Beyond Expected Goals: A Probabilistic Framework for Shot Occurrences in Soccer

ArXiv.org · 2025-11-28

Expected goals (xG) models estimate the probability that a shot results in a goal from its context (e.g., location, pressure), but they operate only on observed shots. We propose xG+, a possession-level framework that first estimates the probability that a shot occurs within the next second and its corresponding xG if it were to occur. We also introduce ways to aggregate this joint probability estimate over the course of a possession. By jointly modeling shot-taking behavior and shot quality, xG+ remedies the conditioning-on-shots limitation of standard xG. We show that this improves predictive accuracy at the team level and produces a more persistent player skill signal than standard xG models.

Estimating player value in American football using plus–minus models

Journal of Quantitative Analysis in Sports · 2021-08-29 · 13 citations

Abstract Calculating the value of football player’s on-field performance has been limited to scouting methods while data-driven methods are mostly limited to quarterbacks. A popular method to calculate player value in other sports are Adjusted Plus–Minus (APM) and Regularized Adjusted Plus–Minus (RAPM) models. These models have been used in other sports, most notably basketball (Rosenbaum, D. T. 2004. Measuring How NBA Players Help Their Teams Win . http://www.82games.com/comm30.htm#_ftn1 ; Kubatko, J., D. Oliver, K. Pelton, and D. T. Rosenbaum. 2007. “A Starting Point for Analyzing Basketball Statistics.” Journal of Quantitative Analysis in Sports 3 (3); Winston, W. 2009. Player and Lineup Analysis in the NBA . Cambridge, Massachusetts; Sill, J. 2010. “Improved NBA Adjusted +/− Using Regularization and Out-Of-Sample Testing.” In Proceedings of the 2010 MIT Sloan Sports Analytics Conference ) to estimate each player’s value by accounting for those in the game at the same time. Football is less amenable to APM models due to its few scoring events, few lineup changes, restrictive positioning, and small quantity of games relative to the number of teams. More recent methods have found ways to incorporate plus–minus models in other sports such as Hockey (Macdonald, B. 2011. “A Regression-Based Adjusted Plus-Minus Statistic for NHL players.” Journal of Quantitative Analysis in Sports 7 (3)) and Soccer (Schultze, S. R., and C.-M. Wellbrock. 2018. “A Weighted Plus/Minus Metric for Individual Soccer Player Performance.” Journal of Sports Analytics 4 (2): 121–31 and Matano, F., L. F. Richardson, T. Pospisil, C. Eubanks, and J. Qin (2018). Augmenting Adjusted Plus-Minus in Soccer with Fifa Ratings . arXiv preprint arXiv:1810.08032). These models are useful in coming up with results-oriented estimation of each player’s value. In American football, many positions such as offensive lineman have no recorded statistics which hinders the ability to estimate a player’s value. I provide a fully hierarchical Bayesian plus–minus (HBPM) model framework that extends RAPM to include position-specific penalization that solves many of the shortcomings of APM and RAPM models in American football. Cross-validated results show the HBPM to be more predictive out of sample than RAPM or APM models. Results for the HBPM models are provided for both Collegiate and NFL football players as well as deeper insights into positional value and position-specific age curves.

VLT/SPHERE imaging survey of the largest main-belt asteroids: Final results and synthesis

