# Predicting the Outcome of NBA Playoffs Based on the Maximum Entropy Principle

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Maximum Entropy Model

#### 2.2. K-Means Clustering

- Let $\{{x}_{1},{x}_{2},\xb7\xb7\xb7,{x}_{n}\}$ be the set of data points and $V=\{{v}_{1},{v}_{2},\xb7\xb7\xb7,{v}_{c}\}$ be the set of centers;
- Randomly select “c” cluster centers and calculate the distance between each data point and cluster centers;
- Assign the data point to the cluster center whose distance from the cluster center is the minimum of all the cluster centers;
- Recalculate the new cluster center using: ${v}_{i}=(1/{c}_{i}){\sum}_{j=1}^{{c}_{i}}{x}_{i}$, where ${c}_{i}$ represents the number of data points in ith cluster;
- Recalculate the distance between each data point and new obtained cluster centers;
- If no data point was reassigned, then stop; otherwise, repeat from step 3.

## 3. Materials and Methods

#### 3.1. Basic Technical Features

#### 3.2. NBAME Model Overview

- $p(y|x)\ge 0$ for all $x,y$;
- ${\sum}_{y}p(y|x)=1$ for all x;
- ${\sum}_{(x,y)}\tilde{p}(x,y){f}_{k}(x,y)={\sum}_{(x,y)}\tilde{p}(x)p(y|x){f}_{k}(x,y)$ for $k\in \{1,2,\dots ,K\}$.

## 4. Results

#### 4.1. Data Collection and Preprocessing

#### 4.2. The Results of the NBAME Model for Predicting the NBA Playoffs

#### 4.3. Comparison of NBAME Model with Some Selected Existing Machine Learning Algorithms

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**The number and accuracy of predictions with different confidence by the NBAME model from the 2007–08 season to the 2014–15 season playoffs.

**Figure 4.**ROC curves and AUC values of prediction using the NBAME model from the 2007–08 season to the 2014–15 season playoffs.

Feature | Abbreviation | Feature | Abbreviation |
---|---|---|---|

Field Goal Made | FGM | Field Goal Attempt | FGA |

Three Point Made | 3PM | Three Point Attempt | 3PA |

Free Throw Made | FTM | Free Throw Attempt | FTA |

Offensive Rebounds | Oreb | Defensive Rebounds | Dreb |

Assists | Ast | Steals | Stl |

Blocks | Blk | Turnover | TO |

Personal Fouls | PF | Points | PTS |

**Table 2.**Sample features’ raw values obtained from http://www.stat-nba.com/ website.

Features | FGM | FGA | 3PM | 3PA | FM | FTA | Oreb | Dreb | Ast | Stl | Blk | TO | PF | PTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Features’ | 32 | 79 | 6 | 24 | 18 | 24 | 8 | 28 | 17 | 10 | 2 | 18 | 15 | 88 |

values of | 45 | 87 | 9 | 24 | 8 | 11 | 5 | 32 | 32 | 8 | 3 | 14 | 23 | 107 |

last | 33 | 85 | 7 | 23 | 22 | 29 | 9 | 36 | 22 | 10 | 4 | 12 | 21 | 95 |

six games | 33 | 83 | 6 | 23 | 12 | 15 | 14 | 28 | 22 | 6 | 4 | 15 | 18 | 84 |

for | 48 | 85 | 8 | 23 | 10 | 14 | 12 | 31 | 29 | 9 | 6 | 13 | 20 | 114 |

home team | 44 | 80 | 7 | 19 | 14 | 18 | 7 | 35 | 25 | 9 | 8 | 14 | 16 | 109 |

Average | 39.17 | 83.17 | 7.17 | 22.67 | 14.00 | 18.50 | 9.17 | 31.67 | 24.50 | 8.67 | 4.50 | 14.33 | 18.83 | 99.50 |

