sports ranking and scheduling with kalman filtersbertozzi/workforce/reu 2013/sports... ·...

Sports Ranking andScheduling with Kalman

Filters

Peter Elliott, Matt Owen,Kyle Shan, Yingjie Yu

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 (Rating)

Introduction

Definition (Rating)

Introduction

Definition (Game)

A game between two teams is a pairwise comparison which yields areal number outcome

Definition (Schedule)

A schedule is a matching of teams to play against each other ineach round.

Goal: schedule games to find the most “informative” ranking.

Introduction

Definition (Game)

Introduction

Definition (Game)

Extended Kalman Filter RankingUnderlying Model

Assume that games follow the Bradley-Terry paired comparisonmodel:

I Teams 1, . . . , n have skill levels x1, . . . , xnI 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 iscomputationally difficult.

Extended Kalman Filter RankingApplying a Filter to Bradley-Terry

Start with the system

xk = xk−1 = x∗

yk = hk(xk) + �k

where {�k} is an uncorrelated random process, and the predictedoutcomes hk(xk) of the games in each round are given by theBradley-Terry prediction.

Then, we get the simple update equations:

Pk = (P†k−1 + σ

−2HTk Hk)†

x̂k = x̂k−1 − σ−2PkHTk (hk(x̂k−1)− yk)

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

Extended Kalman Filter RankingApplying a Filter to Bradley-Terry

Start with the system

xk = xk−1 = x∗

yk = hk(xk) + �k

where {�k} is an uncorrelated random process, and the predictedoutcomes hk(xk) of the games in each round are given by theBradley-Terry prediction.Then, we get the simple update equations:

Pk = (P†k−1 + σ

−2HTk Hk)†

x̂k = x̂k−1 − σ−2PkHTk (hk(x̂k−1)− yk)

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

Extended Kalman Filter RankingSimilarity to Elo

Elo’s method seems a lot like the EKF update:

Elo: xi ← xi − K (h(xi , xj)− yk)EKF: x̂ ← x̂ − σ−2PkHTk (hk(x̂k−1)− yk)

where K is a constant.

EKF ranking has several advantages over Elo.

Extended Kalman Filter RankingSimilarity to Elo

Elo’s method seems a lot like the EKF update:

Elo: xi ← xi − K (h(xi , xj)− yk)EKF: x̂ ← x̂ − σ−2PkHTk (hk(x̂k−1)− yk)

where K is a constant.

EKF ranking has several advantages over Elo.

SchedulingIntroduction

Definition (Fisher Information)

The Fisher Information matrix is the inverse of the covariancematrix of the state estimate.

I = P†

Schedule Design Goal: Choose matchups that maximize the FisherInformation.

Notions of Fisher optimality:

I T-optimality: maximize the trace of the Fisher Informationmatrix

I Any monotone function of the eigenvalues

Definition (Fisher Information)

The Fisher Information matrix is the inverse of the covariancematrix of the state estimate.

I = P†

Schedule Design Goal: Choose matchups that maximize the FisherInformation.

Notions of Fisher optimality:

I T-optimality: maximize the trace of the Fisher Informationmatrix

I Any monotone function of the eigenvalues

Complete schedules for tournaments are often created before atournament begins. This is known as static scheduling.

Dynamic scheduling refers to creating a schedule as thetournament occurs, taking into account the results as they happen.

Goal: dynamically schedule one game for each team, for eachround of play, in a way that maximizes information of the rating.

SchedulingUsing Kalman Filters

The Kalman Filter update equations give a strategy for dynamicscheduling.

Pk = (P†k−1 + σ

−2HTk Hk)†

In terms of Fisher Information:

Ik = Ik−1 + σ−2HTk Hk

Goal: Choose games at each time step that maximize Ik

Important features:

I Maximizing tr(Ik) is equivalent to maximizing tr(HTk Hk)I If team i plays a single game, the i th entry on the diagonal of

HTk Hk is a function of the previous ratings of team i and itsopponent

SchedulingUsing Kalman Filters

The Kalman Filter update equations give a strategy for dynamicscheduling.

