Power Rankings: Math for March Madness

Transcription

1 Power Rankings: Math for March Madness James A. Swenson University of Wisconsin Platteville March 5, 2011 Math Club Madison Area Technical College James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 1 / 23

2 Thanks for coming! It s a pleasure to visit Madison! I hope you ll enjoy the talk; please feel free to get involved! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 2 / 23

3 Outline 1 Goals 2 Mathematics NCAA Division I Men s Basketball 4 Conclusion James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 3 / 23

4 Who would win? James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 4 / 23

5 Why math for sports rankings? Champion Best Team Strength of schedule? Math is objective......and some mathematicians like sports! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 5 / 23

6 Why math for sports rankings? Champion Best Team Strength of schedule? Math is objective......and some mathematicians like sports! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 5 / 23

7 Why math for sports rankings? Champion Best Team Strength of schedule? Math is objective......and some mathematicians like sports! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 5 / 23

8 Why math for sports rankings? Champion 6= Best Team Strength of schedule? Math is objective......and some mathematicians like sports! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 5 / 23

9 The Big Idea Fundamental hypothesis Everything I ll do is based on one assumption: The ranking of team T is in direct proportion to the sum of the rankings of the teams that team T beat. We only consider wins and losses (not margin of victory, date, &c.) Fundamental hypothesis in equations c x i = W i,1 x W i,n x n where x i 0 is the ranking of team i, n is the total number of teams, W i,j is the number of times that team i beat team j, and c > 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 6 / 23

10 The Big Idea Fundamental hypothesis Everything I ll do is based on one assumption: The ranking of team T is in direct proportion to the sum of the rankings of the teams that team T beat. We only consider wins and losses (not margin of victory, date, &c.) Fundamental hypothesis in equations c x i = W i,1 x W i,n x n where x i 0 is the ranking of team i, n is the total number of teams, W i,j is the number of times that team i beat team j, and c > 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 6 / 23

11 The Big Idea Fundamental hypothesis Everything I ll do is based on one assumption: The ranking of team T is in direct proportion to the sum of the rankings of the teams that team T beat. We only consider wins and losses (not margin of victory, date, &c.) Fundamental hypothesis in equations c x i = W i,1 x W i,n x n where x i 0 is the ranking of team i, n is the total number of teams, W i,j is the number of times that team i beat team j, and c > 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 6 / 23

12 Outline 1 Goals 2 Mathematics NCAA Division I Men s Basketball 4 Conclusion James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 7 / 23

13 From hypothesis to algebra c x 1 = W 1,1 x 1 + W 1,2 x W 1,n x n c x 2 = W 2,1 x 1 + W 2,2 x W 2,n x n. c x n = W n,1 x 1 + W n,2 x W n,n x n James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 8 / 23

14 From hypothesis to algebra c x 1 c x 2. c x n = W 1,1 W 1,2... W 1,n W 2,1 W 2,2... W 2,n. W n,1 W n,2... W n,n x 1 x 2. x n James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 8 / 23

15 From hypothesis to algebra c x = W x James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 8 / 23

16 From hypothesis to algebra c x = W x We say x is an eigenvector of W, and c is an eigenvalue of W. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 8 / 23

17 Eigenvalues and Eigenvectors Cauchy Hilbert ( ) ( ) James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 9 / 23

18 Finding eigenvalues c x = W x (ci ) x = W x (I is the identity matrix ) 0 = W x ci x 0 = (W ci ) x If (W ci ) 1 exists, then x = 0. If (W ci ) 1 does not exist, then det(w ci ) = 0. We don t want to give all teams a ranking of zero, so we want to find a value of c for which det(w ci ) = 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 10 / 23

19 Finding eigenvalues c x = W x (ci ) x = W x (I is the identity matrix ) 0 = W x ci x 0 = (W ci ) x If (W ci ) 1 exists, then x = 0. If (W ci ) 1 does not exist, then det(w ci ) = 0. We don t want to give all teams a ranking of zero, so we want to find a value of c for which det(w ci ) = 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 10 / 23

20 Finding eigenvalues c x = W x (ci ) x = W x (I is the identity matrix ) 0 = W x ci x 0 = (W ci ) x If (W ci ) 1 exists, then x = 0. If (W ci ) 1 does not exist, then det(w ci ) = 0. We don t want to give all teams a ranking of zero, so we want to find a value of c for which det(w ci ) = 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 10 / 23

