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Automatic event detection in basketball using HMM with energy based defensive assignment

Suraj Keshri, Min-hwan Oh, Sheng Zhang and Garud Iyengar

Abstract

We propose a unsupervised learning framework for automatically labeling events in a basketball game. Our framework uses the the optical player tracking data in the NBA. We first learn the time series of defensive assignments using a novel player and location dependent attraction based model which uses hidden Markov models (HMMs), Gaussian processes, and a “bond breaking” model for changes in defensive assignments. Next, we use the learned defensive assignments as an input to a set of HMMs that automatically detect events such as ball screens, drives and post-ups. We show that our models provide significant improvements over existing benchmarks both on defensive assignments and event detection.

A Appendix

A.1 Derivation of posterior distribution of Γ

The data likelihood is

P(D|Γp,σD2)=t,j,k[P(Dti|Itij,Γ,σD2)P(Itij|I(t1)i.)]Itijk=1Kj=1Npke12σ2[ZkjTΓpkDkj]T[ZkjTΓpkDkj]e12σ2(kΓpkTWkΓpk2kΓpkTVk)

where Wk=j=1NpkZkjZkjT and Vk=j=1NpkZkjDkj.

Note that we no longer write the likelihood with respect to time. Only thing that time tells us is the assignment of defensive players to offensive players. Once we have estimated the sequence I, then we no longer have time dependence in our data. From the estimation perspective, it is more efficient to write the likelihood in terms of events.

Let V=[V1,,VK] and Γp=[Γp1,,ΓpK]. Also, we define W to be a block diagonal matrix with each block being Wk for k = 1, …, K. Then, we can express the data likelihood as the following:

P(D|Γp,σD2)e12σ2(ΓpTWΓp2ΓpTV)

Using the GP prior on Γp mentioned above

ΓpGP(μΓ,𝒦),

we can compute the posterior distribution

P(ΓpD)P(D|Γp)P(Γp)e12σD2(ΓpTWΓp2ΓpTV)e12(ΓpμΓ)T𝒦1(ΓpμΓ)exp(12[Γpμ]TΣ1[Γpμ])

where μ=(WσD2+𝒦1)1(VσD2+𝒦1μΓ) and Σ=(WσD2+𝒦1)1. Hence, the posterior distribution is

ΓpD,I,σD2N(μ,Σ)with ΓpkT1=1.

A.2 Sampling of multivariate gaussian distribution with linear constraints

We want to simulate zN(μ,Σ) conditioned on Fz = v. Without loss of generality, assume that Fm×n has full row rank. Define:

  1. P=F(FF)1F: projection onto ={Fv:vm}

  2. P=IP: projection onto the linear space orthogonal to

Then, the distribution of z conditioned on Fz = v is given by

zN(v¯+Pμ,PΣ(P))

where v¯=F(FF)1v.

A.3 Convergences

Figure 10: Convergence Of data log-likelihood in defense assignment modeling.

Figure 10:

Convergence Of data log-likelihood in defense assignment modeling.

Figure 11: Convergence Of BFGS algorithm for defense transition modeling.

Figure 11:

Convergence Of BFGS algorithm for defense transition modeling.

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Published Online: 2019-02-28
Published in Print: 2019-06-26

©2019 Walter de Gruyter GmbH, Berlin/Boston