NFL球员PCA分析

Author

Li Zongzhang

Published

October 17, 2025

安装包

install.packages("tidyverse")
install.packages("factoextra")
install.packages("MASS")
install.packages("psych")

加载包

library(tidyverse)
library(factoextra)
library(psych)

数据文件

讲义NFL Play Statistics dataset

Download icon 点击下载数据文件: NFL.xlsx

第1步 评估数据是否适合做PCA

library(readxl)
data <- read_excel("NFL.xlsx")

#提取data中的第5至12列，保存为combine
combine <- data[, 5:12]

library(psych)

KMO(combine)
Kaiser-Meyer-Olkin factor adequacy
Call: KMO(r = combine)
Overall MSA =  0.87
MSA for each item = 
    height     weight      forty   vertical      bench broad_jump three_cone 
      0.80       0.80       0.91       0.91       0.74       0.89       0.91 
   shuttle 
      0.92 
bartlett.test(combine)

    Bartlett test of homogeneity of variances

data:  combine
Bartlett's K-squared = 89371, df = 7, p-value < 2.2e-16

第2步 估计PCA

#方法一：调用prcomp函数
combine.pr <- prcomp(combine, scale = TRUE)
combine.pr
Standard deviations (1, .., p=8):
[1] 2.3679065 0.9227977 0.7890378 0.6134782 0.4681091 0.3717784 0.3483419
[8] 0.2526648

Rotation (n x k) = (8 x 8):
                  PC1         PC2         PC3          PC4         PC5
height      0.2913200 -0.36243690 -0.78456426  0.201411503 -0.10676615
weight      0.3982567 -0.23599332 -0.08084940  0.006795590  0.19904888
forty       0.3967636  0.08177256 -0.02806160  0.007186135  0.47533576
vertical   -0.3467039 -0.37295634 -0.11888605 -0.570750251  0.50876800
bench       0.2433913 -0.73405585  0.56077575  0.100768662 -0.16400782
broad_jump -0.3707226 -0.29425658 -0.20956737 -0.282435248 -0.38806157
three_cone  0.3779002  0.12306906  0.06454279 -0.546224235  0.08716904
shuttle     0.3733848  0.16307221 -0.02114816 -0.495272480 -0.52830179
                   PC6         PC7         PC8
height      0.04131390 -0.25540971 -0.22209123
weight     -0.03852389  0.23485745  0.82634941
forty      -0.10347494  0.61777481 -0.46557052
vertical    0.36727384 -0.07881529 -0.02938450
bench       0.01827782 -0.06658600 -0.21361784
broad_jump -0.51246785  0.48890122  0.00708534
three_cone -0.56866804 -0.45335440 -0.05483418
shuttle     0.51465642  0.20678970 -0.03889983
#方法二：计算combine的相关系数矩阵的特征值和特征向量
combine %>% cor() %>% eigen()
eigen() decomposition
$values
[1] 5.60698132 0.85155552 0.62258070 0.37635549 0.21912616 0.13821921 0.12134208
[8] 0.06383951