21 Finding eigenvalues c x = W x (ci ) x = W x (I is the identity matrix ) 0 = W x ci x 0 = (W ci ) x If (W ci ) 1 exists, then x = 0. If (W ci ) 1 does not exist, then det(w ci ) = 0. We don t want to give all teams a ranking of zero, so we want to find a value of c for which det(w ci ) = 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 10 / 23

22 Finding eigenvalues c x = W x (ci ) x = W x (I is the identity matrix ) 0 = W x ci x 0 = (W ci ) x If (W ci ) 1 exists, then x = 0. If (W ci ) 1 does not exist, then det(w ci ) = 0. We don t want to give all teams a ranking of zero, so we want to find a value of c for which det(w ci ) = 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 10 / 23

23 Finding eigenvalues c x = W x (ci ) x = W x (I is the identity matrix ) 0 = W x ci x 0 = (W ci ) x If (W ci ) 1 exists, then x = 0. If (W ci ) 1 does not exist, then det(w ci ) = 0. We don t want to give all teams a ranking of zero, so we want to find a value of c for which det(w ci ) = 0. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 10 / 23

24 Can we solve it? c x = W x det(w ci ) = 0 Good news: by the Fundamental Theorem of Algebra, there is a value of c that satisfies this equation. Bad news: if there are n teams, there are n values of c that would work... and they re complex numbers. (We want c > 0 to be real.) Ugly news: for each c, there are infinitely many eigenvectors x that solve our original equation c x = W x... how do we pick just one? James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 11 / 23

25 Can we solve it? c x = W x det(w ci ) = 0 Good news: by the Fundamental Theorem of Algebra, there is a value of c that satisfies this equation. Bad news: if there are n teams, there are n values of c that would work... and they re complex numbers. (We want c > 0 to be real.) Ugly news: for each c, there are infinitely many eigenvectors x that solve our original equation c x = W x... how do we pick just one? James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 11 / 23

26 Can we solve it? c x = W x det(w ci ) = 0 Good news: by the Fundamental Theorem of Algebra, there is a value of c that satisfies this equation. Bad news: if there are n teams, there are n values of c that would work... and they re complex numbers. (We want c > 0 to be real.) Ugly news: for each c, there are infinitely many eigenvectors x that solve our original equation c x = W x... how do we pick just one? James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 11 / 23

27 Perron-Frobenius Theorem Frobenius Perron ( ) ( ) James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 12 / 23

28 Perron-Frobenius Theorem Theorem If W is a matrix of non-negative integers, then W has a dominant eigenvalue λ, such that: λ is an eigenvalue of W... λ is a positive real number... λ is the biggest eigenvalue of W... There is an eigenvector for λ, x +, whose entries are non-negative... Every eigenvector for λ is a multiple of x unless W is reducible. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 13 / 23

29 Perron-Frobenius Theorem Theorem If W is a matrix of non-negative integers, then W has a dominant eigenvalue λ, such that: λ is an eigenvalue of W... λ is a positive real number... λ is the biggest eigenvalue of W... There is an eigenvector for λ, x +, whose entries are non-negative... Every eigenvector for λ is a multiple of x unless W is reducible. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 13 / 23

30 Perron-Frobenius Theorem Theorem If W is a matrix of non-negative integers, then W has a dominant eigenvalue λ, such that: λ is an eigenvalue of W... λ is a positive real number... λ is the biggest eigenvalue of W... There is an eigenvector for λ, x +, whose entries are non-negative... Every eigenvector for λ is a multiple of x unless W is reducible. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 13 / 23

31 Perron-Frobenius Theorem Theorem If W is a matrix of non-negative integers, then W has a dominant eigenvalue λ, such that: λ is an eigenvalue of W... λ is a positive real number... λ is the biggest eigenvalue of W... There is an eigenvector for λ, x +, whose entries are non-negative... Every eigenvector for λ is a multiple of x unless W is reducible. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 13 / 23

