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Article

Regression Tree Model for Predicting Game Scores for the Golden State Warriors in the National Basketball Association

Department of Industrial Engineering and Management National Chin-Yi University of Technology, Taichung 411, Taiwan
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(5), 835; https://doi.org/10.3390/sym12050835
Submission received: 28 April 2020 / Revised: 13 May 2020 / Accepted: 16 May 2020 / Published: 19 May 2020

Abstract

:
Data mining is becoming increasingly used in sports. Sport data analyses help fans to understand games and teams’ results. Information provided by such analyses is useful for game lovers. Specifically, the information can help fans to predict which team will win a game. Many scholars have devoted attention to predicting the results of various sporting events. In addition to predicting wins and losses, scholars have explored team scores. Most studies on score prediction have used linear regression models to predict the scores of ball games; nevertheless, studies have yet to use regression tree models to predict basketball scores. Therefore, the present study analyzed game data of the Golden State Warriors and their opponents in the 2017–2018 season of the National Basketball Association (NBA). Strong and weak symmetry requirements were identified for each team. We developed a regression tree model for score prediction. After predicting the scores of each player on two teams, we summed and compared the predicted total scores to obtain the predicted results (lose or win) of the team of interest. The results of this study revealed that the regression tree model can effectively predict the score of each player and the total score of the team. The model achieved a predictive accuracy of 87.5%.

1. Introduction

Advanced statistical methods were commonly used in various studies. A soft computing model used a learning approach for addressing data management over social networks [1]. Dulebenets et al. [2] applied regression models to estimate the effects of various factors on the driving ability of individuals. Andrée et al. [3] estimated a penalized non-parametric model of environmental output across economic development. A multivariate random parameter Tobit model was utilized to determine the factors that drive both the crash occurrence probability and the crash rate of 65+ roadway users [4]. Narasingam et al. [5] applied sparse regression to determine the structure of reduced-order model on a hydraulic fracturing process.
Since the 20th century, sport has grown globally [6,7]. Professional sporting games, such as basketball, baseball, tennis, and golf, and events such as the World Football Championship and the Olympic Games not only attract the attention of many fans but also continue to create extremely high output value for the sport industry. The Internet has enabled sport betting to develop rapidly. In most countries, sport betting is inextricably linked to professional sport. In sport betting, increasing the betting win rate requires—in addition to subjective judgements—predictions from historical data on the game, such as predictions of total results, total over or under, handicaps, and point spreads.
Scholars in most studies have applied linear regression models to predict the scores of various ball games. However, studies have yet to use the regression tree method to predict basketball scores. Regression trees are similar to classification trees and are easy to understand and interpret. Unlike classification trees, the regression tree is suitable for continuous or ordered discrete dependent variables, and the prediction error of a regression tree is usually measured by the square difference between the actual value and the predicted value (see [8]). Regression trees have been used for predictions in several areas; for example, they have been used for predicting short-term algal blooms in the field of environmental engineering [9], for predicting student performance in teaching [10], for predicting the ultimate bearing capacity of shallow foundations in cohesive soils in civil engineering [11], and for demand analysis in economics [12]. Therefore, in the present study, we used the regression tree method to predict the scores of basketball games. Single-season (2017–2018) match data from the National Basketball Association (NBA) were used to predict the scores of each player on two teams. After the predicted scores of the two teams were added and compared, the team’s predicted win or loss results could be obtained. The linear regression and support vector regression were also utilized in this study. Results from regression tree model, linear regression model, and support vector regression model were compared.

2. Related Studies

Several scholars have devoted attention to predicting the results of various sporting events and have conducted research on basketball, baseball, football, cricket, and other ball games [13,14,15,16]. Thabtah et al. [13] applied naive Bayes, neural network–like, and decision tree machine learning methods to various feature sets, in order to construct prediction models. By comparing the respective prediction accuracy rates, they could select the model with superior performance and determine the key factors affecting the results of the game. Valero [14] analyzed 10 years of Major League Baseball (MLB) regular season game data, using four data mining methods, namely lazy learners, artificial neural networks, support vector machines, and decision trees; the goal was to evaluate the abilities of the aforementioned classification- and regression-based methods in predicting game outcomes (home team win or lose) in MLB games. Razali et al. [15] used Bayesian networks to predict home victories, away victories, and draws in the English Premier League. Pathak et al. [16] applied modern classification techniques, namely naive Bayes, support vector machine, and Random Forest, to predict the outcome of the One Day International (ODI) cricket match.
Loeffelholz et al. [17] collected 620 NBA games and used neural networks to predict the success of basketball teams. The selection of features input to the neural networks as the most salient features for prediction from signal-to-noise ratios and expert opinions was also discussed in this study. Cao [18] collected the data of five regular NBA seasons and applied machine learning algorithms to build models for predicting the NBA game outcomes.
In addition to the prediction of victory or defeat, many scholars have studied team scores. For example, Harville [19] proposed a linear model for predicting differences in scores for college basketball or football games, using the difference in team effects plus or minus the home-field or field advantage. To fit a relevant linear model, Harville proposed an improved method of least-squares estimation and applied team estimates, in order to rank teams in the league. The study findings revealed that the results of the playoffs could be effectively predicted. Karlis and Ntzoufras [20] proposed bivariate Poisson models for analyzing goals scored by two teams and adjusted the models to increase the probability of a draw. Adam [21] attempted to extend the bivariate Poisson method through a generalized linear model. The score was modeled as the joint probability of a Poisson distribution representing the total number of goals and a binomial distribution representing a team goal.
Wheeler [22] first used the chi-square test to screen input variables; the threshold was set to 0.05, excluding features not exceeding the threshold, and 16 variables were finally obtained. Linear regression was then used to calculate the average score of each player according to the characteristic variables and to obtain the sum of the predicted scores for the two players on the field. Finally, the results of the two teams’ matches (win/lose) were compared. In addition, to compare the performance of the linear regression model with other benchmark models, feature variables were input into naive Bayes and support vector machine (SVM) classifiers, to obtain classification results. The linear regression output value was converted into two classification values for comparison. The linear regression model predicted a player’s average score, and the converted win–loss classification result had an error rate of 53%. The naive Bayes and SVM classifier error rates were as low as 31%.
Singh et al. [23] proposed two separate models for predicting the match results for the ODI. They used data for the 2013 and 2014 ODI competitions as training and testing sets and conducted 10 cross-validations. Linear regression was used to predict the final score of the first round of ODI, and a naive Bayes classifier was used to estimate the probability of a team winning the second round. The results showed that, for prediction of the final score, the errors in the linear regression classifier were less than those for the current Run Rate method. As the game progressed, the accuracy of the naive Bayes prediction of the game results increased from 70% to 91%.
Wiseman [24] predicted the winning score of events on the PGA Tour, using first-round data. The author used linear regression, neural network regression, Bayesian linear regression, decision forest regression, and boosted decision tree regression models and compared the performance of the methods. Models were constructed by using data from 2004 to 2015 and validated by using the 2016 tournament. Correlation matrix analysis was conducted for various features. The first-round lead score, first-round average score, event, course yardage, and total prize money were selected as forecast indicators, and the R-squared and mean square error (MSE) values were used as evaluation indicators. The results revealed that the linear regression and Bayesian linear regression models were superior to the other models.
Lu et al. [25] analyzed games from 2012 to 2016 and established the least square fir model, using previous game results, team ability, and home advantages, based on data from five seasons, to predict the point difference for each team. Linear regression models depending on total over/under were fit to data before the all-star break and checked for adequacy, to predict final score difference between home and away NBA teams, for a regular season during 2011–2012 [26].

