Bradley-Terry W 03D8103002L 2007 3
Bradley-Terry W Bradley-Terry FIFA Bradley-Terry 1998 W 2002 W 2006 W Bradley-Terry W
1...1 2 Bradley-Terry...2 2.1...2 2.2 BT...3 2.3...4 2.4...5 3...8 3.1...8 3.2 FIFA...8 4 BT...9 4.1 BT FIFA...9 4.2...13 4.3...17 4.4...21 5 BT W...24 5.1...24 5.2 W...26 5.2.1 W...26 5.2.2...27 5.2.3...28 5.3...30 6...32 6.1...32 6.2...32...33...34
1 4 1 W W 2006 6 9 7 9 2006 W 64 330 W 2 W 2006 W 1998 W 2002 W FIFA FIFA Bradley-Terry Bradley-Terry 1998 W 2002 W 2006 W 1
2 Bradley-Terry 2 2 Bradley-Terry [1] 2.1 m ( ;, = 1,2, L, m) (2.1) = 1 ( ) (2.2) + m = 3 12 23 31 < < < 1 2 1 2 1 2 < < < 21 32 13 (2.3) 2 1 3 2 1 3 3 2
m,, 2, 1 L m = ( ) + (2.4) = ( ) (2.5) Bradley and Terry1952Bradley-Terry BT 2.2 BT BT k k k k k k k k k : k k k k k k k k + k k (2.6) k k k k = (2.7) + k k k k 3
k k = (2.8) k k m = ( 1,2,, 1) m ( ) k = L m = 1 (2.9) m m m = = (2.10) m m (2.5) (2.5) = (2.11) k k 3 1 k k (2.11) k k 2.3 m n = n ) ( X ( ;, = 1,2, L, m) (2.12) X + X = n ( ) (2.13) X 2 B n, ) ( n! x x Pr { X = x } = ( x = 0,1, L, n ) (2.14) x! x! 4
n! x { } x Pr X = = = x ; ;, 1,2, L, m (2.15) < x! x! BT (2.15) Pr { X = x ; ;, = 1,2, L, m} = = m < x = < n! x! x! n! x! x! ( + ) ( + ) x 1 x n n m = 1 t (2.16) t = (2.17) (2.16) =,, L, ) ( 1 2 m const. L = const. n ( + ) < m = 1 T (2.18) X m T = (2.19) X = BT ( T, T2,, T m ) ( T, T2,, ) T m 1 L 1 L BT 2.4 =,, L, ) ( 1 2 m =,, L, ) ( 1 2 m (2.4) k m =1 k (2.20) = 5
(2.18) m l = log L = T log n log( + ) + const. (2.21) = 1 λ < l λ m = 1 k = 0 m l λ k = 0 λ = 1 (2.22) (2.23) T n ˆ ˆ + ˆ λ = 0( = 1,2, L, m) (2.24) (2.24) ˆ m =1 ˆ ˆ + ˆ ˆ k (2.25) = = ( ;, = 1,2, L, m) T = n (2.26) ˆ + λ ˆ ( = 1,2, L, m) (2.27) m = 1 T = = = < < < n n n ( ˆ + ˆ ) + λk + λ ˆ (2.28) λ = 0 ˆ n = T ( = 1,2, L, m) (2.29) ˆ + ˆ m =1 ˆ = k (2.30) 6
(0) (0) (0) (0) ˆ ( ˆ, ˆ, L, ˆ ) n n T = ( = 1,2, L, m) (2.31) ˆ + ˆ ˆ 1 2 = 1 m n r (0) = (0) (0 ˆ + ˆ ) ( = 1,2, L, m) (2.32) ˆ (1) T = r (0) (1) = 1 ˆ (1) (2.33) (1) (1) k ˆ ˆ = m ( = 1,2, L, m) (2.34) ˆ ( k ) (2.32)(2.34) ˆ ( = 1,2, L, m) 7
3 FIFA A FIFA [2] A FIFA 2 W A 3.1 1995 1 1 2006 7 24 FIFA 207 A 9,627 3 90 PK BT 0.5 0.5 3.2 FIFA FIFA 3.1 A FIFA FIFA 1995 2 2006 7 129 FIFA 3.1 FIFA 8
4 BT 2 BT BT 4.1 BT FIFA 3BT BT 4.1 2006 7 FIFA 40 4.1 4.2 2 4.1 FIFA 6 FIFA FIFA 2006 7 46 53 52 11 35 10 2003 1 1 2006 7 24 BT (2.28) 0 2003 1 1 2006 7 24 13 194 4.1 1 2 2006 W 2 4.1 9
