日本オペレーションズ・リサーチ学会
2021年 秋季研究発表会

1-D-7

Classification of fielders in Nippon Professional Baseballusing a Gaussian mixture clustering model
申請中
01506960

Juntendo University
Juntendo University

1. Introduction
The purpose of this study is to present a framework forgrouping players in Nippon Professional Baseball (NPB).
In order to group them, we could apply a method proposedby Soto-Valero (2017). He utilized principal component
analysis (PCA) on 7,705 European footballers using datafrom FIFA's official website, and shrank their informationsuch as Dribbling, Handling and Kicking into twovariables. Then, he conducted cluster analysis using a
Gaussian mixture model (GMM). After classifying theminto four clusters consisting of players who have similar
characteristics, excellent players were identified amongthem.
In our study, we use a similar method. We apply PCA to129 baseball players and shrink their performances into asmall number of variables. Then, we conduct cluster
analysis by GMM using 127 indicators such as Slugging
percentage, On-base percentage and On-base plus
slugging in the 2020 season taken from service "1.02
ESSENCE OF BASEBALL" provided by DELTA Co.,
Ltd. In each group, we evaluate the players according totheir principal component scores corresponding to theircharacteristics and discuss the usefulness of this method.

ODA Taishi
HIROTSU Nobuyoshi
different probability distributions.
Now, let G probability density functions be
𝒇𝟏 (𝒙; 𝜽𝟏 ), … , 𝒇𝑮 (𝒙; 𝜽𝑮 ) , and their mixed ratios be
𝝅𝟏 , … , 𝝅𝑮 . oowever, 𝜽𝒈 (𝒈 = 𝟏, … 𝑮) is a vector
consisting of parameters included in the probability(density) function 𝒇𝒈 (𝒙; 𝜽𝒈 ) . oor the mixing ratios𝝅𝟏 , … , 𝝅𝑮 , 𝟎 ≤ 𝝅𝒈 ≤ 𝟏(𝒈 = 𝟏, … , 𝑮), ∑𝑮𝒈=𝟏 𝝅𝒈 = 𝟏
shall be satisfied. At this time, the probability (density)
function of the mixture distribution model is given asfollows.
𝑮𝒇(𝒙; 𝜽) = ∑ 𝝅𝒈 𝒇𝒈 (𝒙; 𝜽𝒈 )𝒈=𝟏

The EM algorithm is used to estimate the parameters𝜃 = (𝜃1𝑇 , … , 𝜃𝐺𝑇 , 𝜋1 , … , 𝜋𝐺−1 )𝑇 included in this model.
Cluster analysis can also be performed using a mixture
distribution model. Conditional expectation used in the E
step of the EM algorithm (see reference [2] P178,179) for
which data each observation belongs to𝛾𝑖𝑔 = 𝐸(𝑍𝑖𝑔 |𝑥𝑖 ) = 𝑃𝑟(𝑍𝑖𝑔 = 1|𝑥𝑖 )𝝅𝒈 𝒇𝒈 (𝒙𝒊 ; 𝜽𝒈 )𝑃𝑟(𝑍𝑖𝑔 = 1|𝑥𝑖 ) = 𝐺∑ℎ=1 𝝅𝒉 𝒇𝒉 (𝒙𝒊 ; 𝜽𝒉 )2. Principal components analysis
One of the most difficult problems in performing
multivariate analysis is reduction and selection ofvariables used in the analysis. This is a method ofsynthesizing a small number of uncorrelated variables
called principal components that best represent the overallvariation from many correlated variables.
Assuming that 𝑋𝑛×𝑝 is a dataset consisting of nindividuals and p variables, the composite variable is thefollowing linear combination equation that contracts pdimensional data to a lower k-dimensional (k ≤ p).
𝑧𝑗 = 𝑎1,𝑗 𝑥1 + 𝑎2,𝑗 𝑥2 + 𝑎3,𝑗 𝑥3 + ⋯ + 𝑎𝑝,𝑗 𝑥𝑝 , (𝑗 = 1, ⋯ , 𝑘)

The coefficient 𝑎𝑖,𝑗 (𝑖 = 1, ⋯ , 𝑝) at this time is called
the main component. In PCA, this principal component is𝑝
obtained under the constraint of ∑𝑖=1 𝑎𝑖,𝑗 = 1 so that thevariance of 𝑧𝑗 is maximized.

3. Gaussian mixture clustering model
Mixture clustering is a method of finding the parametersof the original probability distribution, assuming that thecontinuous variables at hand are generated from several

The i-th observed value is classified into the component
that maximizes the estimated value of these formulas.

