vballOlympicPredictionLondon02.dvi

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1 (2) a 2016 6 28 FIVB ( ) 1 1 ( ) ( ) ( ) ( ) ( ) [1] [2] (point exchange) (Elo rating)[3] [4] (ranking) (rating) a 1-501konaka@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 3

)[15] ( ) 2000 1200 4720 8360 16180 3290 6445 12810 25 16 2 8 64 4 1 2 5/3 ( 1000500 250 ) (1000500250) 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, 17] 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) 4

Probability 1 0.9 0.8 0.2 0.1 0 0 0.05 0.1 0.15 0.2 r 1 Won-lost sets probability 3 0 3 1 3 2 2 3 1 3 0 3 i,j *1 [4] [18] 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) (3) *1 200 76% 5

1 ( ) (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 *2 6

1. r (0) = 0 ǫ th > 0 K > 0 10 3 k = 0 N S i,j,s i,s j 2. 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. 0,+ 2 ( ) r r r (maxr) 1 (8) 0 2.3 Bradley-Terry (BT )[19, 20] BT i j i j *3 p i,j π i,π j > 0 p i,j = π i π i +π j (9) p i,j = π i π i +π j = 1 1+ πj π i = 1+exp 1 1 ( ) = log πj 1+exp(logπ j logπ i ) π i (10) r i = logπ i (11) (1) *3 1 1 7

[1] BT [16, 17] a i b i θ j j j p i,j p i,j = 1 1+exp( 1.7a j (θ i b j )) k i j i j p i,j p i,j = 1 1+exp( a k (r i r j )) a k > 0 ( )BT (1) (12) (13) 3 ( ) 3.1 1 FIVB 1 2011 ( )2011 11 ( )2012 5 ( )2012 5 ( )2012 5 ( )2012 2 ( )2012 5 2012 2012 6 7 8

2012 7 8 2 2015 ( )2015 8 9 ( )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 9

3.2 2 9 ( ) 3, ((1)) ( ) 5 Probability/Scoring rate by set 5 Prediction ALL JPN Probability/Scoring rate 5 5 5 0.25 2016 Konakalab 0 Rating Gap 2 Rate difference and scoring rate in each set (London Olympic 2012, Women) 3 Regression and correlation coefficients based on proposed rating method 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 10

5 5 Probability/Scoring rate by game Probability/Scoring rate 5 5 5 0.25 Prediction ALL JPN 2016 Konakalab 0 Rating Gap 3 Rate difference and scoring rate in each game (London Olympic 2012, Women) 3.3 ( ) (1) ( ) 3 1 ( ) ( ) 11

5 Probability/Scoring rate by set 5 Prediction ALL JPN Probability/Scoring rate 5 5 5 0.25 2016 Konakalab 0 Rating Gap 4 Rate difference and scoring rate in each set (Rio WOQT 2016, Women) 3.3.1 FIVB 4 1( ) FIVB FIVB 10 13 FIVB FIVB 4 ( 1 3) ( 69 92 ) ( ) 3 3*( ) 12

5 5 Probability/Scoring rate by game Prediction ALL JPN Probability/Scoring rate 5 5 5 0.25 2016 Konakalab 0 Rating Gap 5 Rate difference and scoring rate in each game (Rio WOQT 2016, Women) 4 Correlation coefficients: FIVB ranking and scoring rate FIVB ranking gap by Set Correlation coefficient FIVB ranking point gap by Game by Set by Game Dataset 1 074 491 179 983 Dataset 1* 768 417 763 267 Dataset 2 078 578 658 901 Dataset 3 789 105 362 016 Dataset 3* 587 331 731 512 Dataset 4 624 380 171 641 *: except GBR. 3.4 13

5 Probability/Scoring rate by set 5 Prediction ALL Probability/Scoring rate 5 5 5 2016 Konakalab 0.25 0.2 0.1 0 0.1 0.2 Rating Gap 6 Rate difference and scoring rate in each set (London Olympic 2012, Men) ( ) ( ) *4 1 (FIVB 3 ( )) ( 2 3 ) ( 2 ) ( 2 ) 4 ( 3 ) 2 ( 23 ) (3 ) (3 ) 1 2 1 *4 14

