Detectando jugadores de béisbol sobresalientes

Alberto Torrejón Valenzuela

2021/02/18

Categories: Estadística Estadística deportiva Tags: ACP Xaringan Béisbol

“El poder de la estadística en el béisbol es que, a diferencia del precio de la vivienda o la inflación, toma vida”. Bill James.

DIAPOSITIVAS
# Manipulación de datos
library(tidyverse)
library(magrittr)
# Gráficos
library(ggplot2)
library(ggthemes)
library(gridExtra)
# Tabla
library(kableExtra)

A principios de la década de 2000, Billy Beane y Paul DePodesta trabajaron para los Oakland Athletics, un equipo de la MLB estadounidense, revolucionando el mundo del deporte y de la estadística deportiva. Sentaron las bases en el béisbol de lo que más tarde se extendería a otros deportes como el baloncesto o el fútbol y que en la actualidad está causando gran furor en el mundo deportivo. Aplicaron a la hora de constuir su plantilla de jugadores, para lo que disponían de un presupuesto ajustado, los conceptos que años atrás curiosos del juego, como Bill James entre otros, habían formulado mientras se planteaban si el béisbol era un juego que se podía descrifrar con datos y estadística.

Como no es el objetivo de este ejercicio, no se explicará en profundidad ni el juego del béisbol, que tiene su interés, ni se hará un desarrollo de las estadísticas recogidas por la sabermetría. Para lo que nos concierne, la sabermetría recoge un cuantioso número de estadísticas para cada jugador reduciendo el potencial del jugador, y por tanto del equipo, a los números. Todos estos estadísticos se calculan teniendo en cuenta los 4 elementos básicos del juego: el pitcheo o lanzamiento, el hit o golpeo, el strike y el run o carrera, estudiando todas las posibles variables y para las distintas posiciones del juego. El hecho de que en su mayoría estas estadísticas sean combinaciones lineales ponderadas de estás 4 variables sugiere que podría existir relación entre algunas, lo que invita a estudiar una posible reducción de la dimensionalidad. Como ejemplo se pueden mostrar dos ampliamente conocidas como son el porcentaje en base y el slug.

Porcentaje en base o promedio de bateo.

\[OBP = \frac{H+BB+HBP}{AB+BB+HBP+SF}\]

donde:

Slug.

\[SLG = \frac{1B + 2 \times 2B + 3 \times 3B + 4 \times HR}{AB}\]

donde:

API baseballr

Otro problema muy generalizado en el mundo de la estadística deportiva es que cada vez más el acceso a las bases de datos queda restringido por la monetización de la información. No obstante, hay numerosas páginas webs que se dedican de forma altruista a recoger estos datos. El paquete baseballr extrae la información de sitios web, como FanGraphs.com, Baseball-Reference.com y baseballsavant.com. También incluye funciones para calcular métricas, como wOBA, FIP y consistencia a nivel de equipo en períodos de tiempo personalizados.

library(devtools)
# DESCOMENTAR LA SIGUIENTE LÍNEA PARA REPLICAR
# install_github("BillPetti/baseballr") 
library(baseballr)

Vamos a analizar a los bateadores. Obtenemos la información de los estadísticos de los bateadores en la temporada 2020 como se sigue:

bateadores <- daily_batter_bref("2020-06-23", "2020-09-27")
bateadores %>% summary() %>% .[,1:15]
   bbref_id            season              Name                Age       
 Length:588         Length:588         Length:588         Min.   :20.00  
 Class :character   Class :character   Class :character   1st Qu.:25.00  
 Mode  :character   Mode  :character   Mode  :character   Median :27.00  
                                                          Mean   :27.78  
                                                          3rd Qu.:30.00  
    Level               Team                 G               PA        
 Length:588         Length:588         Min.   : 1.00   Min.   :  0.00  
 Class :character   Class :character   1st Qu.:15.00   1st Qu.: 40.75  
 Mode  :character   Mode  :character   Median :33.00   Median :105.00  
                                       Mean   :31.94   Mean   :113.11  
                                       3rd Qu.:49.25   3rd Qu.:184.00  
       AB               R               H              X1B       
 Min.   :  0.00   Min.   : 0.00   Min.   : 0.00   Min.   : 0.00  
 1st Qu.: 35.75   1st Qu.: 4.00   1st Qu.: 6.00   1st Qu.: 4.00  
 Median : 94.50   Median :11.00   Median :21.00   Median :13.00  
 Mean   :100.39   Mean   :14.19   Mean   :24.56   Mean   :15.43  
 3rd Qu.:160.25   3rd Qu.:22.00   3rd Qu.:39.00   3rd Qu.:25.00  
      X2B              X3B               HR        
 Min.   : 0.000   Min.   :0.0000   Min.   : 0.000  
 1st Qu.: 1.000   1st Qu.:0.0000   1st Qu.: 0.000  
 Median : 4.000   Median :0.0000   Median : 2.500  
 Mean   : 4.801   Mean   :0.4099   Mean   : 3.918  
 3rd Qu.: 8.000   3rd Qu.:1.0000   3rd Qu.: 6.000  
 [ reached getOption("max.print") -- omitted 2 rows ]
bateadores %>% head() %>% .[,1:10]
  bbref_id season             Name Age  Level        Team  G  PA  AB  R
1   547989   2020    Marcell Ozuna  29 MLB-NL     Atlanta 60 267 228 38
2   660670   2020 Francisco Lindor  26 MLB-AL   Cleveland 60 266 236 30
3   642715   2020     Cavan Biggio  25 MLB-AL     Toronto 59 265 220 41
4   613534   2020  Whit Merrifield  31 MLB-AL Kansas City 60 265 248 38
5   571431   2020   Dansby Swanson  26 MLB-NL     Atlanta 60 264 237 49
6   666176   2020  Freddie Freeman  30 MLB-NL     Atlanta 60 262 214 51
# Comprobación de que los jugadores no se repiten:
bateadores %>% nrow() == length(bateadores %$% unique(Name))
[1] TRUE

Restringimos nuestra matriz de datos a aquellas variables numéricas.

bateadores %>% names()
 [1] "bbref_id" "season"   "Name"     "Age"      "Level"    "Team"    
 [7] "G"        "PA"       "AB"       "R"        "H"        "X1B"     
[13] "X2B"      "X3B"      "HR"       "RBI"      "BB"       "IBB"     
[19] "uBB"      "SO"       "HBP"      "SH"       "SF"       "GDP"     
[25] "SB"       "CS"       "BA"       "OBP"      "SLG"      "OPS"     
bateadores <- bateadores %>%
  dplyr::select(-c(bbref_id,season,Age,Level,Team)) %>% 
  column_to_rownames('Name')

Hay valores perdidos en las variables BA, OBP, SLG y OPS. Como no es objetivo del ejercicio, vamos a eliminar a estos individuos de nuestra muestra, aunque se podrían imputar.

bateadores %>% dplyr::select(BA,OBP,SLG,OPS) %>% summary()
       BA              OBP              SLG              OPS        
 Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
 1st Qu.:0.1840   1st Qu.:0.2530   1st Qu.:0.2853   1st Qu.:0.5497  
 Median :0.2305   Median :0.3080   Median :0.3850   Median :0.6965  
 Mean   :0.2229   Mean   :0.2982   Mean   :0.3698   Mean   :0.6668  
 3rd Qu.:0.2700   3rd Qu.:0.3520   3rd Qu.:0.4620   3rd Qu.:0.8063  
 Max.   :0.6670   Max.   :1.0000   Max.   :1.3330   Max.   :1.8330  
 NA's   :8        NA's   :7        NA's   :8        NA's   :8       
missing_ids <- list()
missing_cols <- c("BA","OBP","SLG","OPS")
for (i in 1:length(missing_cols)){
  missing_ids[[i]] <- bateadores[,missing_cols[i]] %>% is.na() %>% which()
}
deleted_obs <- c(missing_ids[[1]],missing_ids[[2]],missing_ids[[3]],missing_ids[[4]]) %>% unique()
cat("Las observaciones que tiene valores perdidos son:",deleted_obs)
Las observaciones que tiene valores perdidos son: 581 582 583 584 585 586 587 588
bateadores <- bateadores %>% slice(-deleted_obs)

Construimos el diagrama de cajas y bigotes.

bateadores %>% pivot_longer(everything(), names_to = "item", values_to = "valor") %>% 
    mutate(item = fct_reorder(item, valor, .fun = "median")) %>% ggplot(aes(x = item, 
    y = valor, fill = item)) + geom_boxplot() + xlab("") + ylab("") + theme_pander() + 
    theme(legend.position = "none")

El gráfico sugiere que existen valores atípicos en nuestra muestra. También podemos observar que las variables se miden en escalas bastante diferentes (unidades y rango de valores bastante diferentes), sobre todo las variables AB y PA, por lo que trabajaremos con las variables estandarizadas.

Faltaría estudiar la correlación de las variables, caso en el que el ACP arroja unos resultados más fiables. Calcularemos el determinante de la matriz de correlación y visualizaremos el mapa de calor con ayuda del paquete corrpot.

library(corrplot)
R_bateadores <- cor(bateadores)
R_bateadores %>% corrplot(method = "square")

R_bateadores %>% det()
[1] -8.335519e-53

En nuestra matriz, dado que det(R) está cerca de cero, existe colinealidad y el ACP es apropiado para tratar con este conjunto de datos. Ahora ya estamos en condiciones de realizar nuestro Análisis de Componentes Principales.

ACP

Se usará la función princomp. Se indica cor=TRUE para trabajar con las variables estandarizadas.

pca_beisbol <- princomp(bateadores, cor = TRUE)
pca_beisbol
Call:
princomp(x = bateadores, cor = TRUE)

Standard deviations:
      Comp.1       Comp.2       Comp.3       Comp.4       Comp.5       Comp.6 
3.573095e+00 1.598255e+00 1.289853e+00 1.002606e+00 9.589373e-01 9.151802e-01 
      Comp.7       Comp.8       Comp.9      Comp.10      Comp.11      Comp.12 
8.863202e-01 8.198668e-01 7.484558e-01 7.369167e-01 6.704975e-01 5.902761e-01 
     Comp.13      Comp.14      Comp.15      Comp.16      Comp.17      Comp.18 
5.096326e-01 4.720466e-01 3.621281e-01 3.052015e-01 2.663781e-01 2.506803e-01 
     Comp.19      Comp.20      Comp.21      Comp.22      Comp.23      Comp.24 
1.841724e-01 1.377867e-01 2.501310e-03 1.435223e-03 3.437449e-08 0.000000e+00 

 24  variables and  580 observations.

