1	Mathematics Invades Competitive Sports Kenneth Massey Virginia Tech
2	Goals Objectively measure the performance of a team relative to the schedule faced Correct for disparate schedules (esp. college sports) Predictive vs. Retrodictive Wins, scores, date, stats, homefield, preseason, other ? Seed playoffs
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4	Challenges There is no property of transitivity! Disparate schedules “Mt. Union Syndrome” Separate Divisions Strength vs. Performance vs. Results Environment (venue, homefield, weather, day/night, crowd) Teams don’t always play at full potential (injury, unfavorable matchups, intangible, psychological) The score isn’t always a good indicator (coaching philosophy, chaos “bounce of ball”) Lack of data Connectedness Undefeated / winless teams
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9	Types of Ratings Standings / WL% / Points Polls Tabulated votes, subjective, time sensitive, no corrections, incomplete analysis Formula (RPI) Update (Elo chess) Least Squares MLE Matrix (Markov) Other (Neural nets)
10	Bowl Championship Series (BCS) Polls Computers Schedule Losses Quality Wins
11	Schedule Ratings Average rating of opponents (corrected for homefield) A good team prefers a less distributed schedule; a bad team prefers a more distributed schedule. For example (Florida, Vanderbilt) vs. (Alabama, Arkansas)
12	BCS Computers Anderson / Hester formula Billingsley update Colley matrix Massey MLE (Gaussian) Matthews matrix Rothman MLE (logistic) Sagarin MLE (logistic) Wolfe / Baker least squares
13	Least Squares Model
14	Least Squares Model
15	Rank Deficiency
16	LS Example
17	LS Example
18	LS Example
19	More LS Model Features Preseason ratings Weighting Choice of GOF
20	LS Notes
21	Maximum Likelihood Estimator (MLE) Method Optimization Problem Choose ratings to maximize the probability of reproducing the observed results Game Outcome Function Measures the result of a particular game Game Likelihood Function The probability of a given result given a set of ratings
22	Game Outcome Function Win Indicator Function Score Ratio Rothman Sagarin ? Massey
23	GOF Values
24	GOF Bias
25	Proof
26	Game Likelihood Functions Gaussian Logistic Equivalent to Arctan
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29	MLE Function
30	MLE Optimization (Logistic Model)
31	MLE Issues Non-uniqueness Ideally, wins help and losses hurt Undefeated / Winless Teams Update ratings based on linearization (time dependent, n large)
32	Ratings on the Web Massey Ratings http://www.masseyratings.com College Football Rankings http://www.cae.wisc.edu/~dwilson/rsfc/rate/index.html Bowl Championship Series http://www.collegebcs.com