from the learned model.

A more detailed discussion and examples are found on the learning DSTs for a toy problem page.

Data:

The data was collected by the Computers Watching Football project. The data set is made up of 111 football plays of which there are 23 different types of plays. Each type of play has from 1 to 8 examples in the set. Each player is represented with an a normalized (x,y) position and an orientation. We use subsampled (x,y) positions to model players in our experiments. The orientation was not used because it was not always available in the data set and we wished to try to model only the dynamics of players’ movements and interactions.

Experiment: Modeling Specific Football Plays

In this experiment we look at a specific football plays, comparing the generalization of various models. We train on a collection of the same type of play and then test on held out examples of the same type of play. Higher likelihood in the testing data is indicative of a more accurate model.

Play:

Above are two examples of the first play we model. This is a passing play called “143dig”. The blue lines are offensive players, the red are defensive, the black are the referees, and the yellow circle is the ball. We do not use any of the ball or referee data because we are interested in modeling player dynamics and not all of the plays come with ball and referee information. Additional experiments could easily be run to incorporate this additional information.

There are 7 examples of “143dig” in the dataset. We train our models using 5 of them, and test using the remaining 2.

Above are training log likelihoods for our models. However, testing log likelihood is the true indicator of how well our models fit data.

Model	Log Likelihood Test Play 1	Log Likelihood Test Play 2
Single LDS dim(x)=1	-1.9E5	-1.5E5
Single LDS dim(x)=2	-1.9E5	-1.5E5
Single LDS dim(x)=3	-2.7E7	-3.2E7
Multi LDS dim(x)=1	-8.1E7	-1.1E7
Multi LDS dim(x)=2	-1.0E7	-1.8E7
Single SLDS dim(s)=2	-7.7e5 ± є	-1.5E6 ± є
Single SLDS dim(s)=4	-1.8E5 ± є	-1.5E5 ± є
Multi SLDS dim(s)=2	-1.4E4 ± 3.4E2	-1.6E4 ± 1.0E3
DST w/ team & game states	-5.6E3 ± 1.8E2	-6.1E3 ± 1.4E2
DST w/ game states	-8.7E3 ± 1.7E3	-7.9E3 ± 1.6E3

When the dimensionality of a parameter is unspecified, its dimension is 2.

Both types of DSTs perform better than the other models on testing data.

This webpage will updated incrementally, please check back often.

Thanks to Stephen Intille and his project Computers Watching Football for our data set.

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Models Page Toy Problem Andrew Howard Home Prof. Tony Jebara Home