Resumen del resultado.

resumen_name <- paste0("CP", 1:24)
resumen_eign <- pca_beisbol$sdev^2
resumen_CP <- tibble(CP = resumen_name, Eigen = resumen_eign) %>% 
  mutate(Percentage = 100 *Eigen/sum(Eigen), 
         `Cumulative Percentage` = cumsum(Percentage))
resumen_CP %>% mutate_at(2:4, round, 2) %>% kable()
CP Eigen Percentage Cumulative Percentage
CP1 12.77 53.20 53.20
CP2 2.55 10.64 63.84
CP3 1.66 6.93 70.77
CP4 1.01 4.19 74.96
CP5 0.92 3.83 78.79
CP6 0.84 3.49 82.28
CP7 0.79 3.27 85.55
CP8 0.67 2.80 88.36
CP9 0.56 2.33 90.69
CP10 0.54 2.26 92.95
CP11 0.45 1.87 94.83
CP12 0.35 1.45 96.28
CP13 0.26 1.08 97.36
CP14 0.22 0.93 98.29
CP15 0.13 0.55 98.83
CP16 0.09 0.39 99.22
CP17 0.07 0.30 99.52
CP18 0.06 0.26 99.78
CP19 0.03 0.14 99.92
CP20 0.02 0.08 100.00
CP21 0.00 0.00 100.00
CP22 0.00 0.00 100.00
CP23 0.00 0.00 100.00
CP24 0.00 0.00 100.00

En nuestro caso la primera componente principal ya explica más de la mitad de la variabilidad total, \(53.20\%\) de los datos. Veamos cuántas componentes seleccionaríamos observando el scree plot.

resumen_CP %>% ggplot(aes(x = fct_reorder(CP, -Eigen), y = Eigen)) + 
  geom_bar(stat = "identity", width = 0.01) + 
  geom_point() + 
  geom_hline(yintercept = 1, linetype = "dashed", color = "red") + 
  theme_pander() + xlab("") + theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1))

Si seguimos la regla de regla de Kaiser-Guttman seleccionaríamos \(4\) componentes. Si decidiésemos tomar aquellas que expliquen más del \(80\%\) de la variabilidad total nos quedaríamos con \(6\) componentes.

Observemos la matriz de cargas de las componentes y las puntuaciones de cada individuo que se calculan en el ACP.