$vectors
           [,1]        [,2]        [,3]         [,4]        [,5]        [,6]
[1,] -0.2913200 -0.36243690  0.78456426 -0.201411503 -0.10676615  0.04131390
[2,] -0.3982567 -0.23599332  0.08084940 -0.006795590  0.19904888 -0.03852389
[3,] -0.3967636  0.08177256  0.02806160 -0.007186135  0.47533576 -0.10347494
[4,]  0.3467039 -0.37295634  0.11888605  0.570750251  0.50876800  0.36727384
[5,] -0.2433913 -0.73405585 -0.56077575 -0.100768662 -0.16400782  0.01827782
[6,]  0.3707226 -0.29425658  0.20956737  0.282435248 -0.38806157 -0.51246785
[7,] -0.3779002  0.12306906 -0.06454279  0.546224235  0.08716904 -0.56866804
[8,] -0.3733848  0.16307221  0.02114816  0.495272480 -0.52830179  0.51465642
            [,7]        [,8]
[1,]  0.25540971  0.22209123
[2,] -0.23485745 -0.82634941
[3,] -0.61777481  0.46557052
[4,]  0.07881529  0.02938450
[5,]  0.06658600  0.21361784
[6,] -0.48890122 -0.00708534
[7,]  0.45335440  0.05483418
[8,] -0.20678970  0.03889983
combine.pr$sdev^2
[1] 5.60698132 0.85155552 0.62258070 0.37635549 0.21912616 0.13821921 0.12134208
[8] 0.06383951
#查看主成分的方差贡献率、累计方差贡献率
summary(combine.pr)
Importance of components:
                          PC1    PC2     PC3     PC4     PC5     PC6     PC7
Standard deviation     2.3679 0.9228 0.78904 0.61348 0.46811 0.37178 0.34834
Proportion of Variance 0.7009 0.1064 0.07782 0.04704 0.02739 0.01728 0.01517
Cumulative Proportion  0.7009 0.8073 0.88514 0.93218 0.95957 0.97685 0.99202
                           PC8
Standard deviation     0.25266
Proportion of Variance 0.00798
Cumulative Proportion  1.00000
#查看第1主成分载荷
combine.pr$rotation[,1:2]
                  PC1         PC2
height      0.2913200 -0.36243690
weight      0.3982567 -0.23599332
forty       0.3967636  0.08177256
vertical   -0.3467039 -0.37295634
bench       0.2433913 -0.73405585
broad_jump -0.3707226 -0.29425658
three_cone  0.3779002  0.12306906
shuttle     0.3733848  0.16307221

  • PC1含义解释

    • PC1对所有变量的系数（载荷）多数为正，主要由体重（weight）和速度/敏捷相关（forty、three_cone、shuttle）指标主导。

    • broad_jump和vertical为负系数，表明跳跃能力好的个体PC1得分低。

    • PC1可以理解为“体型+速度/敏捷主成分”，与跳跃能力呈负相关。

  • PC2含义解释

    • PC2主要由卧推（bench），其次是vertical、height、broad_jump，且系数均为负，说明这些能力高的人在PC2得分低。

    • PC2可以理解为“力量+弹跳能力逆向主成分”。

第3步 确定保留的主成分的个数

# 计算各个主成分的方差
pr.var <- combine.pr$sdev^2
pr.var
[1] 5.60698132 0.85155552 0.62258070 0.37635549 0.21912616 0.13821921 0.12134208
[8] 0.06383951
#  计算各个主成分的方差贡献率
pve <- pr.var/sum(pr.var)
pve
[1] 0.700872665 0.106444440 0.077822587 0.047044437 0.027390770 0.017277401
[7] 0.015167760 0.007979939
#绘制各个主成分的方差贡献率
plot(pve, xlab = "Principal Component", 
     ylab = "Proportion of Variance Explained", 
     ylim = c(0, 1), type = "b")

# 绘制累计方差贡献率
plot(cumsum(pve), xlab = "Principal Component", 
     ylab = "Cumulative Proportion of Variance Explained", 
     ylim = c(0, 1), type = "b")

#  绘制碎石图
library(factoextra)
fviz_eig(combine.pr)

第4步 可视化

# 绘制主成分1和主成分2的分组散点图
combine.pcscore <- cbind(data,combine.pr$x)

combine.pcscore %>% ggplot(aes(PC1,PC2, col= position))+
  geom_point()

combine.pcscore %>% ggplot(aes(PC2,PC3, col= position))+
  geom_point()

#Graph of individuals. 
#Individuals with a similar profile are grouped together.
fviz_pca_ind(combine.pr,
             col.ind = "cos2", # Color by the quality of representation
             geom = c("point"),
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE     # Avoid text overlapping
)