3. Materials and Methods

3.1. Data

In this study, we considered the Golden State Warriors (GSW), one of the 30 NBA teams, for analysis. All teams competing with the GSW were regarded as opponents. The reason for choosing the GSW as the research object is that, in the traditional basketball concept, the closer to the basket score, the more solid the game will win, and the three-pointer is just a way to assist in scoring. However, the presence and rise of Stephen Curry subverts this traditional concept and opens up the modern basketball of “three-pointers”. In addition, the Golden State Warriors are the champions of the NBA’s first season, and the team won a total of six league championships. Therefore, this study used the Golden State Warriors as the object of analysis. Other teams can also model and predict according to the proposed method of this article.
We executed the data collection step by capturing GSW player match data for the 2017–2018 season from the Basketball Reference website [27]. Because the final purpose was to obtain prediction results through prediction scores, GSW opponents’ data must also be collected. After the collection of the data, the missing values were deleted. The missing data in this dataset comprised only two major items: Inactive and Player Suspended. We manually removed data fields indicating “Inactive” and “Player Suspended”. The number of records is not fixed for each player on the field. Some teams change players frequently. Each team has a total of 30 database fields, as shown in Table 1. The first eight items are related to the event and were not considered in the prediction model in this study. For the ninth item (i.e., Games Started), players in the starting lineup are usually the best players on the team (see [28]). If the best player acts as a starter but does not contribute to the team, the player is considered to have hindered the team. The corresponding variable was therefore selected to establish whether it is relevant to the score. Regarding the 10th item (i.e., Minutes Played), Martínez and Martínez [29] indicated that no linear correlation exists between score and playing time. However, we used the M5Prime model tree algorithm (M5P) [30], which predicts nonlinear continuous data; therefore, this variable was selected and used in the prediction model. The 11th to 28th items pertain to personal data contributed by players to the team in the game. Among them, Field Goals, 3-Point Field Goals, and Free Throws are absolutely linearly related to the score (2 × FG + 3P + FT = PTS). Therefore, the above three items were excluded in the prediction model. In addition, the 28th item represents the player’s score, which we used as an output item. Game Score (29th item) and Plus/Minus (30th item) are the player efficiency level and personal goal difference. Both items are calculated based on the player’s personal data; therefore, they were not used in the prediction model. Finally, items considered to be removed from the dataset were player uniform number, rank, season game, date, age, team, home/away, opponent, field foals, 3-point field goals, free throws, game score, and plus/minus.
Table 1 presents a summary of the variables considered in the regression tree model for this study, including the variable fields and variable abbreviations. The last column in the table indicates whether the variables were included in the model in this study. Sixteen input variables were used, and the output variables were the actual points.
The dataset was divided into a training set, validation set, and test set, and the ratio of the three sets was 6:2:2. In 82 games, the total number of players playing in each game is different. To avoid a scenario in which data of the 3rd player in the 50th game are assigned to the training set and data of the 4th player in the same game are assigned to the verification set, we divided the dataset according to Season Game, with the first 50 fields representing the training set, the 51st–66th fields representing the validation set, and the 67th–82nd fields representing the test set.

3.2. Methods

The flowchart of the study procedure is illustrated in Figure 1. We considered three regression methods, namely regression tree, linear regression, and support vector regression models, for modeling, prediction, and comparison. After training three regression models through the training set, we used the three constructed models to predict the validation dataset and used the root mean square error (RMSE) as an error index (loss function) for the models. We determined the superior of the three aforementioned models and used it to predict player scores. This step was executed by using the test set. After predicting and summing up the scores of players in each field, we obtained the team’s predicted total score. By comparing the two teams’ predicted total scores, we could obtain the predicted match result; finally, we could compare the result with the actual result and calculate the accuracy rate of the predicted match result.

3.3. Regression Tree

The overall process of the regression tree method is similar to that of the classification tree method, and a prediction value is obtained at each node. Classification trees are used to process discrete data, whereas regression trees are used to process continuous data.
We constructed the model employed in this study by using the M5P tree regression algorithm in Weka software. M5P is a machine learning algorithm published by Wang and Witten in 1996 [31]. Its predecessor was M5, which was developed by Quinlan in 1992. Compared with traditional linear regression algorithms, M5P can accurately predict nonlinear data, and the rules and regression models are easy to interpret.
M5P is a binary regression tree model. The last node in the regression tree is a linear regression function that produces continuous numerical attributes. The M5P algorithm includes four main steps: The first step entails dividing the input space into several subspaces, to create a tree. The variability in the subspace from root to node can be minimized by using segmentation criteria. The standard deviation of the value reaching this node is used to measure variability. The construction of the tree is completed by using a reduced standard deviation range (SDR) factor, which maximizes the expected reduction in errors on the nodes, as expressed in the following equation:
S D R = s d S i S i S × s d S i
where S is the set of data records arriving at the node, S i is the set obtained by dividing the node according to a given attribute, and s d is the standard deviation. The second step entails developing a linear regression model in each subspace, using the data associated with that subspace. The third step involves applying pruning techniques to overcome the problem of overtraining. However, the pruning process may cause a sharp interruption between adjacent linear models. The final step entails performing a smoothing process to compensate for the sharp interruption. The smoothing process combines all models from leaf to root to create the final model of the leaf. In the process, the predicted values of the leaves are filtered when they return to the root. The filtered values are combined with the predicted values through a linear regression of the node, as follows:
E = n e + k a n + k
where E is the estimated value passed to the next highest node, e is the estimated value passed from below to the current node, a is the predicted value of the model at this node, n is the number of training examples that have reached the node, and k is a constant (see [31,32]).

3.4. Linear Regression

Linear regression is the simplest and most commonly used prediction model. A linear regression model predicts the linear relationship between continuous target variables and predicted variables, and many data items fulfil the basic assumptions of normal distribution and linear relationship.
Linear regression models can be divided into simple linear regression and multiple linear regression models. Simple linear regression models entail the use of a single independent variable (X) to predict a dependent variable (Y). The regression equation can be expressed as follows:
Y i = β 0 + β 1 X i + ε i ,   i = 1 , , n
where Y i is the actual observation value (variable) for the ith observation value of the dependent variable, Y; X i is the ith observation (variable) of the independent variable, X; β 0 is the parameter of the regression mode (termed the intercept or constant term); β 1 is the parameter of the regression mode (termed the regression coefficient or slope); n is the number of observations; and ε i is a random variable of the ith observation and belongs to a random error.
Multiple linear regression models entail the use of two or more independent variables to predict a dependent variable (Y). The regression equation can be expressed as follows:
Y i = β 0 + β 1 X 1 i + β 2 X 2 i + + β k X k i + ε i ,   i = 1 ,   ,   n
where Y i is the actual observation value of the ith observation value for the dependent variable, Y; X k i is the ith observation for the kth independent variable, X; β 0 is a parameter of the regression mode (termed the intercept); β 1 , …, β k is a parameter of the multiple regression mode (termed the regression coefficient); ε i is a random variable of the ith observation value; n is the number of observations; k is the number of independent variables; and k > 0 is a positive integer.

3.5. Support Vector Regression

Support vector regression (SVR) solves binary classification problems and has been proven to be an effective tool in real-value function estimation [33]. Like a regression method, the output of SVR is a real number. SVR finds an optimal hyperplane that balancing the model complexity and prediction error. The main advantages of SVR include that its computational complexity does not depend on the dimensionality of the input space, and it has excellent generalization capability, with high prediction accuracy. The prediction function of SVR is defined as follows [34]:
f x = w , x + b
where X denotes the space of the input patterns; and (w,x) denotes the dot product in X. If we minimize w and b, the optimization problem is defined as follows:
R s v m C = C 1 n i = 1 n L ε d i , y i + 1 2 w 2
L ε d i , y i = | d i y i ε , d i y i ε 0 , o t h e r w i s e
where C 1 n i = 1 n L ε d i , y i is empirical error risk, which could be obtained from an ε-insensitive loss function in Equation (7); 1 2 w 2 is a regularization term; and C is a regularization.
By introducing positive slack variables ξ i and ξ i into Equation (6), we get the following:
Minimize   R s v m w , ξ = 1 2 w 2 + C i = 1 n ( ξ i + ξ i ) Subject   to   d i w ϕ x i b i ε + ξ w ϕ x i + b i d i ε + ξ i   , ξ i 0
Lagrange multiplier is used to solve the optimization problem, and Equation (5) becomes the following:
f x , a i , a i = i = 1 n ( a i a i ) K x , x i + b
where a i and a i are Lagrange multipliers, satisfying   a i * a i = 0, a i 0 and a i 0 for i = 1, …, n, and the dual optimization problem is as follows:
Max   R a i , a i = i = 1 n d i a i a i ε i = 1 n a i + a i 1 2 i = 1 n j = 1 n a i a i ( a j a j ) K x i x j Subject   to   i = 1 n a i a i = 0 , 0 a i   C ,     i = 1 , 2 , , n 0 a i C ,     i = 1 , 2 , , n

3.6. Performance Evaluation

Several loss functions are used for regression, with commonly used functions being the MSE and mean absolute error (MAE). The MAE is an absolute value of the deviation between target and output value; therefore, positive and negative phases cannot cancel out. The MAE can thus effectively reflect the reality of the prediction error. Nevertheless, the MAE value is not differentiable at 0, and no method exists for determining the correction direction of a model through differentiation. The MSE overcomes this disadvantage but cannot easily be used to interpret data to obtain interpretable units. The solution is to use the RMSE to obtain an interpretable unit.
Accordingly, we used the RMSE to evaluate predictive performance. The RMSE is the square root of the ratio of the sum of all squared deviations of the predicted value from the actual values to the number of observations, n. To explain the degree of dispersion of a sample, the RMSE can be minimized for nonlinear fitting. The RMSE formula is as follows:
RMSE = 1 n i = 1 n Y ^ i Y i 2
where Y ^ i is the predicted value, and Y i is the actual value.