pca_beisbol %$% loadings %>% .[,1:6] %>% kable() %>% scroll_box(width = "100%", height = "500px")
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
G 0.2594857 0.1378893 0.0046197 0.0750033 0.0874311 0.0199150
PA 0.2713424 0.1197659 -0.0267112 0.0560914 0.0371727 -0.0108127
AB 0.2689364 0.1207215 -0.0170634 0.1022724 0.0203450 -0.0468006
R 0.2673369 0.0633075 0.0027768 -0.0536285 -0.0404179 -0.0224804
H 0.2689850 0.0425295 0.0146431 0.1317977 -0.0052926 -0.1243490
X1B 0.2501315 0.0568159 0.0775908 0.2210321 0.0355906 -0.1010041
X2B 0.2364637 0.0122838 -0.0471772 0.1034711 -0.0107242 -0.1578839
X3B 0.1215184 0.0565229 0.3591231 -0.0801236 -0.2251420 -0.3741882
HR 0.2324540 0.0041800 -0.1819224 -0.1407911 -0.0775290 -0.0417497
RBI 0.2578520 0.0283526 -0.1443192 0.0229230 -0.0232428 -0.1068401
BB 0.2369586 0.0820832 -0.0994723 -0.3432352 0.1364549 0.0879893
IBB 0.1244151 -0.0060262 -0.1620108 -0.5697133 0.0801553 -0.1762287
uBB 0.2352945 0.0867916 -0.0869633 -0.2989227 0.1345811 0.1113001
SO 0.2343060 0.1520614 -0.0516574 -0.0245002 0.0083419 0.0925232
HBP 0.1399490 0.0508833 -0.0443863 0.0228213 -0.1823027 0.8093427
SH -0.0003036 0.1201986 0.3707600 0.0521724 0.8531428 0.0812668
SF 0.1363911 0.0741089 -0.1287193 0.4656238 0.0241048 -0.0138416
GDP 0.1897789 0.0661702 -0.1952078 0.2560573 0.0001246 -0.0249824
SB 0.1223084 0.1299711 0.5119899 -0.1620118 -0.2171467 -0.0016621
CS 0.1036299 0.1560982 0.5162089 0.0047777 -0.2487654 0.1800112
BA 0.1477842 -0.4697618 0.1496483 0.1552142 0.0492613 -0.0474390
OBP 0.1546336 -0.4457658 0.1070933 -0.0700609 0.1077405 0.1806365
SLG 0.1733345 -0.4384912 0.0289025 -0.0046668 -0.0467080 -0.0353682
OPS 0.1775693 -0.4707478 0.0618645 -0.0307954 0.0113684 0.0477918
pca_beisbol %$% scores   %>% .[,1:6] %>% kable() %>% scroll_box(width = "100%", height = "500px")
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
Marcell Ozuna 8.3136847 -1.2621956 -2.5137078 -1.9871045 1.2409938 -1.5912620
Francisco Lindor 6.1511734 1.3981353 -0.0140493 0.3514227 -0.5071217 0.6233584
Cavan Biggio 6.1636386 0.8453899 -0.5014682 -1.6410395 0.4339342 0.8003743
Whit Merrifield 5.4913348 0.9786588 1.7374533 1.0525069 -1.2510838 0.3846823
Dansby Swanson 6.1208623 0.6412711 -0.3974105 0.1434527 -0.1747973 0.5922904
Freddie Freeman 9.6100869 -1.3664616 -2.4292364 -4.8237452 0.9812713 -1.6343181
José Abreu 7.9608660 -0.6234731 -2.7602742 1.2810530 0.1066485 -0.6368167
Cesar Hernandez 5.0030030 0.5144520 -0.4407774 0.7300916 1.9998305 -0.1369850
Trea Turner 8.4309950 0.5966447 4.8161429 0.6033656 -2.7768715 -1.6971991
Trevor Story 7.4368612 1.3729438 4.8344346 -1.4784962 -2.4695373 -2.4785556
Fernando Tatis Jr.  8.2650351 1.0601221 2.3035923 -1.0936127 -2.1438076 0.6030328
Carlos Santana 5.0380406 1.3527342 -2.3685233 -1.8614322 1.3589495 0.1886158
Jose Ramirez 7.7187837 0.2672994 2.1864958 -1.0198423 -1.5324890 0.3209782
Manny Machado 8.3388416 0.4557484 0.0893707 -0.2675180 -0.6679428 -1.9645394
Trent Grisham 6.1146819 1.5008807 2.9650553 -1.3173824 -0.0181955 -0.2560082
Starling Marte 5.9491042 1.2151606 2.0434093 0.7749812 -2.0014577 2.1934924
Rafael Devers 5.5489788 0.4695396 -1.5488269 1.3555000 -0.3116686 -0.6625101
Kyle Seager 6.5372407 1.0669386 -2.1356619 0.2474989 0.1457180 1.5723574
Max Muncy 5.0749140 1.3934498 -2.3737187 -1.9916613 0.7316053 1.2850951
Charlie Blackmon 6.3731206 0.4915938 -1.0053035 0.8746220 0.1271816 -1.5170596
Christian Yelich 6.0114523 1.5725365 -0.0197996 -3.5759412 0.0988449 0.1730385
Mookie Betts 6.5107946 0.0454973 1.2355908 -1.1104498 -0.9756877 -0.6672156
Keston Hiura 5.7078172 2.0112541 -1.0142979 0.2343833 -1.3416349 4.0225778
Matt Olson 4.9324643 1.2279693 -1.7384650 -2.8063685 0.4868313 -0.6389181
Bryce Harper 8.1465314 0.4483695 0.0435091 -6.0173617 1.5664108 -1.4173163
Mark Canha 6.5187092 1.1889556 -0.7356946 0.2529984 -0.4929630 2.7290254
Christian Walker 5.7245633 0.6970817 -0.7405362 2.1505558 -0.3313836 -0.3196492
Vladimir Guerrero Jr.  5.1247497 0.3288419 -0.6531259 -0.0987665 -0.2280967 -1.4400797
Cody Bellinger 4.9715747 0.5524778 -0.8609741 -2.0753882 0.3977870 -1.0730652
Maikel Franco 4.9354967 0.2692985 -1.7636596 2.2373465 0.5584921 -1.6719794
Anthony Rizzo 5.3034324 0.9381319 -2.2540580 -1.7135311 -0.1118523 2.9887038
Kyle Lewis 5.7085317 0.9841106 -0.3630385 -0.1274973 0.3587094 -0.0708020
Nicholas Castellanos 5.6189114 1.2094902 -0.2031909 -0.1807668 -1.0931376 0.0613629
Mike Trout 7.4710947 -0.2618854 -1.2508398 -2.1219268 -0.0618952 -0.8472823
Franmil Reyes 5.6270365 0.6356707 -2.2711084 2.7020333 0.7604929 -0.4600392
Andrew McCutchen 4.6306257 0.6328816 -0.8492147 0.3936680 0.1889214 -0.5080081
Pete Alonso 5.4281931 0.4400980 -2.5504057 -2.2481228 0.1311405 1.1483454
Didi Gregorius 5.6540638 0.4199166 0.6034648 -0.1404499 0.4929154 -0.8105359
J.D. Martinez 4.4171231 1.1918496 -2.3513432 -0.4468307 0.5443343 -0.6519598
Marcus Semien 3.6365348 1.1022090 -0.1156257 -0.6193340 0.0904462 -1.0952691
Javier Baez 3.8480120 2.0290663 -1.0300679 2.1593319 -0.7573124 0.1403906
Luke Voit 5.7840626 -0.6117315 -2.1408369 0.4452051 -0.1248746 0.1388008
Michael Conforto 6.5252643 -0.0118719 0.8648279 0.4406715 -0.9964393 2.6809117
Adalberto Mondesi 6.1135194 3.7922228 10.7572222 -0.4592465 -2.1849092 -0.1566113
Nick Solak 4.5185825 1.4442914 0.3386236 2.5724162 -0.2465495 0.0267849
Corey Seager 5.9718625 -0.7043712 -1.4260169 1.7467922 -0.0988710 -1.4572233
Anthony Rendon 6.1398207 -0.4570308 -1.9847401 -1.3965004 0.5219846 0.7476177
J.P. Crawford 4.6387332 1.7724707 2.5276797 0.7357436 -1.3383034 0.2271609
Paul Goldschmidt 5.2518053 -0.4579733 -1.1546746 -0.1969197 1.0962000 -0.0451554
Ian Happ 5.1231464 0.4324038 0.8003219 -1.6008773 1.0792519 0.2436866
Randal Grichuk 4.6914418 0.1450093 -1.6371132 1.2010334 0.2776088 -1.4541867
Eddie Rosario 4.5960342 0.1947467 -1.4625977 -0.2648468 0.3383271 -1.4671914
Eugenio Suarez 4.9346686 0.8990412 -2.1788328 -1.1951670 0.3626408 0.1968786
Miguel Cabrera 4.2955770 0.5589653 -1.6233676 0.0881029 0.6121902 -0.4229834
Yoan Moncada 4.1793559 1.1674735 -0.1380592 -0.1948004 -0.0795058 -1.3382551
Hanser Alberto 3.8301324 0.8633777 0.5871156 3.1018489 2.3871557 -0.1052350
David Fletcher 4.1553430 -0.2050746 0.0774257 1.8472265 1.9779985 -1.0082840
Adam Frazier 3.1969858 1.4926731 1.3642403 0.4558834 0.6241991 1.1801540
Yuli Gurriel 3.7925680 1.3145926 -0.9247590 3.3765132 -0.3272244 -0.6978067
Brian Anderson 4.9393435 0.3089421 -1.3516574 -0.3981803 -0.1659811 1.2206458
J.D. Davis 4.8899144 0.6705557 -2.1103723 -0.0801746 0.4345757 2.4306478
Kole Calhoun 5.8590936 0.6352784 -2.0620223 1.0970604 -0.3103911 2.0591314
Kyle Tucker 5.8752266 0.7172801 3.1439426 -1.5153932 -2.0497790 -4.1290163
Tommy La Stella 4.7747532 0.1159389 0.1088543 1.0602549 1.3925916 -0.9346189
Isiah Kiner-Falefa 4.4550386 1.8141208 4.9688386 1.0703857 -2.4662794 -0.4005066
Luis Robert 4.9247121 1.6135742 1.0284801 0.3445092 -0.9000290 0.1016800
Tommy Edman 4.4670027 1.7037392 1.6746646 1.5666996 -1.4797159 1.5911784
Eloy Jimenez 4.8046286 -0.6660997 -1.5781806 1.3908014 0.2305573 -1.4241488
Joey Gallo 3.6981250 1.3512662 -1.5604393 -2.5970107 0.3197720 1.1558208
Jorge Polanco 3.3887105 1.5674176 1.8501614 1.8509839 2.1779600 0.0295695
Mike Yastrzemski 6.4721448 -0.4368737 1.0888053 -2.2825530 -0.9102737 -1.4783435
Brandon Nimmo 5.2144409 0.0604337 1.6435993 -1.3670122 -1.0308997 1.2940629
Xander Bogaerts 5.0791830 -0.2406003 0.0344413 -0.9303344 0.0995141 -1.3040306
Willson Contreras 5.4052368 1.0102179 -1.3968178 0.8157866 -0.9971212 5.2609813
Brandon Lowe 5.6910356 -0.0982065 -0.3525753 -0.1053609 -0.5780074 0.0547487
Lourdes Gurriel Jr.  5.1596227 -0.4280538 -1.1252928 2.0519876 0.1772818 -1.3768954
Kyle Schwarber 3.8184569 1.1803779 -1.7806651 -1.7667007 0.4433704 0.9056133
Joey Votto 4.2603876 0.2434683 -1.9665198 -1.8175409 1.0476418 -0.4042909
Kevin Pillar 4.8291181 0.4964236 2.3776817 -0.0465082 -1.5393017 -1.5348189
Josh Bell 3.8704463 1.1052019 -2.4164148 -0.5432316 0.8291088 -0.3741154
George Springer 5.3115577 0.0903370 0.4747207 0.1408635 -1.1245108 0.9091091
Ramon Laureano 4.9338690 1.6417606 -0.9087023 1.3121175 -1.0983690 4.3689681
Eduardo Escobar 3.3881157 1.4577253 -0.5751478 -0.9246735 -0.3604485 -2.1446868
Tim Anderson 5.0830036 -0.3447249 1.0198208 1.2920155 -0.8943086 -0.6927284
Alex Verdugo 4.6408513 -0.3734143 -0.3722829 0.5692440 0.1445385 -0.4512182
Carlos Correa 3.6472836 0.3751798 -1.5091515 0.2065234 0.4504300 -0.0128901
Wil Myers 5.2858111 -0.6656608 0.5742728 -0.5771792 -0.8904653 -0.9700792
David Peralta 3.9113984 -0.0617231 -0.5160735 2.0092809 0.1846561 -1.6775184
Jackie Bradley Jr.  4.4964320 0.1653595 0.2978832 -0.4277550 -0.0677320 0.1339440
Jean Segura 4.4788912 0.8254317 1.3883860 -1.0314546 0.7888711 -0.9583520
Nick Ahmed 3.6835612 0.4477173 0.2186144 0.1648605 -0.0235771 -1.3905401
DJ LeMahieu 5.4039683 -1.6684939 0.6765428 0.7763280 -0.4091107 -1.2651923
Renato Nunez 4.2223894 0.0393063 -1.6817813 0.7024195 0.0893778 0.3967032
Jesus Aguilar 4.7245965 0.2091356 -1.3312096 2.0399762 0.3576429 -0.2860734
Nelson Cruz 6.1146175 -1.1171580 -2.7203514 -2.6373318 0.5606432 0.1102902
Chris Taylor 4.9267429 0.5567808 1.8912588 -0.6234664 0.5169809 -0.1530518
Wilmer Flores 4.1628630 -0.2496318 -1.0733550 0.4982550 -0.1474140 -1.4468213