4. Decision tree
Decision tree analysis is a method of dividing data instages and outputting tree-like analysis results. It is analgorithm that sets a branch according to the condition,
traces from the root, and divides into the one that best
meets the condition. When the result of cluster analysis isused as the objective variable, it is possible to know how
much the explanatory variables such as slugging
percentage and on-base percentage affect theclassification result.

5. Data
We used the data of 127 results recorded by fielders whocome up to bat 157 or more times among the players whoparticipated in the official games in 2020. The number ofplayers who come up to 157 or more times is a maximum
value that does not exceed the number of indicators. Whenperforming PCA, the number of observations (number ofplayers) must be larger than the number of originalvariables.
As shown in Table1, the results of the decision tree
analysis are described with the highest features, alongwith the mean and standard deviation. For missing values,“0” is substituted for all values this time.

Then, we apply to the classification results as theobjective variable and the other variables as theexplanatory variables. Figure 1 shows the variable
importance. We found wRC and OPS influenced the resultof cluster analysis.

Table 1: Examples of typical indicators
wRCOPS
Batting
OffenseWPA/LIwRAA
SLv
1000
1200
FieldingUZR
CTv+WPABB/KPARBI

Mean41.29
0.73
4.62
9.97
1.18
9.73
128.9
7.4
8.88
4.64
4.64
137.796.19
0.53
334.93
37.64

SD
23.61
0.11
13.66
11.11
1.26
10.69
1.3
6.55
7.86
4.53
4.53
1.17
2.9
0.29
119.7322.42

Replacement
FAv
CB%
RngR
BB%
CBv
Soft%
O-Swing%
BABIP
FT%
Zone%
BB
K%
OBP
SL%
DefenseMean
9.39
144.97
7.41
3.6
9.17
117.92
23.19
28.53
0.3
6.71
44.63
32.3
18.64
0.33
16.1
7.86

SD
3.46
0.85
2.39
3.58
3.38
1.71
5.59
5.65
0.04
1.71
2.97
19.7
4.93
0.05
3.15
5.96

6. Results
Looking at Table 2, the contribution ratio up to the 10thprincipal component is about 60%. Since the previous
study Soto-Valero uses up to the second principal
component which has the contribution ratio of about 60%,
we use up to the 10th principal component in this study.
Table2: Results of PCA
Standard
deviation
Proportionof Variance
Cumulative
Proportion
Standard
deviation
Proportionof Variance
Cumulative
Proportion

Comp1
5.198

Comp2
3.791

Comp3
2.773

Comp4
2.428

Comp5
2.355

0.213

0.113

0.061

0.046

0.044

0.213

0.326

0.386

0.433

0.477

Comp62.125

Comp7
1.991

Comp8
1.779

Comp9
1.736

Comp10

0.036

0.031

0.025

0.024

0.022

0.512

0.543

0.568

0.592

0.614

1.685

Table 3: Means of typical indicators in each cluster
Cluster Number of players wRC
OPS
1
37
34.8
0.731
235
41.3
0.713
3
13
89.9
0.951
4
30
40.5
0.718
5
13
13.7
0.593
6
1
28.7
0.625
The results of applying a mixture model using theseprincipal component scores are as follows. The advantage
of using gaussian mixture model is that, unlike othermethods, the number of clusters can be determined using
BIC. In this study, 6 clusters are selected.

Figure 1: Variable Importance7. Conclusions and further study
We were able to present a framework for grouping
players using PCA, Gaussian mixture model, and decision
tree analysis, and to clarify the differences in the types ofplayers in NPB. This time we used the indicators recordedby the fielders, but we would like to analyze the indicators
recorded by the pitchers as well. “0” is substituted to makeup for the missing values in the dataset this time, but wewould like to try other methods such as the average
substitution method and the listwise method. In
calculating the features, we used decision tree, the simple
model, but next time we would like to use the gradientboosting decision tree as Soto-Valero.

References[1] DELTA Co., Ltd. “Online Baseball Analysis Course
Baseball Analytics Now”, https://peatix.com/event/1561186/view (viewed on August 5, 2020).
[2] O Matsui, K Koizumi, “Statistical model and guess”,
Koudansha (2020). (in Japanese)[3] S Miyamoto “Introduction to Cluster Analysis-Theory
and Application of Fuzzy Clustering”, Morikita
Publishing (2006). (in Japanese)[4] K Nishiuchi “With Yasuhito Endo, the team will have
117% of points”, SoftBank Shinsho (2012). (in Japanese)[5] C Soto-Valero “A Gaussian mixture clustering model
for characterizing football players using the EA Sports’FIFA video game system”, RICYDE. Revista
International de Ciencias del Deporte (2017).