Astronomy and Astrophysics · 2021 · 102 citations

• Physics
• Astrophysics
• Astronomy

Context. Until recently, the 3D shape, and therefore density (when combining the volume estimate with available mass estimates), and surface topography of the vast majority of the largest ( D ≥ 100 km) main-belt asteroids have remained poorly constrained. The improved capabilities of the SPHERE/ZIMPOL instrument have opened new doors into ground-based asteroid exploration. Aims. To constrain the formation and evolution of a representative sample of large asteroids, we conducted a high-angular-resolution imaging survey of 42 large main-belt asteroids with VLT/SPHERE/ZIMPOL. Our asteroid sample comprises 39 bodies with D ≥ 100 km and in particular most D ≥ 200 km main-belt asteroids (20/23). Furthermore, it nicely reflects the compositional diversity present in the main belt as the sampled bodies belong to the following taxonomic classes: A, B, C, Ch/Cgh, E/M/X, K, P/T, S, and V. Methods. The SPHERE/ZIMPOL images were first used to reconstruct the 3D shape of all targets with both the ADAM and MPCD reconstruction methods. We subsequently performed a detailed shape analysis and constrained the density of each target using available mass estimates including our own mass estimates in the case of multiple systems. Results. The analysis of the reconstructed shapes allowed us to identify two families of objects as a function of their diameters, namely “spherical” and “elongated” bodies. A difference in rotation period appears to be the main origin of this bimodality. In addition, all but one object (216 Kleopatra) are located along the Maclaurin sequence with large volatile-rich bodies being the closest to the latter. Our results further reveal that the primaries of most multiple systems possess a rotation period of shorter than 6 h and an elongated shape ( c ∕ a ≤ 0.65). Densities in our sample range from ~1.3 g cm −3 (87 Sylvia) to ~4.3 g cm −3 (22 Kalliope). Furthermore, the density distribution appears to be strongly bimodal with volatile-poor ( ρ ≥ 2.7 g cm −3 ) and volatile-rich ( ρ ≤ 2.2 g cm −3 ) bodies. Finally, our survey along with previous observations provides evidence in support of the possibility that some C-complex bodies could be intrinsically related to IDP-like P- and D-type asteroids, representing different layers of a same body (C: core; P/D: outer shell). We therefore propose that P/ D-types and some C-types may have the same origin in the primordial trans-Neptunian disk.

IHS Climate in Context – “Thinking Historically About the Future of Energy and Climate,” by Paul Sabin, Yale University

2021-02-17

Index

2019-12-31

Environmental Activism on the Ground

University of Calgary Press eBooks · 2019-01-21 · 3 citations

Environmental Activism on the Ground draws upon a wide range of interdisciplinary scholarship to examine small scale, local environmental activism, paying particular attention to Indigenous experiences. It illuminates the questions that are central to the ongoing evolution of the environmental movement while reappraising the history and character of late twentieth and early twenty-first environmentalism in Canada, the United States, and beyond. This collection considers the different ways in which Indigenous and non-Indigenous activists have worked to achieve significant change. It examines attempts to resist exploitative and damaging resource developments, and the establishment of parks, heritage sites, and protected areas that recognize the indivisibility of cultural and natural resources. It pays special attention to the thriving environmentalism of the 1960s through the 1980s, an era which saw the rise of major organizations such as Greenpeace along with the flourishing of local and community-based environmental activism. Environmental Activism on the Ground emphasizes the effects of local and Indigenous activism, offering lessons and directions from the ground up. It demonstrates that the modern environmental movement has been as much a small-scale, ordinary activity as a large-scale, elite one. With Contributions By: Jonathan Clapperton, Jessica M. DeWitt, Sterling Evans, Zoltán Grossman, Tobasonakwut Peter Kinew, Mark Leeming, Mark J. McLauchlin, Liza Piper, John R. Welsch, Anna J. Willow, and Frank Zelko.

CHAPTER FIVE. Polarizing Politics

Yale University Press eBooks · 2017-12-31

Index

Yale University Press eBooks · 2017-12-31

A Bayesian simulation approach for supply chain synchronization

2017 Winter Simulation Conference (WSC) · 2017-12-01 · 3 citations

While simulation has been used extensively to model supply chain processes, the use of a Bayesian approach has been limited. However, Bayesian modeling brings key advantages, especially in cases of uncertainty. In this paper, we develop a data informatics model that could be used to realize a digital synchronized supply chain. To realize this model, we take a hybrid approach that combines Bayesian modeling with discrete-event simulation and apply it to the supply chain process of a Procter & Gamble (P&G) manufacturing and distribution facility. We use approximately one year of transactional data to inform our model, including information on customer orders, production, raw materials, inventory, and shipments. A driving force for creating this model is to better understand and manage the balance between inventory, profit, and service.

CHAPTER SIX. Betting the Future of the Planet

Yale University Press eBooks · 2017-12-31