A new range of statistical analysis has emerged in sports after the introduction of the high-resolution player tracking technology, specifically in basketball. However, this high dimensional data is often challenging for statistical inference and decision making. In this article, we employ Hidalgo, a state-of-the-art Bayesian mixture model that allows the estimation of heterogeneous intrinsic dimensions (ID) within a dataset and propose some theoretical enhancements. ID results can be interpreted as indicators of variability and complexity of basketball plays and games. This technique allows classification and clustering of NBA basketball player's movement and shot charts data. Analyzing movement data, Hidalgo identifies key stages of offensive actions such as creating space for passing, preparation/shooting and following through. We found that the ID value spikes reaching a peak between 4 and 8 seconds in the offensive part of the court after which it declines. In shot charts, we obtained groups of shots that produce substantially higher and lower successes. Overall, game-winners tend to have a larger intrinsic dimension which is an indication of more unpredictability and unique shot placements. Similarly, we found higher ID values in plays when the score margin is small compared to large margin ones. These outcomes could be exploited by coaches to obtain better offensive/defensive results.
In this paper, we develop a group learning approach to analyze the underlying heterogeneity structure of shot selection among professional basketball players in the NBA. We propose a mixture of finite mixtures (MFM) model to capture the heterogeneity of shot selection among different players based on Log Gaussian Cox process (LGCP). Our proposed method can simultaneously estimate the number of groups and group configurations. An efficient Markov Chain Monte Carlo (MCMC) algorithm is developed for our proposed model. Simulation studies have been conducted to demonstrate its performance. Ultimat…
Every shot in basketball has an opportunity cost; one player's shot eliminates all potential opportunities from their teammates for that play. For this reason, player-shot efficiency should ultimately be considered relative to the lineup. This aspect of efficiency---the optimal way to allocate shots within a lineup---is the focus of our paper. Allocative efficiency should be considered in a spatial context since the distribution of shot attempts within a lineup is highly dependent on court location. We propose a new metric for spatial allocative efficiency by comparing a player's field goal pe…
We propose a Bayesian nonparametric matrix clustering approach to analyze the latent heterogeneity structure in the shot selection data collected from professional basketball players in the National Basketball Association (NBA). The proposed method adopts a mixture of finite mixtures framework and fully utilizes the spatial information via a mixture of matrix normal distribution representation. We propose an efficient Markov chain Monte Carlo algorithm for posterior sampling that allows simultaneous inference on both the number of clusters and the cluster configurations. We also establish larg…
What Influences the Field Goal Attempts of Professional Players? Analysis of Basketball Shot Charts via Log Gaussian Cox Processes with Spatially Varying Coefficients
We propose a multidimensional tensor clustering approach for studying how professional basketball players' shooting patterns vary over court locations and game time. Unlike most existing methods that only study continuous-valued tensors or have to assume the same cluster structure along different tensor directions, we propose a Bayesian nonparametric model that deals with count-valued tensors and projects the heterogeneity among players onto tensor dimensions while allowing cluster structures to be different over directions. Our method is fully probabilistic; hence allows simultaneous inferenc…
It is common to be interested in rankings or order relationships among entities. In complex settings where one does not directly measure a univariate statistic upon which to base ranks, such inferences typically rely on statistical models having entity-specific parameters. These can be treated as random effects in hierarchical models characterizing variation among the entities. In this paper, we are particularly interested in the problem of ranking basketball players in terms of their contribution to team performance. Using data from the United States National Basketball Association (NBA), we …
In recent years, analytics has started to revolutionize the game of basketball: quantitative analyses of the game inform team strategy, management of player health and fitness, and how teams draft, sign, and trade players. In this review, we focus on methods for quantifying and characterizing basketball gameplay. At the team level, we discuss methods for characterizing team strategy and performance, while at the player level, we take a deep look into a myriad of tools for player evaluation. This includes metrics for overall player value, defensive ability, and shot modeling, and methods for un…
For professional basketball, finding valuable and suitable players is the key to building a winning team. To deal with such challenges, basketball managers, scouts and coaches are increasingly turning to analytics. Objective evaluation of players and teams has always been the top goal of basketball analytics. Typical statistical analytics mainly focuses on the box score and has developed various metrics. In spite of the more and more advanced methods, metrics built upon box score statistics provide limited information about how players interact with each other. Two players with similar box sco…
In this paper we model basketball plays as episodes from team-specific non-stationary Markov decision processes (MDPs) with shot clock dependent transition probabilities. Bayesian hierarchical models are employed in the modeling and parametrization of the transition probabilities to borrow strength across players and through time. To enable computational feasibility, we combine lineup-specific MDPs into team-average MDPs using a novel transition weighting scheme. Specifically, we derive the dynamics of the team-average process such that the expected transition count for an arbitrary state-pair…
In basketball and hockey, state-of-the-art player value statistics are often variants of Adjusted Plus-Minus (APM). But APM hasn't had the same impact in soccer, since soccer games are low scoring with a low number of substitutions. In soccer, perhaps the most comprehensive player value statistics come from video games, and in particular FIFA. FIFA ratings combine the subjective evaluations of over 9000 scouts, coaches, and season-ticket holders into ratings for over 18,000 players. This paper combines FIFA ratings and APM into a single metric, which we call Augmented APM. The key idea is reca…
