Power method, colley's method를 이용한 sports ranking 보고서 (선형대수학)(영재고생)

문서 내 토픽

• 1. Sports Ranking Methods

  스포츠 경기에서 리그전이 이루어질 때, 일반적으로 단순히 승률을 통해 순위를 나열하는 방식을 주로 사용하지만 여러 선형대수적 방법들을 통해 조금 더 타당한 순위를 도출할 수 있다. 본 보고서는 리그 경기에서 순위를 매기는 여러 선형대수적 방법들을 소개하고, 이를 확장하여 한국 프로농구(KBL) 경기 전적에 적용해보고 새로운 순위표를 작성하였다.

• 2. Power Method

  Power method는 행렬의 고유벡터를 이용해 순위를 매기는 방법이다. 각 팀의 probability r_i를 정의하여, 이 값이 클수록 팀의 순위가 높도록 정의한다. r_i값들의 합은 1이 되도록 하며, i번째 팀이 경기에서 이겼던 팀들의 r값들의 합에 비례하도록 r_i값을 정의한다.

• 3. Colley's Method

  Colley's method는 Wes Colley가 제안한 팀 랭킹 매기는 새로운 방법이다. 기본적인 아이디어는 각 팀의 순위를 최종적으로 매길 때 사용하는 랭킹 점수 r_i를 r_i = (w_i - l_i)/2 + sum(r_j)/(t_i+2)와 같이 정의하는 것이다. 이때 w_i는 i번째 팀이 이긴 횟수, l_i는 i번째 팀이 패배한 횟수, t_i는 i번째 팀이 치른 총 경기 횟수이다.

• 4. 실제 데이터 적용

  보고서에서는 KBL 프로농구 22-23 시즌 1라운드 결과와 21-22 전체 시즌 결과 데이터를 사용하여 Power method와 Colley's method를 적용하였다. 각 방법을 통해 도출된 순위와 실제 KBL 규정을 통해 매긴 순위를 비교하였으며, 두 방법이 동일한 순위를 나타내는 것을 확인하였다.

• 5. Power Method와 Colley's Method의 관계

  보고서에서는 Power method에서 A를 계속 제곱하여 사용한 결과가 Colley's method의 결과와 수렴한다는 것을 보였다. 이를 통해 두 방법이 근본적으로 유사한 접근법을 사용하고 있음을 확인하였다.

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• 1. Sports Ranking Methods

  Sports ranking methods are an important topic in the field of sports analytics. These methods aim to objectively evaluate and compare the performance of teams or players in a given sport. Some common ranking methods include the Power Method, Colley's Method, and the Massey Method, each with its own strengths and weaknesses. The choice of ranking method can have a significant impact on the final rankings, and it is important to understand the underlying assumptions and mathematical principles behind these methods. Ultimately, the goal of sports ranking methods is to provide a more accurate and unbiased assessment of team or player performance, which can inform decision-making in areas such as team selection, tournament seeding, and fan engagement.

• 2. Power Method

  The Power Method is a widely used sports ranking algorithm that assigns a numerical score to each team based on their performance against other teams. The method works by iteratively updating the scores of each team based on the scores of their opponents, with the assumption that a team's score is proportional to the sum of their opponents' scores. This approach tends to favor teams that have played against stronger opponents, as their scores will be higher than teams that have played against weaker opponents. The Power Method is relatively simple to implement and has been successfully applied to a variety of sports, including college football, basketball, and soccer. However, it does have some limitations, such as its sensitivity to the initial starting values and its inability to account for factors such as home-field advantage or player injuries.

• 3. Colley's Method

  Colley's Method is another popular sports ranking algorithm that takes a different approach from the Power Method. Instead of iteratively updating team scores, Colley's Method uses a system of linear equations to directly calculate the ranking of each team. The method is based on the principle that a team's ranking should be proportional to the difference between the number of games they have won and the number of games they have lost. Colley's Method is known for its simplicity and stability, as it is less sensitive to changes in the underlying data than the Power Method. Additionally, Colley's Method can be used to rank teams in sports with different numbers of games played, making it a versatile tool for sports analytics. However, some critics argue that Colley's Method may not capture certain nuances of team performance, such as the strength of opponents faced or the margin of victory.

• 4. Relationship between Power Method and Colley's Method

  The Power Method and Colley's Method are two of the most widely used sports ranking algorithms, and understanding the relationship between them is important for sports analytics. While the two methods have different mathematical foundations, they share some similarities in their underlying principles and the information they use to generate team rankings. Both the Power Method and Colley's Method rely on the concept of team performance, as measured by wins and losses, to determine the relative strength of teams. However, the Power Method places more emphasis on the strength of a team's opponents, while Colley's Method focuses more on the net difference between a team's wins and losses. Interestingly, research has shown that the rankings generated by the Power Method and Colley's Method are often highly correlated, suggesting that the two methods may be capturing similar aspects of team performance. This relationship can be particularly useful when analyzing sports data, as it allows researchers to cross-validate their findings and gain a more comprehensive understanding of team dynamics. At the same time, the differences between the two methods can also provide valuable insights. For example, in situations where the Power Method and Colley's Method produce significantly different rankings, it may indicate the presence of factors that are not being adequately accounted for by one or both of the methods, such as home-field advantage, player injuries, or other contextual factors. By exploring these discrepancies, researchers can refine and improve the sports ranking algorithms, ultimately leading to more accurate and reliable assessments of team performance.