Abstract
The deployment of sensors that characterize the trajectory of pitches and batted balls in three dimensions provides the opportunity to assign an intrinsic value to a pitch that depends on its physical properties and not on its observed outcome. We exploit this opportunity by using a Bayesian framework to learn a set of mappings from five-dimensional velocity, movement, and location vectors to intrinsic pitch values. A kernel method generates nonparametric estimates for the component probability density functions in Bayes theorem while nonparametric regression is used to derive a batted ball weight function that is invariant to the defense, ballpark, and atmospheric conditions. Cross-validation is used to determine the parameters of the model. We use Cronbach’s alpha to show that intrinsic pitch values have a significantly higher reliability than outcome-based pitch values. We also develop a method to combine intrinsic values at the individual pitch level into a statistic that captures the value of a pitcher’s collection of pitches over a period of time. We use this statistic to show that pitchers who outperform their intrinsic values during a season tend to perform worse the following year. We also show that this statistic provides better predictive value for future Earned Run Average (ERA) than either current ERA or Fielding Independent Pitching (FIP).
Appendix I: strike zone transformation
For each pitch recorded by the PITCHf/x system the height of the top and bottom of the batter’s strike zone is specified manually. Since these specifications are error-prone and can vary over time, we use the average of the specified top and bottom for each individual batter over the full season to represent his strike zone in the vertical dimension. For a given season, let Bt and Bb denote the average height of the top and bottom of the strike zone specified for batter B and let Lt and Lb denote the average height of the top and bottom of the strike zone for the league. For the 2014 data set, we used Lt = 3.403 feet and Lb = 1.571 feet. For a pitch with a measured vertical height of
Thus, a pitch at the bottom of any batter’s strike zone (z = Bb) maps to lz = Lb and a pitch at the top of any batter’s strike zone (z = Bt) maps to lz = Lt.
Appendix II: distribution of pitches across counts and outcomes
Table 4 presents the distribution of pitches thrown in the RHP vs. RHB configuration in 2014 for each count and with each outcome. As described in Section 2.2 the outcomes are R0 = ball in play, R1 = called ball, R2 = called strike, R3 = swinging strike, R4 = foul ball, and R5 = batter hit-by-pitch where foul tips that are caught for strikeouts are classified as R3 and not R4.
Number of pitches for each count and with each outcome, RHP vs. RHB, 2014.
| Count | R0 | R1 | R2 | R3 | R4 | R5 |
|---|---|---|---|---|---|---|
| 0–0 | 7090 | 25,233 | 22,682 | 4683 | 6994 | 151 |
| 1–0 | 4315 | 7920 | 6023 | 2484 | 4145 | 41 |
| 2–0 | 1468 | 2296 | 2064 | 639 | 1315 | 14 |
| 3–0 | 118 | 656 | 1328 | 40 | 117 | 1 |
| 0–1 | 6250 | 13,552 | 4054 | 4215 | 6079 | 125 |
| 1–1 | 5581 | 8481 | 3085 | 3317 | 5390 | 78 |
| 2–1 | 3102 | 3417 | 1491 | 1494 | 2825 | 39 |
| 3–1 | 1208 | 1235 | 808 | 446 | 1125 | 11 |
| 0–2 | 3141 | 7817 | 739 | 2438 | 3394 | 70 |
| 1–2 | 5265 | 9057 | 1106 | 3705 | 5464 | 98 |
| 2–2 | 5141 | 5517 | 964 | 2763 | 5153 | 82 |
| 3–2 | 3485 | 2260 | 582 | 21 | 3243 | 38 |
Appendix III: optimal bandwidths for kernel density estimates
In Tables 5–10, we present the optimal bandwidths derived for different configurations using the process described in Section 2.5.
Optimal bandwidths for each outcome, RHP vs. RHB, 0–0 count.
| Outcome Rj | |||||
|---|---|---|---|---|---|
| ball in play | 1.45 | 1.30 | 1.30 | 0.215 | 0.255 |
| called ball | 1.55 | 1.30 | 1.35 | 0.260 | 0.290 |
| called strike | 1.45 | 1.20 | 1.20 | 0.180 | 0.185 |
| swinging strike | 1.55 | 1.30 | 1.40 | 0.280 | 0.345 |
| foul ball | 1.45 | 1.30 | 1.20 | 0.255 | 0.285 |
| hit-by-pitch | 2.50 | 1.60 | 2.45 | 0.365 | 0.360 |
Optimal bandwidths for each outcome, RHP vs. RHB, 0–1 count.
| Outcome Rj | |||||
|---|---|---|---|---|---|
| ball in play | 1.55 | 1.25 | 1.30 | 0.255 | 0.280 |
| called ball | 1.60 | 1.40 | 1.40 | 0.295 | 0.375 |
| called strike | 1.75 | 1.55 | 1.55 | 0.190 | 0.225 |
| swinging strike | 1.75 | 1.40 | 1.40 | 0.280 | 0.345 |
| foul ball | 1.55 | 1.25 | 1.40 | 0.260 | 0.300 |
| hit-by-pitch | 2.45 | 2.65 | 2.00 | 0.285 | 0.350 |
Optimal bandwidths for each outcome, RHP vs. RHB, 1–0 count.