FIFA FIFA 2 4.2 2 y = 0.9473x + 5.1201 (4.1) R 2 0.8956 2 FIFA 2 BT 2 BT FIFA 2 FIFA 10
4.1 FIFA 11
4.1 4.2 FIFA 12
4.2 BT FIFA 1 1 2 4.1 BT FIFA 4.3 4.8 1 FIFA FIFA FIFA BT 60 4.8 4.9 1995 1 1 2006 7 24 4 1995 1 6 1997 10 4 48 1997 10 11 1998 6 26 17 1998 10 28 2002 6 18 55 2002 10 16 2006 6 22 72 1995 1 1 2006 7 24 4.1 5 FIFA FIFA 1998 4.32 1998 2002 FIFA 4 15 2 1999 2002 13
W 2006 4.4 FIFA 2 1998 2000 4.5 FIFA 2002 1 4.62 2001 2005 FIFA 2 4.5 4.7 FIFA 1998 1999 2006 4.8 4.9 4.8 2002 W 16 1998 W 4.3 FIFA 14
4.4 4.5 4.6 15
4.7 4.8 16
4.8 4.9 4.3 17
4.2 1 3 BT 4.1 3 130 4.1 18 4.10 4.27 4.10 4.13 4.12 4.14 4.22 4.11 4.1 1 H A N 6 4.10 4.28 4.10 45.4 59 1.5 2 130 116 4.28 BT 18
4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 19
4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 20
4.26 4.27 4.10 4.28 4.4 FIFA 4.1 FIFA 1995 1 1 21
2006 7 24 BT FIFA 2005 2006 1 2 2 FIFA 4.11 4.29 4.11 1 1 2 3 2 1 1 2 FIFA 1 2 4.29 4.11 FIFA 22
4.29 23
5 BT W BT 1998 2002 2006 W 5.1 W BT 2 2 BT (2.4) BT 2 (5.1) + S = (5.1) 2 S = S ) 2 2 ( (2.2)(5.2) H = (log( )) (log( )) (5.2) H = H ) 2 = 0 = 1 ( = 1 = 0 H = 0 = 0. 5 H = 1 = 2 24
(5.1)(5.2) I = I ) ( I = S H (5.3) 4.1 5 5.1 5.1 1 3 2 1 3 3 5 2 2 3 5 (5.3) 2 5.1 5.1 25
5.2 W 5.2.1 W 1998 2002 2006 3 W 32 4 A H 8 2 16 1 1 A 1 B 2 1998 W 2002 W 2006 W 32 5.2 5.4 4.2 3 53 1995 1998 1999 2002 2003 2006 BT 5.2 1998 W 5.3 2002 W 26
5.4 2006 W 5.2.2 (5.3) I 1 W 1 1 15 m 1 8 m = 1,2, L, 8 2 4 m = 9, L, 12 2 m = 13, 14 1 m = 15 w 1 w = 1 2 w = 2 w = 4 w = 8 m I m ( = I ) 1 8 4 2 2 2 2 = 16, 384 t I t 15 I = w ( t = 1,2, L,16384) (5.4) t I m m= 1 2 t 1 I 16384 = I t I t (5.5) 1998 W 2002 W 2006 W 1 I Real I max t= 1 27
I mn I 5.2 5.2 I Real 1998 W 157.82006 W 120.32002 W 70.9 I I Real 1998 W 128.82006 W 99.12002 W 66.5 1998 W 2006 W 2002 W (5.1) IReal I max I mn I 5.2 W 5.2.3 1 (5.5) I 16 I 16 16 1998 W 2002 W 2006 W I 16 1998 W 2002 W 2006 W I 1 12 8 = 429,981,696 1 f I E ( f = 1,2, L, 429981696 ) f 28
f BT f f E f E f E f E = f E f (5.6) 1998 W 2002 W 2006 W 1 E Real 1 E max 1 E mn E f E 5.3 5.3 E 1998 W 118.82006 W 94.62002 W 75.0 2002 W E Real E 3 1998 W 2006 2002 W f EReal Emax Emn E 5.3 29
5.3 W 1 W 1 1 1 1 1 2 1 1 2 16 1 1,625,702,400 [3] 1 4 2 4 8 1 3 1 1 4.1 2006 W 1 8 1 2 5.5 1 5.4 2 5.5 1 63.8 66.4 8 24 16 8 1 264,600 16 30
5.5 5.4 31
6 6.1 BT W FIFA 2 1998 2002 2006 W 1998 W 2002 W 2002 W 6.2 BT W 8 W 16 W 32
33
[1] 1988 [2] <htt://www.ffa.com> [3] 2000 [4] 2000 [5] W 2006 6 319327 34