32 Perron-Frobenius Theorem Theorem If W is a matrix of non-negative integers, then W has a dominant eigenvalue λ, such that: λ is an eigenvalue of W... λ is a positive real number... λ is the biggest eigenvalue of W... There is an eigenvector for λ, x +, whose entries are non-negative... Every eigenvector for λ is a multiple of x unless W is reducible. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 13 / 23

33 Perron-Frobenius Theorem Theorem If W is a matrix of non-negative integers, then W has a dominant eigenvalue λ, such that: λ is an eigenvalue of W... λ is a positive real number... λ is the biggest eigenvalue of W... There is an eigenvector for λ, x +, whose entries are non-negative... Every eigenvector for λ is a multiple of x unless W is reducible. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 13 / 23

34 Is W reducible? Definition In this context, W is reducible if the teams can be separated into two (non-empty) groups, A and B, so that no team from Group B has beaten any team from Group A. Strategy If some team from Group A has beaten some team from Group B, then rank the teams in each group against each other, and rank the Group-A teams above the Group-B teams. Otherwise, no team from Group A has played any team from Group B: don t try to rank these teams against each other! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 14 / 23

35 Is W reducible? Definition In this context, W is reducible if the teams can be separated into two (non-empty) groups, A and B, so that no team from Group B has beaten any team from Group A. Strategy If some team from Group A has beaten some team from Group B, then rank the teams in each group against each other, and rank the Group-A teams above the Group-B teams. Otherwise, no team from Group A has played any team from Group B: don t try to rank these teams against each other! James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 14 / 23

36 Reducibility Test Theorem Let W be an n n matrix of non-negative integers. W is reducible (I + W ) n 1 contains a zero. Lemma If row i of (I + W ) n 1 has zeros in columns j 1,..., j r (but not in other columns), then you can put teams j 1,... j r in Group A, and the rest in Group B. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 15 / 23

37 Reducibility Test Theorem Let W be an n n matrix of non-negative integers. W is reducible (I + W ) n 1 contains a zero. Lemma If row i of (I + W ) n 1 has zeros in columns j 1,..., j r (but not in other columns), then you can put teams j 1,... j r in Group A, and the rest in Group B. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 15 / 23

38 Outline 1 Goals 2 Mathematics NCAA Division I Men s Basketball 4 Conclusion James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 16 / 23

39 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

40 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

41 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

42 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

43 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

44 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

45 Implementation In , 344 NCAA Division-I basketball teams played 4766 games (as of 1 March) so λ is a root of a degree-344 polynomial, and we have to solve a system of 344 equations after we write down the right ( ) matrix to represent the 4766 game results. We are certainly going to do this with a computer! It is vital to have accurate, uniform data: I get mine from Ken Pomeroy [ I feed the data directly into a Perl script I wrote. (Perl is an open-source programming lanugage optimized for text data manipulation [ The Perl script extracts the results in a form that can be read by GNU Octave. (Octave is an open-source numerical matrix algebra system [ James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 17 / 23

46 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

47 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

48 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

49 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

50 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

51 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

52 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

53 Hunting the rankings vector Assume W is irreducible, λ k is an eigenvalue for each k with W v k = λ k v k, and λ 1 is the dominant eigenvalue. Then it is known that: 1, 1,..., 1 = c 1 v 1 + c 2 v c n v n W 1, 1,..., 1 = W (c 1 v 1 + c 2 v c n v n ) W 1, 1,..., 1 = c 1 W v 1 + c 2 W v c n W v n W 1, 1,..., 1 = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n W 2 1, 1,..., 1 = c 1 λ 2 1 v 1 + c 2 λ 2 2 v c n λ 2 n v n W 3 1, 1,..., 1 = c 1 λ 3 1 v 1 + c 2 λ 3 2 v c n λ 3 n v n. W q 1, 1,..., 1 = c 1 λ q 1 v 1 + c 2 λ q 2 v c n λ q n v n [ λ q 1 W q c2 λ q ] [ 2 cn λ q ] n 1, 1,..., 1 = c 1 v 1 + v v n λ q 1 λ q 1 James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 18 / 23