| Outcome Rj | |||||
|---|---|---|---|---|---|
| ball in play | 1.55 | 1.25 | 1.25 | 0.240 | 0.270 |
| called ball | 1.80 | 1.45 | 1.50 | 0.280 | 0.335 |
| called strike | 1.65 | 1.50 | 1.45 | 0.220 | 0.235 |
| swinging strike | 1.65 | 1.40 | 1.45 | 0.330 | 0.360 |
| foul ball | 1.40 | 1.45 | 1.35 | 0.275 | 0.285 |
| hit-by-pitch | 2.15 | 2.70 | 2.00 | 0.315 | 0.255 |
Optimal bandwidths for each outcome, RHP vs. LHB, 0–0 count.
| Outcome Rj | |||||
|---|---|---|---|---|---|
| ball in play | 1.60 | 1.25 | 1.30 | 0.220 | 0.260 |
| called ball | 1.55 | 1.30 | 1.35 | 0.250 | 0.300 |
| called strike | 1.50 | 1.25 | 1.25 | 0.170 | 0.190 |
| swinging strike | 1.85 | 1.50 | 1.40 | 0.290 | 0.315 |
| foul ball | 1.50 | 1.30 | 1.35 | 0.240 | 0.285 |
| hit-by-pitch | 2.85 | 2.95 | 2.75 | 0.520 | 0.465 |
Optimal bandwidths for each outcome, RHP vs. LHB, 0–1 count.
| Outcome Rj | |||||
|---|---|---|---|---|---|
| ball in play | 1.35 | 1.30 | 1.55 | 0.245 | 0.290 |
| called ball | 1.75 | 1.45 | 1.50 | 0.270 | 0.355 |
| called strike | 1.95 | 1.55 | 1.70 | 0.175 | 0.240 |
| swinging strike | 1.75 | 1.45 | 1.50 | 0.305 | 0.360 |
| foul ball | 1.45 | 1.35 | 1.35 | 0.275 | 0.300 |
| hit-by-pitch | 2.80 | 2.65 | 2.90 | 0.260 | 1.035 |
Optimal bandwidths for each outcome, RHP vs. LHB, 1–0 count.
| Outcome Rj | |||||
|---|---|---|---|---|---|
| ball in play | 1.60 | 1.35 | 1.25 | 0.260 | 0.280 |
| called ball | 1.75 | 1.50 | 1.50 | 0.275 | 0.325 |
| called strike | 1.60 | 1.40 | 1.55 | 0.195 | 0.235 |
| swinging strike | 1.85 | 1.45 | 1.55 | 0.300 | 0.370 |
| foul ball | 1.55 | 1.35 | 1.35 | 0.260 | 0.290 |
| hit-by-pitch | 1.55 | 1.80 | 1.55 | 0.455 | 0.795 |
Appendix IV: data used to evaluate intrinsic pitch statistics
In Tables 11 and 12, we present the data for the 34 pitchers who were used to evaluate the intrinsic pitch statistics as described in Section 4.
RHP with at least 162 innings pitched in 2014 and 2015.
| Pitcher | 2014 | 2014 | 2015 | ERA |
|---|---|---|---|---|
| OMI ⋅ 1000 | ERA | ERA | Difference | |
| Chris Archer | −11.1 | 3.33 | 3.23 | −0.10 |
| A.J. Burnett | 8.8 | 4.59 | 3.18 | −1.41 |
| Bartolo Colon | 8.0 | 4.09 | 4.16 | 0.07 |
| Johnny Cueto | −6.1 | 2.25 | 3.44 | 1.19 |
| R.A. Dickey | 4.7 | 3.71 | 3.91 | 0.20 |
| Yovani Gallardo | 16.1 | 3.51 | 3.42 | −0.09 |
| Kyle Gibson | 7.0 | 4.47 | 3.84 | −0.63 |
| Sonny Gray | −10.8 | 3.08 | 2.73 | −0.35 |
| Zack Greinke | 10.7 | 2.71 | 1.66 | −1.05 |
| Jason Hammel | 8.0 | 3.47 | 3.74 | 0.27 |
| Aaron Harang | 6.3 | 3.57 | 4.86 | 1.29 |
| Dan Haren | 13.2 | 4.02 | 3.60 | −0.42 |
| Felix Hernandez | −13.3 | 2.14 | 3.53 | 1.39 |
| Ian Kennedy | 3.9 | 3.63 | 4.28 | 0.65 |
| Corey Kluber | 3.5 | 2.44 | 3.49 | 1.05 |
| Tom Koehler | −2.8 | 3.81 | 4.08 | 0.27 |
| John Lackey | 17.3 | 3.82 | 2.77 | −1.05 |