Asdrubal Cabrera 4.1515527 0.6619850 -0.6960577 1.9218904 -0.2870441 -2.3281747
Victor Reyes 3.4012676 0.6978803 1.7805906 0.6660494 -1.0260006 -1.0738946
Aaron Hicks 4.4981484 0.6845147 0.5078335 -2.5341810 0.0105509 -0.5334233
AJ Pollock 4.7591847 -0.0839793 -0.4877503 1.0000454 -0.5012976 -0.9924017
Corey Dickerson 2.8179049 0.2382525 -0.5337536 0.0161005 0.1580650 -1.4586491
Josh Reddick 3.1914894 0.6247495 -0.7030580 0.7651917 0.3042193 -1.2665955
Jose Altuve 2.8773132 1.5083646 0.8857028 0.3472118 -0.6742939 0.2439420
Jeff McNeil 4.1790263 -0.5419325 -0.8784874 0.9171877 0.3800416 0.0756645
Adam Duvall 3.8969838 -0.0583831 -1.6990981 0.2495525 -0.1113841 0.3900506
Evan Longoria 4.0244686 0.7928658 -1.0286459 2.9005452 -0.4188745 -0.6291278
Kolten Wong 3.2511802 1.3965009 3.4119858 -0.2614969 1.6070346 0.7208323
Bryan Reynolds 2.6055203 1.4938398 0.3807132 -0.7705353 -0.4984653 -0.6912864
Teoscar Hernandez 4.8549051 -0.3234831 0.1576896 0.4226191 -0.6540506 -0.3327784
Avisail Garcia 3.1757505 1.2090664 0.7189001 -0.9650255 -0.6456491 2.2536597
Jonathan Villar 3.3562379 2.9085219 5.4275770 -0.3199224 -0.5780419 0.4801801
Jeimer Candelario 4.5276955 -0.3465090 1.1158836 -0.1065170 -0.7019952 -1.6106502
Raimel Tapia 4.1784061 0.6980763 3.4462519 1.9028964 0.0721317 -0.4782425
Austin Riley 3.2435340 0.4876093 -1.2117052 0.2783867 0.1072815 -1.0389486
Willy Adames 4.2363033 0.2985972 0.0585050 -0.0862151 -0.2047975 -1.0177399
Miguel Sano 3.6513855 0.6694914 -1.9199188 -0.9601237 0.2186746 -0.3719730
Rio Ruiz 3.1554281 0.7324695 -1.0354084 0.0908849 1.6740511 -0.9787840
Donovan Solano 3.6435251 -0.8077620 -0.1866199 1.7405830 0.0097644 -0.9431121
Ronald Acuna Jr.  5.9834011 -0.2509914 -0.0629885 -3.4652905 -0.3271798 1.3910173
Jurickson Profar 3.5196299 0.4775155 1.5231852 0.8970694 0.6133549 1.1753641
Evan White 2.7165266 2.0345902 -0.3832986 0.1191340 -0.3332326 0.3994028
Nolan Arenado 3.4495866 0.1563893 -2.3772105 1.0361977 0.6365645 -1.6704512
Colin Moran 3.9037518 0.1393280 -1.9774514 1.0531373 0.2720808 0.0822284
Dominic Smith 5.2325013 -1.3250371 -1.1680711 1.8539822 -0.4027808 0.2324493
Marwin Gonzalez 2.0593208 1.1636313 -1.4769048 1.7335860 0.3209916 0.7139405
Juan Soto 7.5088059 -2.3487177 -0.7729967 -8.6853725 0.7659052 -1.8629893
Max Kepler 2.8358168 0.1901760 -0.7178761 -0.3850392 0.0912410 0.2654012
J.T. Realmuto 3.9382321 -0.0828557 0.0312949 -0.1425460 -0.8543978 2.0159492
Ketel Marte 2.5363803 -0.1690059 -0.1985636 2.5535176 -0.3879662 -0.0598000
Yasmani Grandal 3.5801073 0.3286579 -1.8250797 -0.0271310 0.7458814 0.1928942
Brandon Crawford 3.9287158 0.3163671 -0.4374939 0.1076377 -0.4867727 0.8165562
Ryan McMahon 3.1043422 0.9071692 -0.6063886 0.2491768 -0.3324747 0.0829743
Erik Gonzalez 2.5957636 1.8539559 1.6792052 2.4705007 0.3127726 -0.8309557
Jake Cronenworth 3.8777668 -0.1356395 1.4752335 0.5119476 -0.8708332 -1.7182438
Robbie Grossman 3.8976727 0.2619627 1.5729311 -2.2560967 -1.3212177 0.3974688
Nicky Lopez 1.5512350 2.4290209 3.2810682 0.2234772 3.2830327 1.5950198
Christian Vazquez 3.3704261 0.0318417 0.7827829 0.5141966 -0.4244660 -0.4253642
Orlando Arcia 3.2441632 0.3283170 -0.7800258 1.6379257 -0.0625978 -0.9998869
Victor Robles 2.2032734 1.6320248 1.4278091 1.0935250 -0.0614094 3.0989288
Michael Brantley 2.9576003 -0.9445500 -0.3951448 0.3994371 0.3621540 -1.0029560
Hunter Dozier 2.7764802 0.5078645 0.5639841 -1.3081970 -0.2312951 -0.5901307
Rhys Hoskins 3.8431366 -0.5569520 -1.4125530 -1.0458339 0.1193297 1.8240803
Yoshi Tsutsugo 2.9550389 0.7625768 -1.4913518 -0.9228707 0.3442834 -0.4677061
Travis d’Arnaud 4.1040019 -1.1331513 -1.3590147 1.4102428 0.2223140 -0.0380540
Austin Nola 3.3689440 -0.4821488 -1.0291153 0.3857423 0.0351377 -0.1464433
Joey Wendle 3.3292360 0.3408030 2.8477387 -0.0467668 -1.8091940 0.5285723
Garrett Hampson 2.4056946 1.7272116 4.0372695 0.1002491 2.7527906 -1.2762813
Alex Gordon 0.9347269 0.7064139 -0.7164501 -0.3638934 0.2821429 0.9607065
Jesse Winker 3.9412123 -0.7545791 -1.4120914 -0.6455499 0.1800708 1.9129744
Shogo Akiyama 2.2121205 1.0638661 2.4228482 -0.9452268 -0.6503212 0.6295348
Robinson Cano 3.4467114 -1.3163373 -1.4484011 1.3095109 0.1996892 -1.0041794
Jason Heyward 3.4813445 -0.3876440 -0.0640477 -1.0703334 0.1299139 -0.4205760
Edwin Encarnacion 1.5238995 1.1950595 -1.7415610 -0.0426838 -0.1648668 1.3044177
Alec Bohm 3.6063443 -1.0493854 -0.1374100 1.6867100 0.0903076 0.1565170
Alex Bregman 2.6389962 -0.4165684 -0.9358943 -0.7786416 0.2434299 -0.3657895
Travis Shaw 2.1291494 0.1839497 -1.2478635 0.7357773 0.4600449 -0.6566236
Brandon Belt 4.4768475 -1.7139457 -1.0737039 -1.2014810 0.6469938 -1.1674639
Niko Goodrum 2.3775752 1.9776598 0.5911488 0.6603720 -0.6600623 -0.6716585
Pedro Severino 1.9207781 0.2734368 0.3117870 0.1368923 1.3562477 -0.4702865
Gary Sanchez 1.8970792 1.4536972 -2.0251644 -0.3445199 -0.1133456 1.4072323
Jonathan Schoop 3.1709004 -0.2291316 -0.6443941 2.0305679 -0.7447248 -0.1463699
Mauricio Dubon 2.6882875 0.6340284 1.5488853 1.3334864 0.8090412 -0.1154667
Adam Eaton 1.7112014 1.1565894 2.1502522 0.0318616 4.9893324 -0.5773931
Justin Turner 3.1157877 -1.0259145 -0.1632073 0.5010028 -0.3103725 1.4092345
Shohei Ohtani 2.1492853 1.2380798 0.5274266 -1.1141604 -0.3773526 -0.1272628
Gio Urshela 3.5372240 -0.7702882 -1.4975361 2.3185106 0.4142280 -0.5136569
Jorge Soler 2.4398607 0.0252269 -1.4688571 0.1183384 0.1762991 0.8560011
Paul DeJong 2.4035665 0.5910317 -1.3455940 2.0770178 0.4606917 -0.1575826
Gregory Polanco 1.2205193 2.0683125 -0.5180248 0.8178082 -0.2820964 -0.2961765
Todd Frazier 1.4833835 0.3746813 0.5393115 0.1628567 -0.7185600 0.7867570
Kevin Newman 0.4042506 1.0754464 0.4520920 1.3572595 1.4128805 0.1273964
Brad Miller 2.9391474 -0.1460587 -0.8851094 -1.1197781 0.1042945 0.3496135
Stephen Piscotty 1.7273146 1.0370447 0.0638066 1.0182161 1.1040373 -0.2630177
Alex Dickerson 3.7865928 -1.4428066 -1.1669663 -0.3840539 0.0590379 -0.8449829
Matt Carpenter 1.8664731 0.8414008 -1.3925154 -1.4111986 0.1344026 2.2106233
Justin Upton 2.1006625 0.5881869 -1.1457378 0.8898311 -0.8138529 2.5577534
Anthony Santander 2.6893640 -1.2682202 -1.1533143 -1.5271384 -0.2853803 -1.3101597
Martin Maldonado 1.8775173 0.5478125 0.2129168 -1.1298133 3.2311322 0.6813770
Brian Goodwin 2.2671923 0.6019754 -0.0400809 -0.4617521 -0.2736041 -0.7162041
Mike Moustakas 2.5673902 -0.2167294 -1.7126375 0.2722144 0.0447919 0.9153279
Albert Pujols 1.1323668 0.1411036 -1.3985294 0.5386789 0.0938938 -0.7059604
Clint Frazier 3.3570269 -0.7906128 -0.4692697 -0.4049945 -0.1437249 0.5660305
Gleyber Torres 1.6801062 -0.1486813 -0.9017058 -0.1446681 0.4078079 0.4933392
Dylan Moore 3.6304742 0.6780513 4.0567223 -0.5620302 -2.8881076 3.8714459
Freddy Galvis 1.5762243 0.2822885 -0.4648895 0.2572233 -0.5563650 1.8021011
Kevin Kiermaier 2.2169237 0.9887025 2.3731700 -1.7746530 -1.2212240 -1.2307113
Manuel Margot 1.8676641 1.0116569 4.3104344 0.0088235 -1.6087828 0.1444238
Brett Gardner 1.8888096 0.2473682 0.6798808 -1.0610223 -0.1224125 0.1316520
Michael Chavis 1.6496750 0.9988810 -0.0406436 1.3367782 -0.8495524 -0.7830588
Tyler O’Neill 1.2832771 1.2594062 -0.2862258 -0.0380805 -0.4399243 0.6393508
Salvador Perez 2.5916111 -2.1981522 -0.4744625 1.1747722 -0.3468836 -0.4280426
Eric Hosmer 2.4940016 -0.8948708 -0.4626946 1.1938872 -0.3802062 -0.1354898
Yadier Molina 0.9123567 0.1357534 -0.4743394 1.9541326 1.2657774 0.6042009
Ty France 2.1712480 -1.1337166 -0.2015975 0.6137166 -0.1961963 0.0372095
Wilson Ramos 1.0527334 0.0310920 -1.1210247 1.1752577 0.0874331 0.1646971
Cedric Mullins 1.5082125 1.0857506 5.9594870 0.0145957 3.5120287 -0.6309980
Mitch Moreland 2.1217758 -1.4024167 -1.1891054 -1.5557412 0.2764898 -0.6400406
Matt Chapman 1.9234034 -0.3774842 -0.3546351 0.3479208 -0.7175699 -1.3205655
Austin Meadows 1.2276681 0.7190181 0.3770119 0.0405261 -0.4045975 -0.1670011
Tim Lopes 0.8029883 0.3382599 0.3195077 0.5269394 -0.4547364 -0.0694771
José Iglesias 2.2512162 -2.5130131 -0.2209844 1.9586259 -0.1722206 0.3242043
Pat Valaika 1.4718971 -0.4658739 0.4781365 0.6493733 1.2097725 -0.3722051
Jon Berti 2.0784083 0.7626688 3.3206665 -0.6246032 1.8466354 1.8684977
Nomar Mazara 0.3049101 0.6061206 0.0933132 0.2327163 -0.2370098 0.9839480
Matthew Joyce 1.1942099 0.0340755 -0.6645300 0.1957963 0.6367687 -0.1823929
Enrique Hernandez 1.3364755 0.3703610 -0.1563973 1.2845193 -0.6809774 -0.3125277
Amed Rosario 0.7715650 0.3538895 0.2806680 1.0875626 -0.4873135 -1.0589277
Rougned Odor 0.6815397 1.0292868 -1.0627951 0.8449162 -0.5118684 -0.1120949
Danny Jansen 0.7513670 0.9173495 0.5223723 -0.3374035 4.2901338 1.1051025
Kris Bryant 0.4682487 0.3355147 -0.3145778 -0.2763608 -0.4344066 0.8569799
Ji-Man Choi 1.6081069 -0.2101710 -1.5097613 -0.3024324 0.7342702 -0.8404959
David Bote 1.7261805 0.4402351 -0.8631365 0.1293610 -0.2703379 0.0109329
Miguel Rojas 2.2856959 -1.2974202 1.2367272 -1.5390979 -0.5422427 -0.4002169
Jacob Stallings 0.8536213 0.1427015 0.3081804 0.5943986 3.1191541 -0.1871870
Ryan Braun 1.3445512 -0.3587338 -0.8049661 1.0320965 -0.5601401 -0.4994597
Nick Markakis 0.7961974 -0.4471868 -0.8558150 0.8967393 0.1620413 -0.4939279
Joe Panik 0.5087413 0.1466584 -0.8195480 -0.2018821 0.5469201 0.2448170
Tyler Naquin 0.7466867 0.6412595 -0.1215018 1.5050491 -0.6270360 -0.6963544