**Table 3.**Sample records of the experimental dataset obtained by getting averages of the previous six games.

Home teams’ features | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$FG{M}_{h}$ | $FG{A}_{h}$ | $3P{M}_{h}$ | $3P{A}_{h}$ | $FT{M}_{h}$ | $FT{A}_{h}$ | $Ore{b}_{h}$ | $Dre{b}_{h}$ | $As{t}_{h}$ | $St{l}_{h}$ | $Bl{k}_{h}$ | $T{O}_{h}$ | $P{F}_{h}$ | $PT{S}_{h}$ | ||

39.17 | 83.17 | 7.17 | 22.67 | 14.00 | 18.50 | 9.17 | 31.67 | 24.50 | 8.67 | 4.50 | 14.33 | 18.83 | 99.50 | ||

38.33 | 83.67 | 6.83 | 18.00 | 12.83 | 18.33 | 7.83 | 35.00 | 23.67 | 7.00 | 6.17 | 12.83 | 22.00 | 96.33 | ||

37.50 | 84.67 | 10.67 | 26.50 | 18.83 | 25.33 | 10.83 | 32.83 | 24.33 | 8.50 | 6.17 | 12.17 | 19.67 | 104.50 | ||

37.17 | 79.67 | 8.50 | 25.67 | 17.17 | 23.00 | 9.33 | 29.50 | 23.50 | 7.50 | 4.50 | 15.00 | 18.33 | 100.00 | ||

37.83 | 85.50 | 11.50 | 33.33 | 16.67 | 23.83 | 11.83 | 31.83 | 22.00 | 9.67 | 3.33 | 16.50 | 22.00 | 103.83 | ||

39.00 | 78.50 | 7.50 | 20.83 | 16.33 | 20.67 | 7.67 | 31.17 | 24.00 | 7.17 | 3.83 | 16.33 | 21.33 | 101.83 | ||

40.67 | 88.17 | 6.83 | 19.33 | 17.33 | 24.67 | 13.83 | 36.00 | 20.33 | 7.17 | 6.00 | 12.83 | 21.17 | 105.50 | ||

Away Teams’ Features | Home Team | ||||||||||||||

$FG{M}_{a}$ | $FG{A}_{a}$ | $3P{M}_{a}$ | $3P{A}_{a}$ | $FT{M}_{a}$ | $FT{A}_{a}$ | $Ore{b}_{a}$ | $Dre{b}_{a}$ | $As{t}_{a}$ | $St{l}_{a}$ | $Bl{k}_{a}$ | $T{O}_{a}$ | $P{F}_{a}$ | $PT{S}_{a}$ | Win | |

41.00 | 82.33 | 7.50 | 18.33 | 21.00 | 27.83 | 10.33 | 31.17 | 22.17 | 6.67 | 4.00 | 16.33 | 22.17 | 110.50 | 1 | |

36.67 | 75.33 | 7.17 | 20.17 | 17.33 | 23.50 | 8.33 | 28.83 | 19.33 | 8.83 | 3.67 | 13.50 | 21.17 | 97.83 | 1 | |

38.00 | 87.00 | 5.83 | 19.00 | 17.17 | 21.17 | 12.67 | 28.33 | 21.50 | 7.50 | 5.00 | 12.33 | 20.67 | 99.00 | 1 | |

38.33 | 80.33 | 9.17 | 23.00 | 15.33 | 20.00 | 7.50 | 32.83 | 24.67 | 8.00 | 5.83 | 17.83 | 23.17 | 101.17 | 0 | |

35.83 | 85.33 | 7.50 | 22.50 | 18.33 | 24.33 | 10.83 | 35.33 | 19.17 | 7.17 | 5.50 | 11.83 | 19.17 | 97.50 | 1 | |

37.33 | 85.17 | 5.33 | 17.33 | 16.67 | 21.00 | 11.50 | 32.33 | 21.00 | 7.17 | 5.33 | 12.17 | 16.33 | 96.67 | 1 | |

41.67 | 86.67 | 10.17 | 25.17 | 17.17 | 22.50 | 12.17 | 31.33 | 20.67 | 8.33 | 6.00 | 12.50 | 19.17 | 110.67 | 1 |