# Graph of variables. 
# Positive correlated variables point to the same side of the plot. 
# Negative correlated variables point to opposite sides of the graph.
fviz_pca_var(combine.pr,
             col.var = "contrib", # Color by contributions to the PC
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE     # Avoid text overlapping
)

#Biplot of individuals and variables
fviz_pca_biplot(combine.pr, repel = TRUE,
                geom = c("point"),
                col.var = "#2E9FDF", # Variables color
                col.ind = "#696969"  # Individuals color
)

table(data$position)

   C   CB   DE   DT EDGE   FB   FS  ILB   LB   LS   OG   OL  OLB   OT   QB   RB 
 115  311  279  253    8   77  123  130    1    2  232    1  240  273   12  245 
   S   SS   TE   WR 
   7  107  194  275
缩写 英文全称 中文解释
C Center 中锋，进攻线中央负责开球给四分卫
CB Cornerback 角卫，防守组主要负责盯防外接手
DE Defensive End 防守端锋，防守线两侧，冲传和防跑
DT Defensive Tackle 防守截锋，防守线中间，堵截跑动
EDGE Edge Rusher (OLB/DE) 边锋冲传手，专注于冲击四分卫
FB Fullback 近卫跑卫，主要负责掩护和短码冲球
FS Free Safety 游动安全卫，防守后场，负责深区
ILB Inside Linebacker 内线卫，防守中路，兼顾跑动和传球
LB Linebacker 线卫，防守二线，分内外线卫
LS Long Snapper 长传手，专门负责长开球（开球、射门）
OG Offensive Guard 进攻护锋，中锋两侧，保护四分卫
OL Offensive Lineman 进攻线球员总称，包括C、OG、OT等
OLB Outside Linebacker 外线卫，负责边路防守和冲传
OT Offensive Tackle 进攻截锋，进攻线两侧，保护四分卫
QB Quarterback 四分卫，进攻组织核心，传球主力
RB Running Back 跑卫，负责持球冲跑或接球
S Safety 安全卫，泛指FS和SS
SS Strong Safety 强侧安全卫，靠近防跑，兼顾传球
TE Tight End 近端锋，既能接球也参与阻挡
WR Wide Receiver 外接手，主要负责接球推进 
table(data$position) %>% 
  as.data.frame() %>%
  arrange(desc(Freq))
   Var1 Freq
1    CB  311
2    DE  279
3    WR  275
4    OT  273
5    DT  253
6    RB  245
7   OLB  240
8    OG  232
9    TE  194
10  ILB  130
11   FS  123
12    C  115
13   SS  107
14   FB   77
15   QB   12
16 EDGE    8
17    S    7
18   LS    2
19   LB    1
20   OL    1
combine.pcscore %>%
  filter(position %in% c("CB","DE","WR")) %>%
  ggplot(aes(PC1, fill= position))+
  geom_histogram() +
  facet_wrap(~position, ncol = 1)

PC1: “体型+速度/敏捷主成分”

DE Defensive End 防守端锋，防守线两侧，冲传和防跑

CB Cornerback 角卫，防守组主要负责盯防外接手

WR Wide Receiver 外接手，主要负责接球推进

第5步 针对主成分得分的进一步分析

不同位置球员的主成分得分比较

# 绘制PC1主成分得分按位置分组的箱线图
combine.pcscore %>%
  ggplot(aes(x = fct_reorder(position, PC1, .fun = median, .desc = TRUE), 
             y = PC1, fill = position)) +
  scale_fill_brewer(palette = "Set3") +
  geom_boxplot() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
  labs(title = "不同位置球员PC1主成分得分分布", x = "position", y = "PC1得分")

# 绘制PC2主成分得分按位置分组的箱线图
combine.pcscore %>%
  ggplot(aes(x = fct_reorder(position, PC2, .fun = median, .desc = TRUE), 
             y = PC2, fill = position)) +
  scale_fill_brewer(palette = "Pastel1") +
  geom_boxplot() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
  labs(title = "不同位置球员PC2主成分得分分布", y = "PC2得分")