4. Results

In this study, we used the training set to construct three regression models—regression tree, linear regression, and support vector regression models—through Weka software. We subsequently used the three regression models to predict the validation data and then calculated the RMSE values of the models, in order to determine the optimal model. Finally, the optimal model obtained from the training and validation sets was employed for prediction, using the test set to measure model performance. Results obtained by using data for the GSW as an example are described in the following sections.

4.1. Information on Opponents

Table 2 lists the opponent data corresponding to Season Game for the test set. The total scores of the last 16 GSW games were predicted; subsequently, the total scores of the 11 teams in Table 2 were also predicted. By comparing the predicted total scores of the two teams, we obtained the predicted outcomes. Finally, we compared the actual results with the predicted results and calculated the accuracy of win or loss predictions.

4.2. Model Validation Results

We applied the training set to construct the model and used the validation set to establish the best model; we then obtained the prediction results for the two models, as well as the equations for the regression tree, linear regression, and support vector regression models. The modeling results are presented in Table 3. Figure 2 illustrates the regression tree, and Table 4 presents the regression equation. The linear regression equation is represented by Equation (12).

4.2.1. Regression Tree

According to Table 3, the regression tree model with the lowest RMSE was the optimal model. Although constructing the model would require a longer time than that required for the linear regression model, the RMSE value of the regression tree model was the lowest among the three models. Therefore, the regression tree model was used to predict player scores in the subsequent step.

4.2.2. Linear Regression

The linear regression model for conducting predictions for the GSW can be expressed as follows:
PTS = 0.716 GS + 0.001 MP + 1.049 FGA + 6.03 FG % + 0.133 3 PA + 4.635 3 P %   + 0.933 FTA + 0.219 ORB + 0.164 PF + 2.055

4.3. Model Test Results

Consider, for example, the test set data of GSW players for March 11, 2018; each row in Figure 3 contains information on players who played for the GSW on that day. According to the rules derived from the training set in Figure 2, we used the program to determine the rules (LM1–LM6) for each row of test set data. After judgement, the input variables were entered into the corresponding equations, to obtain the predicted PTS in the last column of Figure 4. The actual total score of the GSW team on March 11, 2018, was 103 points, and the team’s total score based on predictions was 108 points (Figure 4).
We used the data in the remaining 15 test sets for the GSW and the data in the test sets for the opponent teams, to obtain the total score from each prediction. Each team input into the M5P training model was subject to several rules, with each rule having a corresponding regression equation. The relevant form of the opponent is presented in the Appendix A to Appendix C. After obtaining the predicted scores of both parties in all test sets, we could predict the GSW match results by comparing the predicted GSW scores with the opponent scores. Finally, we compared the predicted scores with the actual win or loss results and then calculated the accuracy of the predicted match results. Table 5 lists the predicted results, on the basis of the test set, of the last 16 games of the GSW, and their opponents in the 2017–2018 season. Figure 5 presents a line chart comparing the actual and predicted scores of the GSW and their opponents.
Except for the prediction errors for the 67th and 77th fields, all predictions were accurate, and the prediction accuracy was 87.5% (Table 5). In addition, the actual scores for some games were not significantly different from the corresponding predicted scores. For some games, the actual scores were accurately predicted; for example, the predictions for the GSW in game 74, for both teams in game 81, and for the opponents in game 82 were accurate. Therefore, the regression tree model can effectively predict team scores.

5. Discussion

Table 6 provides results from relevant studies. Several scholars have devoted attention to predicting game wins or losses. The use of machine learning models in earlier studies, to predict competition results, and the development of models based on other principles in recent years have engendered an increase in the accuracy of predictions. Miljkovic et al. [35] used four machine learning algorithms to predict the competition results and reported that the naive Bayes classifier had the best prediction accuracy rate (67%). Moreover, Cao [18] used four machine learning methods for prediction, including a naive Bayes classifier, and revealed that the logistic regression model achieved higher prediction accuracy than did the naive Bayes classifier. Cheng et al. [36] developed an NBAME model based on the principle of maximum entropy, to predict the outcome of games. They compared the performance of the NBAME model with traditional machine learning classifiers and reported that the NBAME model achieved a higher prediction accuracy rate (74.4%). To solve the shortcomings of SVMs that lack rule generation, Pai et al. [37] and Kaur et al. [38] used SVMs to combine decision rules and fuzzy rules, respectively, to develop new predictive match outcome models. The results demonstrated that the models achieved higher accuracy than did the conventional SVMs. Linear regression was used in another study [22], and the accuracy achieved was 47%. Although it is much lower than our result from our linear regression model, it is important to know that the proposed methodology is not comparable with this study, due to the use of a different database.
Due to the uncertainty of game results, linear regression cannot be used to generate a regression equation that illustrates the linear relationship between variables and scores. The M5P regression tree algorithm used in this study could establish multiple regression models based on the distribution of data, and the prediction accuracy was determined to be higher than those of the linear regression model and support vector regression model. The present study differs from other studies in that it applied a regression tree model to predict the scores of players in two opposing teams for each match and then summed and compared the scores, to obtain the game results of the team of interest.

6. Conclusions

We conducted this study to develop regression tree and linear regression models by using data from two competing teams in a single NBA season. We predicted the scores of each player on the two teams and summed and compared the predicted scores of the two competing teams; thus, the win or loss results of the team of interest could be obtained. The results reveal that the regression tree model could predict player scores more accurately when compared with the linear regression model.
Any game is a complex system. The model proposed in this study yields favorable results for the prediction of the outcome of NBA games. This model can thus provide valuable prediction information for NBA team leaders and players. The limitation and future study of our method includes the following:
(1)
The procedures to determine the data rules and to obtain the corresponding equations for obtaining the prediction scores for each team were manual and must be debugged to avoid errors, which was time-consuming.
(2)
Other factors may not have been considered in this study. For example, if a team’s key player does not play due to injury, the team may score lower in the relevant match. Factors that have been overlooked in the present study can be looked at by future studies, to determine whether the inclusion of such factors can further improve predictive accuracy.
(3)
The proposed models were established and tested for GSW and its opponents. More teams should be tested to evaluate the generality of the proposed model.
(4)
The dataset included GSW player match data for the 2017–2018 season only; we encourage researchers to analyze a larger dataset in the future.
(5)
The proposed models were established and tested for NBA games only. It can be used in other ball leagues; nevertheless, we do not guarantee that its predictive accuracy can be transferred to other ball leagues. Application of the proposed models on other sport events needs to be verified.

Author Contributions

M.-L.H. and Y.-J.L. conceived and planned the experiments; Y.-J.L. carried out the experiment; M.-L.H. and Y.-J.L. wrote the paper; M.-L.H. directed the project and verified the experimental results. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Figure A1 presents the comparison of the prediction performance of two regression models of 11 opponents of GSW.
Figure A1. Comparison of the prediction performance of two regression models of 11 opponents of GSW.
Figure A1. Comparison of the prediction performance of two regression models of 11 opponents of GSW.
Symmetry 12 00835 g0a1

Appendix B

Figure A2 and Figure A3 show the regression tree diagram of 11 opponents of GSW.
Figure A2. Regression tree diagram of GSW opponents (A).
Figure A2. Regression tree diagram of GSW opponents (A).
Symmetry 12 00835 g0a2
Figure A3. Regression tree diagram of GSW opponents (B).
Figure A3. Regression tree diagram of GSW opponents (B).
Symmetry 12 00835 g0a3