Willi Castro 2.1981847 -1.5894001 1.6808413 1.2976572 0.6414666 -0.9177710
Ryan Mountcastle 1.7304332 -1.5739107 -0.0504500 1.4150356 0.0474651 -0.0304701
Sean Murphy 1.7127372 -0.6427689 -1.3166239 -0.6045228 0.6205085 -0.0700333
Eric Thames 0.3059279 0.4623098 -0.8172314 -0.7431886 0.0818826 0.8162438
Luis Garcia 0.4994080 -0.1948759 0.3436864 0.9779368 -0.2029973 -0.6240599
Hunter Renfroe 0.6539623 0.8081509 -1.1710936 -0.1618222 -0.2203240 0.4257344
Yandy Díaz 1.4546418 -1.3673246 -0.8581090 -0.5414797 0.8110976 0.1256199
Joc Pederson 0.8433895 0.3413458 -1.1418245 0.0760991 -0.4343619 1.6885756
Will Smith 2.3379259 -2.0219357 -1.1575383 -0.6958793 0.3604360 0.2048347
Daniel Vogelbach 0.7742242 -0.2008929 -1.2209341 -1.2455374 0.4511878 0.1251568
Byron Buxton 1.3745019 -0.7268157 -0.2439516 1.2063696 -0.7862026 -0.2915408
Jedd Gyorko 1.6386921 -0.8128040 -1.4721485 0.1427736 0.3050930 -0.1054598
Jason Kipnis 1.2044116 -0.2121562 -0.3255967 0.2494419 0.1129265 -0.3149376
Austin Romine -0.0395588 0.4086536 -0.7803324 1.3657643 0.0902245 -0.7322817
Austin Hays 0.8124646 -0.0741556 1.6369756 1.1339489 -0.9439375 0.9410684
Leody Taveras 0.5772202 0.3313040 2.0748743 -1.0063040 0.6858082 -0.6075856
Garrett Cooper 1.4620274 -1.3434582 -0.9305664 0.5131860 0.0264497 0.1860957
Matt Kemp 1.1997292 -0.3444911 -1.0756715 -0.5817008 0.3400955 -0.5218331
Victor Caratini 0.7353843 0.0799413 -0.3922001 -0.0485435 -0.1759256 1.2569949
Justin Smoak 0.1491372 0.6878743 -1.3522402 0.5732729 -0.0995871 0.3263035
Daniel Murphy -0.1122204 0.0531404 -1.0685870 -0.0203404 0.3189648 -1.0360598
Ryan O’Hearn 0.5459137 0.5762652 -1.6301751 -0.7437242 0.6626639 -0.6461336
Jo Adell -0.7662315 1.5947433 -0.3940371 0.2853006 -0.3248968 0.2378985
Andres Gimenez 1.3837433 0.0886213 2.6231958 0.0618106 -1.8087762 1.3483389
Ender Inciarte -0.2348134 1.6005611 1.6581694 0.3833673 0.8032167 -0.4592296
Kurt Suzuki 1.0748633 -0.6051251 -0.9110850 1.8371266 -0.0802797 1.0548397
Carson Kelly -0.0625235 -0.0045166 -0.9471744 0.6328359 -0.0682047 -0.2864338
Bo Bichette 1.1562203 -1.3896876 0.5973479 -0.1781680 -0.5290491 -1.5970362
Eric Sogard -0.6627126 0.4578982 -0.6397472 0.5059732 0.2508095 -0.0143417
Shed Long -0.8225271 1.0220617 0.0002855 -0.4990965 -0.1652011 -0.3597388
Rowdy Tellez 1.3805493 -1.4438155 -0.4220316 0.2420320 -0.1082875 -0.1869943
Shin-Soo Choo 0.8373218 0.0432515 0.6889141 -0.3132874 -0.5667519 0.5501186
Ben Gamel 0.7625121 0.0346442 0.8006977 0.0874864 -0.5110101 -0.4611025
Andrelton Simmons 0.3456437 -0.8034787 -0.0342802 0.9424518 0.0356687 -0.2839216
Johan Camargo -0.5778795 0.2487226 -0.6714559 0.1875653 -0.1194139 -0.2176652
Nico Hoerner 0.5649920 1.0828229 0.8278209 0.9919191 -0.6363677 1.3051591
Omar Narvaez -0.5708181 0.7180208 -0.7991087 -0.5676493 0.2439900 0.7733591
Harrison Bader 1.5694559 0.0253392 1.1454653 -0.1239153 -1.2134829 1.1220489
Ozzie Albies 0.2956633 -0.9836226 0.1886425 0.0628722 -0.3744789 -0.2956869
Tommy Pham -0.1593273 0.2925687 0.4581150 -0.7239107 -0.2152425 0.2335538
Scott Kingery -1.2182341 1.1870180 -0.1968628 -0.0941526 1.3108706 0.1742485
Jake Cave 0.4456874 0.4167926 1.5188158 -0.0365455 -1.5635007 1.1618171
Carter Kieboom 0.1412629 0.8990266 -0.5052788 0.3807160 -0.1824871 2.3508608
Luis Arraez 0.2492533 -1.4887161 -0.2087785 1.2336014 0.3432313 -0.7702174
Chance Sisco 0.3446903 -0.4270344 -0.7746598 -0.7010830 -0.0979956 2.5999791
Delino DeShields -0.2910053 1.0462801 4.8441840 0.0729400 4.2456631 -0.3460519
Jose Peraza -0.2424679 0.3023323 0.5168423 1.0065956 -0.8469883 0.3000270
Luis Urias 0.1395122 0.6225642 1.3836033 0.3246714 -0.7533492 -0.0173085
Yan Gomes 0.4423493 -1.0559095 -0.0263937 1.5952617 -0.3071615 -0.7412802
Dylan Carlson -0.0954289 0.5956367 0.2151311 0.5606908 -0.5642886 -0.8635594
Roman Quinn -0.3002391 1.1661473 2.9776996 -0.5585246 -0.0495235 0.0996694
Cole Tucker -0.8434355 0.8279827 0.1015861 1.1038363 1.3464475 -0.4225284
Jose Marmolejos -0.4605994 -0.1536658 -0.8620453 0.0636893 -0.1173606 -0.1141976
Daulton Varsho -0.0093534 0.5367039 1.3647154 -0.8006826 -1.0743781 -0.1208645
Aaron Judge 1.1128536 -1.3170477 -0.5043012 0.0927715 -0.4496300 0.5845541
Sam Hilliard -0.2075768 -0.0873252 0.9121916 -0.7918137 -0.7421468 -1.2028770
Tony Kemp 0.1009867 0.1759144 1.2844870 0.4728792 1.1392881 1.4606945
Danny Mendick -0.4176445 -0.1352885 0.4183211 0.8662589 -0.4545520 -0.7987590
DJ Stewart 0.0802484 -0.4382685 0.2969018 -1.2510463 2.8655036 1.2660364
Lewis Brinson -0.3650008 0.1241750 0.1899930 0.1490155 -0.2349132 -0.5843156
James McCann 1.0644277 -1.4761331 -0.0121434 0.9470000 -0.5983916 1.4729574
Mike Tauchman 0.6780810 0.1020481 -0.0468274 -1.5089029 0.1014853 -0.4520974
Joey Bart -0.4694402 0.1546066 0.5513710 0.3664459 -0.9858451 0.7240330
Elvis Andrus -0.8826390 0.5599458 0.2608042 0.2390210 -0.4809794 -0.3408062
Tucker Barnhart -0.6012274 -0.2387073 -0.7815071 -0.3209229 0.2101147 -0.2672370
Roberto Perez -1.5081863 1.0481291 -0.7481319 -0.1861935 0.0605297 0.7412310
Nick Madrigal -0.0926789 -1.3766769 0.8244243 1.2854380 -0.3631434 0.3936024
Tony Wolters -1.1875464 0.6081975 0.8349813 0.5612258 2.7593809 0.2523122
Jared Walsh 1.0611019 -2.1047951 0.0890618 1.1560924 -0.7172301 -1.2839327
JaCoby Jones 0.6241984 -1.2143802 0.1433848 0.4996530 -0.5897761 0.8983253
Isaac Paredes -1.4427561 0.0766523 -0.3624354 0.1398083 0.1996118 -0.3831056
Willie Calhoun -1.6155849 0.7991844 -0.3242144 1.0750280 -0.3143954 -0.5930989
Brock Holt -0.9388894 0.4145966 -0.5171991 0.6833142 0.0767489 0.0220837
Luis Rengifo -1.4648870 1.7526855 1.6053128 -0.5291579 2.4426588 0.4382634
Max Stassi 0.8158352 -1.5744528 -1.2479225 1.5215119 0.1547685 0.0024941
Tyler Wade -0.2867533 1.1107841 1.0699022 -0.1740544 0.7194376 1.4264639
Austin Slater 0.9381589 -1.2241640 2.7554096 -1.0713253 0.3538197 0.7248742
Austin Barnes -0.5655653 -0.0310330 1.3316962 0.0110786 2.6244853 0.9805043
Josh Naylor -1.0149421 -0.2652617 0.3780423 0.3754612 -0.3179966 -0.5455051
Josh Fuentes -0.0324895 -1.2358125 -0.2233003 1.6317773 -0.1194269 -0.3624932
Jay Bruce -0.6113113 -0.4294944 0.0408150 -0.9659536 -0.5808524 -1.6131404
Josh Donaldson 0.3496642 -1.1894214 -1.2221310 -0.8084136 0.2863492 0.6998243
Phil Gosselin -0.2017399 -0.8072715 -0.7427672 -0.4003904 0.3106861 -0.5584800
Taylor Ward -0.3094669 -0.7849524 1.1929641 -0.0778882 -0.4679466 -1.3176577
Darin Ruf 0.2344650 -1.7501069 -0.2902075 -0.4139280 0.2427183 -0.3076818
Dexter Fowler -0.4069561 -0.4183662 0.1598806 -0.1892848 -0.2870217 0.3492584
Cameron Maybin -0.4299482 -0.4730716 0.4561365 0.3019864 -0.4673438 -0.5606010
Edward Olivares -0.8731371 0.0298473 1.1662225 0.6800579 -0.8293312 -0.5688304
Ehire Adrianza -1.0987540 0.4823652 -0.4698632 0.0114126 0.1361882 0.1435647
Howie Kendrick -0.2534784 -0.9151421 -1.0682175 0.4039216 0.3179888 -0.9482375
Jorge Alfaro -0.6775017 -0.0300151 -0.3899000 -0.2845815 -0.3932891 0.6314525
Michael A. Taylor -0.8207841 -0.2236936 -0.8318723 0.0710342 -0.1577792 -0.1343304
Jake Lamb -1.1050503 0.0658369 -0.0528035 -0.1297850 -0.4854938 1.1365773
Khris Davis -0.6242515 0.0610540 -1.0860223 0.6847720 -0.0768952 0.9596223
Christin Stewart -1.5614809 0.7972174 -0.9120307 0.6026989 -0.2086976 0.4268146
David Dahl -1.8110776 0.9978390 0.5763860 0.3046652 -0.7688792 -0.9597863
Mike Brosseau 0.6266301 -2.0309756 0.4287478 0.2715330 -0.5121903 0.4744328
Jose Martinez -0.9509559 0.5035021 -1.2310839 -0.2437618 0.2228734 -0.5291975
Abraham Toro -1.4867680 1.0895082 0.0065104 0.0334536 -1.1647988 2.7859008
Ke’Bryan Hayes 1.3471422 -3.7382665 0.7695391 -1.1498897 -0.3670647 -1.3066199
Giancarlo Stanton 0.2666802 -1.7203440 -0.7428910 -0.9566463 0.1180804 0.5297015
Jorge Bonifacio -1.2197552 -0.0764966 -0.6408653 0.6592702 -0.1414689 0.3714326
Pablo Sandoval -1.5546603 0.2265327 -0.6571369 0.6949482 0.1415063 0.1624790
Curt Casali -0.1942921 -1.4161369 -0.5018767 -0.7379048 -0.0902721 1.3013546
Adam Engel -0.4074720 -1.6096654 0.4983993 0.3686943 -0.5035713 -0.1899117
Michael Perez -1.5806068 1.0403123 -0.9728139 0.6491774 0.0089688 0.0362844
Oscar Mercado -2.5915329 2.1567991 0.4176545 0.2133421 1.0501959 -0.2290561
Bobby Dalbec 0.0197678 -2.2176970 -0.5003390 -0.6301200 -0.1294673 0.6744136
Adam Haseley -0.5452055 -0.3486574 1.0895991 0.7196020 4.1665119 0.6175140
Jason Castro -1.0705877 -0.2491370 -0.7614677 -0.3717107 0.1760599 -0.2784460
Jordan Luplow -0.8309026 0.0079156 0.1664887 0.0971738 -0.4188669 -0.0573889
Josh Harrison -0.1138085 -0.8903965 0.7686541 1.1761413 -0.8316755 1.7025490
Kevin Plawecki -0.0686949 -2.2304912 0.4192129 0.7384963 -0.3543492 -0.1710616
Andrew Knapp -0.0732779 -1.7291395 0.0145763 -0.1624198 0.1225486 -0.1032518
Phil Ervin -1.7788616 0.9805070 -0.5240085 -0.6070276 0.1847452 0.4441247
Ronald Guzman -1.1501368 -1.0794242 0.3864983 -0.4184199 -0.3964432 -0.2704950
Myles Straw -1.7108010 1.0003353 2.0598272 -0.1628721 -1.0357164 -0.0607144
Ryan McBroom -0.9483733 -1.2425488 -0.5087750 0.1959001 -0.0700219 -0.4342414
Mike Zunino -1.7639621 0.3563088 -0.7549832 -0.4266682 -0.4005617 1.0026824
Domingo Santana -1.6877247 0.2479188 -0.8324693 -0.6173499 0.1176256 0.3665717
Mike Ford -1.9265414 0.9677908 -1.1825283 0.5440938 -0.2082632 0.4366662
Edwin Rios -0.1760661 -2.0687546 -0.7878881 0.5149148 -0.3655303 0.2352801
