A Bayesian Hierarchical Model of Pitch Framing in Major League Baseball

Transcription

1 A Bayesian Hierarchical Model of Pitch Framing in Major League Baseball Sameer K. Deshpande and Abraham J. Wyner Statistics Department, The Wharton School University of Pennsylvania 1 August 2016

2 Introduction Framing: ability of a catcher to affect likelihood a taken pitch is called a strike Most estimates: top framers can save runs per season wins above replacement! ESPN The Magazine: If you have confidence in the additional 2 WAR that framing would have given Lucroy, his 2014 season would have been worth about $56M dollars on free agent market this offseason Deshpande & Wyner Catcher Framing JSM / 16

3 Overview Hierarchical Bayesian logistic regression model of called strike probability Estimate of each catcher s effect on each umpire over and above factors like pitch location, count, and other pitch participants Translate these effects to estimates of the impact framing (i.e. runs saved) has on the game, along with natural uncertainty quantification We use PITCHf/x data from Horizontal and vertical coordinates of pitch as it crosses home plate Main focus is on 2014 season: 300k called pitches Deshpande & Wyner Catcher Framing JSM / 16

4 Model Let y = 1 for called strike, y = 0 for ball. For umpire u: ( ) P(y = 1) log = θ0 u + θb u P(y = 0) + θu c + θp u + θcount u + f u (x, z) where θ u b, θu c, θ u p: partial effect of batter b, catcher c and pitcher p on umpire u s log-odds of calling strike θ u count: partial effect of count on umpire u f u (x, z): function of pitch location θ u 0 : intercept Deshpande & Wyner Catcher Framing JSM / 16

5 Incorporating Pitch Location Direct Parametrization: Each coordinate as linear predictor: f u (x, z) = θ u x x + θ u z z Polar coordinates: f u (x, z) = θ u r r(x, z) + θ u φ φ Indirect Parametrization: 1. Fit a Generalized Additive Model of called strike probability as smooth function of location: Uses data from Separate GAM for each combination of batter and pitcher handedness 2. Use forecasted log-odds of called strike a linear predictor in model: f u (x, z) = θlp u log-odds ˆ Deshpande & Wyner Catcher Framing JSM / 16

6 Hierarchical Model For each of the 93 umpires u 1,..., u 93 ( P(y u ) log i = 1) P(yi u = x u i Θ u = 0) Θ u 1,..., Θ u 93 µ N ( µ, σ 2 I ) µ N ( 0, τ 2 I ) σ = 1: less than 0.3% chance that one umpire would call the same pitch strike 99% of time and other umpire calls strike 1% of time τ = 0.5: replacing player by baseline player unlikely to change called strike probability from 75% to 25% Model fit using Stan Deshpande & Wyner Catcher Framing JSM / 16

7 Posterior Densities of Player Effects (a) Hank Conger (b) Jonathan Lucroy Deshpande & Wyner Catcher Framing JSM / 16

8 Impact of Framing For each catcher, look at all of the called pitches he received: ˆp: fitted probability of strike ˆp 0 : fitted probability of strike with catcher replaced by baseline catcher ˆp ˆp 0 : catcher s framing effect Sum ρ (ˆp ˆp 0 ) over all called pitches received Value of a called strike, ρ, depends on the count: Deshpande & Wyner Catcher Framing JSM / 16

9 Value of a called strike on an 0 1 pitch Between 2011 and 2014: 182, pitches taken: 140,667 balls, 41,738 called strikes Avg. # runs allowed in rest of inning after called ball: Avg. # runs allowed in rest of inning after called strike: Conditional on an 0 1 pitch being taken: called strike saves ρ = runs, on average Deshpande & Wyner Catcher Framing JSM / 16

10 Estimated Runs Saved, On Average Catcher Runs Saved (SD) 95% Interval P(> 0) N BP Miguel Montero 25.1 (7.1) [11.3, 38.8] (8172) Mike Zunino 19.9 (7.3) [5.4, 34.1] (7457) Jonathan Lucroy 19.5 (8.1) [3.8, 35.3] (8241) Rene Rivera 18.9 (5.3) [8.6, 29.2] (5182) Hank Conger 17.6 (4.5) [8.8, 26.4] (4768) Russell Martin 15.4 (5.9) [3.6, 27.2] (6502) Buster Posey 15.0 (6.1) [3.1, 26.9] (6190) Travis d Arnaud 13.5 (6.1) [1.8, 25.7] (6276) Brian McCann 12.9 (5.4) [2.2, 23.2] (6471) Christian Vazquez 12.4 (3.4) [5.9, 18.9] (3370) Deshpande & Wyner Catcher Framing JSM / 16

11 Spatially Aggregate Framing Effect Is Montero really a better pitch framer than Vazquez? 8086 pitches vs 3198 Results further confounded by other pitch participants, location, counts We can integrate ρ (ˆp ˆp 0 ) over all batter, pitcher, umpire, count, location combinations. Framing analog of Spatial Aggregate Fielding Evaluation of Jensen, Shirley, and Wyner (2008) SAFE2: Estimate how many runs catcher saves through framing on 4000 average pitches Deshpande & Wyner Catcher Framing JSM / 16

12 SAFE2 Rank Player Mean (SD) 95% Interval P(> 0) 1. Rene Rivera 15.1 (4.4) [6.5, 23.6] Hank Conger 14.7 (4.4) [6.1, 23.3] Christian Vazquez 14.6 (4.9) [5.0, 24.3] Miguel Montero 12.8 (3.7) [5.5, 19.9] Yasmani Grandal 12.5 (4.5) [3.8, 21.4] Mike Zunino 11.5 (4.1) [3.6, 19.5] Martin Maldonado 11.4 (5.9) [0.1, 23.3] Chris Stewart 11.1 (5.6) [0.2, 22.2] Russell Martin 10.3 (4.0) [2.4, 18.0] Drew Butera 10.1 (5.2) [0.1, 20.3] Deshpande & Wyner Catcher Framing JSM / 16

13 Conclusions SAFE2 year-to-year correlation encouraging: There are some players with statistically distinguishable effects on some umpires Even with these effects, out-of-sample performance similar to that of underlying GAM s: non-stationarity between seasons Our estimate of framing s impact similar to others, but considerable uncertainty in our estimates! Deshpande & Wyner Catcher Framing JSM / 16

14 Thanks!

15 Fitted Called Strike Probabilities 0 1 pitch, Yasiel Puig, Madison Bumgarner, Buster Posey (a) Angel Hernandez (b) Average Umpire (c) Scott Barry Deshpande & Wyner Catcher Framing JSM / 16

16 Average # Runs Given Up and Value of Strike Count Ball Strike Value of strike, ρ (0.002) (0.002) (0.002) (0.002) (0.004) (0.004) (0.003) (0.007) (0.008) (0.003) (0.003) (0.005) (0.003) (0.004) (0.005) (0.003) (0.006) (0.006) (0.007) (0.006) (0.009) (0.005) (0.006) (0.008) (0.004) (0.006) (0.007) (0.013) (0.008) (0.015) (0.010) (0.009) (0.014) (0.008) (0.008) (0.011) Table : Standard errors in parentheses Deshpande & Wyner Catcher Framing JSM / 16