5 5 Probability/Scoring rate by game Probability/Scoring rate 5 5 5 Prediction ALL 2016 Konakalab 0.25 0.2 0.1 0 0.1 0.2 Rating Gap 7 Rate difference and scoring rate in each game (London Olympic 2012, Men) FIVB ( ) (12) (24) (12) (28) 5 ( ) ( ) 15

5 Probability/Scoring rate by set 5 Prediction ALL JPN Probability/Scoring rate 5 5 5 2016 Konakalab 0.25 0.2 0.1 0 0.1 0.2 Rating Gap 8 Rate difference and scoring rate in each set (Rio WOQT 2016, Men) 16

5 5 Probability/Scoring rate by game Prediction ALL JPN Probability/Scoring rate 5 5 5 0.25 2016 Konakalab 0.2 0.1 0 0.1 0.2 Rating Gap 9 Rate difference and scoring rate in each game (Rio WOQT 2016, Men) 17

5 Probability/Scoring rate by set 5 Probability/Scoring rate 5 5 5 0.25 20 15 10 5 0 5 10 15 20 Rankig Gap 10 FIVB ranking difference and scoring rate in each set (Rio WOQT 2016, Men) 5 Probability/Scoring rate by set 5 Probability/Scoring rate 5 5 5 0.25 300 200 100 0 100 200 300 Rankig Point Gap 11 FIVB ranking point difference and scoring rate in each set (Rio WOQT 2016, Men) 18

5 Probability/Scoring rate by game 5 Probability/Scoring rate 5 5 5 0.25 20 15 10 5 0 5 10 15 20 Rankig Gap 12 FIVB ranking difference and scoring rate in each game (Rio WOQT 2016, Men) 5 Probability/Scoring rate by game 5 Probability/Scoring rate 5 5 5 0.25 300 200 100 0 100 200 300 Rankig Point Gap 13 FIVB ranking point difference and scoring rate in each game (Rio WOQT 2016, Men) 19

5 Numer of teams Competition name Teams Men Women Basketball FIBA Basketball World Cup 24 Basketball FIBA Women s Basketball World Cup 16 Basketball 2016 Summer Olympics 12 12 Hockey Hockey World Cup 12 12 Hockey 2016 Summer Olympics 12 12 Football FIFA World Cup 32 Football FIFA Women s World Cup 24 Football 2016 Summer Olympics 16 12 Handball World Men s Handball Championship 24 Handball World Women s Handball Championship 24 Handball 2016 Summer Olympics 12 12 Rugby (sevens) Rugby World Cup Sevens 24 16 Rugby (sevens) 2016 Summer Olympics 12 12 Ice Hockey IIHF Ice Hockey World Championships 16 Ice Hockey IIHF Ice Hockey Women s World Championship 8 Ice Hockey 2014 Winter Olympics 12 8 Curling World Curling Championships 12 Curling World Women s Curling Championship 12 Curling 2014 Winter Olympics 10 10 20

A IOC 3 [21] 6 Rating: Dataset 1 (just befor London Olympic Games, Women) RUS 358 1 USA 264 2 NED 0.2432 16 ARG 0.2072 17 CAN 0.0231 31 EGY 0.0147 32 BRA 129 3 PER 0.2011 18 GBR 0.0000 33 ITA 862 4 PUR 0.1871 19 FRA -0.0611 34 CHN 814 5 ALG 0.1860 20 SVK -0.1297 35 SEY -120 46 TUR 807 6 ROM 0.1832 21 UKR -0.1299 36 DEN -145 47 POL 546 7 GER 440 8 BUL 0.1533 22 TPE 0.1451 23 AUT -0.1931 37 MEX -0.2062 38 POR -326 48 SWE -717 49 JPN 174 9 KEN 0.1380 24 HUN -008 39 CHI -675 50 THA 165 10 AZE 0.0768 25 ISR -298 40 HON -034 51 SRB 957 11 CRO 0.0427 26 GRE -024 41 GEO -1.2756 52 CUB 775 12 ESP 0.0377 27 CRC -241 42 KOR 342 13 DOM 0.2939 14 CZE 0.0256 28 VEN 0.0246 29 BIH -341 43 TTO -431 44 BEL 0.2503 15 COL 0.0244 30 URU -896 45 [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. [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. 21