| Mike Leake | 16.3 | 3.70 | 3.70 | 0.00 |
| Colby Lewis | 25.1 | 5.18 | 4.66 | −0.52 |
| Lance Lynn | −4.9 | 2.74 | 3.03 | 0.29 |
| Shelby Miller | 7.9 | 3.74 | 3.02 | −0.72 |
| Jake Odorizzi | −1.3 | 4.13 | 3.35 | −0.78 |
| Rick Porcello | 0.5 | 3.43 | 4.92 | 1.49 |
| Garrett Richards | −7.3 | 2.61 | 3.65 | 1.04 |
| Tyson Ross | −5.8 | 2.81 | 3.26 | 0.45 |
| Jeff Samardzija | 6.1 | 2.99 | 4.96 | 1.97 |
| Max Scherzer | 2.1 | 3.15 | 2.79 | −0.36 |
| James Shields | −0.2 | 3.21 | 3.91 | 0.70 |
| Alfredo Simon | −3.0 | 3.44 | 5.05 | 1.61 |
| Julio Teheran | −5.4 | 2.89 | 4.04 | 1.15 |
| Chris Tillman | 3.2 | 3.34 | 4.99 | 1.65 |
| Yordano Ventura | −8.4 | 3.20 | 4.08 | 0.88 |
| Edinson Volquez | −13.3 | 3.04 | 3.55 | 0.51 |
| Jordan Zimmermann | −4.9 | 2.66 | 3.66 | 1.00 |
RHP with at least 162 innings pitched in 2014 and 2015.
| Pitcher | 2014 | 2014 | 2014 | ERA |
|---|---|---|---|---|
| ERA | FIP | ERA – FIP | Difference | |
| Chris Archer | 3.33 | 3.39 | −0.06 | −0.10 |
| A.J. Burnett | 4.59 | 4.14 | 0.45 | −1.41 |
| Bartolo Colon | 4.09 | 3.57 | 0.52 | 0.07 |
| Johnny Cueto | 2.25 | 3.30 | −1.05 | 1.19 |
| R.A. Dickey | 3.71 | 4.32 | −0.61 | 0.20 |
| Yovani Gallardo | 3.51 | 3.94 | −0.43 | −0.09 |
| Kyle Gibson | 4.47 | 3.80 | 0.67 | −0.63 |
| Sonny Gray | 3.08 | 3.46 | −0.38 | −0.35 |
| Zack Greinke | 2.71 | 2.97 | −0.26 | −1.05 |
| Jason Hammel | 3.47 | 3.92 | −0.45 | 0.27 |
| Aaron Harang | 3.57 | 3.57 | 0.00 | 1.29 |
| Dan Haren | 4.02 | 4.09 | −0.07 | −0.42 |
| Felix Hernandez | 2.14 | 2.56 | −0.42 | 1.39 |
| Ian Kennedy | 3.63 | 3.21 | 0.42 | 0.65 |
| Corey Kluber | 2.44 | 2.35 | 0.09 | 1.05 |
| Tom Koehler | 3.81 | 3.84 | −0.03 | 0.27 |
| John Lackey | 3.82 | 3.78 | 0.04 | −1.05 |
| Mike Leake | 3.70 | 3.88 | −0.18 | 0.00 |
| Colby Lewis | 5.18 | 4.46 | 0.71 | −0.52 |
| Lance Lynn | 2.74 | 3.35 | −0.61 | 0.29 |
| Shelby Miller | 3.74 | 4.54 | −0.80 | −0.72 |
| Jake Odorizzi | 4.13 | 3.75 | 0.38 | −0.78 |
| Rick Porcello | 3.43 | 3.67 | −0.24 | 1.49 |
| Garrett Richards | 2.61 | 2.60 | 0.01 | 1.04 |
| Tyson Ross | 2.81 | 3.24 | −0.43 | 0.45 |
| Jeff Samardzija | 2.99 | 3.20 | −0.21 | 1.97 |
| Max Scherzer | 3.15 | 2.85 | 0.30 | −0.36 |
| James Shields | 3.21 | 3.59 | −0.38 | 0.70 |
| Alfredo Simon | 3.44 | 4.33 | −0.89 | 1.61 |
| Julio Teheran | 2.89 | 3.49 | −0.60 | 1.15 |
| Chris Tillman | 3.34 | 4.01 | −0.67 | 1.65 |
| Yordano Ventura | 3.20 | 3.60 | −0.40 | 0.88 |
| Edinson Volquez | 3.04 | 4.15 | −1.11 | 0.51 |
| Jordan Zimmermann | 2.66 | 2.68 | −0.02 | 1.00 |
Acknowledgement
I am grateful to Sportvision and MLB Advanced Media for providing the HITf/x data which made this work possible. I am also happy to acknowledge the assistance of Qi Shi and Jason Wang in the preparation of this document.
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