Appendix C

Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10 and Table A11 present the regression equations of several regression trees of 11 GSW opponents.
Table A1. Eight regression equations of the regression tree model of MIN.
Table A1. Eight regression equations of the regression tree model of MIN.
RuleRegression Equation
LM1PTS = 0.1136 * GS − 0.0001 * MP + 0.1766 * FGA + 1.0854 * FG% + 0.5169 * 3P% + 0.2683 * FTA + 0.2759 * FT% + 0.0278 * DRB − 0.0262 * AST − 0.0096 * STL − 0.058 * BLK − 0.3728
LM2PTS = 0.1136 * GS − 0.0001 * MP + 0.1766 * FGA + 1.0854 * FG% + 0.5169 * 3P% + 0.8043 * FTA + 1.2538 * FT% + 0.0278 * DRB − 0.0262 * AST − 0.0096 * STL − 0.058 * BLK − 1.2327
LM3PTS = 0.0641 * GS − 0.0001 * MP + 1.3903 * FGA + 5.1844 * FG% + 0.1773 * 3PA + 1.2693 * 3P% + 0.2176 * FTA + 1.6953 * FT% + 0.0386 * DRB − 0.0281 * AST − 0.0096 * STL − 0.0723 * BLK − 0.0305 * TOV − 3.8398
LM4PTS = 0.0641 * GS − 0.0001 * MP + 0.9832 * FGA + 10.0824 * FG% + 0.1815 * 3PA + 1.7658 * 3P% + 0.7227 * FTA + 0.7961 * FT% + 0.1687 * ORB + 0.0299 * DRB − 0.0211 * AST − 0.0096 * STL − 0.0523 * BLK − 0.1202 * TOV − 5.3827
LM5PTS = 0.0231 * GS − 0.0002 * MP + 0.8761 * FGA + 22.6725 * FG% + 0.2429 * 3PA + 2.3976 * 3P% + 0.6056 * FTA + 1.36 * FT% − 0.0084 * ORB + 0.017 * DRB + 0.0226 * AST − 0.0093 * STL − 0.0145 * BLK + 0.0078 * PF − 10.3598
LM6PTS = 0.0231 * GS − 0.0006 * MP + 1.1301 * FGA + 21.8965 * FG% + 0.2616 * 3PA + 1.9132 * 3P% + 0.7687 * FTA + 0.8672 * FT% − 0.0084 * ORB + 0.017 * DRB + 0.0757 * AST − 0.0093 * STL − 0.0145 * BLK + 0.0078 * PF − 12.1217
LM7PTS = 0.0231 * GS − 0.0002 * MP + 0.85 * FGA + 30.7728 * FG% + 0.4108 * 3PA + 2.1747 * 3P% + 0.2773 * FTA + 2.3993 * FT% − 0.0114 * ORB + 0.0194 * DRB + 0.0342 * AST − 0.0093 * STL − 0.0717 * BLK + 0.0424 * PF − 14.1381
LM8PTS = 0.0231 * GS − 0.0002 * MP + 0.9548 * FGA + 32.9434 * FG% + 0.3487 * 3PA + 2.3784 * 3P% + 0.8931 * FTA + 4.6337 * FT% − 0.0114 * ORB + 0.0194 * DRB + 0.0371 * AST − 0.0093 * STL − 0.079 * BLK + 0.0468 * PF − 20.7739
Table A2. Seven regression equations of the regression tree model of LAL.
Table A2. Seven regression equations of the regression tree model of LAL.
RuleRegression Equation
LM1PTS = −0.041 * GS + 0.1584 * FGA + 0.9103 * FG% − 0.0292 * 3PA + 0.5522 * 3P% + 0.1344 * FTA + 1.9649 * FT% − 0.0137 * DRB + 0.0263 * TRB − 0.0058 * AST − 0.3066
LM2PTS = −0.1506 * GS + 0.0001 * MP + 0.8048 * FGA + 2.1766 * FG% − 0.0222 * 3PA + 1.782 * 3P% + 0.2419 * FTA + 0.4397 * FT% + 0.1166 * DRB + 0.0199 * TRB − 0.0058 * AST + 0.0508 * STL − 0.1423 * TOV − 0.8115
LM3PTS = −0.0976 * GS + 0.9177 * FGA + 6.8786 * FG% + 0.0752 * 3PA + 1.7277 * 3P% + 0.5734 * FTA + 0.9142 * FT% − 0.0088 * DRB + 0.0199 * TRB − 0.0058 * AST + 0.1641 * STL − 3.534
LM4PTS = 0.0332 * GS + 0.6652 * FGA + 17.0546 * FG% + 0.1436 * 3PA + 2.9707 * 3P% + 0.5066 * FTA + 0.968 * FT% + 0.0039 * DRB − 0.0168 * AST − 0.0097 * STL − 0.0193 * BLK − 5.9762
LM5PTS = 0.0518 * GS + 0.8974 * FGA + 20.9579 * FG% + 0.2801 * 3PA + 2.3938 * 3P% + 0.6731 * FTA + 1.2043 * FT% + 0.0039 * DRB − 0.0094 * AST − 0.0097 * STL − 0.0287 * BLK − 0.0092 * PF − 10.2445
LM6PTS = 0.0527 * GS + 1.125 * FGA + 15.3185 * FG% + 0.3924 * 3PA + 1.7757 * 3P% + 0.6069 * FTA + 1.4276 * FT% + 0.0039 * DRB − 0.0094 * AST − 0.0097 * STL − 0.1476 * BLK − 0.0093 * PF − 9.355
LM7PTS = −0.0238 * GS + 0.8634 * FGA + 33.1303 * FG% + 0.3647 * 3PA + 2.4466 * 3P% + 0.6734 * FTA + 1.9331 * FT% + 0.0039 * DRB − 0.0033 * AST − 0.025 * STL − 16.0197
Table A3. Eleven regression equations of the regression tree model of SAC.
Table A3. Eleven regression equations of the regression tree model of SAC.
RuleRegression Equation
LM1PTS = −0.0129 * GS + 0.2146 * FGA + 1.5615 * FG% + 0.0132 * 3PA + 1.6046 * 3P% + 0.1703 * FTA + 0.8129 * FT% + 0.0034 * TRB − 0.0036 * AST − 0.0062 * STL − 0.0228 * PF − 0.7113
LM2PTS = −0.0129 * GS + 0.2146 * FGA + 1.5615 * FG% + 0.0132 * 3PA + 1.6046 * 3P% + 0.2985 * FTA + 1.4463 * FT% + 0.0034 * TRB − 0.0036 * AST − 0.0062 * STL − 0.0228 * PF − 0.7983
LM3PTS = −0.0129 * GS + 0.2146 * FGA + 1.5615 * FG% + 0.0132 * 3PA + 1.6046 * 3P% + 0.3021 * FTA + 1.4463 * FT% + 0.0034 * TRB − 0.0036 * AST − 0.0062 * STL − 0.0228 * PF − 0.7939
LM4PTS = −0.0129 * GS + 0.2146 * FGA + 1.5615 * FG% + 0.0132 * 3PA + 1.6046 * 3P% + 0.278 * FTA + 1.4463 * FT% + 0.0034 * TRB − 0.0036 * AST − 0.0062 * STL − 0.0228 * PF − 0.6649
LM5PTS = −0.0129 * GS + 0.2609 * FGA + 1.5615 * FG% + 0.0132 * 3PA + 5.3769 * 3P% + 0.1705 * FTA + 2.0271 * FT% + 0.0034 * TRB − 0.0036 * AST − 0.0062 * STL − 0.2509 * PF + 0.0913
LM6PTS = −0.0129 * GS + 0.5289 * FGA + 3.4292 * FG% + 0.0257 * 3PA + 2.3564 * 3P% + 0.2095 * FTA + 1.5175 * FT% − 0.0371 * ORB + 0.0034 * TRB − 0.0036 * AST − 0.0189 * STL + 0.0472 * BLK − 1.0459