Although basketball is a dynamic process sport, with 5 plus 5 players competing on both offense and defense simultaneously, learning some static information is predominant for professional players, coaches and team mangers. In order to have a deep understanding of field goal attempts among different players, we propose a zero inflated Poisson model with clustered regression coefficients to learn the shooting habits of different players over the court and the heterogeneity among them. Specifically, the zero inflated model recovers the large proportion of the court with zero field goal attempts,…
Basketball players' performance measurement is of critical importance for a broad spectrum of decisions related to training and game strategy. Despite this recognized central role, the main part of the studies on this topic focus on performance level measurement, neglecting other important characteristics, such as variability. In this paper, shooting performance variability is modeled with a Markov Switching dynamic, assuming the existence of two alternating performance regimes. Then, the relationships between each player's variability and the lineup composition is modeled as an ARIMA process …
In the domain of Sport Analytics, Global Positioning Systems devices are intensively used as they permit to retrieve players' movements. Team sports' managers and coaches are interested on the relation between players' patterns of movements and team performance, in order to better manage their team. In this paper we propose a Cluster Analysis and Multidimensional Scaling approach to find and describe separate patterns of players movements. Using real data of multiple professional basketball teams, we find, consistently over different case studies, that in the defensive clusters players are clo…
Analysis of player tracking data for American football is in its infancy, since the National Football League (NFL) released its Next Gen Stats tracking data publicly for the first time in December 2018. While tracking datasets in other sports often contain detailed annotations of on-field events, annotations in the NFL's tracking data are limited. Methods for creating these annotations typically require extensive human labeling, which is difficult and expensive. We begin tackling this class of problems by creating annotations for pass coverage types by defensive backs using unsupervised learni…
Global Positioning Systems (GPS) are nowadays intensively used in Sport Science as they permit to capture the space-time trajectories of players, with the aim to infer useful information to coaches in addition to traditional statistics. In our application to basketball, we used Cluster Analysis in order to split the match in a number of separate time-periods, each identifying homogeneous spatial relations among players in the court. Results allowed us to identify differences in spacing among players, distinguish defensive or offensive actions, analyze transition probabilities from a certain gr…
In the sports of soccer, hockey and basketball the most commonly used statistics for player performance assessment are divided into two categories: offensive statistics and defensive statistics. However, qualitative assessments of playmaking (for example making "smart" passes) are difficult to quantify. It would be advantageous to have available a single statistic that can emphasize the flow of a game, rewarding those players who initiate and contribute to successful plays more. In this paper we will examine a model based on Google's PageRank. Other papers have explored ranking teams, coaches,…
I address the difficult challenge of measuring the relative influence of competing basketball game strategies, and I apply my analysis to plays resulting in three-point shots. I use a glut of SportVU player tracking data from over 600 NBA games to derive custom position-based features that capture tangible game strategies from game-play data, such as teamwork, player matchups, and on-ball defender distances. Then, I demonstrate statistical methods for measuring the relative importance of any given basketball strategy. In doing so, I highlight the high importance of teamwork based strategies in…
Although basketball is a dualistic sport, with all players competing on both offense and defense, almost all of the sport's conventional metrics are designed to summarize offensive play. As a result, player valuations are largely based on offensive performances and to a much lesser degree on defensive ones. Steals, blocks and defensive rebounds provide only a limited summary of defensive effectiveness, yet they persist because they summarize salient events that are easy to observe. Due to the inefficacy of traditional defensive statistics, the state of the art in defensive analytics remains qu…
A new approach in team sports analysis consists in studying positioning and movements of players during the game in relation to team performance. State of the art tracking systems produce spatio-temporal traces of players that have facilitated a variety of research aimed to extract insights from trajectories. Several methods borrowed from machine learning, network and complex systems, geographic information system, computer vision and statistics have been proposed. After having reviewed the state of the art in those niches of literature aiming to extract useful information to analysts and expe…
We develop a machine learning approach to represent and analyze the underlying spatial structure that governs shot selection among professional basketball players in the NBA. Typically, NBA players are discussed and compared in an heuristic, imprecise manner that relies on unmeasured intuitions about player behavior. This makes it difficult to draw comparisons between players and make accurate player specific predictions. Modeling shot attempt data as a point process, we create a low dimensional representation of offensive player types in the NBA. Using non-negative matrix factorization (NMF),…
Basketball shot location data provide valuable summary information regarding players to coaches, sports analysts, fans, statisticians, as well as players themselves. Represented by spatial points, such data are naturally analyzed with spatial point process models. We present a novel nonparametric Bayesian method for learning the underlying intensity surface built upon a combination of Dirichlet process and Markov random field. Our method has the advantage of effectively encouraging local spatial homogeneity when estimating a globally heterogeneous intensity surface. Posterior inferences are pe…
We consider a study of players employed by teams who are members of the National Basketball Association where units of observation are functional curves that are realizations of production measurements taken through the course of one's career. The observed functional output displays large amounts of between player heterogeneity in the sense that some individuals produce curves that are fairly smooth while others are (much) more erratic. We argue that this variability in curve shape is a feature that can be exploited to guide decision making, learn about processes under study and improve predic…
