Abstract
In the analysis of qualification stage data from FIRST Robotics Competition (FRC) championships, the ratio (1.67–1.68) of the number of observations (110–114 matches) to the number of parameters (66–68 robots) in each division has been found to be quite small for the most commonly used winning margin power rating (WMPR) model. This usually leads to imprecise estimates and inaccurate predictions in such three-on-three matches that FRC tournaments are composed of. With the recognition of a clustering feature in estimated robot strengths, a more flexible model with latent clusters of robots was proposed to alleviate overparameterization of the WMPR model. Since its structure can be regarded as a dimension reduction of the parameter space in the WMPR model, the identification of clusters of robot strengths is naturally transformed into a model selection problem. Instead of comparing a huge number of competing models \((7.76\times 10^{67}\) to \(3.66\times 10^{70})\), we develop an effective method to estimate the number of clusters, clusters of robots and robot strengths in the format of qualification stage data from the FRC championships. The new method consists of two parts: (i) a combination of hierarchical and non-hierarchical classifications to determine candidate models; and (ii) variant goodness-of-fit criteria to select optimal models. In contrast to existing hierarchical classification, each step of our proposed non-hierarchical classification is based on estimated robot strengths from a candidate model in the preceding non-hierarchical classification step. A great advantage of the proposed methodology is its ability to consider the possibility of reassigning robots to other clusters. To reduce overestimation of the number of clusters by the mean squared prediction error criteria, corresponding Bayesian information criteria are further established as alternatives for model selection. With a coherent assembly of these essential elements, a systematic procedure is presented to perform the estimation of parameters. In addition, we propose two indices to measure the nested relation between clusters from any two models and monotonic association between robot strengths from any two models. Data from the 2018 and 2019 FRC championships and a simulation study are also used to illustrate the applicability and superiority of our proposed methodology.
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Source: FIRST (2019)
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References
Bondell HD, Reich BJ (2008) Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR. Biometrics 64:115–123
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3:1–122
Bradley RA (1953) Some statistical methods in taste testing and quality evaluation. Biometrics 9:22–38
Bradley RA, Terry ME (1952) Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika 39:324–345
Cattelan M, Varin C, Firth D (2013) Dynamic Bradley–Terry modelling of sports tournaments. J R Stat Soc C 62:135–150
Clark N, Macdonald B, Kloo I (2020) A Bayesian adjusted plus-minus analysis for the esport Dota 2. J Quant Anal Sports 16:325–341
Clarke SR, Norman JM (1995) Home ground advantage of individual clubs in English soccer. Statistician 44:509–521
DeSarbo WS, Cron WL (1988) A maximum likelihood methodology for clusterwise linear regression. J Classif 5:249–282
Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96:1348–1360
Fang E (2017) The math behind OPR—an introduction. The Blue Alliance Blog. https://blog.thebluealliance.com/2017/10/05/the-math-behind-opr-an-introduction/
FIRST (2019) 2019 FRC game season manual. https://firstfrc.blob.core.windows.net/frc2019/Manual/2019FRCGameSeasonManual.pdf
FIRST (2022) 2022 FIRST robotics competition game manual. https://firstfrc.blob.core.windows.net/frc2022/Manual/2022FRCGameManual.pdf
Gardner W (2015) An overview and analysis of statistics used to rate FIRST robotics teams. https://www.chiefdelphi.com/uploads/default/original/3X/e/b/ebd6f208f128bcc2d44267c7b6d25a4ae53aa8a4.pdf
Han AK (1987) Non-parametric analysis of a generalized regression model: the maximum rank correlation estimator. J Econom 35:303–316
Hass Z, Craig B (2018) Exploring the potential of the plus/minus in NCAA women’s volleyball via the recovery of court presence information. J Syst Archit 4:285–295
Hosmer DW Jr, Lemeshow S, Sturdivant RX (2013) Applied logistic regression. Wiley, New York
Hvattum LM (2019) A comprehensive review of plus–minus ratings for evaluating individual players in team sports. Int J Comput Sci Sport 18:1–23
Law E (2008) New scouting database from team 2834. https://www.chiefdelphi.com/forums/showpost.php?p=835222 &postcount=48
LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521:436–444
Lim A, Chiang C-T, Teng J-C (2021) Estimating robot strengths with application to selection of alliance members in FIRST robotics competitions. Comput Stat Data Anal 158:107181
Lin C-C, Ng S (2012) Estimation of panel data models with parameter heterogeneity when group membership is unknown. J Econ Methods 1:42–55
Macdonald B (2011) A regression-based adjusted plus–minus statistic for NHL players. J Quant Anal Sports 7:4
Mosteller F (1951) Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations. Psychometrika 16:3–9
Rosenbaum DT (2004) Measuring how NBA players help their teams win. https://www.82games.com/comm30.htm
Rosenblatt F (1962) Principles of neurodynamics. Spartan, New York
Sæbø OD, Hvattum LM (2015) Evaluating the efficiency of the association football transfer market using regression based player ratings. In: NIK-2015 conference, vol 4, pp 285–295
Schuckers ME, Lock DF, Wells C, Knickerbocker C, Lock RH (2011) National Hockey League skater ratings based upon all on-ice events: an adjusted minus/plus probability (AMPP) approach. Unpublished Manuscript. http://myslu.stlawu.edu/~msch/sports/LockSchuckersProbPlusMinus113010.pdf
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464
Sokal RR (1958) A statistical method for evaluating systematic relationships. Univ Kansas Sci Bull 38:1409–1438
Späth H (1979) Algorithm 39: clusterwise linear regression. Computing 22:367–373
Stefani RT (1977) Football and basketball predictions using least squares. IEEE Trans Syst Man Cyber 7:117–21
Stefani RT (1980) Improved least squares football, basketball, and soccer predictions. IEEE Trans Syst Man Cyber 10:116–123
Thurstone LL (1927) A law of comparative judgment. Psychol Rev 34:273–286
Weingart S (2006) Offense/defense rankings for 1043 teams. https://www.chiefdelphi.com/forums/showpost.php?p=484220 &postcount=19
Wikipedia (2018) FIRST robotics competition. https://en.wikipedia.org/wiki/FIRST_Robotics_Competition. Accessed on 30 Aug 2018
Zhang C-H (2010) Nearly unbiased variable selection under minimax concave penalty. Ann Stat 38:894–942
Acknowledgements
Chin-Tsang Chiang’s research was partially supported by the National Science and Technology Council grant 109-2118-M-002-002-MY2 (Taiwan). The authors would like to thank Alejandro Lim, who introduced us to the original version of this research problem and co-authored Lim et al. (2021) with the first two authors. We also thank the editor and a reviewer for their constructive comments.
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Teng, JC., Chiang, CT. & Lim, A. An effective method for identifying clusters of robot strengths. Comput Stat (2023). https://doi.org/10.1007/s00180-023-01442-5
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DOI: https://doi.org/10.1007/s00180-023-01442-5