Pk = (P†k−1 + σ

−2HTk Hk)†

In terms of Fisher Information:

Ik = Ik−1 + σ−2HTk Hk

Goal: Choose games at each time step that maximize Ik

Important features:

I Maximizing tr(Ik) is equivalent to maximizing tr(HTk Hk)I If team i plays a single game, the i th entry on the diagonal of

HTk Hk is a function of the previous ratings of team i and itsopponent

SchedulingA New Dynamic Scheduling Method

Our proposed method:

I For each pair of teams i and j ,calculate the potentialinformation gain

I Create a complete weightedgraph G , with edge weightbetween i and j correspondingto the potential information gain

I Choose the maximum weightedperfect matching on G

I The weight of the matching isexactly tr(HTk Hk)

Edmonds’ Blossom Algorithm (1965) solves the perfect matchingproblem in polynomial time.

This method is trace-optimal for EKF ranking.

SchedulingTrace-Optimality Graph

SchedulingEffect of Optimal Scheduling

Scheduling HeatmapAll Games Played

randomallweeks.aviMedia File (video/avi)

blossomallweeks.aviMedia File (video/avi)

swissallweeks.aviMedia File (video/avi)

EKF Convergence

Linear Kalman Filter RankingIntroduction

Consider a system where the expected outcome in a game betweenteams i and j in round k is simply the difference of their skills:

hk(xk) = xik − x

jk

One way to find ratings is using least squares to solve

x̂ = arg minx‖Hx − y‖2

where yi is team i ’s cumulative point differential, and H is amatrix where each row represents a game between two teams i andj , with 1 in the winner’s slot and -1 in the loser’s slot.

Linear Kalman Filter RankingIntroduction

Consider a system where the expected outcome in a game betweenteams i and j in round k is simply the difference of their skills:

hk(xk) = xik − x

jk

One way to find ratings is using least squares to solve

x̂ = arg minx‖Hx − y‖2

where yi is team i ’s cumulative point differential, and H is amatrix where each row represents a game between two teams i andj , with 1 in the winner’s slot and -1 in the loser’s slot.

Linear Kalman Filter RankingLinear System

The Kalman Filter is an online algorithm solving the sameproblem. Assume a linear system for game results:

xk = xk−1 = x∗

yk = Hx∗ + �k

where H is defined as before, and {�k} is a Gaussian white noiseprocess with variance σ2.

Linear Kalman Filter RankingUpdate Equations

Applying the Kalman Filter yields

Pk = (P†k−1 + σ

−2HTH)†

x̂k = x̂k−1 − σ−2PkHT (Hx̂k−1 − yk).

We use the initial values of P0 =∞ and x̂0 = x̄ where x̄ is theaverage rating.

If every team plays every other team each round, we can simplify to

Pk =σ2

n2kHTH

x̂k = x̂k−1 −1

nkHT (Hx̂k−1 − yk).

Linear Kalman Filter RankingUpdate Equations

Applying the Kalman Filter yields

Pk = (P†k−1 + σ

−2HTH)†

x̂k = x̂k−1 − σ−2PkHT (Hx̂k−1 − yk).

We use the initial values of P0 =∞ and x̂0 = x̄ where x̄ is theaverage rating.If every team plays every other team each round, we can simplify to

Pk =σ2

n2kHTH

x̂k = x̂k−1 −1

nkHT (Hx̂k−1 − yk).

Convergence of Kalman Filter RankingProof of Convergence

Theorem:In the case that every team plays every other team each round, weprove that

E(Hx̂k) = Hx∗ and ‖Cov(Hx̂k)‖ =σ2

k.

The mean does not deviate from its initial value, so Hx̂k describesthe entire state.

Convergence of Kalman Filter RankingTwo Teams

Convergence of Kalman Filter RankingMore Teams, Large Range

Convergence of Kalman Filter RankingMore 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 informationto achieve more accurate rankings

Purpose

Questions:

1. Which is more important, rushing or passing, in theNFL?

2. How well can singular statistics predict game outcomes?

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?I More rushing yards and longer passes

Questions:

K-Fold Cross Validation Method

Questions:

2. How well can singular statistics predict game outcomes?I Total points is most predictiveI Aggregating other stats together almost improves

predictive powerI Total points is highly representative of team strength

Future Directions

I Investigate convergence for the extended Kalman Filter

I Experiment with other nonlinear filters

I Examine predictive power of nonlinear filter rankings

I Further optimize rank aggregation

Thanks!

Data

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