Home Teams’ Features | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$FG{M}_{h}$ | $FG{A}_{h}$ | $3P{M}_{h}$ | $3P{A}_{h}$ | $FT{M}_{h}$ | $FT{A}_{h}$ | $Ore{b}_{h}$ | $Dre{b}_{h}$ | $As{t}_{h}$ | $St{l}_{h}$ | $Bl{k}_{h}$ | $T{O}_{h}$ | $P{F}_{h}$ | $PT{S}_{h}$ | ||

37.63 | 83.31 | 7.85 | 22.66 | 14.35 | 19.02 | 9.50 | 31.68 | 24.26 | 8.82 | 4.18 | 14.49 | 18.65 | 98.6 | ||

37.63 | 83.31 | 7.85 | 18.94 | 14.35 | 19.02 | 7.17 | 35.01 | 24.26 | 6.94 | 6.50 | 12.42 | 22.07 | 94.48 | ||

37.63 | 84.75 | 10.74 | 26.11 | 17.80 | 25.44 | 11.27 | 32.59 | 24.26 | 8.82 | 6.50 | 12.42 | 19.87 | 106.59 | ||

37.63 | 80.20 | 7.85 | 26.11 | 17.80 | 22.35 | 9.50 | 29.60 | 22.97 | 7.90 | 4.18 | 15.14 | 18.65 | 98.60 | ||

37.63 | 86.12 | 10.74 | 34.05 | 17.80 | 23.80 | 12.30 | 31.68 | 21.97 | 9.64 | 3.36 | 16.09 | 22.07 | 102.39 | ||

37.63 | 77.95 | 7.85 | 20.87 | 17.80 | 20.76 | 7.17 | 30.77 | 24.26 | 6.94 | 4.18 | 16.09 | 20.98 | 102.39 | ||

40.85 | 87.78 | 7.85 | 18.94 | 17.80 | 25.44 | 13.39 | 36.94 | 19.96 | 6.94 | 5.66 | 12.42 | 20.98 | 106.59 | ||

Away Teams’ Features | Home Team | ||||||||||||||

$FG{M}_{a}$ | $FG{A}_{a}$ | $3P{M}_{a}$ | $3P{A}_{a}$ | $FT{M}_{a}$ | $FT{A}_{a}$ | $Ore{b}_{a}$ | $Dre{b}_{a}$ | $As{t}_{a}$ | $St{l}_{a}$ | $Bl{k}_{a}$ | $T{O}_{a}$ | $P{F}_{a}$ | $PT{S}_{a}$ | Win | |

40.36 | 82.6 | 7.77 | 18.40 | 20.61 | 26.54 | 10.34 | 32.16 | 22.40 | 6.59 | 4.08 | 16.19 | 21.65 | 108.43 | 1 | |

36.76 | 73.84 | 7.77 | 21.63 | 16.85 | 24.42 | 8.26 | 28.68 | 19.34 | 9.00 | 3.76 | 13.17 | 21.65 | 100.06 | 1 | |

37.73 | 86.81 | 5.39 | 18.40 | 16.85 | 20.34 | 12.48 | 28.68 | 21.66 | 7.76 | 5.15 | 12.42 | 20.45 | 100.06 | 1 | |

38.88 | 79.5 | 10.49 | 21.63 | 15.58 | 20.34 | 7.59 | 32.16 | 24.13 | 7.76 | 5.71 | 17.36 | 22.79 | 100.06 | 0 | |

35.66 | 85.13 | 7.77 | 21.63 | 17.90 | 24.42 | 10.83 | 35.50 | 19.34 | 7.18 | 5.71 | 11.74 | 19.09 | 100.06 | 1 | |

37.73 | 85.13 | 5.39 | 18.40 | 16.85 | 20.34 | 11.48 | 32.16 | 20.70 | 7.18 | 5.15 | 12.42 | 16.32 | 100.06 | 1 | |

42.48 | 86.81 | 10.49 | 24.73 | 16.85 | 22.41 | 12.48 | 32.16 | 20.70 | 8.40 | 5.71 | 12.42 | 19.09 | 108.43 | 1 |

