Dan Cervone
New York University
dcervone@nyu.edu
JSM 2016
Chicago, IL
August 3, 2016
Space is a valuable commodity in basketball.
During a possession:
How do we quantify:
SportVu player-tracking data:
Player's real estate investment portfolio value:
“Weighted Voronoi” model for real estate portfolios:
Specifically: divide the court into \( M \) equally sized cells
\( w^i_j(t) = \text{dist}(\text{player } i, \text{cell } j)^2 \) at time \( t \).
\( x_j^i(t) \) gives player \( i \)'s investiment in cell \( j \) at time \( t \):
\[ x^i_j(t) = \begin{cases} \frac{1}{1 + w^i_j(t)} & i = \text{argmin}_h w^h_j(t) \\ 0 & \text{otherwise} \end{cases} \]
\( \beta \in \mathbb{R}^M \), where \( \beta_j \) is price/value of region \( j \)
\( X^i(t) \) row vector, entry \( j \) is \( x^i_j(t) \).
\( X^i(t) \beta \) is the total property value of player \( i \) at time \( t \)
“Dominant regions” generalize Voronoi partitions
Idea: Passes move the ball to more valuable real estate
Idea: Passes move the ball to more valuable real estate
\( \beta \) is estimated using a penalized Plackett-Luce model.
Player- (team-) specific property values (\( \beta + \alpha^i \))
\[ \begin{align} L_{\lambda_1, \lambda_2}(\beta, \alpha^1, \ldots, \alpha^K) = & \\ & \hspace{-6cm} \Bigg[ \sum_{i, j} \sum_{t : \text{pass } i \rightarrow j} X^j(t)(\beta + \alpha^j) - \log\Big( \exp \big(X^i(t)(\beta + \alpha^i) \big) + \exp \big( X^j(t)(\beta + \alpha^j) \big) \Big) \Bigg] \nonumber \\ & \hspace{2cm} - \frac{1}{2}\lambda_1 ||\beta||_2^2 - \frac{1}{2}\lambda_2 \sum_i ||\alpha^i||_2^2 \\ & \text{ subject to } \beta_k \geq 0, \: k=1, \ldots, M. \nonumber \end{align} \]