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1 An Intelligence Approach to Evaluation of Sports Teams by Edward Kambour, Ph.D.

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Agenda I.College Football II.Linear Model III.Generalized Linear Model IV.Intelligence (Bayesian) Approach V.Results VI.Other Sports VII.Future Work

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General Background Goals Forecast winners of future games Beat the Bookie! Estimate the outcome of unscheduled games Whats the probability that Iowa would have beaten Ohio St? Generate reasonable rankings

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Major College Football No playoff system Computer rankings are an element of the BCS 114 teams 12 games for each in a season

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Linear Model Rothman (1970s), Harville (1977), Stefani (1977), …, Kambour (1991), …, Sagarin??? Response, Y, is the net result (point-spread) Parameter,, is the vector of ratings For a game involving teams i and j, E[Y] = i - j

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Linear Model (cont.) Let X be a row vector with E[Y]=X

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Regression Model Notes Least Squares Normality, Homogeneity College Football Estimate 100 parameters Sample size for a full season is about 600 Design Matrix is sparse and not full rank

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Home-field Advantage Generic Advantage (Stefani, 1980) Force i to be home team and j the visiting team Add an intercept term to X Adds one more parameter to estimate UAB = Alabama Rice = Texas A&M Team Specific Advantage Doubles the number of parameters to estimate

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Linear Model Issues Normality Homogeneity Lots of parameters, with relatively small sample size Overfitting The bookie takes you to the cleaners!

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Linear Model Issues (cont.) Should we model point differential A and B play twice A by 34 in first, B by 14 in the second A by 10 each time Running up the score (or lack thereof) BCS: Thou shalt not use margin of victory in thy ratings!

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Logistic Regression Rothman (1970s) Linear Model Use binary variable Winning is all that matters Avoid margin of victory Coin Flips

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Logistic Regression Issues Still have sample size issues Throw away a lot of information Undefeated teams

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Transformations Transform the differentials to normality Power transformations Rothman logistic transform Transforms points to probabilities for logistic regression Diminishing returns transforms Downweights runaway scores

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Power Transforms Transform the point-spread Y = sign(Z)|Z| a a = 1 straight margin of victory a = 0 just win baby a = 0 Poisson or Gamma ish

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Maximum Likelihood Transform seasons MLE = 0.98 Power-2ln(likelihood)

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Predicting the Score Model point differential Y 1 = S i – S j Additionally model the sum of the points scored Y 2 = S i + S j Fit a similar linear model (different parameter estimates) Forecast home and visitors score H = (Y 1 + Y 2 )/2, V = (Y 2 - Y 1 )/2

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Another Transformation Idea Scores (touchdowns or field goals) are arrivals, maybe Poisson Final score = 7 times a Poisson + 3 times a Poisson + … Transform the scores to homogeneity and normality first The differences (and sums) should follow suit

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Square Root Transform Since the score is similar to a linear combination of Poissons, square root should work Transformation Why k? For small Poisson arrival rates, get better performance (Anscombe, 1948)

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Likelihood Test LRT: No transformation vs. square root with fitted k Used College Football results from k = 21 Transformation was significantly better p-value = , chi-square = 9.26

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Predicting the Score with Transform Model point differential Additionally model the sum of the points scored Forecast home and visitors score H = ((Y 1 + Y 2 )/2) 2, V = ((Y 2 - Y 1 )/2) 2 Note the point differential is the product

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Unresolved Linear Model Issues Overfitting History Going into the season, we have a good idea as to how teams will do The best teams tend to stay the best The worst teams tend to stay the worst Changes happen Kansas State

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Intelligence Model Concept The ratings and home-ads for year t are similar to those of year t-1. There is some drift from one year to the next. Model

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Intelligence Model (Details) Notation L teams M seasons of data N i games in the ith season X i : the N i by 2L X matrix for season i Y i : the N i vector of results for season i i : the N i vector of results for season I

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Details (cont.) Data Distribution: For all i = 1, 2, …, M

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Details (cont.) Prior Distribution

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Details (finally, the end) The Posterior Distribution of M and -2 is closed form and can be calculated by an iterative method The Predictive Distribution for future results (transformed sum or difference) is straight- forward correlated normal (given the variance)