Austin Hedges -1.9327721 1.8003136 2.0798694 0.7695290 6.3377559 1.2285190
Tim Locastro -0.3367720 -1.6401745 1.7802035 -0.4022906 0.5855015 1.1815944
Jose Trevino -1.2348986 -0.8762348 0.0438150 1.3284794 1.1832790 -0.1173175
Jose Osuna -1.8608665 -0.2659251 0.0792612 0.0429423 -0.4169827 -0.2964618
Dee Strange-Gordon -1.9361265 0.9494642 1.4509749 -0.0298038 -0.9163156 0.9701309
Robinson Chirinos -2.0492893 0.9077369 -1.0699404 0.8378986 -0.0811744 0.0371985
Sandy Leon -2.1755215 0.5403598 -0.7852384 -0.7821391 0.1708611 0.5410625
Stephen Vogt -2.1915844 0.4654779 -0.7040384 0.2116519 0.0651431 -0.3423367
Mitch Garver -2.1040274 0.6817698 -0.7230897 0.1697331 -0.0647394 0.2744871
Tyler Flowers -1.4896027 -0.6131600 -0.4253526 -0.1580596 -0.1395194 1.1126953
Jonathan Arauz -1.7825134 -0.7938373 -0.1425908 -0.1692904 0.1949718 -0.2026226
Josh VanMeter -2.4542808 0.9393769 -0.4671551 -0.5440172 -0.3273937 0.6216180
Luis Torrens -1.7506366 -0.9758102 0.4021177 0.1322501 1.4776986 -0.1936473
Nick Senzel -1.7360449 0.1938812 0.2299523 0.3313972 -0.4983967 -0.2785428
Anderson Tejeda -1.1299198 -0.6712817 1.4574194 0.0703923 -1.0235144 -0.7572722
Andrew Velazquez -1.6710215 2.1751243 4.1247732 -0.3362949 4.3567263 0.4318522
Randy Arozarena 0.1451597 -2.5498410 0.2236981 0.0720775 -0.8056263 1.9079112
Nate Lowe -1.0711043 -1.0537876 -0.6740528 -1.4781450 0.1183952 -0.5449797
Derek Dietrich -0.8933932 -1.0916972 -0.6475493 -0.2903764 -0.4779202 2.0941849
Elias Diaz -1.8594505 -0.6437530 -0.5518634 0.4270290 0.0298204 -0.3768484
Dwight Smith Jr.  -1.6501916 -0.6887126 -0.4285344 -0.3494761 -0.0340880 0.0023688
Greg Garcia -2.1726069 0.8739923 1.4469558 0.4395186 4.0486492 0.1166808
Vimael Machin -2.2822015 0.0294207 -0.5048723 0.1480719 0.1523138 -0.1856023
Kyle Farmer -2.0825249 -0.9385800 0.2319692 0.0755763 0.0060139 0.1727199
Josh Rojas -2.7115277 0.8013285 -0.4271857 0.6202972 0.0155533 -0.1768229
Gavin Lux -2.4781452 -0.0915729 -0.2725212 -0.4833761 -0.1914001 -0.2819577
Jeff Mathis -2.3109525 0.3978246 -0.0223424 -0.0887310 -0.4945748 -0.7627988
JT Riddle -3.4033918 1.0893139 -0.2138933 -0.0795173 -0.3100298 -0.4568693
Luis Guillorme -0.5603200 -2.3347651 0.1728212 0.5250021 0.2745453 -0.1598912
José García -3.0464398 1.0606997 0.4743923 0.2903888 -0.4861181 -0.1613130
Santiago Espinal -1.8951693 -0.6585299 0.5621908 0.7252198 1.3279568 -0.2514016
Jarrod Dyson -2.8504195 1.2607730 1.4611273 -0.3334563 0.8248956 -0.1709978
Miguel Andujar -2.4252140 -0.7920004 0.3094665 0.1024580 -0.3330763 -0.8558123
Chadwick Tromp -1.8628609 -0.3373393 -0.8628669 1.2616196 -0.2406163 -0.5468732
Bubba Starling -2.7646532 0.8269504 -0.5271117 0.3255628 -0.0155145 -0.2540784
Leury Garcia -2.2098947 -1.7790045 -0.0236507 -0.0690799 -0.0418958 -0.3119275
Starlin Castro -2.0860971 -1.6060900 0.4690989 -0.0764087 -0.3701316 -0.8326466
Adeiny Hechavarria -2.4365875 -0.7713553 -0.0572078 0.2171949 0.0757421 -0.3023701
Danny Santana -2.5964145 0.6452212 -0.3726003 -0.0675629 -0.2188443 -0.2560457
John Ryan Murphy -3.0239836 0.9840424 0.1498138 0.0758332 1.2741548 -0.1180245
Francisco Cervelli -1.5177227 -1.6916615 -0.2249111 -0.2264484 -0.0998559 0.3067112
Ryan Jeffers -1.7900301 -1.8284254 -0.0345313 -0.1498565 -0.0961830 0.7575732
Alex Avila -2.0778848 -0.4525771 -0.4906592 -0.5387878 0.0698641 1.0083690
Jazz Chisholm -2.3308878 0.6176442 1.5727098 -0.6044287 -1.2268571 0.1514209
Jace Peterson -1.6956524 -1.0775232 -0.2189971 -0.7363676 0.3475194 0.3221482
Chad Pinder -1.9938294 -0.9769073 -0.4514105 0.1729619 -0.0175823 -0.3351643
Cam Gallagher -1.8685727 -1.7936812 0.6716823 -0.0393613 1.4530898 -0.0968583
Anthony Bemboom -1.9246143 -0.6261767 1.3614820 0.0798580 2.2260672 1.2455935
Aledmys Díaz -2.2990970 -1.3849065 -0.2448381 0.2068469 -0.2612537 -0.5635301
Brett Phillips -1.5535191 -0.1912680 2.4108786 -1.1329673 -1.2293388 -0.7429716
Daz Cameron -3.0448408 0.3791347 0.3268228 -0.0019777 -0.5219024 -0.8471190
Jon Jay -3.0751907 0.7856178 -0.7698850 1.1875777 -0.2054022 0.0630802
Tzu-Wei Lin -3.5341075 1.6596046 0.5497090 0.7311112 2.5146816 -0.1267176
Aristides Aquino -2.3455099 -0.2545545 -0.3138013 -0.4254382 -0.3714022 1.1997717
Hunter Pence -3.3542486 1.3937056 -0.4878110 -0.1461998 -0.6926439 -0.4021001
Grayson Greiner -2.9844610 0.6795623 -0.8451284 -0.1698057 -0.4289053 0.0326490
Chris Davis -3.9854206 1.4593568 -0.5288256 -0.2160333 -0.2386794 -0.3722495
Harold Castro -1.7674557 -2.7124648 0.2966733 0.2300063 0.2302082 -0.1704572
Christian Arroyo -2.3112938 -1.4281940 -0.3190298 -0.0000109 -0.1107169 -0.2925970
Jacob Nottingham -2.1418692 -0.9773603 -0.5374922 -0.3714197 -0.2507245 0.1999398
Sam Haggerty -2.2376395 -1.3143039 0.7441475 -0.3280347 -0.3675643 -0.2954521
Matt Beaty -2.4016622 -0.6769043 -0.4444836 0.3067524 -0.3231998 0.5382714
Ildemaro Vargas -2.8010778 0.1133065 -0.0591773 0.5655172 -0.4249369 -0.8646889
Scott Heineman -3.1682932 0.8199481 0.0408144 -0.1394065 -0.4913916 -0.4946042
Magneuris Sierra -1.8922655 -0.4930861 2.8127392 0.1078568 1.7341626 0.2025955
C.J. Cron -1.7327033 -1.9263402 -0.5307692 -0.6596261 -0.1165474 0.3996068
Thairo Estrada -3.1548543 0.6283410 -0.1541833 -0.0857359 -0.5401656 1.0265359
Eli White -3.1216596 0.6275655 0.4035127 0.3179072 -0.4532525 -0.1113183
Andrew Benintendi -3.1914503 1.2638810 0.7564180 -0.8544652 1.0354117 0.9725887
Matt Adams -2.8667889 -0.0602254 -0.6511264 0.2278899 -0.2557420 -0.4403434
Monte Harrison -2.7295714 0.7668118 0.8134680 -0.6156496 -0.5509727 -0.1673377
Bradley Zimmer -2.2670286 0.0539962 0.3422694 -0.0277799 -0.7677814 2.4757852
Tyler Heineman -3.4130752 0.3124584 1.1747792 -0.1130229 2.3613596 0.9708399
Brandon Drury -3.8893972 1.1190157 -0.4701442 0.3886940 -0.1681760 -0.3709838
Kyle Higashioka -2.5646015 -1.7009357 -0.1415857 0.0357134 -0.3221055 -0.4809803
Chad Wallach -2.8772661 -0.7433163 0.3531265 0.0975448 1.2092104 -0.2001836
Jack Mayfield -3.2946857 0.7039092 0.1125238 0.6186450 1.1278857 0.2545311
Andrew Stevenson -0.5792943 -4.5018366 1.1636418 -0.3483003 -0.4680318 -0.2765331
Matt Davidson -2.8412450 -0.3406769 -0.6434156 -0.1762233 -0.2086551 -0.3116347
Mallex Smith -3.9438236 1.3293348 -0.1603624 -0.1410477 -0.4266893 -0.4071887
Dustin Garneau -3.1158352 0.3597994 1.1892217 -0.3843699 2.3247245 -0.2598211
Phillip Evans -1.6269415 -3.2789117 0.8699270 0.0425455 -0.2004989 0.5536247
Yairo Munoz -2.1424487 -2.6210541 0.5967876 0.3332819 -0.2863424 -0.4534397
Manny Pina -2.6357206 -1.5605151 -0.0963161 -0.2137261 -0.3561933 1.1574470
Reese McGuire -4.7040067 3.1143987 1.5246021 0.2987063 4.8351008 -0.0102656
Chris Owings -2.5134235 -1.7843477 0.2413985 -0.1955146 -0.1574332 -0.2105267
Pavin Smith -2.3086840 -1.6241602 0.4953297 0.4531578 -0.2934934 -0.5673574
LaMonte Wade Jr -2.9264740 -0.7759123 0.7589682 -0.2148944 -0.4474203 0.4783775
Joseph Odom -3.9568962 1.5194858 0.0371067 -0.1531361 1.2071227 -0.0245162
Mike Freeman -2.8034583 -0.8218287 -0.1625816 0.4115541 -0.0787990 0.1978253
Drew Butera -3.3852086 1.2384775 -0.1961362 0.8108698 1.1695997 -0.2871295
Franchy Cordero -2.6961527 -1.3222309 -0.0746891 -0.2745027 -0.2297299 -0.2987653
Francisco Mejia -4.2066692 1.7393740 -0.7410612 -0.1476137 -0.5739489 0.4556980
Tyrone Taylor -2.4490692 -1.7706378 -0.1764229 0.0210517 -0.2579327 0.0779061
Neil Walker -3.2121399 -0.4320164 -0.2058368 0.5522939 -0.1431463 -0.3920465
Rangel Ravelo -3.3199209 0.0366747 -0.5644185 0.5453085 -0.1153884 -0.2726192
Matt Wieters -3.2757980 -0.0626375 0.3249050 0.0387999 1.1084260 0.8413569
Eddy Alvarez -3.4604697 0.1532562 0.1826149 -0.3346745 -0.3641994 0.2685637
Jonah Heim -3.6104081 -0.1254695 -0.2102662 -0.0132304 -0.0850573 -0.2529733
Lewin Diaz -3.8517995 0.8030279 -0.5431566 0.1001139 -0.2305151 -0.3747298
Lane Thomas -3.7489884 0.8259475 -0.5942284 -0.3523799 -0.2204230 -0.2443572
Derek Fisher -2.1731625 -1.6295320 0.8478351 -0.2248704 -0.4772052 -0.2538392
Kevan Smith -2.3372200 -2.2406612 -0.0676014 -0.2126714 -0.0357787 0.3657628
Logan Forsythe -4.0380018 0.6785476 -0.4639141 -0.4844394 -0.2135729 -0.1881955
Guillermo Heredia -3.1475249 -1.0537077 0.0573393 -0.3697318 -0.2407453 -0.1805803
Mark Mathias -3.2202356 -1.3295093 0.3062562 0.1851732 -0.2248293 -0.3993980
Steven Duggar -3.6308201 0.3959857 -0.1616394 0.0279196 -0.3722346 0.0912735
Billy Hamilton -3.2730088 2.0124838 2.3454838 -0.0785347 0.0413752 0.2056694
Jake Marisnick -2.2684333 -3.3946015 0.3086013 0.0797901 -0.0986713 -0.2619920
Jonathan Davis -2.4065026 -2.0429728 0.0295410 0.2823148 -0.2813976 0.7593380
Andy Young -2.8655136 -1.6275279 -0.1228533 -0.5439192 -0.2717377 1.3745393
Yoenis Céspedes -3.4580145 -0.4763792 -0.4254030 -0.2378251 -0.3962315 0.1845765
Rob Refsnyder -3.5950475 -0.0747618 -0.2638391 0.3187058 -0.1939642 0.2328732
Braden Bishop -3.8338889 0.3963578 0.4788175 -0.1725101 0.9549122 0.2955961
Albert Almora -3.6762212 0.2997559 -0.2319751 -0.2305689 -0.1418213 0.3062410
Sam Huff -1.8607681 -4.5101752 0.4123861 -0.0929960 -0.0918078 -0.1604170
Wyatt Mathisen -2.8263517 -1.9995606 -0.0457279 -0.5598686 -0.1253989 0.5003009
Travis Demeritte -3.6335214 0.1539897 -0.2589835 -0.2776331 -0.1858215 0.3066688
Bryan Holaday -3.8934236 0.6758791 -0.4077319 -0.0489661 -0.1905993 -0.3032516
Greg Allen -3.2655620 -0.1719053 -0.0401350 -0.0695431 -0.4816180 0.7376895
