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 expected possession value (EPV), derives from a stochastic process model for the evolution of a basketball possession. We model this process at multiple levels of resolution, differentiating between continuous, infinitesimal movements of players, and discrete events such as shot attempts and turnovers. Transition kernels are estimated using hierarchical spatiotemporal models that share information across players while remaining computationally tractable on very large data sets. In addition to estimating EPV, these models reveal novel insights on players' decision-making tendencies as a function of their spatial strategy. A data sample and R code for further exploration of our model/results are available in the repository https://github.com/dcervone/EPVDemo.

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 qualitative, based on expert intuition and analysis that can be prone to human biases and imprecision.Fortunately, emerging optical player tracking systems have the potential to enable a richer quantitative characterization of basketball performance, particularly defensive performance. Unfortunately, due to computational and methodological complexities, that potential remains unmet. This paper attempts to fill this void, combining spatial and spatio-temporal processes, matrix factorization techniques and hierarchical regression models with player tracking data to advance the state of defensive analytics in the NBA. Our approach detects, characterizes and quantifies multiple aspects of defensive play in basketball, supporting some common understandings of defensive effectiveness, challenging others and opening up many new insights into the defensive elements of basketball.

One primary utility of animated maps is their ability to depict change over time and space; unfortunately, recent research suggests that humans frequently fail to perceive changes within dynamic graphics. However, different types of dynamic graphics include different manifestations of change. For example, an animated proportional-symbol map possesses different change properties than an animated choropleth map. This article examines issues of change on animated choropleth maps. We identify relevant limitations of the human visual system that pertain to animated map reading, including change blindness and foveal versus peripheral attention, and introduce methods to quantify the magnitude of change that separates individual scenes within choropleth animations. These methods are useful for measuring and describing changes that confront users of animated choropleth maps. We also characterize the transitional behaviours of enumeration units and discuss the influences of data classification and other cartographic controls on change within animated choropleth maps. RésuméUne des principales utilités des cartes animées est qu'elles permettent de suivre les changements en fonction du temps et de l'espace. Malheureusement, selon de récentes recherches, il arrive souvent que les humains ne perçoivent pas ces changements dans les graphiques dynamiques. De plus, chaque type de graphique dynamique inclut des manifestations différentes de l'évolution. Par exemple, les propriétés d'une carte animée avec symboles proportionnels sont différentes de celles d'une carte choroplèthe animée. Dans l'article, on examine les problèmes liés au changement dans les cartes choroplèthes animées. On définit les limitations pertinentes des systèmes visuels humains qui sont associées à la lecture de cartes animées, y compris la cécité au changement et l'attention fovéale/périphérique, et on présente des méthodes pour quantifier la magnitude du changement qui sépare des scènes individuelles dans des animations choroplèthes. Ces méthodes sont utiles pour mesurer et décrire les changements auxquels se heurtent les utilisateurs de cartes choroplèthes animées. On caractérise aussi les comportements transitionnels des unités de dénombrement et on parle des effets de la classification des données et d'autres mesures cartographiques sur les changements apportés aux cartes choroplèthes animées.

Animated choropleth maps enable cartographers to visualize time-series data in a way that congruently depicts change over time. However, users have difficulty apprehending information encoded within these displays, and often fail to detect important changes between adjacent scenes. Failures of visual experience, such as change blindness, threaten the effectiveness of dynamic geovisual displays, in which several important changes can occur simultaneously throughout the display. Animated choropleth maps require viewers not only to notice changes but also understand symbolic meanings encoded in rapid transitions between scenes. Graphic interpolation between key frames, also known as "in-betweening" or "tweening", smoothes transitions and lengthens the duration of the transition between scenes in a dynamic sequence. Previous cartographic literature suggests tweening could be a potential solution for change blindness in the cartographic context. This article examines this issue of change blindness in the cartographic context and reports on a human subjects investigation designed to evaluate the influence of cartographic design variables on map readers' change detection abilities. Our results indicate that 1) map readers have difficulty detecting changes in animated choropleth maps, 2) map readers overestimate their own change detection abilities, and 3) tweening influences the legibility of change in animated choropleth maps.

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