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Forecasts For Scores Simply untransform E[Z 2 ] = Var[Z] + E[Z] 2 For the point-spread Product of two normals Simulate results

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Enhanced Model Fit the prior parameters Hierarchical models Drifts and initial variances No closed form for posterior and predictive distributions (at least as far as I know) The complete conditionals are straight-forward, so Gibbs sampling will work (eventually)

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Results (www.geocities.com/kambour/football.html) 2002 Final Rankings TeamRatingHome Miami72.23 (1.03)0.21 (0.04) Kansas St72.04 (1.04)0.44 (0.03) USC71.95 (1.03)0.04 (0.03) Oklahoma71.85 (1.02)0.18 (0.03) Texas71.57 (1.03)0.36 (0.03) Georgia71.49 (1.03)0.02 (0.03) Alabama71.45 (1.03)-0.09 (0.03) Iowa71.30 (1.03)0.21 (0.04) Florida St71.29 (1.02)0.43 (0.03) Virginia Tech71.25 (1.03)0.12 (0.03) Ohio St71.18 (1.03)0.27 (0.03)

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Results 2002 Final Rankings TeamRatingHome Miami Kansas St USC Oklahoma Texas Georgia Alabama Iowa Florida St Virginia Tech Ohio St

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Results 2002 Final Rankings TeamRatingHome Miami Kansas St USC Oklahoma Texas Georgia Alabama Iowa Florida St Virginia Tech Ohio St

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Bowl Predictions Ohio St 17 Miami Fl (-13) Washington St 21 Oklahoma (-6.5) Iowa 21 USC (-6) NC State (E) 20 Notre Dame Florida St (+4) 24 Georgia

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2002 Final Record Picking Winners 522 – Against the Vegas lines 367 – 307 – Best Bets 9 – In 2001,

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ESPN College Pickem (http://games.espn.go.com/cpickem/leader) 1. Barry Schultz Jim Dobbs Michael Reeves Fup Biz Joe * Rising Cream Intelligence Ratings 5559

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Ratings System Comparison (http://tbeck.freeshell.org/fb/awards2002.html) Todd Beck Ph.D. Statistician Rush Institute Intelligence Ratings – Best Predictors

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College Football Conclusions Can forecast the outcome of games Capture the random nature High variability Sparse design Scientists should avoid BCS Statistical significance is impossible Problem Complexity Other issues

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NFL Similar to College Football Square root transform is applicable Drift is a little higher than College Football Better design matrix Small sample size Playoff

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NFL Results (www.geocities.com/kambour/NFL.html) 2002 Final Rankings (after the Super Bowl) TeamRatingHome Tampa Bay Oakland Philadelphia New England Atlanta NY Jets Pittsburgh Green Bay Kansas City Denver Miami

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2002 Final NFL Record Picking Winners 162 – 104 – Against the Vegas lines 135 – 128 – Best Bets 9 –

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NFL Europe Similar to College and NFL Square root transform Dramatic drift Teams change dramatically in mid-season Few teams Better design matrix

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College Basketball Transform? Much more normal (Central Limit Theorem) A lot more games Intersectional games Less emphasis on programs than in College Football More drift NCAA tournament

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NCAA Basketball Pre-tournament Ratings TeamRatingHome Arizona Kentucky Kansas Texas Duke Oklahoma Florida Wake Forest Syracuse Xavier Louisville

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NBA Similar to College Basketball Normal – No transformation A lot more games – fewer teams Playoffs are completely different from regular season Regular season – very balanced, strong home court Post season – less balanced, home court lessened

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Hockey Transform Rare events = Poissonish Square root with k around 1 A lot more games History matters Playoffs seem similar to regular season Balance

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Soccer Similar to hockey Transform Square root with low k Not a lot of games Friendlys versus cup play Home pitch is pronounced Varies widely

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Soccer Results Correctly forecasted 2002 World Cup final Brazil over Germany Correctly forecasted US run to quarter-finals Won the PROS World Cup Soccer Pool

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Future Enhancements Hierarchical Approaches Conferences More complicated drift models Correlations Individual drifts Drift during the season Mean correcting drift More informative priors

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