# 方差分析：不同位置的PC1得分是否有显著差异
anova_pc1 <- aov(PC1 ~ position, data = combine.pcscore)
summary(anova_pc1)
              Df Sum Sq Mean Sq F value Pr(>F)    
position      19  14133   743.9    1046 <2e-16 ***
Residuals   2865   2037     0.7                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# 方差分析：不同位置的PC2得分是否有显著差异
anova_pc2 <- aov(PC2 ~ position, data = combine.pcscore)
summary(anova_pc2)
              Df Sum Sq Mean Sq F value Pr(>F)    
position      19  497.8  26.198   38.33 <2e-16 ***
Residuals   2865 1958.1   0.683                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

基于主成分得分的聚类分析

# k-means聚类（以2类为例）
set.seed(123)
km <- kmeans(combine.pcscore[, c("PC1", "PC2")], centers = 3)
combine.pcscore$cluster <- as.factor(km$cluster)

# 可视化聚类结果
ggplot(combine.pcscore, aes(PC1, PC2, color = cluster)) +
  geom_point() +
  labs(title = "基于主成分得分的球员聚类")

聚类与位置的对应关系

prop.table(table(combine.pcscore$position, combine.pcscore$cluster), 1)
      
                 1           2           3
  C    0.930434783 0.000000000 0.069565217
  CB   0.000000000 0.996784566 0.003215434
  DE   0.132616487 0.021505376 0.845878136
  DT   0.790513834 0.000000000 0.209486166
  EDGE 0.000000000 0.000000000 1.000000000
  FB   0.051948052 0.116883117 0.831168831
  FS   0.000000000 0.991869919 0.008130081
  ILB  0.007692308 0.153846154 0.838461538
  LB   0.000000000 1.000000000 0.000000000
  LS   0.000000000 0.000000000 1.000000000
  OG   0.969827586 0.000000000 0.030172414
  OL   1.000000000 0.000000000 0.000000000
  OLB  0.004166667 0.262500000 0.733333333
  OT   0.945054945 0.000000000 0.054945055
  QB   0.083333333 0.250000000 0.666666667
  RB   0.000000000 0.881632653 0.118367347
  S    0.000000000 1.000000000 0.000000000
  SS   0.000000000 0.962616822 0.037383178
  TE   0.051546392 0.061855670 0.886597938
  WR   0.000000000 0.956363636 0.043636364

  • 1号聚类（Cluster 1）

    • 主要特征： 高度集中于内线和进攻线球员： OL（进攻线）: 100% OG（进攻护锋）: 97% OT（进攻截锋）: 95% C（中锋）: 93% DT（防守截锋）: 79%

    • 这些位置典型特征是体型大、力量强，符合PCA能力分型的“体型/力量主导”类别。

  • 2号聚类（Cluster 2）

    • 主要特征： 集中于速度型和二线防守球员： CB（角卫）: 99.7% FS（游动安全卫）: 99% LB（线卫）: 100% S（安全卫）: 100% SS（强侧安全卫）: 96% WR（外接手）: 96% RB（跑卫）: 88%

    • 这些位置通常身材相对较小，速度、灵活性、爆发力强，对应“敏捷/速度主导”能力分型。

  • 3号聚类（Cluster 3）

    • 主要特征： 集中于力量+爆发型或多面手球员： EDGE（冲传手）: 100% LS（长传手）: 100% DE（防守端锋）: 85% FB（近卫跑卫）: 83% ILB（内线卫）: 84% OLB（外线卫）: 73% TE（近端锋）: 89% QB（四分卫）: 67% DE、FB、TE、QB、OLB、ILB等为多功能或力量/爆发型球员，显示这些球员的身体素质在主成分空间中更接近第三类。

    • 这些位置球员兼具力量、体型和一定灵活性，属于“力量/多面手型”分型。