LM7PTS = −0.0129 * GS +1.3501 * FGA + 5.7764 * FG% + 0.0831 * 3PA + 0.8934 * 3P% + 0.5419 * FTA + 1.0467 * FT% − 0.016 * ORB + 0.0034 * TRB − 0.0036 * AST − 0.0189 * STL + 0.0673 * BLK − 4.0174
LM8PTS = −0.0129 * GS + 1.2549 * FGA + 8.1146 * FG% + 0.2546 * 3PA + 1.3603 * 3P% + 0.4042 * FTA + 1.3464 * FT% − 0.016 * ORB + 0.0367 * DRB + 0.0034 * TRB − 0.0036 * AST − 0.1139 * STL + 0.1569 * BLK − 5.5553
LM9PTS = −0.0129 * GS + 0.8811 * FGA + 12.7822 * FG% + 0.2248 * 3PA + 1.92 * 3P% + 0.5038 * FTA + 1.066 * FT% + 0.0431 * DRB + 0.0034 * TRB − 0.0036 * AST − 0.0196 * STL − 6.0382
LM10PTS = −0.0177 * GS + 0.7554 * FGA + 21.0721 * FG% + 0.3239 * 3PA + 2.1617 * 3P% + 0.5608 * FTA + 1.2428 * FT% + 0.0124 * ORB + 0.0046 * TRB − 0.0049 * AST − 0.0271 * STL − 8.648
LM11PTS = −0.0177 * GS + 1.0667 * FGA + 23.761 * FG% + 0.519 * 3PA + 1.9583 * 3P% + 0.6918 * FTA + 1.0464 * FT% + 0.0106 * ORB − 0.0938 * DRB + 0.087 * TRB − 0.0049 * AST − 0.1599 * STL − 13.9301
Table A4. Ten regression equations of the regression tree model of SAS.
Table A4. Ten regression equations of the regression tree model of SAS.
RuleRegression Equation
LM1PTS = 0.1117 * FGA + 0.9506 * FG% + 0.0409 * 3PA + 0.4507 * 3P% + 0.2321 * FTA + 0.4317 * FT% − 0.0134 * AST − 0.0288 * TOV − 0.2691
LM2PTS = 0.1117 * FGA + 0.9506 * FG% + 0.0409 * 3PA + 0.4507 * 3P% + 0.2924 * FTA + 0.4856 * FT% − 0.0134 * AST − 0.0288 * TOV − 0.2767
LM3PTS = 0.1117 * FGA + 0.9506 * FG% + 0.0409 * 3PA + 0.4507 * 3P% + 0.6312 * FTA + 1.2838 * FT% − 0.0134 * AST − 0.0288 * TOV − 0.7792
LM4PTS = 0.0002 * MP + 1.1678 * FGA + 5.1841 * FG% + 0.052 * 3PA + 0.5047 * 3P% + 0.247 * FTA + 1.4588 * FT% − 0.0079 * AST − 0.0279 * BLK − 0.0431 * TOV + 0.0142 * PF − 3.4654
LM5PTS = 0.0007 * MP + 0.8117 * FGA + 8.8607 * FG% + 0.052 * 3PA + 0.5047 * 3P% + 0.5133 * FTA + 1.0659 * FT% − 0.0079 * AST − 0.0279 * BLK − 0.035 * TOV + 0.0142 * PF − 4.4419
LM6PTS = 0.0008 * MP + 1.024 * FGA + 8.438 * FG% + 0.4651 * 3PA + 0.6746 * 3P% + 0.4815 * FTA + 1.244 * FT% − 0.0079 * AST − 0.0553 * BLK − 0.0199 * TOV + 0.0281 * PF − 5.5065
LM7PTS = 0.0642 * GS + 0.6585 * FGA + 17.1977 * FG% + 0.123 * 3PA + 2.8478 * 3P% + 0.5512 * FTA + 0.3376 * FT% − 0.0062 * TOV − 5.7993
LM8PTS = -0.1892 * GS + 0.6895 * FGA + 19.3371 * FG% + 0.1374 * 3PA + 1.1288 * 3P% + 0.9205 * FTA + 3.8532 * FT% − 0.0062 * TOV − 10.0091
LM9PTS = 0.9738 * FGA + 18.7541 * FG% + 0.3873 * 3PA + 1.8752 * 3P% + 0.6744 * FTA + 0.9939 * FT% − 0.1228 * TOV+ 0.0941 * PF − 9.9508
LM10PTS = 0.9505 * FGA + 33.2296 * FG% + 0.4135 * 3PA + 0.5176 * 3P% + 0.6855 * FTA + 1.3068 * FT% − 0.0062 * TOV − 16.1274
Table A5. Thirteen regression equations of the regression tree model of PHX.
Table A5. Thirteen regression equations of the regression tree model of PHX.
RuleRegression Equation
LM1PTS = 0.1417 * FGA + 2.3515 * FG% + 0.0438 * 3PA + 0.5717 * 3P% + 0.3088 * FTA + 0.4956 * FT% + 0.0123 * AST − 0.0114 * STL − 0.0186 * PF − 0.5119
LM2PTS = 0.1417 * FGA + 2.3515 * FG% + 0.0438 * 3PA + 0.5717 * 3P% + 0.5335 * FTA + 1.5342 * FT% + 0.0123 * AST − 0.0114 * STL − 0.0186 * PF − 0.9602
LM3PTS = 0.1417 * FGA + 2.3515 * FG% + 0.0438 * 3PA + 0.5717 * 3P% + 0.7157 * FTA + 1.8756 * FT% + 0.0123 * AST − 0.0114 * STL − 0.0186 * PF − 1.6306
LM4PTS = 0.544 * FGA + 10.6236 * FG% + 0.2015 * 3PA + 1.5694 * 3P% + 0.5165 * FTA + 0.3576 * FT% + 0.0086 * AST − 0.0114 * STL − 0.0186 * PF − 3.1167
LM5PTS = 0.0007 * MP + 0.2844 * FGA + 5.1577 * FG% + 0.1501 * 3PA + 1.7466 * 3P% + 0.505 * FTA + 0.4932 * FT% + 0.0269 * ORB + 0.0086 * AST − 0.0114 * STL − 0.0186 * PF − 0.6677
LM6PTS = 0.0009 * MP + 0.2844 * FGA + 5.1577 * FG% + 0.1501 * 3PA + 1.7207 * 3P% + 0.505 * FTA + 0.4932 * FT% + 0.0086 * AST − 0.0114 * STL − 0.0186 * PF − 0.7115
LM7PTS = 0.0009 * MP + 0.2844 * FGA + 5.1577 * FG% + 0.1501 * 3PA + 1.7207 * 3P% + 0.505 * FTA + 0.4932 * FT% + 0.0086 * AST − 0.0114 * STL − 0.0186 * PF − 0.6864
LM8PTS = 0.0004 * MP + 0.2844 * FGA + 5.1577 * FG% + 0.1638 * 3PA + 2.3301 * 3P% + 0.5467 * FTA + 0.4932 * FT% + 0.0269 * AST − 0.0114 * STL − 0.0186 * PF + 0.0274
LM9PTS = 0.0004 * MP + 0.2844 * FGA + 5.1577 * FG% + 0.1638 * 3PA + 2.5019 * 3P% + 0.5467 * FTA + 0.4932 * FT% + 0.0086 * AST − 0.0114 * STL − 0.0186 * PF + 0.1507
LM10PTS = −0.3212 * GS + 0.0005 * MP + 1.2472 * FGA + 7.1835 * FG% + 0.3778 * 3PA + 1.2633 * 3P% + 0.569 * FTA + 1.0027 * FT% − 0.01 * AST − 0.0114 * STL + 0.022 * TOV − 0.0175 * PF − 5.3952
LM11PTS = −0.4592 * GS + 0.0007 * MP + 0.9651 * FGA + 12.8293 * FG% + 0.3184 * 3PA + 1.8388 * 3P% + 0.594 * FTA + 1.3671 * FT% − 0.0122 * AST − 0.0114 * STL + 0.0267 * TOV − 0.0175 * PF − 7.5166
LM12PTS = 0.0004 * MP + 0.8434 * FGA + 23.068 * FG% + 0.3742 * 3PA + 1.8547 * 3P% + 0.6369 * FTA + 1.2095 * FT% − 0.0187 * STL − 0.0246 * PF − 11.2644
LM13PTS = 0.9557 * FGA + 35.1348 * FG% + 0.4416 * 3PA + 2.0119 * 3P% + 0.7654 * FTA + 1.7094 * FT% − 0.0187 * STL − 0.0293 * PF − 18.4038