Basketball games evolve continuously in space and time as players constantly interact with their teammates, the opposing team, and the ball. However, current analyses of basketball outcomes rely on discretized summaries of the game that reduce such interactions to tallies of points, assists, and similar events. In this paper, we propose a framework for using optical player tracking data to estimate, in real time, the expected number of points obtained by the end of a possession. This quantity, called \textit{expected possession value} (EPV), derives from a stochastic process model for the evol…
In this paper, we describe an approach to rank sport players based on their efficiency. Although is extremely useful to analyze the performance of team games there is no unanimity on the use of a single index to perform such a ranking. We propose a method to summarize the huge amount of information collected at different aspects of a sport team which is almost daily publicly available. The tool will allow agents involved in a player's negotiation to show the strengths (and weaknesses) of the player with respect to other players. The approach is based on applying a multicriteria outranking meth…
Big Data Analytics help team sports' managers in their decisions by processing a number of different kind of data. With the advent of Information Technologies, collecting, processing and storing big amounts of sport data in different form became possible. A problem that often arises when using sport data regards the need for automatic data cleaning procedures. In this paper we develop a data cleaning procedure for basketball which is based on players' trajectories. Starting from a data matrix that tracks the movements of the players on the court at different moments in the game, we propose an …
This paper develops metrics from a social network perspective that are directly translatable to the outcome of a basketball game. We extend a state-of-the-art multi-resolution stochastic process approach to modeling basketball by modeling passes between teammates as directed dynamic relational links on a network and introduce multiplicative latent factors to study higher-order patterns in players' interactions that distinguish a successful game from a loss. Parameters are estimated using a Markov chain Monte Carlo sampler. Results in simulation experiments suggest that the sampling scheme is e…
Data analytics in sports is crucial to evaluate the performance of single players and the whole team. The literature proposes a number of tools for both offence and defence scenarios. Data coming from tracking location of players, in this respect, may be used to enrich the amount of useful information. In basketball, however, actions are interleaved with inactive periods. This paper describes a methodological approach to automatically identify active periods during a game and to classify them as offensive or defensive. The method is based on the application of thresholds to players kinematic p…
How much is an on-ball screen worth? How much is a backdoor cut away from the ball worth? Basketball is one of a number of sports which, within the past decade, have seen an explosion in quantitative metrics and methods for evaluating players and teams. However, it is still challenging to evaluate individual off-ball events in terms of how they contribute to the success of a possession. In this study, we develop an end-to-end deep learning architecture DeepHoops to process a unique dataset composed of spatio-temporal tracking data from NBA games in order to generate a running stream of predict…
The success rate of a basketball shot may be higher at locations where a player makes more shots. For a marked spatial point process, this means that the mark and the intensity are associated. We propose a Bayesian joint model for the mark and the intensity of marked point processes, where the intensity is incorporated in the mark model as a covariate. Inferences are done with a Markov chain Monte Carlo algorithm. Two Bayesian model comparison criteria, the Deviance Information Criterion and the Logarithm of the Pseudo-Marginal Likelihood, were used to assess the model. The performances of the…
The present work aims to improve the existing analysis on the performance of professional basketball players presenting a methodology to measure its performance and regularity.
Regression-based adjusted plus-minus statistics were developed in basketball and have recently come to hockey. The purpose of these statistics is to provide an estimate of each player's contribution to his team, independent of the strength of his teammates, the strength of his opponents, and other variables that are out of his control. One of the main downsides of the ordinary least squares regression models is that the estimates have large error bounds. Since certain pairs of teammates play together frequently, collinearity is present in the data and is one reason for the large errors. In hoc…
Soccer is undeniably the most popular sport world-wide and everyone from general managers and coaching staff to fans and media are interested in evaluating players' performance. Metrics applied successfully in other sports, such as the (adjusted) +/- that allows for division of credit among a basketball team's players, exhibit several challenges when applied to soccer due to severe co-linearities. Recently, a number of player evaluation metrics have been developed utilizing optical tracking data, but they are based on proprietary data. In this work, our objective is to develop an open framewor…
Most historical National Football League (NFL) analysis, both mainstream and academic, has relied on public, play-level data to generate team and player comparisons. Given the number of oft omitted variables that impact on-field results, such as play call, game situation, and opponent strength, findings tend to be more anecdotal than actionable. With the release of player tracking data, however, analysts can better ask and answer questions to isolate skill and strategy. In this article, we highlight the limitations of traditional analyses, and use a decades-old punching bag for analysts, fourt…
The release of NBA player tracking data greatly enhances the granularity and dimensionality of basketball statistics used to evaluate and compare player performance. However, the high dimensionality of this new data source can be troublesome as it demands more computational resources and reduces the ability to easily interpret findings. Therefore, we must find a way to reduce the dimensionality of the data while retaining the ability to differentiate and compare player performance. In this paper, Principal Component Analysis (PCA) is used to identify four principal components that account for …