Threshold | 2007–08 | 2008–09 | 2009–10 | 2010–11 | 2011–12 | 2012–13 | 2013–14 | 2014–15 |
---|---|---|---|---|---|---|---|---|

0.5 | 74.4 | 68.2 | 68.3 | 66.7 | 69.0 | 67.1 | 65.2 | 62.5 |

0.6 | 77.1 | 74.5 | 75.0 | 69.8 | 73.0 | 71.4 | 66.7 | 70.4 |

0.7 | 100.0 | 80.0 | 100.0 | 100.0 | 100.0 | 75.0 | 100.0 | 100.0 |

Threshold | 2007–08 | 2008–09 | 2009–10 | 2010–11 | 2011–12 | 2012–13 | 2013–14 | 2014–15 |
---|---|---|---|---|---|---|---|---|

0.5 | 86 | 85 | 82 | 81 | 84 | 85 | 89 | 80 |

0.6 | 48 | 55 | 44 | 53 | 26 | 42 | 36 | 27 |

0.7 | 3 | 5 | 2 | 0 | 1 | 4 | 1 | 6 |

**Table 7.**Prediction accuracy (in percentages) of selected algorithms for NBA playoffs for seasons between 2007 and 2015.

Algorithm | 2007–08 | 2008–09 | 2009–10 | 2010–11 | 2011–12 | 2012–13 | 2013–14 | 2014–15 |
---|---|---|---|---|---|---|---|---|

Naive Bayes | 54.7 | 61.5 | 56.1 | 59.3 | 53.6 | 58.8 | 59.3 | 55.0 |

Logistic Regression | 61.6 | 57.1 | 61.0 | 61.7 | 60.7 | 64.7 | 62.6 | 60.0 |

BP Neural Networks | 59.3 | 60.4 | 52.4 | 67.9 | 56.0 | 63.5 | 57.1 | 57.5 |

Random Forest | 64.0 | 60.4 | 64.6 | 64.2 | 58.3 | 70.6 | 62.6 | 56.3 |

NBAME model | 74.4 | 68.2 | 68.3 | 66.7 | 69.0 | 67.1 | 65.2 | 62.5 |

**Table 8.**AUC (in percentages) values of selected algorithms for NBA playoffs for seasons between 2007 and 2015.

Algorithm | 2007–08 | 2008–09 | 2009–10 | 2010–11 | 2011–12 | 2012–13 | 2013–14 | 2014–15 |
---|---|---|---|---|---|---|---|---|

Naive Bayes | 50.0 | 61.6 | 51.9 | 55.6 | 51.6 | 61.2 | 59.4 | 54.7 |

Logistic Regression | 51.8 | 61.7 | 53.2 | 56.4 | 51.9 | 63.1 | 58.7 | 59.6 |

BP Neural Networks | 50.6 | 56.0 | 52.8 | 61.1 | 51.2 | 66.0 | 58.5 | 54.6 |

Random Forest | 51.8 | 58.3 | 50.5 | 50.8 | 52.4 | 66.7 | 59.0 | 58.3 |

NBAME model | 57.2 | 62.3 | 54.1 | 61.7 | 52.9 | 61.7 | 57.9 | 60.4 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Cheng, G.; Zhang, Z.; Kyebambe, M.N.; Kimbugwe, N.
Predicting the Outcome of NBA Playoffs Based on the Maximum Entropy Principle. *Entropy* **2016**, *18*, 450.
https://doi.org/10.3390/e18120450

**AMA Style**

Cheng G, Zhang Z, Kyebambe MN, Kimbugwe N.
Predicting the Outcome of NBA Playoffs Based on the Maximum Entropy Principle. *Entropy*. 2016; 18(12):450.
https://doi.org/10.3390/e18120450

**Chicago/Turabian Style**

Cheng, Ge, Zhenyu Zhang, Moses Ntanda Kyebambe, and Nasser Kimbugwe.
2016. "Predicting the Outcome of NBA Playoffs Based on the Maximum Entropy Principle" *Entropy* 18, no. 12: 450.
https://doi.org/10.3390/e18120450