Sports Ranking and Scheduling with Kalman Filters Peter Elliott, Matt Owen, Kyle Shan, Yingjie Yu
Introduction
Introduction
Introduction Definition (Rating) A rating is a numerical evaluation of an item. In sports rating systems, higher rated teams perform better.
Introduction Definition (Rating) A rating is a numerical evaluation of an item. In sports rating systems, higher rated teams perform better. Definition (Ranking) A ranking is an ordering of items.
Introduction Definition (Rating) A rating is a numerical evaluation of an item. In sports rating systems, higher rated teams perform better. Definition (Ranking) A ranking is an ordering of items. It is easy to turn ratings into rankings
Introduction Definition (Game) A game between two teams is a pairwise comparison which yields a real number outcome
Introduction Definition (Game) A game between two teams is a pairwise comparison which yields a real number outcome Definition (Schedule) A schedule is a matching of teams to play against each other in each round.
Introduction Definition (Game) A game between two teams is a pairwise comparison which yields a real number outcome Definition (Schedule) A schedule is a matching of teams to play against each other in each round. Goal: schedule games to find the most “informative” ranking.
Extended Kalman Filter Ranking Underlying Model Assume that games follow the Bradley-Terry paired comparison model: ◮ Teams 1 , . . . , n have skill levels x 1 , . . . , x n ◮ The expected outcome in a game between teams i and j is 1 h ( x i , x j ) = 1 + 10 − ( x i − x j ) /ξ Ratings can be computed using maximum likelihood, but this is computationally difficult.
Extended Kalman Filter Ranking Applying a Filter to Bradley-Terry Start with the system x k = x k − 1 = x ∗ y k = h k ( x k ) + ǫ k where { ǫ k } is an uncorrelated random process, and the predicted outcomes h k ( x k ) of the games in each round are given by the Bradley-Terry prediction.
Extended Kalman Filter Ranking Applying a Filter to Bradley-Terry Start with the system x k = x k − 1 = x ∗ y k = h k ( x k ) + ǫ k where { ǫ k } is an uncorrelated random process, and the predicted outcomes h k ( x k ) of the games in each round are given by the Bradley-Terry prediction. Then, we get the simple update equations: P k = ( P † k − 1 + σ − 2 H T k H k ) † x k − 1 − σ − 2 P k H T x k = ˆ ˆ k ( h k (ˆ x k − 1 ) − y k ) where P k is the covariance of ˆ x k and H k = Dh k ( x k − 1 ).
Extended Kalman Filter Ranking Similarity to Elo Elo’s method seems a lot like the EKF update: Elo: x i ← x i − K ( h ( x i , x j ) − y k ) x − σ − 2 P k H T EKF: ˆ x ← ˆ k ( h k (ˆ x k − 1 ) − y k ) where K is a constant.
Extended Kalman Filter Ranking Similarity to Elo Elo’s method seems a lot like the EKF update: Elo: x i ← x i − K ( h ( x i , x j ) − y k ) x − σ − 2 P k H T EKF: ˆ x ← ˆ k ( h k (ˆ x k − 1 ) − y k ) where K is a constant. EKF ranking has several advantages over Elo.
Scheduling Introduction Definition (Fisher Information) The Fisher Information matrix is the inverse of the covariance matrix of the state estimate. I = P † Schedule Design Goal: Choose matchups that maximize the Fisher Information.
Scheduling Introduction Definition (Fisher Information) The Fisher Information matrix is the inverse of the covariance matrix of the state estimate. I = P † Schedule Design Goal: Choose matchups that maximize the Fisher Information. Notions of Fisher optimality: ◮ T-optimality: maximize the trace of the Fisher Information matrix ◮ Any monotone function of the eigenvalues
Scheduling Introduction Complete schedules for tournaments are often created before a tournament begins. This is known as static scheduling . Dynamic scheduling refers to creating a schedule as the tournament occurs, taking into account the results as they happen. Goal: dynamically schedule one game for each team, for each round of play, in a way that maximizes information of the rating.
