1 ( 1 ) a 2016 6 14 FIVB ( ) 1 1 ( ) ( ) ( ) ( ) ( ) [1] [2] (point exchange) (Elo rating)[3] [4] (ranking) (rating) a 1-501 konaka@meijo-u.ac.jp
2 1. 2. 1. 1. 2. 2. 1. ( ) ( [5, 6, 7, 8, 9, 10] ) [11] ( ) Massey [12] [13] [14] 1 150 200 ( ) 1 1 ( ) ( ) 2
2 2.1 FIVB (FIVB) [2] 1 1 FIVB Ranking Point System Competition name Standing Olympic World Cup World Championship Men Women 1 100 100 100 100 2 90 90 90 90 3 80 80 80 80 4 70 70 70 70 5 50 50 62 58 6 40 56 7 30 50 50 8 25 9 30 5 45 45 10 5 11 20 5 40 40 12 5 13 Tie 36 36 15 Tie 33 33 17 Tie 30 30 21 Tie 25 25 ( ) ( 4 ) 1 4 10 5 4 5 20 8( ),12( ) (ATP )[15] ( ) 3
2000 1200 4 720 8 360 16 180 32 90 64 45 128 10 25 16 2 8 64 4 1 2 5/3 ( 1000 500 250 ) (1000 500 250) 2.2 FIVB ATP FIVB i ( i ) r i i j i p i,j 1 p i,j = 1+e (ri rj). (1) [3] [16] 0.1 1 25 1 25 22.6 2 r = r i r j 0,0.01,0.02,,0.20 10 4 1 3-2, 2-3 r i r j = 0.05 2-3 12.4% 2.2.1 [3] (1) p i,j = 1 1+10 (r i r j ) 400 (2) i,j *1 *1 200 76% 4
1 0.9 0.8 3 0 3 1 3 2 2 3 1 3 0 3 Probability 0.2 0.1 0 0 0.05 0.1 0.15 0.2 r 1 Won-lost sets probability [4] [17] i,j s i,j = { 1 i win. 0 i lose. r i = r i +K(s i,j p i,j ) (4) r j (4) 1 ( ) (3) 5
(4) K 16 K r i (4) r i *2 2.2.2 1 1 ( (4) K) 1 (4) 2 2 Notations N T r = (r 1,,r NT ) T Number of teams Rating vector N S i,j,s i,s j ǫ th K k Number of sets Result of one set. Team i and j scored s i and s j points in a set. N S tuples are stored in database. Threshold value Parameter used in rating update Iteration index 0, 1 Column vector composed of zeros and ones with suitable dimensions x Euclidean norm of vector x 1. r (0) = 0 ǫ th > 0 K > 0 10 3 k = 0 N S i,j,s i,s j 2. *2 6
3. i,j,s i,s j 4. r i r j p = 1 ( ),s = s i, (5) 1+e r (k) i r (k) j s i +s j r (k+1) i = r i (k)+k(s p), (6) r (k+1) j = r j (k)+k((1 s) (1 p)). (7) 5. r (k+1) r (k) < ǫ th r (k+1) k k +1 2. 3 ( ) 3.1 1 FIVB 1 2011 ( ) 2011 11 ( ) 2012 5 ( ) 2012 5 ( ) 2012 5 ( ) 2012 2 ( ) 2012 5 2012 2012 6 7 2012 7 8 2 2015 ( ) 2015 8 9 7
( ) 2016 1 ( ) 2016 1 ( ) 2016 1 ( ) 2016 2 1( ) 2016 5 3 2011 ( ) 2011 11 12 ( ) 2012 5 ( ) 2012 5 ( ) 2012 5 ( ) 2012 1 ( ) 2012 6 2012 2012 5 7 2012 7 8 4 2015 2015 5 7 2015 ( ) 2015 9 ( ) 2016 1 ( ) 2015 10 ( ) 2016 1 ( ) 2016 1 1( ) 2016 5 3.2 2 9 ( ) 3, ( (1)) 8
( ) 5 by set 5 JPN 5 5 5 0.25 0 2 Rate difference and scoring rate in each set (London Olympic 2012, Women) 3 Regression and correlation coefficients Regression coefficient Correlation coeficcient by Set by Game by Set by Game Dataset 1 0.9983 1.0025 110 909 Dataset 2 0.9956 0.9947 549 244 Dataset 3 1.0009 1.0010 313 775 Dataset 4 0.9980 0.9965 764 413 3.3 ( ) (1) ( ) 9
5 by game 5 5 5 5 0.25 JPN 0 3 Rate difference and scoring rate in each game (London Olympic 2012, Women) 3 1 ( ) ( ) 3.4 ( ) ( ) *3 *3 10
