Sports Ranking and Scheduling with Kalman Filters Peter Elliott, - - PowerPoint PPT Presentation

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 x1, . . . , xn ◮ The expected outcome in a game between teams i and j is

h(xi, xj) = 1 1 + 10−(xi−xj)/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 xk = xk−1 = x∗ yk = hk(xk) + ǫk where {ǫk} is an uncorrelated random process, and the predicted

• utcomes hk(xk) 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 xk = xk−1 = x∗ yk = hk(xk) + ǫk where {ǫk} is an uncorrelated random process, and the predicted

• utcomes hk(xk) of the games in each round are given by the

Bradley-Terry prediction. Then, we get the simple update equations: Pk = (P†

k−1 + σ−2HT k Hk)†

ˆ xk = ˆ xk−1 − σ−2PkHT

k (hk(ˆ

xk−1) − yk) where Pk is the covariance of ˆ xk and Hk = Dhk(xk−1).

Extended Kalman Filter Ranking

Similarity to Elo

Elo’s method seems a lot like the EKF update: Elo: xi ← xi − K (h(xi, xj) − yk) EKF: ˆ x ← ˆ x − σ−2PkHT

k (hk(ˆ

xk−1) − yk) where K is a constant.

Extended Kalman Filter Ranking

Similarity to Elo

Elo’s method seems a lot like the EKF update: Elo: xi ← xi − K (h(xi, xj) − yk) EKF: ˆ x ← ˆ x − σ−2PkHT

k (hk(ˆ

xk−1) − yk) 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. Pk = (P†

k−1 + σ−2HT k Hk)†

In terms of Fisher Information: Ik = Ik−1 + σ−2HT

k Hk

Goal: Choose games at each time step that maximize Ik

Scheduling

Using Kalman Filters

The Kalman Filter update equations give a strategy for dynamic scheduling. Pk = (P†

k−1 + σ−2HT k Hk)†

In terms of Fisher Information: Ik = Ik−1 + σ−2HT

k Hk

Goal: Choose games at each time step that maximize Ik Important features:

◮ Maximizing tr(Ik) is equivalent to maximizing tr(HT k Hk) ◮ If team i plays a single game, the ith entry on the diagonal of

HT

k Hk is a function of the previous ratings of team i and its

• pponent

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(HT

k Hk)

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(HT

k Hk)

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(HT

k Hk)

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: hk(xk) = xi

k − xj 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: hk(xk) = xi

k − xj k

One way to find ratings is using least squares to solve ˆ x = arg min

x Hx − y2

where yi 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:

xk = xk−1 = x∗ yk = Hx∗ + ǫk where H is defined as before, and {ǫk} is a Gaussian white noise process with variance σ2.

Linear Kalman Filter Ranking

Update Equations

Applying the Kalman Filter yields Pk = (P†

k−1 + σ−2HTH)†

ˆ xk = ˆ xk−1 − σ−2PkHT(Hˆ xk−1 − yk). We use the initial values of P0 = ∞ and ˆ x0 = ¯ x where ¯ x is the average rating.

Linear Kalman Filter Ranking

Update Equations

Applying the Kalman Filter yields Pk = (P†

k−1 + σ−2HTH)†

ˆ xk = ˆ xk−1 − σ−2PkHT(Hˆ xk−1 − yk). We use the initial values of P0 = ∞ and ˆ x0 = ¯ x where ¯ x is the average rating. If every team plays every other team each round, we can simplify to Pk = σ2 n2k HTH ˆ xk = ˆ xk−1 − 1 nk HT(Hˆ xk−1 − yk).

Convergence of Kalman Filter Ranking

Proof of Convergence

Theorem: In the case that every team plays every other team each round, we prove that E(Hˆ xk) = Hx∗ and Cov(Hˆ xk) = σ2 k . The mean does not deviate from its initial value, so Hˆ xk 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

Convergence of Kalman Filter Ranking

More Teams, Small Range

Convergence of Kalman Filter Ranking

More Teams, Small Range

Ranking NFL Statistics

Motivation

Why rank statistics in the National Football League?

• 1. Many different strategies - which is the best one?
• 2. Parity in the NFL means there is a need for more information

to achieve more accurate rankings

Purpose

Questions:

• 1. Which is more important, rushing or passing, in the

NFL?

• 2. How well can singular statistics predict game outcomes?

Method - Passing vs. Rushing

Method - Passing vs. Rushing

Data - Passing vs. Rushing

Figure : Yards Per Game

Data - Passing vs. Rushing

Figure : Yards / Attempt Per Game

Questions:

• 1. Which is more important, rushing or passing, in the NFL?

◮ More rushing yards and longer passes

• 2. How well can singular statistics predict game outcomes?

Questions:

• 1. Which is more important, rushing or passing, in the NFL?

◮ More rushing yards and longer passes

• 2. How well can singular statistics predict game outcomes?

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

K-Fold Cross Validation Method

Questions:

• 1. Which is more important, rushing or passing, in the NFL?

◮ More rushing yards and longer passes

• 2. How well can singular statistics predict game outcomes?

◮ Total points is most predictive ◮ Aggregating other stats together almost improves

predictive power

◮ Total points is highly representative of team strength

Future Directions

◮ Investigate convergence for the extended Kalman Filter ◮ Experiment with other nonlinear filters ◮ Examine predictive power of nonlinear filter rankings ◮ Further optimize rank aggregation

Thanks!