# Chapter 9 Appendix A

## 9.1 Model Replication

The initial approach to understand how to best capture passing networks sought to replicate Daniel Cervone, Alex, D’Amour, Luke Bornn, and Kirk Goldsberry’s paper,“A Multiresolution Stochastic Process Model for Predicting Basketball Possession Outcomes.” They attempt to capture the game wholelistically via a new measure called Expected Possession Value (EPV). This new metric uses three models–a Microtransition Model, Macrotransition Entrance Model, and a Macrotransition Exit Model–to capture the spatial biases of each player and the in-game effects of pressure, so that it can measure the likelihood of a successful play (made shot) given the previous sequence of events. To compare players against the league-average scores, they also calculated Expected Possession Value -Adjusted as an application for teams. Below is a brief overview of each model.

This paper is particularly interesting because EPV utilizes the spatio-temporal elements of the game, so it models the NBA game dynamically. Given Duke Basketball data, the motivation is to replicate “A Multiresolution Stochastic Process Model for Predicting Basketball Possession Outcomes,” to better understand the Duke Men’s team, as well as to compare professional basketball to collegiate basketball individual and team playing styles. Below is a brief overview of each model used in the paper to calculate EPV.

### 9.1.1 Microtransition Model

$$x^{l}(t+\epsilon) = x^{l}(t) + \alpha^{l}_{x}[x^{l}(t) - x^{l}(t-\epsilon)] + \eta^{l}_{x}(t)$$ where $$\eta^{l}_{x}(t) \sim N(\mu^{l}_{x}(z^{l}(t)), (\sigma^{l}_{x})^{2})$$

The microtransition model models the defensive conditions of the game based on the $$(x,y)$$ coordinates of a player and their acceleration effects ($$\alpha^{l}_{x}(t)$$). It is also assumed that a player’s spatial location is normally distributed. Since players play differently, each microtransition model is specifically fitted to the player.

### 9.1.2 Macrotransition Entrance Model

$$P(M(t)|F_{t}^{(Z)}$$ The macrotransition entrance model predicts whether the next move will be a pass (4 options), shot attempt, or turnover. The model is disjoint.

### 9.1.3 Macrotransition Exit Model

$$P(C_{\delta_{t}}|M(t), F_{t}^{(Z)})$$ Given the Macrotransition Entrance Model predicts a shot attempt, it indexes to a logistic regression model to calculate player $$l$$’s successful shot probability. Given the Macrotransition Entrance Model predicts a pass,it indexes to a model that predicts where the pass will take place. Otherwise, a turnover is assumed.

### 9.1.4 Fall Backs on the Implementation of this Model

Currently, the implementation of the model has yet to be completed due to setbacks of incompatible R code. The implementation of this paper is currently still in progress.

### 9.1.5 Proposal

Regardless, we hypothesize that since both metrics are calculated via a semi-Markov process, EPV fails to capture the full nature of the possession because it only uses the last posession as a prior. The model would be more robust if it captured the entirety of the posession in its prior–however, the computational time of such an ordeal would prevent any real-time analyses. Thus, this paper proposes that a simpler model may perform more quickly and potentially just as robustly to allow for game-time analyses.