Table A6. Eight regression equations of the regression tree model of ATL.
Table A6. Eight regression equations of the regression tree model of ATL.
RuleRegression Equation
LM1PTS = −0.0001 * MP + 0.1764 * FGA + 1.2818 * FG% + 0.0302 * 3PA + 0.3982 * 3P% + 0.2116 * FTA + 0.2799 * FT% − 0.0063 * ORB − 0.0039 * AST + 0.0066 * TOV − 0.4257
LM2PTS = −0.0001 * MP + 0.1764 * FGA + 1.2818 * FG% + 0.0302 * 3PA + 0.3982 * 3P% + 0.6367 * FTA + 1.3491 * FT% − 0.0063 * ORB − 0.0039 * AST + 0.0066 * TOV − 1.08
LM3PTS = 0.0004 * MP + 1.2723 * FGA + 5.901 * FG% + 0.0727 * 3PA + 1.4136 * 3P% + 0.1406 * FTA + 0.3975 * FT% − 0.0063 * ORB − 0.0101 * DRB − 0.0039 * AST + 0.0066 * TOV − 4.2519
LM4PTS = 0.0002 * MP + 1.3525 * FGA + 5.4025 * FG% + 0.2633 * 3PA + 1.8194 * 3P% + 0.1406 * FTA + 0.5444 * FT% − 0.0063 * ORB − 0.1234 * DRB − 0.0039 * AST + 0.1613 * TOV − 3.1576
LM5PTS = 0.2564 * FGA + 10.7834 * FG% + 0.3757 * 3PA + 1.4551 * 3P% + 0.576 * FTA + 1.2406 * FT% − 0.0063 * ORB − 0.0039 * AST + 0.0066 * TOV − 2.0274
LM6PTS = −0.0558 * GS + 0.6383 * FGA + 19.7827 * FG% + 0.268 * 3PA + 2.5233 * 3P% + 0.7214 * FTA + 0.6825 * FT% − 0.0058 * ORB + 0.0199 * DRB − 0.017 * TRB − 0.0036 * AST − 0.0131 * STL + 0.1189 * TOV − 7.0797
LM7PTS = −0.3244 * GS + 1.0839 * FGA + 19.3213 * FG% + 0.4449 * 3PA + 2.1591 * 3P% + 0.6807 * FTA + 1.0698 * FT% − 0.0058 * ORB + 0.0231 * DRB − 0.0163 * TRB − 0.0036 * AST − 0.0079 * STL + 0.1128 * TOV − 11.7177
LM8PTS = −0.1213 * GS + 0.9832 * FGA + 28.6595 * FG% + 0.4575 * 3PA + 3.2159 * 3P% + 0.6941 * FTA + 1.0059 * FT% − 0.0058 * ORB + 0.1812 * DRB − 0.2023 * TRB − 0.0036 * AST − 0.0079 * STL + 0.0431 * TOV − 15.7175
Table A7. Ten regression equations of the regression tree model of UTH.
Table A7. Ten regression equations of the regression tree model of UTH.
RuleRegression Equation
LM1PTS = 0.0879 * GS + 0.2072 * FGA + 1.2399 * FG% + 0.0347 * 3PA + 0.5013 * 3P% + 0.2015 * FTA + 0.3589 * FT% + 0.0115 * DRB − 0.4674
LM2PTS = −0.0195 * GS + 0.305 * FGA + 1.2399 * FG% + 0.0347 * 3PA + 0.5013 * 3P% + 0.2015 * FTA + 0.3589 * FT% + 0.0115 * DRB − 0.5628
LM3PTS = −0.0534 * GS + 0.3246 * FGA + 1.2399 * FG% + 0.0347 * 3PA + 0.5013 * 3P%+ 0.2015 * FTA + 0.3589 * FT% + 0.0115 * DRB − 0.438
LM4PTS = 0.0879 * GS + 0.2789 * FGA + 1.2399 * FG% + 0.0347 * 3PA + 0.5013 * 3P% + 0.355 * FTA + 1.5366 * FT% + 0.0115 * DRB − 0.736
LM5PTS = 0.2084 * GS + 0.2692 * FGA + 1.2399 * FG% + 0.0347 * 3PA + 0.5013 * 3P% + 0.4238 * FTA + 1.3183 * FT% + 0.0115 * DRB − 0.514
LM6PTS = 0.0519 * GS + 1.076 * FGA + 6.512 * FG% + 0.1729 * 3PA + 1.8056 * 3P% + 0.7639 * FTA + 0.1474 * FT% + 0.0049 * DRB − 0.0194 * STL − 4.1035
LM7PTS = 0.0519 * GS + 0.9832 * FGA + 9.8844 * FG% + 0.0892 * 3PA + 2.7334 * 3P% + 0.6035 * FTA + 1.0813 * FT% + 0.0132 * DRB + 0.0488 * AST − 0.0168 * STL + 0.0159 * TOV − 0.0707 * PF − 5.3049
LM8PTS = 0.0519 * GS + 1.1435 * FGA + 9.7413 * FG% + 0.4609 * 3PA + 1.436 * 3P% + 0.7036 * FTA + 0.5026 * FT% + 0.0148 * DRB − 0.0168 * STL + 0.0189 * TOV − 6.3336
LM9PTS = 0.0552 * GS − 0.0005 * MP + 0.933 * FGA + 19.8717 * FG%+ 0.3163 * 3PA + 2.8292 * 3P% + 0.6613 * FTA + 0.7231 * FT% + 0.1377 * ORB + 0.0706 * AST + 0.0995 * PF − 9.7557
LM10PTS = 0.0949 * GS − 0.0001 * MP + 0.8701 * FGA + 34.7682 * FG% + 0.4849 * 3PA + 0.7035 * 3P% + 0.8098 * FTA + 0.854 * FT% − 16.5234
Table A8. Eleven regression equations of the regression tree model of IND.
Table A8. Eleven regression equations of the regression tree model of IND.
RuleRegression Equation
LM1PTS = 0.0373 * GS + 0.0001 * MP + 0.1732 * FGA + 1.3719 * FG% + 0.0186 * 3PA + 0.5554 * 3P% + 0.4754 * FTA + 0.1408 * FT% − 0.0137 * ORB − 0.4115
LM2PTS = 0.0373 * GS − 0 * MP + 1.1834 * FGA + 4.4244 * FG% + 0.2057 * 3PA + 0.8084 * 3P% + 0.2278 * FTA + 1.1137 * FT% − 0.0137 * ORB − 0.0526 * DRB + 0.0372 * TRB − 2.7073
LM3PTS = 0.0373 * GS + 1.636 * FGA + 4.5897 * FG% + 0.4005 * 3PA + 0.8084 * 3P% + 0.2278 * FTA + 0.7968 * FT% − 0.0137 * ORB − 0.0526 * DRB + 0.0372 * TRB − 3.3137
LM4PTS = 0.0373 * GS + 1.7152 * FGA + 4.5897 * FG% + 0.3666 * 3PA + 0.8084 * 3P% + 0.2278 * FTA + 0.7968 * FT% − 0.0137 * ORB − 0.0526 * DRB + 0.0372 * TRB − 3.2551
LM5PTS = 0.0373 * GS + 1.7152 * FGA + 4.5897 * FG% + 0.3666 * 3PA + 0.8084 * 3P% + 0.2278 * FTA + 0.7968 * FT% − 0.0137 * ORB − 0.0526 * DRB + 0.0372 * TRB − 3.2495
LM6PTS = 0.0373 * GS + 0.9708 * FGA + 7.3572 * FG% + 0.2765 * 3PA + 1.4484 * 3P% + 0.4488 * FTA + 1.1871 * FT% − 0.0137 * ORB − 0.026 * DRB + 0.0184 * TRB − 3.8555
LM7PTS = 0.0576 * GS + 0.0004 * MP + 0.6212 * FGA + 15.8628 * FG% + 0.2172 * 3PA + 2.6106 * 3P% + 0.6639 * FTA + 0.7769 * FT% − 0.0066 * ORB − 0.0034 * TRB − 6.2503
LM8PTS = 0.0428 * GS − 0.0003 * MP + 0.917 * FGA + 17.8927 * FG% + 0.3595 * 3PA + 1.7654 * 3P% + 0.6567 * FTA + 0.9399 * FT% − 0.0066 * ORB − 0.0034 * TRB − 8.5359