Austin Allen -3.7760404 -0.2780883 -0.2132668 -0.1050195 -0.2537162 -0.3015050
Steven Souza Jr.  -3.5376803 -0.2585285 -0.1579204 -0.5635546 -0.2671364 -0.1414458
Erik Kratz -2.9338562 -2.4119098 0.3140474 0.0600129 0.0411279 -0.1405877
Anthony Alford -2.9335443 -1.2143220 0.9039181 -0.4680655 -0.8146937 -0.7716111
Jake Fraley -3.7139212 0.3533028 1.1773949 -0.5306185 -1.0586240 -0.0058444
Jesus Sanchez -4.7512352 2.1297752 -0.7906065 -0.4972080 -0.2694301 -0.1908636
Yadiel Hernández -3.4065305 -0.6954832 -0.3719685 0.3032322 -0.3026193 -0.4071144
Logan Morrison -4.1609474 0.3183024 -0.4061595 -0.4783460 -0.2782640 -0.2079115
Jorge Mateo -3.8476481 0.7462964 0.3446767 0.0218222 0.9665667 -0.2628531
Franklin Barreto -4.7352489 2.5404427 -0.4705130 -0.2240303 -0.5737015 0.0318084
Ramon Urias -2.4240674 -3.7727109 0.3956235 0.0998757 0.0185608 -0.0885091
Tomas Nido -2.5973261 -3.0632185 0.0207110 0.0222626 -0.1496458 -0.1727589
Alejandro Kirk -2.5343844 -3.9953835 0.5395092 0.0726691 -0.0262269 -0.1348413
Matt Thaiss -3.8997984 -0.2799949 -0.3434759 -0.4665905 -0.1546087 -0.0260961
Daniel Robertson -3.0740595 -2.5645160 0.4091011 -0.1211824 0.1386013 0.0868306
Meibrys Viloria -3.9178496 -0.3825213 -0.1060060 -0.2349156 -0.1921378 0.3528228
Christian Colon -4.5211566 1.1309781 -0.2978852 -0.1461324 -0.4175652 -0.3889224
Beau Taylor -5.2252550 2.6075027 -0.1230811 -0.3634538 0.9445265 -0.1746794
Sergio Alcantara -3.9797979 -0.3333576 0.1948988 -0.5076134 -0.5912830 -0.6768831
Kyle Garlick -4.5780501 0.9589419 -0.4026996 -0.1064518 -0.4357898 0.0567427
Taylor Jones -3.7543065 -0.6835505 -0.3515760 0.0075252 -0.3166150 -0.3142195
Jordy Mercer -4.1992971 -0.2122041 -0.1641692 -0.1153238 -0.1352144 -0.1441062
Nicky Delmonico -4.4707699 0.6461522 -0.4804042 -0.0638307 -0.2257328 -0.2394210
Isan Diaz -4.5977932 0.5913876 -0.2357477 -0.0586799 -0.3250750 -0.3565951
Yolmer Sanchez -2.0434638 -4.6059495 0.3980855 -0.4031110 0.1038029 0.1954456
Brent Rooker -2.7058265 -3.5028729 0.3295205 -0.0172803 -0.3287062 0.7173705
Lorenzo Cain -2.9316236 -3.0199596 0.1945882 -0.6206872 0.1560657 -0.1297284
Sherten Apostel -4.9410856 1.4988246 -0.5018057 -0.2831239 -0.3879995 -0.3492133
Brendan Rodgers -5.1244280 1.8649899 -0.5808387 -0.1700897 -0.4829681 -0.5070550
Tyler Stephenson -2.5332617 -3.8083639 0.2884785 -0.3261252 -0.1918250 0.4701173
Mark Payton -4.2642027 0.0708354 -0.0862131 -0.2430716 -0.2775159 -0.1720222
Joe Hudson -4.4610041 0.2683479 0.4176141 -0.2094223 1.1722892 0.0208882
Kevin Cron -5.3956030 2.7926773 -0.7817195 -0.4792690 -0.5863570 0.6295100
Eric Haase -4.4625791 0.5059834 -0.3670397 0.2698250 -0.2393963 -0.2726924
Mickey Moniak -3.8551557 -1.1670202 0.0732827 -0.5009861 0.0597800 0.1944903
Nick Heath -3.6847149 0.3064485 1.9545312 -0.4109370 0.3879892 0.9463645
Luis Alexander Basabe -4.1335579 0.0935860 0.1993830 -0.7213983 -0.2059264 0.1094978
Josh Phegley -4.6468674 1.0561421 -0.5543316 -0.4546514 -0.5203842 0.1436372
Mason Williams -4.6943187 1.5400476 0.5761224 -0.2836964 -1.0166647 -0.6725358
Wilmer Difo -4.7833112 1.5887014 -0.6258330 -0.0917113 -0.1859081 -0.1067984
Zack Collins -5.0255982 1.6504613 -0.5595626 -0.4584528 -0.3424036 -0.2523968
Jake Noll -3.3448790 -2.7470463 0.3653899 0.2398589 -0.0670578 -0.1489446
Andrew Knizner -3.7785734 -0.7229581 -0.3222719 0.7063793 -0.2321809 -0.3725462
Max Schrock -4.3300093 -0.1961925 -0.2257595 -0.1517555 -0.3821500 -0.3505023
Travis Jankowski -4.6830551 2.2358028 0.3325386 -0.4400795 -0.7290219 0.0008005
Willians Astudillo -3.6477292 -1.9060870 0.0641564 -0.0918600 -0.3163187 -0.3198691
Jared Oliva -4.7069982 0.4761652 -0.0096266 -0.1429326 -0.3976658 -0.3200380
Brian Dozier -4.8332622 1.0602853 -0.4423604 -0.1526854 -0.3076513 -0.2669150
Erick Mejia -5.2758692 2.3927107 0.1270039 -0.2128063 0.7171694 -0.3776335
Jorge Oña -3.0147098 -3.3173823 0.2514424 -0.4143420 -0.1990266 0.5279149
Donovan Walton -4.3901792 0.4254333 0.2144919 -0.1434484 -0.5892428 -0.0720517
Brandon Dixon -5.0409375 1.5574417 -0.6459621 -0.2423730 -0.4228140 -0.3697796
Yu Chang -4.1857669 -0.2568248 -0.2699413 -0.1204962 -0.1024171 -0.0274413
Abraham Almonte -4.8491181 1.3911847 0.3721619 -0.5331329 -0.6067569 0.1052801
Sean Rodriguez -5.0479294 0.9981194 -0.3364632 -0.1484828 -0.3841501 -0.3597292
Daniel Johnson -5.1979414 1.7537953 -0.6781220 -0.9056750 -0.3118079 -0.4753023
Jaylin Davis -4.3918113 -0.4665273 -0.2315612 -0.2390586 -0.4583018 -0.3580906
Kelvin Gutierrez -4.6612058 0.2380944 -0.2380710 -0.6227257 -0.0928826 0.1449772
Derek Hill -5.0223853 1.6875750 -0.4847300 -0.3347007 -0.3225149 -0.2495739
Cesar Puello -3.2591526 -3.6120039 1.2350046 -0.1168501 1.5070232 0.4135884
Ryan Lavarnway -3.7198646 -2.7502870 0.4800355 0.0773336 -0.0422957 -0.0653206
Dawel Lugo -4.3601396 -0.2773833 -0.1609622 -0.1187485 -0.1829480 -0.1443795
Harold Ramirez -4.5422274 -0.3501006 -0.0756151 -0.2577057 -0.2076244 -0.1326293
Jason Martin -5.4789366 2.4839953 -0.6947652 -0.6280787 -0.3454725 -0.1406904
Matt Reynolds -6.0701213 3.7217491 -0.9316822 -0.3374233 -0.6443699 -0.5769646
William Contreras -3.2427923 -3.8831804 0.6283368 0.0894032 -0.0223030 -0.0365123
Michael Hermosillo -4.0987910 -0.9466875 0.1387168 0.2231836 -0.1926211 -0.1211183
Ali Sanchez -4.9500878 1.1801912 -0.5638796 -0.1221867 -0.3142442 -0.2293516
Garrett Stubbs -4.9649087 2.0638347 0.5731067 0.3942818 0.6513091 -0.1314444
Ryan Goins -5.6312423 3.0811509 -0.7844397 -0.4620646 -0.4522224 -0.3548490
Joe McCarthy -6.1103727 3.7047448 -0.9256783 -0.3284139 -0.6378411 -0.5828764
Rafael Marchan -1.8363106 -7.4127292 1.1454881 -0.0653211 0.1289491 0.2584519
Yordan Álvarez -3.3138542 -3.1514525 0.1268579 -0.1420120 -0.3698172 0.3215893
Caleb Joseph -4.2320047 -1.0710522 -0.2127464 -0.4794826 -0.4213441 -0.2085827
Keibert Ruiz -3.7482630 -2.6220576 0.1489319 -0.2001868 -0.3683007 -0.2379083
Ryan Cordell -5.2749941 1.5236187 -0.2478199 -0.2604657 -0.5153797 -0.4024904
Austin Dean -3.1058902 -4.2236769 0.5432527 -0.6489012 0.1695271 0.5380712
Elliot Soto -3.3605833 -3.7307632 0.5365445 -0.2003482 -0.0162582 0.1092608
Jahmai Jones -3.3376830 -3.9271290 0.7082727 0.0827873 0.0377086 0.0424848
Alex Jackson -3.8811124 -2.0556022 0.1291588 0.0539873 -0.2160711 -0.1927813
Zach McKinstry -3.8848921 -2.0657032 0.1318478 0.0466719 -0.2246654 -0.2006055
Charlie Culberson -4.7712364 0.4901845 -0.3366652 -0.1964361 -0.4367456 -0.4065727
Ronald Torreyes -4.8474111 0.4436539 -0.4051429 -0.0912500 -0.4568174 -0.4320819
Ryon Healy -5.2423516 1.1358826 -0.3704514 -0.1893951 -0.4105248 -0.3778611
Adolis García -5.7535247 2.6888732 -0.7314968 -0.5384322 -0.4379761 -0.2466486
Nate Orf -6.1303777 3.6835568 -0.9123900 -0.3261346 -0.6357765 -0.5995945
Jose Briceno -4.5541835 -0.8238421 0.0223288 -0.3431511 -0.1353683 0.0256997
Justin Williams -4.5278313 -0.8083958 0.0198743 -0.3403638 -0.1301866 0.0316258
Hernan Perez -5.1145339 0.7006420 -0.2800150 -0.1757534 -0.3777486 -0.3395164
Dilson Herrera -5.6774516 2.5253429 -0.6992109 -0.4822807 -0.5333116 0.2110581
Joey Rickard -5.7007568 2.5055772 -0.6871432 -0.5591822 -0.4087138 -0.2078660
Brian O’Grady -3.1479237 -4.3907112 0.8774087 -0.0626545 -0.1546330 -0.0184566
Brian Navarreto -3.6586896 -3.4552549 0.6076067 0.0347950 -0.0090202 0.0183574
Seth Brown -6.1410382 3.6872269 -0.9139247 -0.3231295 -0.6286920 -0.5901705
Yadiel Rivera -6.1493413 3.7059228 -0.7185210 -0.3952720 -0.7254100 -0.5988861
Luis Campusano -1.6037965 -8.3830852 0.9711432 -0.4186410 -0.2981039 0.6889927
Travis Blankenhorn -2.9783875 -5.1016078 0.7111879 -0.2412841 -0.1029960 0.6803972
Oscar Hernandez -2.9032793 -5.1853108 0.8957254 0.2362459 0.1560513 0.1697486
Deivy Grullon -3.6453782 -3.4390392 0.5619577 -0.2714324 0.1113323 0.3081288
Andrew Romine -4.0978343 -2.0370189 0.1357418 -0.1547949 -0.3248731 -0.2447051
John Nogowski -4.6502059 -0.8061962 0.0361917 -0.1188204 -0.2565900 -0.2104023
Cristian Pache -4.6238537 -0.7907499 0.0337371 -0.1160331 -0.2514083 -0.2044762
Rene Rivera -4.6115833 -0.7827866 0.0310319 -0.1173162 -0.2509715 -0.1996309
Andrew Susac -4.9112822 0.1345416 -0.2158021 -0.9447154 0.0232950 0.5495424
Will Craig -6.2309383 3.6386328 -0.9118958 -0.3443532 -0.6536076 -0.5996197
Billy McKinney -1.9959297 -8.2083864 1.6169406 0.2468848 0.4119487 0.4450601
Estevan Florial -4.1259534 -2.2789262 0.3469121 -0.0541577 -0.1243132 -0.0702360
Emilio Bonifacio -6.0713707 3.8337465 -0.3264313 -0.3428153 -0.9355256 -0.3900571
Rob Brantly -6.2645107 3.6199710 -0.9088641 -0.3492956 -0.6595441 -0.6047458
Eduardo Nunez -3.0551213 -5.2079707 1.1752442 0.0265409 0.0633920 0.1913531
Luis Gonzalez -4.8860669 0.1487965 -0.2004923 -0.7966388 -0.1248647 0.9378616
Patrick Wisdom -6.2576488 3.6242386 -0.9080362 -0.3473804 -0.6555543 -0.6028651
Austin Adams -6.2666805 3.6215034 -0.9104150 -0.3548889 -0.6606171 -0.5983006
Cheslor Cuthbert -6.2789509 3.6135401 -0.9077098 -0.3536059 -0.6610540 -0.6031459
Tim Hill -6.2666805 3.6215034 -0.9104150 -0.3548889 -0.6606171 -0.5983006
Yermin Mercedes -6.2789509 3.6135401 -0.9077098 -0.3536059 -0.6610540 -0.6031459
Alec Mills -6.2666805 3.6215034 -0.9104150 -0.3548889 -0.6606171 -0.5983006
Scott Schebler -6.2789509 3.6135401 -0.9077098 -0.3536059 -0.6610540 -0.6031459
Aaron Whitefield -6.2282144 3.6338516 -0.9069739 -0.3499933 -0.6549771 -0.6028826

