Mathematics Invades Competitive Sports
| Kenneth Massey | |
| Virginia Tech | |
| 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 |
| 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 |
| 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) |
Bowl Championship Series (BCS)
| Polls | |
| Computers | |
| Schedule | |
| Losses | |
| Quality Wins |
| 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) | |
| Anderson / Hester formula | |
| Billingsley update | |
| Colley matrix | |
| Massey MLE (Gaussian) | |
| Matthews matrix | |
| Rothman MLE (logistic) | |
| Sagarin MLE (logistic) | |
| Wolfe / Baker least squares |
| Preseason ratings | |
| Weighting | |
| Choice of GOF |
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 |
| Win Indicator Function | |
| Score Ratio | |
| Rothman | |
| Sagarin ? | |
| Massey |
| Gaussian | |
| Logistic | |
| Equivalent to | |
| Arctan | |
MLE Optimization (Logistic Model)
| Non-uniqueness | |
| Ideally, wins help and losses hurt | |
| Undefeated / Winless Teams | |
| Update ratings based on linearization | |
| (time dependent, n large) | |
| 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 | |