The Annals of Applied Statistics

Characterizing the spatial structure of defensive skill in professional basketball

Alexander Franks, Andrew Miller, Luke Bornn, and Kirk Goldsberry

Full-text: Open access


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.

Article information

Ann. Appl. Stat., Volume 9, Number 1 (2015), 94-121.

First available in Project Euclid: 28 April 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Basketball hidden Markov models nonnegative matrix factorization Bayesian hierarchical models


Franks, Alexander; Miller, Andrew; Bornn, Luke; Goldsberry, Kirk. Characterizing the spatial structure of defensive skill in professional basketball. Ann. Appl. Stat. 9 (2015), no. 1, 94--121. doi:10.1214/14-AOAS799.

Export citation


  • Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer, New York.
  • Böhning, D. (1992). Multinomial logistic regression algorithm. Ann. Inst. Statist. Math. 44 197–200.
  • Brunet, J.-P., Tamayo, P., Golub, T. R. and Mesirov, J. P. (2004). Metagenes and molecular pattern discovery using matrix factorization. Proc. Natl. Acad. Sci. USA 101.12 4164–9.
  • Cervone, D., D’Amour, A., Bornn, L. and Goldsberry, K. (2014). POINTWISE: Predicting Points and Valuing Decisions in Real Time with NBA Optical Tracking Data.
  • Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
  • Franks, A., Miller, A., Bornn, L. and Goldsberry, K. (2015a). Supplement to “Characterizing the spatial structure of defensive skill in professional basketball.” DOI:10.1214/14-AOAS799SUPPA.
  • Franks, A., Miller, A., Bornn, L. and Goldsberry, K. (2015b). Supplement to “Characterizing the spatial structure of defensive skill in professional basketball.” DOI:10.1214/14-AOAS799SUPPB.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–472.
  • Goldsberry, K. (2012). Courtvision: New visual and spatial analytics for the NBA. MIT Sloan Sports Analytics Conference.
  • Goldsberry, K. (2013). The Dwight Effect: A new ensemble of interior defense analytics for the NBA. MIT Sloan Sports Analytics Conference.
  • Kingman, J. F. C. (1992). Poisson Processes. Oxford Univ. Press, London.
  • Kubatko, J., Oliver, D., Pelton, K. and Rosenbaum, D. T. (2007). A starting point for analyzing basketball statistics. J. Quant. Anal. Sports 3 1–22.
  • Lee, D. D. and Seung, H. S. (1999). Learning the parts of objects by non-negative matrix factorization. Nature 401 788–791.
  • Lee, D. D. and Seung, H. S. (2001). Algorithms for non-negative matrix factorization. Adv. Neural Inf. Process. Syst. 13 556–562.
  • Limnios, N. and Oprisan, G. (2001). Semi-Markov Processes and Reliability. Springer, Berlin.
  • Macdonald, B. (2011). A regression-based adjusted plus-minus statistic for NHL players. J. Quant. Anal. Sports 7 4.
  • Maruotti, A. and Rydén, T. (2009). A semiparametric approach to hidden Markov models under longitudinal observations. Stat. Comput. 19 381–393.
  • Miller, A. C., Bornn, L., Adams, R. and Goldsberry, K. (2014). Factorized Point Process Intensities: A Spatial Analysis of Professional Basketball. In Proceedings of the 31st International Conference on Machine Learning (ICML). Beijing, China.
  • Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Stat. 25 451–482.
  • Murphy, K. (2012). Machine Learning: A Probabilistic Perspective. MIT Press, Cambridge, MA.
  • National Basketball Association (2014). A Glossary of NBA Terms. Available at
  • Rosenbaum, D. T. (2004). Measuring how NBA players help their teams win. Available at ( 4–30.
  • Sill, J. (2010). Improved NBA adjusted plus-minus using regularization and out-of-sample testing. In Proceedings of the 2010 MIT Sloan Sports Analytics Conference. Boston, MA.
  • Stan Development Team (2014). Stan: A C++ Library for Probability and Sampling, Version 2.2.
  • Thomas, A. C., Ventura, S. L., Jensen, S. T. and Ma, S. (2013). Competing process hazard function models for player ratings in ice hockey. Ann. Appl. Stat. 7 1497–1524.
  • Yu, S.-Z. (2010). Hidden semi-Markov models. Artificial Intelligence 174 215–243.

Supplemental materials

  • Supplement A: Additional methods, figures and tables.: We describe detailed methodology related to the shot type parameterizations and include additional graphics. We also include tables ranking players' impact on shot frequency and efficiency (offense and defense) in all court regions.
  • Supplement B: Animations.: We provide GIF animations illustrating the "who's guarding whom" algorithm on different NBA possessions.