Visualizaremos el círculo de correlación.

library(factoextra)
fviz_pca_var(pca_beisbol, col.var = "salmon", ggtheme = theme_pander())

Se observa que, con respecto de la CP1 todas las variables están positivamente correlacionales, mientras que con respecto a la de la segunda se mantiene la asociación que se advirtió en el gráfico de correlaciones, estando las variables BA, OBP, SLG y OPS correlacionadas negativamente con esta componente.

Por último, como nos hemos quedado con las \(4\) CP primeras, vamos a representar los \({4\choose2} = \frac{24}{4} = 6\) gráficos de dispersión de las puntuaciones de los individuos.

puntuaciones_4 <- pca_beisbol %$% scores %>% .[,1:4]
nombre_jugadores <- rownames(puntuaciones_4)
puntuaciones_4 <-  puntuaciones_4 %>% as.tibble() 

q1 <- puntuaciones_4 %>%  
  ggplot(aes(x = Comp.1, y = Comp.2, label = nombre_jugadores)) + geom_text() + theme_pander()
q2 <- puntuaciones_4 %>%  
  ggplot(aes(x = Comp.1, y = Comp.3, label = nombre_jugadores)) + geom_text() + theme_pander()
q3 <- puntuaciones_4 %>%  
  ggplot(aes(x = Comp.1, y = Comp.4, label = nombre_jugadores)) + geom_text() + theme_pander()
q4 <- puntuaciones_4 %>%  
  ggplot(aes(x = Comp.2, y = Comp.3, label = nombre_jugadores)) + geom_text() + theme_pander()
q5 <- puntuaciones_4 %>%  
  ggplot(aes(x = Comp.2, y = Comp.4, label = nombre_jugadores)) + geom_text() + theme_pander()
q6 <- puntuaciones_4 %>%  
  ggplot(aes(x = Comp.3, y = Comp.4, label = nombre_jugadores)) + geom_text() + theme_pander()
q1; q2; q3; q4; q5; q6

En nuestros datos, con las componentes principales que hemos construido, los jugadores Adalberto Mondesi, Bryce Harper, Juan Soto y Freddie Freeman pueden considerarse valores atípicos.

Y ahora, para terminar el ejercicio, traemos a colación la frase de Bill James anterior: “El poder de la estadística en el béisbol es que, a diferencia del precio de la vivienda o la inflación, toma vida”. Y es que resulta que Freddie Freeman, uno de nuestros outliers, ha sido elegido Most Valuable Player (MVP) de su liga en la temporada 2020, lo que encaja con los resultados obtenidos. El resto de jugadores atípicos también han sido los mejores en los rankings de alguna o varias de estás métricas.

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