54 Hunting the rankings vector Convergence! lim q λ q 1 W q x 0 = c 1 v 1 (There s a convergence-accelerating trick, if this doesn t work.) v 1 was our vector of team rankings... but c 1 v 1 is equivalent. So, starting from 1, 1,..., 1, if we repeatedly multiply by W and divide by λ 1, we get increasingly accurate approximations to our solution. In fact, rather than dividing by λ 1 at every step, it s good enough to divide by the greatest entry in the approximation vector. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 19 / 23

55 Hunting the rankings vector Convergence! lim q λ q 1 W q x 0 = c 1 v 1 (There s a convergence-accelerating trick, if this doesn t work.) v 1 was our vector of team rankings... but c 1 v 1 is equivalent. So, starting from 1, 1,..., 1, if we repeatedly multiply by W and divide by λ 1, we get increasingly accurate approximations to our solution. In fact, rather than dividing by λ 1 at every step, it s good enough to divide by the greatest entry in the approximation vector. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 19 / 23

56 Hunting the rankings vector Convergence! lim q λ q 1 W q x 0 = c 1 v 1 (There s a convergence-accelerating trick, if this doesn t work.) v 1 was our vector of team rankings... but c 1 v 1 is equivalent. So, starting from 1, 1,..., 1, if we repeatedly multiply by W and divide by λ 1, we get increasingly accurate approximations to our solution. In fact, rather than dividing by λ 1 at every step, it s good enough to divide by the greatest entry in the approximation vector. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 19 / 23

57 Hunting the rankings vector Convergence! lim q λ q 1 W q x 0 = c 1 v 1 (There s a convergence-accelerating trick, if this doesn t work.) v 1 was our vector of team rankings... but c 1 v 1 is equivalent. So, starting from 1, 1,..., 1, if we repeatedly multiply by W and divide by λ 1, we get increasingly accurate approximations to our solution. In fact, rather than dividing by λ 1 at every step, it s good enough to divide by the greatest entry in the approximation vector. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 19 / 23

58 Hunting the rankings vector Convergence! lim q λ q 1 W q x 0 = c 1 v 1 (There s a convergence-accelerating trick, if this doesn t work.) v 1 was our vector of team rankings... but c 1 v 1 is equivalent. So, starting from 1, 1,..., 1, if we repeatedly multiply by W and divide by λ 1, we get increasingly accurate approximations to our solution. In fact, rather than dividing by λ 1 at every step, it s good enough to divide by the greatest entry in the approximation vector. James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 19 / 23

59 NCAA Division I Men s Basketball Computer scripts: [Perl script] [Octave script] Complete Rankings (March 1): [XLS] [TIF] [PDF] James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 20 / 23

60 Outline 1 Goals 2 Mathematics NCAA Division I Men s Basketball 4 Conclusion James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 21 / 23

61 What I Learned The Big East was very, very competitive this year. Data entry is non-trivial. The Fundamental Hypothesis yields numerical results. The theorems are older than the application! The purpose of computing is insight, not numbers. Richard Hamming James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 22 / 23

62 What I Learned The Big East was very, very competitive this year. Data entry is non-trivial. The Fundamental Hypothesis yields numerical results. The theorems are older than the application! The purpose of computing is insight, not numbers. Richard Hamming James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 22 / 23

63 What I Learned The Big East was very, very competitive this year. Data entry is non-trivial. The Fundamental Hypothesis yields numerical results. The theorems are older than the application! The purpose of computing is insight, not numbers. Richard Hamming James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 22 / 23

64 What I Learned The Big East was very, very competitive this year. Data entry is non-trivial. The Fundamental Hypothesis yields numerical results. The theorems are older than the application! The purpose of computing is insight, not numbers. Richard Hamming James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 22 / 23

65 What I Learned The Big East was very, very competitive this year. Data entry is non-trivial. The Fundamental Hypothesis yields numerical results. The theorems are older than the application! The purpose of computing is insight, not numbers. Richard Hamming James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 22 / 23

66 Thanks! Photo Credits Sullinger: Phil Sandlin / Associated Press [ Fredette: Douglas C. Piza / US PRESSWIRE [ Cauchy, Frobenius, Hilbert, Perron: MacTutor History of Mathematics Archive [ Taylor: UWBadgers.com [ Wolfie: MadisonCollegeAthletics.com [ Wolfie News.aspx?] James A. Swenson (UWP) Power Rankings: Math for March Madness 3/5/11 23 / 23