An Intelligence Approach to Evaluation of Sports Teams

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

Agenda College Football Linear Model Generalized Linear Model
Intelligence (Bayesian) Approach Results Other Sports Future Work

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

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

Linear Model Rothman (1970’s), 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

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

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

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

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

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!

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

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

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

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”

Maximum Likelihood Transform
seasons MLE = 0.98 Power -2ln(likelihood) 0.1 52487 0.3 41213 0.5 35128 0.67 32597 0.8 31418 1 31193

Predicting the Score Model point differential
Y1 = Si – Sj Additionally model the sum of the points scored Y2 = Si + Sj Fit a similar linear model (different parameter estimates) Forecast home and visitors score H = (Y1 + Y2 )/2, V = (Y2 - Y1)/2

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

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)

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

Predicting the Score with Transform
Model point differential Additionally model the sum of the points scored Forecast home and visitors score H = ((Y1 + Y2 )/2)2 , V = ((Y2 - Y1)/2)2 Note the point differential is the product

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

Intelligence Model Concept Model
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

Intelligence Model (Details)
Notation L teams M seasons of data Ni games in the ith season Xi : the Ni by 2L “X” matrix for season i Yi : the Ni vector of results for season i i : the Ni vector of results for season I

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

Details (cont.) Prior Distribution

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)

Forecasts For Scores For the point-spread Simply untransform
E[Z2] = Var[Z] + E[Z]2 For the point-spread Product of two normals Simulate results

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)

Results (www.geocities.com/kambour/football.html)
2002 Final Rankings Team Rating Home Miami 72.23 (1.03) 0.21 (0.04) Kansas St 72.04 (1.04) 0.44 (0.03) USC 71.95 (1.03) 0.04 (0.03) Oklahoma 71.85 (1.02) 0.18 (0.03) Texas 71.57 (1.03) 0.36 (0.03) Georgia 71.49 (1.03) 0.02 (0.03) Alabama 71.45 (1.03) -0.09 (0.03) Iowa 71.30 (1.03) Florida St 71.29 (1.02) 0.43 (0.03) Virginia Tech 71.25 (1.03) 0.12 (0.03) Ohio St 71.18 (1.03) 0.27 (0.03)

Results 2002 Final Rankings Team Rating Home Miami 72.23 0.21
Kansas St 72.04 0.44 USC 71.95 0.04 Oklahoma 71.85 0.18 Texas 71.57 0.36 Georgia 71.49 0.02 Alabama 71.45 -0.09 Iowa 71.30 Florida St 71.29 0.43 Virginia Tech 71.25 0.12 Ohio St 71.18 0.27

Results 2002 Final Rankings Team Rating Home Miami 72.23 0.21
Kansas St 72.04 0.44 USC 71.95 0.04 Oklahoma 71.85 0.18 Texas 71.57 0.36 Georgia 71.49 0.02 Alabama 71.45 -0.09 Iowa 71.30 Florida St 71.29 0.43 Virginia Tech 71.25 0.12 Ohio St 71.18 0.27

Bowl Predictions Ohio St 17 Miami Fl (-13) 31 0.8255 0.5228
Washington St Oklahoma (-6.5) Iowa USC (-6) NC State (E) Notre Dame Florida St (+4) Georgia

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

1. Barry Schultz 2. Jim Dobbs 3. Michael Reeves 4. Fup Biz 5. Joe * 6. Rising Cream 7. Intelligence Ratings

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

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

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

NFL Results (www.geocities.com/kambour/NFL.html)
2002 Final Rankings (after the Super Bowl) Team Rating Home Tampa Bay 70.72 0.29 Oakland 70.57 0.28 Philadelphia 70.55 0.10 New England 70.16 0.12 Atlanta 70.13 0.20 NY Jets 70.10 -0.01 Pittsburgh 69.95 Green Bay 69.92 Kansas City 69.90 0.51 Denver 69.89 0.50 Miami 0.49

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

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

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

Team Rating Home Arizona 100.06 3.97 Kentucky 99.33 4.32 Kansas 95.89 3.85 Texas 93.42 4.44 Duke 92.90 4.66 Oklahoma 90.19 4.31 Florida 90.65 3.99 Wake Forest 88.70 3.65 Syracuse 88.50 3.49 Xavier 87.89 3.37 Louisville 87.88 4.16

NBA Similar to College Basketball A lot more games – fewer teams
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

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

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

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

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