Scheduling Using Kalman Filters The Kalman Filter update equations give a strategy for dynamic scheduling. P k = ( P † k − 1 + σ − 2 H T k H k ) † In terms of Fisher Information: I k = I k − 1 + σ − 2 H T k H k Goal: Choose games at each time step that maximize I k
Scheduling Using Kalman Filters The Kalman Filter update equations give a strategy for dynamic scheduling. P k = ( P † k − 1 + σ − 2 H T k H k ) † In terms of Fisher Information: I k = I k − 1 + σ − 2 H T k H k Goal: Choose games at each time step that maximize I k Important features: ◮ Maximizing tr ( I k ) is equivalent to maximizing tr ( H T k H k ) ◮ If team i plays a single game, the i th entry on the diagonal of H T k H k is a function of the previous ratings of team i and its opponent
Scheduling A New Dynamic Scheduling Method Our proposed method: ◮ For each pair of teams i and j , calculate the potential information gain
Scheduling A New Dynamic Scheduling Method Our proposed method: ◮ For each pair of teams i and j , calculate the potential information gain ◮ Create a complete weighted graph G , with edge weight between i and j corresponding to the potential information gain
Scheduling A New Dynamic Scheduling Method Our proposed method: ◮ For each pair of teams i and j , calculate the potential information gain ◮ Create a complete weighted graph G , with edge weight between i and j corresponding to the potential information gain ◮ Choose the maximum weighted perfect matching on G
Scheduling A New Dynamic Scheduling Method Our proposed method: ◮ For each pair of teams i and j , calculate the potential information gain ◮ Create a complete weighted graph G , with edge weight between i and j corresponding to the potential information gain ◮ Choose the maximum weighted perfect matching on G ◮ The weight of the matching is exactly tr ( H T k H k )
Scheduling A New Dynamic Scheduling Method Our proposed method: ◮ For each pair of teams i and j , calculate the potential information gain ◮ Create a complete weighted graph G , with edge weight between i and j corresponding to the potential information gain ◮ Choose the maximum weighted perfect matching on G ◮ The weight of the matching is exactly tr ( H T k H k ) Edmonds’ Blossom Algorithm (1965) solves the perfect matching problem in polynomial time.
Scheduling A New Dynamic Scheduling Method Our proposed method: ◮ For each pair of teams i and j , calculate the potential information gain ◮ Create a complete weighted graph G , with edge weight between i and j corresponding to the potential information gain ◮ Choose the maximum weighted perfect matching on G ◮ The weight of the matching is exactly tr ( H T k H k ) Edmonds’ Blossom Algorithm (1965) solves the perfect matching problem in polynomial time. This method is trace-optimal for EKF ranking.
Scheduling Trace-Optimality Graph
Scheduling Effect of Optimal Scheduling
Scheduling Heatmap All Games Played
EKF Convergence
Linear Kalman Filter Ranking Introduction Consider a system where the expected outcome in a game between teams i and j in round k is simply the difference of their skills: h k ( x k ) = x i k − x j k
Linear Kalman Filter Ranking Introduction Consider a system where the expected outcome in a game between teams i and j in round k is simply the difference of their skills: h k ( x k ) = x i k − x j k One way to find ratings is using least squares to solve x � Hx − y � 2 ˆ x = arg min where y i is team i ’s cumulative point differential, and H is a matrix where each row represents a game between two teams i and j , with 1 in the winner’s slot and -1 in the loser’s slot.
Linear Kalman Filter Ranking Linear System The Kalman Filter is an online algorithm solving the same problem. Assume a linear system for game results: x k = x k − 1 = x ∗ y k = Hx ∗ + ǫ k where H is defined as before, and { ǫ k } is a Gaussian white noise process with variance σ 2 .
Linear Kalman Filter Ranking Update Equations Applying the Kalman Filter yields P k = ( P † k − 1 + σ − 2 H T H ) † x k − 1 − σ − 2 P k H T ( H ˆ ˆ x k = ˆ x k − 1 − y k ) . We use the initial values of P 0 = ∞ and ˆ x 0 = ¯ x where ¯ x is the average rating.
Linear Kalman Filter Ranking Update Equations Applying the Kalman Filter yields P k = ( P † k − 1 + σ − 2 H T H ) † x k − 1 − σ − 2 P k H T ( H ˆ ˆ x k = ˆ x k − 1 − y k ) . We use the initial values of P 0 = ∞ and ˆ x 0 = ¯ x where ¯ x is the average rating. If every team plays every other team each round, we can simplify to P k = σ 2 n 2 k H T H x k − 1 − 1 nk H T ( H ˆ ˆ x k = ˆ x k − 1 − y k ) .
Convergence of Kalman Filter Ranking Proof of Convergence Theorem: In the case that every team plays every other team each round, we prove that x k ) � = σ 2 x k ) = Hx ∗ E ( H ˆ and � Cov( H ˆ k . The mean does not deviate from its initial value, so H ˆ x k describes the entire state.
Convergence of Kalman Filter Ranking Two Teams
Convergence of Kalman Filter Ranking More Teams, Large Range
Convergence of Kalman Filter Ranking More Teams, Large Range
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