LM9PTS = 0.018 * GS + 1.2138 * FGA + 18.004 * FG% + 0.4471 * 3PA + 1.971 * 3P% + 0.6639 * FTA + 0.6885 * FT% − 0.0066 * ORB − 0.0398 * TRB − 11.5673
LM10PTS = 0.018 * GS + 0.86 * FGA + 28.3833 * FG% + 0.2731 * 3PA + 3.349 * 3P% + 0.8045 * FTA + 0.0848 * FT% − 0.0066 * ORB − 0.0084 * TRB + 0.0604 * AST − 13.2198
LM11PTS = 0.018 * GS + 0.9991 * FGA + 34.5189 * FG% + 0.1635 * 3PA + 6.3365 * 3P% + 0.8812 * FTA + 0.0848 * FT% − 0.0066 * ORB − 0.0084 * TRB − 18.7932
Table A9. Nine regression equations of the regression tree model of MIL.
Table A9. Nine regression equations of the regression tree model of MIL.
RuleRegression Equation
LM1PTS = 0.1275 * FGA + 0.744 * FG% − 0.0091 * 3PA + 0.3932 * 3P% + 0.2197 * FTA + 0.5097 * FT% − 0.0094 * ORB − 0.0116 * STL + 0.0017 * PF − 0.2317
LM2PTS = 0.1275 * FGA + 0.744 * FG% − 0.0091 * 3PA + 0.3932 * 3P% + 0.1947 * FTA + 0.4477 * FT% − 0.0094 * ORB − 0.0116 * STL + 0.0017 * PF − 0.2315
LM3PTS = 0.1275 * FGA + 0.744 * FG% − 0.0091 * 3PA + 0.3932 * 3P% + 0.5317 * FTA + 1.5753 * FT% − 0.0094 * ORB − 0.0116 * STL + 0.0017 * PF − 0.8865
LM4PTS = 0.1211 * GS + 0.8761 * FGA + 7.1345 * FG% + 0.1493 * 3PA + 1.5771 * 3P% + 0.2827 * FTA + 1.2688 * FT% − 0.0094 * ORB − 0.0116 * STL + 0.0561 * PF − 3.4599
LM5PTS = 1.5315 * FGA + 6.1029 * FG% + 0.5123 * 3PA + 0.8355 * 3P% + 0.3973 * FTA + 1.1581 * FT% − 0.0094 * ORB − 0.0116 * STL + 0.0222 * PF − 5.1388
LM6PTS = 0.6704 * FGA + 16.1226 * FG% + 0.0548 * 3PA + 4.5211 * 3P% + 0.7321 * FTA + 0.4145 * FT% + 0.0239 * ORB + 0.0612 * AST − 0.0107 * STL − 0.0074 * PF − 5.732
LM7PTS = 1.0576 * FGA + 17.4133 * FG% + 0.2653 * 3PA + 3.2639 * 3P% + 0.7026 * FTA + 1.1675 * FT% + 0.0023 * ORB − 0.0696 * AST − 0.1496 * STL − 0.0074 * PF − 9.8628
LM8PTS = 0.9827 * FGA + 32.7185 * FG% + 0.3273 * 3PA + 2.747 * 3P% + 0.5761 * FTA + 1.4136 * FT% + 0.122 * ORB + 0.0165 * DRB − 0.0086 * AST − 0.0107 * STL − 0.0074 * PF − 17.2515
LM9PTS = 1.0699 * FGA + 34.8959 * FG% + 0.3141 * 3PA + 1.5965 * 3P% + 0.7351 * FTA + 7.6551 * FT% − 0.0086 * ORB + 0.0149 * DRB − 0.0086 * AST − 0.0107 * STL − 0.0074 * PF − 24.755
Table A10. Seven regression equations of the regression tree model of OCT.
Table A10. Seven regression equations of the regression tree model of OCT.
RuleRegression Equation
LM1PTS = 0.0413 * GS − 0.0001 * MP + 0.1552 * FGA + 0.8205 * FG% + 0.3668 * 3P% + 0.7021 * FTA + 0.2154 * FT% − 0.0063 * DRB − 0.0047 * AST + 0.0219 * TOV − 0.261
LM2PTS = 0.0413 * GS − 0.0001 * MP + 0.1552 * FGA + 0.8205 * FG% + 0.3668 * 3P% + 0.3342 * FTA + 0.2154 * FT% − 0.0063 * DRB − 0.0047 * AST + 0.0219 * TOV − 0.2706
LM3PTS = 0.0413 * GS − 0.0001 * MP + 1.4643 * FGA + 5.6642 * FG% + 0.1354 * 3PA + 1.5398 * 3P% + 0.3504 * FTA + 1.6443 * FT% − 0.0063 * DRB − 0.0047 * AST + 0.1064 * STL + 0.0216 * TOV − 4.5824
LM4PTS = 0.0413 * GS − 0.0001 * MP + 0.9679 * FGA + 11.2032 * FG% + 0.2884 * 3PA + 1.7817 * 3P% + 0.5292 * FTA + 1.1448 * FT% − 0.0063 * DRB − 0.0047 * AST + 0.0105 * STL + 0.1496 * TOV − 5.9879
LM5PTS = 0.0733 * GS − 0.0008 * MP + 0.8959 * FGA + 24.0439 * FG% + 0.4825 * 3PA + 0.8487 * 3P% + 0.7238 * FTA + 1.4474 * FT% − 0.0343 * ORB − 0.0111 * DRB − 0.0656 * TRB − 0.0163 * AST + 0.0239 * STL − 10.3867
LM6PTS = 0.0733 * GS − 0.0002 * MP + 0.6956 * FGA + 35.8693 * FG% + 0.2857 * 3PA + 3.7184 * 3P% + 0.7753 * FTA + 0.6452 * FT% − 0.3115 * ORB − 0.0111 * DRB − 0.099 * AST + 0.0205 * STL − 12.4609
LM7PTS = 0.0733 * GS − 0.0002 * MP + 0.9346 * FGA + 38.5068 * FG% + 0.4206 * 3PA + 3.3224 * 3P% + 0.8132 * FTA + 0.59 * FT% − 0.0863 * ORB − 0.0971 * DRB − 0.0354 * AST + 0.0205 * STL − 19.0252
Table A11. Six regression equations of the regression tree model of NOP.
Table A11. Six regression equations of the regression tree model of NOP.
RuleRegression Equation
LM1PTS = 0.2048 * FGA + 1.25 * FG% + 0.7276 * 3P% + 0.4824 * FTA + 0.1768 * FT% − 0.0656 * ORB − 0.0031 * DRB − 0.0102 * PF − 0.4649
LM2PTS = −0.0339 * GS + 1.5162 * FGA + 6.3018 * FG% + 0.0185 * 3PA + 1.7818 * 3P% + 0.5911 * FTA + 0.7567 * FT% − 0.0523 * ORB − 0.0031 * DRB + 0.0019 * PF − 4.9028
LM3PTS = −0.2885 * GS + 0.7331 * FGA + 11.9836 * FG% + 0.1983 * 3PA + 2.6125 * 3P% + 0.4927 * FTA + 1.1207 * FT% − 0.0639 * ORB − 0.0097 * DRB + 0.0252 * AST + 0.0126 * TOV − 0.0026 * PF − 4.5582
LM4PTS = −0.0506 * GS + 1.1112 * FGA + 13.576 * FG% + 0.3853 * 3PA + 2.3329 * 3P% + 0.7029 * FTA + 0.2593 * FT% − 0.066 * ORB − 0.0103 * DRB − 0.0247 * AST + 0.0137 * TOV − 0.0026 * PF − 8.2262
LM5PTS = 0.9919 * FGA + 25.6773 * FG% + 0.2983 * 3PA + 2.7463 * 3P% + 0.6605 * FTA + 1.2326 * FT% − 0.0118 * DRB − 0.1516 * STL − 0.0327 * PF − 13.4103
LM6PTS = 0.0013 * MP + 0.9668 * FGA + 39.4122 * FG% + 0.3625 * 3PA + 0.8249 * 3P% + 0.7043 * FTA + 1.8209 * FT% − 0.0169 * DRB − 0.0445 * PF − 22.6992