5 by set 5 JPN 5 5 5 0.25 0 4 Rate difference and scoring rate in each set (Rio WOQT 2016, Women) 1 (FIVB 3 ( )) ( 2 3 ) ( 2 ) ( 2 ) 4 2 ( 2 3 ) (3 ) (3 ) 1 2 1 [1] Stefani Ray. The methodology of officially recognized international sports rating systems. Journal of Quantitative Analysis in Sports, 7(4), 2011. [2] FIVB. FIVB volleyball world rankings. http://www.fivb.org/en/volleyball/rankings.asp. referred in 2016/6/14. 11
5 5 by game JPN 5 5 5 0.25 0 5 Rate difference and scoring rate in each game (Rio WOQT 2016, Women) [3] Arpad E. Elo. Ratings of Chess Players Past and Present. HarperCollins Distribution Services, hardcover edition, 1979. [4] World Rugby. Rankings explanation. http://www.worldrugby.org/rankings/explanation. referred in 2016/6/14. [5] Han Joo Eom and Robert W. Schutz. Statistical analyses of volleyball team performance. Research Quarterly for Exercise and Sport, 63(1):11 18, 1992. PMID: 1574656. [6] Eleni Zetou, Athanasios Moustakidis, Nikolaos Tsigilis, and Andromahi Komninakidou. Does effectiveness of skill in complex i predict win in men s olympic volleyball games? Journal of Quantitative Analysis in Sports, 3(4), 2007. [7] Lindsay W. Florence, Gilbert W. Fellingham snd Pat R. Vehrs, and Nina P. Mortensen. Skill evaluation in women s volleyball. Journal of Quantitative Analysis in Sports, 4(2), 2008. [8] Rui Manuel Araújo, José Castro, Rui Marcelino, and Isabel R Mesquita. Relationship between the opponent block and the hitter in elite male volleyball. Journal of Quantitative Analysis in Sports, 6(4), 2010. [9] Marco Ferrante and Giovanni Fonseca. On the winning probabilities and mean durations of volleyball. Journal of Quantitative Analysis in Sports, 10(2), 2014. [10] Tristan Burton and Scott Powers. A linear model for estimating optimal service error fraction in volleyball. Journal of Quantitative Analysis in Sports, 11(2), 2015. [11] Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 604 613. ACM, 1998. [12] Ken Massey. Massey rating. http://www.masseyratings.com/. referred in 2016/6/14. [13] Hope McIlwain Elizabeth Knapper. Predicting wins and losses: A volleyball case study. The College Mathematics Journal, 46(5):352 358, 2015. 12
5 by set 5 5 5 5 0.25 0.2 0.1 0 0.1 0.2 6 Rate difference and scoring rate in each set (London Olympic 2012, Men) [14] Sam Glasson, Brian Jeremiejczyk, and Stephen R. Clarke. Simulation of women s beach volleyball tournaments. Australian Society for Operations Research, 20(2):2 7, 2001. [15] ATP World Tour. Rankings FAQ. http://www.atpworldtour.com/en/rankings/rankings-faq. referred in 2016/6/14. [16]. 2.., 2012. [17] World Rugby. World rankings confirm japan s victory as biggest shock. http://www.rugbyworldcup.com/news/111746, 10 2015. referred in 2016/6/14. 13
5 by game 5 5 5 5 0.25 0.2 0.1 0 0.1 0.2 7 Rate difference and scoring rate in each game (London Olympic 2012, Men) 5 by set 5 JPN 5 5 5 0.25 0.2 0.1 0 0.1 0.2 8 Rate difference and scoring rate in each set (Rio WOQT 2016, Men) 14
5 5 by game JPN 5 5 5 0.25 0.2 0.1 0 0.1 0.2 9 Rate difference and scoring rate in each game (Rio WOQT 2016, Men) 15