Ranking National Football League teams using Google s PageRank.

Transcription

1 Ranking National Football League teams using Google s PageRank. A. Govan C. Meyer Department of Mathematics North Carolina State University A.A. Markov Anniversary Meeting, 2006

2 Outline Google s ranking algorithm. Summary

3 Google s ranking algorithm. Google s ranking. Think of the internet as a graph. Webpages are nodes of the graph, n nodes. Hyperlinks are directed edges. Basic Idea: r(p ) = r(q) Q B P Q where r(p ) is the rank of a webpage P, B P is the set of webpages pointing to P, and Q is the outdegree of a webpage Q.

4 Google s ranking algorithm. Google Matrix. Hyperlink Matrix H { 1/ i there is a link from i to j H(i, j) = 0 otherwise Stochastic matrix S Obtained by modifying matrix H. Replace the zero rows of H with (1/n)e T, where e is a column vector of ones. Google Matrix G. Convex combination: G = αs + (1 α)ev T, α (0, 1) and v T > 0 Personalization vector v.

5 Google s ranking algorithm. PageRank vector π. G is the transition probability matrix. G is irreducible (and aperiodic). π is the stationary probability distribution vector. π is unique (up to a scalar multiple).

6 NFL web. Each NFL team is a node in a graph.

7 NFL web. Each NFL team is a node in a graph. A regular season game results in a directed edge from the loser to the winner.

8 NFL web. Each NFL team is a node in a graph. A regular season game results in a directed edge from the loser to the winner. The edges are weighted, the weight is the score difference of the corresponding game.

9 NFL web. Each NFL team is a node in a graph. A regular season game results in a directed edge from the loser to the winner. The edges are weighted, the weight is the score difference of the corresponding game.

10 Variations on PageRank. H(i, j) = t w t ij /( j ( t w t ij )) where w ij is the weight on the edge from team i to team j during week t. Dealing with the ith zero row (undefeated team i)

11 Variations on PageRank. H(i, j) = t w t ij /( j ( t w t ij )) where w ij is the weight on the edge from team i to team j during week t. Dealing with the ith zero row (undefeated team i) (1/32)e T, equally likely to lose to any other team.

12 Variations on PageRank. H(i, j) = t w t ij /( j ( t w t ij )) where w ij is the weight on the edge from team i to team j during week t. Dealing with the ith zero row (undefeated team i) (1/32)e T, equally likely to lose to any other team. e T i, where e i is the standard basis vector.

13 Variations on PageRank. H(i, j) = t w t ij /( j ( t w t ij )) where w ij is the weight on the edge from team i to team j during week t. Dealing with the ith zero row (undefeated team i) (1/32)e T, equally likely to lose to any other team. e T i, where e i is the standard basis vector. π T t 1, using the ranking vector from previous week.

14 Ranking NFL with Keener (SIAM Review,1993) Nonnegative matrix A ( ) Sij + 1 A(i, j) = h, Laplace s rule of succession S ij + S ji + 2 where S ij is the amount of points scored by team i against team j. h(x) = sgn(x 1 2 ) 2x 1 Rank vector r is the Perron vector of A.

15 Ranking NFL with Colley (Colley s Bias Free Matrix Rankings) Colley matrix C { 2 + ntot,i i = j C(i, j) = n j,i i j Laplace s rule of succession where n tot,i is the total number of games played by team i, and n j,i is the number of times team i played team j. Ranking vector r is the solution to the linear system Cr = b where b(i, 1) = 1 + (n w,i n l,i )/2, given that n w,i is the number of games lost by team i and n l,i is the number of games won by team i.

16 alpha=0.65, v^t=(1/32)e^t Correct predictions weeks Colley Google1 Google2 Google3 Keener

17 alpha=0.65, v^t=(1/32)e^t Correct predictions weeks Google1 Google2 Google3

18 Regular season 2005, α=0.65, v T = (1/32)e T Colley Google1 Google2 Google3 Keener Correct Spread Correct Spread Correct Spread Correct Spread Correct Spread games week week week week week week week week week week week week week week week week week Total

19 Results. PageRank based ranking method: depends on α and the personalization vector v T. does not appear to depend on the method of adjusting the zero rows.

20 Summary Summary Based on preliminary simulations PageRank variation method outperforms Keener s and Colley method. Future Personalization vector(s) using season statistics data. Automate the selection of the best α for a specified v T. Extend the convex combination to include more then one personalization vector.