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Figure 1. Research flowchart.
Figure 1. Research flowchart.
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Figure 2. Regression tree diagram of GSW.
Figure 2. Regression tree diagram of GSW.
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Figure 3. Regression tree rules derived through Excel.
Figure 3. Regression tree rules derived through Excel.
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Figure 4. Predictions of GSW total team scores on 11 March 2018.
Figure 4. Predictions of GSW total team scores on 11 March 2018.
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Figure 5. GSW actual and predicted score (left) vs. opponent’s actual and predicted score (right).
Figure 5. GSW actual and predicted score (left) vs. opponent’s actual and predicted score (right).
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Table 1. Variables for each field in the database.
Table 1. Variables for each field in the database.
ItemsVariable NameAbbreviationWhether to Choose
1Player Uniform NumberNo.X
2RankRkX
3Season GameGX
4DateDateX
5AgeAgeX
6TeamTmX
7Home/AwayH/AX
8OpponentOppX
9Games StartedGSV
10Minutes PlayedMPV
11Field GoalsFGX
12Field Goal AttemptsFGAV
13Field Goal PercentageFG%V
143-Point Field Goals3PX
153-Point Field Goal Attempts3PAV
163-Point Field Goal Percentage3P%V
17Free ThrowsFTX
18Free Throw AttemptsFTAV
19Free Throw PercentageFT%V
20Offensive ReboundsORBV
21Defensive ReboundsDRBV
22Total ReboundsTRBV
23AssistsASTV
24StealsSTLV
25BlocksBLKV
26TurnoversTOVV
27Personal FoulsPFV
28PointsPTSV
(Output)
29Game ScoreGmScX
30Plus/Minus+/−X
Table 2. Opponent data corresponding to GSW test set data.
Table 2. Opponent data corresponding to GSW test set data.
Season GameDateOpponentShort Name
672018/3/11Minnesota TimberwolvesMIN
682018/3/14LA LakersLAL
692018/3/16Sacramento KingsSAC
702018/3/17Phoenix SunsPHX
712018/3/19SA SpursSAS
722018/3/23Atlanta HawksATL
732018/3/25Utah JazzUTH
742018/3/27Indiana PacersIND
752018/3/29Milwaukee BucksMIL
762018/3/31Sacramento KingsSAC
772018/4/1Phoenix SunsPHX
782018/4/3Oklahoma City ThunderOCT
792018/4/5Indiana PacersIND
802018/4/7New Orleans PelicansNOP
812018/4/8Phoenix SunsPHX
822018/4/10Utah JazzUTH
Table 3. Comparison of predictive performance of regression models, using GSW data.
Table 3. Comparison of predictive performance of regression models, using GSW data.
Regression ModelsModel Building Time (Seconds)RMSE
Regression tree (M5P)0.2100.9645
Linear regression0.0011.3081
Support vector regression0.3102.4904
Table 4. Six regression equations for regression tree model for prediction of GSW scores.
Table 4. Six regression equations for regression tree model for prediction of GSW scores.
RuleRegression Equation
LM1PTS = 0.0696 * GS − 0.0003 * MP + 0.4028 * FGA + 0.8397 * FG% − 0.0269 * 3PA + 0.5822 * 3P% + 0.3586 * FTA + 1.5292 * FT% − 0.0075 * ORB + 0.0149 * DRB − 0.0168 * TRB − 0.018 * BLK − 0.0056 * PF − 0.4345
LM2PTS = 0.0375 * GS − 0.0001 * MP + 1.5187 * FGA + 5.7312 * FG% − 0.0045 * 3PA + 2.1884 * 3P% + 0.5562 * FTA + 0.8675 * FT% − 0.0075 * ORB − 0.0407 * DRB − 0.0048 * TRB − 0.0156 * BLK − 0.0056 * PF − 4.5329
LM3PTS = 0.0375 * GS − 0.0003 * MP + 1.1779 * FGA + 11.5318 * FG% + 0.1554 * 3PA + 1.7568 * 3P% + 0.5515 * FTA + 0.8305 * FT% − 0.0075 * ORB + 0.0399 * DRB − 0.0581 * TRB + 0.0477 * AST − 0.0837 * BLK + 0.0804 * TOV − 0.0056 * PF − 6.7484
LM4PTS = 0.3764 * GS − 0.0004 * MP + 1.0899 * FGA + 22.5574 * FG% + 0.3394 * 3PA + 3.1752 * 3P% + 0.755 * FTA + 1.0507 * FT% − 0.0176 * ORB − 0.0132 * PF − 13.0728
LM5PTS = 0.2379 * GS − 0.0007 * MP + 0.8717 * FGA + 32.494 * FG% + 0.1633 * 3PA + 6.1986 * 3P% + 0.9576 * FTA + 0.0606 * FT% − 0.0176 * ORB − 0.0954 * DRB − 0.0132 * PF − 13.2189
LM6PTS = 0.2379 * GS − 0.0015 * MP + 1.0394 * FGA + 32.1141 * FG% + 0.4884 * 3PA + 5.535 * 3P% + 0.8918 * FTA + 0.5658 * FT% − 0.0176 * ORB − 0.0132 * PF − 17.2302
Table 5. Final prediction of test set.
Table 5. Final prediction of test set.
Season GameDataGSW PTSOpponentsOpponents PTSGSW WIN/LOSE
ActualPredictedActualPredictedActualPredicted
672018/3/11103108MIN109106LOSEWIN
682018/3/14117118LAL106106WINWIN
692018/3/169396SAC98100LOSELOSE
702018/3/17124122PHX109111WINWIN
712018/3/197578SAS8986LOSELOSE
722018/3/23106104ATL9499WINWIN
732018/3/259186UTH110109LOSELOSE
742018/3/278181IND9293LOSELOSE
752018/3/29107107MIL116114LOSELOSE
762018/3/31112115SAC9697WINWIN
772018/4/1117114PHX107114WINTIE
782018/4/3111106OCT107104WINWIN
792018/4/5106107IND126131LOSELOSE
802018/4/7120116NOP126129LOSELOSE
812018/4/8117117PHX100100WINWIN
822018/4/107981UTH119119LOSELOSE
Accuracy: 87.5%
Note: Bold font represents that the predicted game result is different from the actual game result.
Table 6. Comparison with results in the relevant literature.
Table 6. Comparison with results in the relevant literature.
Author, YearFeaturesMethodAccuracy
Miljković et al., 2010 [35]FG, FGA, FG%, 3P, 3PA, 3PA%, FT, FTA, FT%, ORB, DRB, TRB, AST, STL, BLK, TOV, PF, PTS, W, L, Pct, Homewon, Homelost, Roadwon, Roadlost, Divwon, Divlost, confwon, conflost, streak, L10won, L10lostNaive Bayes
Decision tree
KNN
SVM
67.00%
Cao, 2012 [18]G, Opp, MP, FG, FGA, 3P, 3PA, FT, FTA, ORB, TRB, AST, STL, BLK, TOV, PF, PTS, PER, TS%, eFG%, ORB%, DRB%, TRB%, AST%, STL%, BLK%, TOV%, USG%, Ortg, DRtg, OWS, DWS, WS, WS/48Logistic regression
ANN
SVM
Naive Bayes
69.67%
Wheeler, 2012 [22]Pace, OPTS, OFG%, OTOR, DRB%, O%Rim, O%Short, OXeFG%, OeFG%, TRB%, MP, FGA, FTA, eFG%, TS%, ORtgLinear regression47.00%
Cheng et al., 2016 [36]FG, FGA, 3P, 3PA, FT, FTA, ORB, DRB, AST, STL, BLK, TOV, PF, PTSNBAME model
(Using the principle of maximum entropy)
74.40%
Pai et al., 2016 [37]2P%, 3P%, FT, DRB, TRB, STL, ASTHSVMDT
(SVM + decision rules)
85.25%
Kaur and Jain, 2017 [38]FG, FGA, FG%, 3P, 3PA%, FT, FT%, DRB, TRB, AST, TOV, PF, PTS, TS%, eFG%, ORB%, TRB%, BLK%, TOV%, ORtg, DRtgHFSVM model
(SVM + fuzzy rules)
88.26%
This studyGS, MP(s), FGA, FG%, 3PA, 3P%, FTA, FT%, ORB, DRB, TRB, AST, STL, BLK, TOV, PFRegression tree (M5P)87.50%
Note: The accuracy listed in this table was the highest one in each related study from the method in the bold font.

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Huang, M.-L.; Lin, Y.-J. Regression Tree Model for Predicting Game Scores for the Golden State Warriors in the National Basketball Association. Symmetry 2020, 12, 835. https://doi.org/10.3390/sym12050835

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Huang M-L, Lin Y-J. Regression Tree Model for Predicting Game Scores for the Golden State Warriors in the National Basketball Association. Symmetry. 2020; 12(5):835. https://doi.org/10.3390/sym12050835

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Huang, Mei-Ling, and Yi-Jung Lin. 2020. "Regression Tree Model for Predicting Game Scores for the Golden State Warriors in the National Basketball Association" Symmetry 12, no. 5: 835. https://doi.org/10.3390/sym12050835

APA Style

Huang, M. -L., & Lin, Y. -J. (2020). Regression Tree Model for Predicting Game Scores for the Golden State Warriors in the National Basketball Association. Symmetry, 12(5), 835. https://doi.org/10.3390/sym12050835

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