Predicting Basketball Possession Outcomes Using Optical Tracking Data

Based on “box score” data:

• Made/missed shots.
• Assists.
• Rebounds, blocks, steals, turnovers, fouls.

Limitations:

• Data misses meaningful actions and events.
• Players depend on teammates.
• Actions analyzed out of context.

Example possession

What's in the box score:

• Made shot by LeBron James.
• Assist by Rashard Lewis.

What's missing:

• Norris Cole's dribble penetration.
• Characterization of defense.
• LeBron James' dribble penetration.

Let \( \Omega \) be the space of all possible basketball possessions. For \( \omega \in \Omega \)

• \( X(\omega) \in \{0, 2, 3\} \): point value of possession \( \omega \).
• \( T(\omega) \): possession length
• \( Z_t(\omega), 0 \leq t \leq T(\omega) \): time series of optical tracking data.
• \( \mathcal{F}^{(Z)}_t = \sigma(\{Z_s^{-1}: 0 \leq s \leq t\}) \): natural filtration.

• EPV provides an instantaneous snapshot of the possession's value, given its full spatiotemporal history.
• \( \nu_t \) is a Martingale: \( \mathbb{E}[\nu_{t + h} | \mathcal{F}^{(Z)}_t] = \nu_t \) for all \( h > 0 \).

Regression-type prediction (points versus possession features):

Finite collection of states, \( \mathcal{C} \)

• \( C_t \in \mathcal{C} \): state of the possession at time \( t \).
• Observable states: \( C_t = h(Z_t) \).
• \( C^{(0)}, C^{(1)}, \ldots, C^{(K)} \): discrete sequence of distinct states.

Some useful stopping times:

\[ \begin{align*} \color{blue}{\tau_t} &= \begin{cases} \text{min} \{ s : s > t, C_s \in \mathcal{C}_{\text{trans}}\} & \text{if } C_t \in \mathcal{C}_{\text{poss}} \\ t & \text{if } C_t \not \in \mathcal{C}_{\text{poss}} \end{cases} \\ \color{red} {\delta_t} &= \text{min}\{s : s \geq \tau_t, C_s \not \in \mathcal{C}_{\text{trans}} \} \end{align*} \]

\[ \begin{align*} \color{blue}{\tau_t} &= \begin{cases} \text{min} \{ s : s > t, C_s \in \mathcal{C}_{\text{trans}}\} & \text{if } C_t \in \mathcal{C}_{\text{poss}} \\ t & \text{if } C_t \not \in \mathcal{C}_{\text{poss}} \end{cases} \\ \color{red} {\delta_t} &= \text{min}\{s : s \geq \tau_t, C_s \not \in \mathcal{C}_{\text{trans}} \} \end{align*} \]
Key assumptions:

• (A1) \( C_t \) is marginally semi-Markov.
• (A2) For all \( s > \delta_t \), \( \mathbb{P}(C_s | C_{\delta_t}, \mathcal{F}^{(Z)}_t) = \mathbb{P}(C_s | C_{\delta_t}) \).

Assume (A1)–(A2), then for all \( 0 \leq t < T \),

\[ \nu_t = \sum_{c \in \{ \mathcal{C}_{\text{trans}} \cup \mathcal{C}_{\text{end}} \}} \mathbb{E}[X | C_{\delta_t} = c]\mathbb{P}(C_{\delta_t} = c | \mathcal{F}^{(Z)}_t). \]

\[ \nu_t = \sum_{c \in \{ \mathcal{C}_{\text{trans}} \cup \mathcal{C}_{\text{end}} \}} \mathbb{E}[X | C_{\delta_t} = c]\mathbb{P}(C_{\delta_t} = c | \mathcal{F}^{(Z)}_t). \] Let \( M(t) \) be the event \( \{\tau_t \leq t + \epsilon\} \).

• (M1)   \( \mathbb{P}(Z_{t + \epsilon} | M(t)^c, \mathcal{F}^{(Z)}_t) \): the microtransition model.
• (M2)   \( \mathbb{P}(M(t) | \mathcal{F}^{(Z)}_t) \): the macrotransition entry model.
• (M3)   \( \mathbb{P}(C_{\delta_t} | M(t), \mathcal{F}^{(Z)}_t) \): the macrotransition exit model.
• (M4)   \( \mathbf{P} \), with   \( P_{qr} = \mathbb{P}(C^{(n+1)} = c_r | C^{(n)} = c_q) \): the Markov transition probability matrix.

Monte Carlo computation of \( \nu_t \):

• Draw   \( C_{\delta_t} | \mathcal{F}^{(Z)}_t \) using (M1)–(M3).
• Calculate \( \mathbb{E}[X | C_{\delta_t}] \) using (M4).

Player \( \ell \)'s position at time \( t \) is \( \mathbf{z}^{\ell}(t) = (x^{\ell}(t), y^{\ell}(t)) \).

\[ x^{\ell}(t + \epsilon) = x^{\ell}(t) + \alpha^{\ell}_x[x^{\ell}(t) - x^{\ell}(t - \epsilon)] + \eta^{\ell}_x(t) \]

• \( \eta^{\ell}_x(t) \sim \mathcal{N} (\mu^{\ell}_x(\mathbf{z}^{\ell}(t)), (\sigma^{\ell}_x)^2) \).
• \( \mu_x \) has Gaussian Process prior.
• \( y^{\ell}(t) \) modeled analogously (and independently).
• Different parameters for all players, offense (ball carrier or not) and defense.

Recall   \( M(t) = \{ \tau_t \leq t + \epsilon \} \):

• Six different “types”“, based on entry state   \( C_{\tau_t}, \cup_{j=1}^6 M_j(t) = M(t) \).
• Hazards:   \( \lambda_j(t) = \lim_{\epsilon \rightarrow 0} \frac{\mathbb{P}(M_j(t) | \mathcal{F}^{(Z)}_t)}{ \epsilon} \).

\[ \log(\lambda_j(t)) = [\mathbf{W}_j^{\ell}(t)]'\boldsymbol{\beta}_j^{\ell} + \xi_j^{\ell}\left(\mathbf{z}^{\ell}(t)\right) \]

• \( \mathbf{W}_j^{\ell}(t), \boldsymbol{\beta}_j^{\ell} \): time-varying covariates and coefficients.
• \( \xi_j^{\ell} \): spatial effect on log-hazard.

Spatial effects \( \xi^{\ell}_j \)

“Partially Bayes'' estimation of all model parameters:

Our EPV framework can be extended to better incorporate unique basketball strategies:

• Luke Bornn, Alex D'Amour, Alex Franks, Kirk Goldsberry, Andrew Miller.
• Moore/Sloan Foundations.