7 Rating: Dataset 2 (just before Rio WOQT, Women) BRA 283 1 CHN 605 2 USA 890 3 SRB 334 4 RUS 096 5 NED 035 6 GER 825 7 THA 691 8 JPN 613 9 ITA 555 10 TUR 0.2556 11 BEL 0.2306 12 CZE 0.1880 13 POL 0.1879 14 KOR 0.1647 15 PUR 0.1418 16 BUL 0.0839 17 DOM 0.0455 18 CAN 0.0407 19 ARG 0.0197 20 KAZ 0.0000 21 CRO -0.0466 22 CUB -0.0693 23 PER -0.1154 24 KEN -0.1893 25 VEN -0.2251 26 COL -0.2584 27 CMR -104 28 EGY -134 29 TUN -698 30 ALG -819 31 CHI -979 32 BOT -0.9086 33 UGA -0.9653 34 [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. [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] R. Hambleton. Fundamentals of Item Response Theory (Measurement Methods for the Social Science). Sage Publications, Incorporated, new. edition, 9 1991. [17] R. J. de Ayala. The Theory and Practice of Item Response Theory (Methodology in the Social Sciences). Guilford Pr, 1 edition, 12 2008. [18] World Rugby. World rankings confirm Japan s victory as biggest shock. http://www.rugbyworldcup.com/news/111746, 10 2015. referred in 2016/6/14. [19] L. L. Kupper P. V. Rao. Ties in paired-comparison experiments: A generalization of the bradley-terry model. Journal of the American Statistical Association, 62(317):194 204, 1967. [20] Roger R. Davidson. On extending the bradley-terry model to accommodate ties in paired comparison experiments. Journal of the American Statistical Association, 65(329):317 328, 1970. [21] International Olympic Committee. List of all national olympic committees in IOC protocol order. https://stillmed.olympic.org/media/document%20library/olympicorg/documents/national-olympic- 22

8 Rating: Dataset 3 (just befor London Olympic Games, Men) GER 141 1 POL 100 2 IRI 0.1628 16 SVK 0.1408 17 GBR 0.0000 31 GRE -0.0176 32 BRA 010 3 LAT 0.1279 18 NED -0.0378 33 AUT -0.1541 46 FRA 0.2678 4 FIN 0.1224 19 HUN -0.0432 34 ISR -0.1655 47 USA 0.2669 5 KOR 0.1073 20 PUR -0.0482 35 COL -0.2412 48 RUS 0.2551 6 JPN 0.1056 21 VEN -0.0537 36 MKD -0.2463 49 BEL 0.2541 7 ITA 0.2523 8 CHN 0.1017 22 EGY 0.0949 23 CMR -0.0576 37 PAK -0.0589 38 DOM -046 50 CHI -101 51 CZE 0.2420 9 POR 0.0758 24 UKR -0.0807 39 BIH -218 52 CUB 0.2373 10 ESP 0.0654 25 MNE -0.0955 40 TOT -786 53 AUS 0.2276 11 SLO 0.0496 26 DEN -0.1075 41 GHA -402 54 BUL 0.2140 12 EST 0.0430 27 ALG -0.1134 42 CRC -905 55 ARG 0.2076 13 SRB 0.1952 14 TUN 0.0391 28 TUR 0.0183 29 ROM -0.1307 43 CRO -0.1331 44 GEO -895 56 CAN 0.1632 15 IND 0.0043 30 MEX -0.1465 45 Committees/List-of-National-Olympic-Committees-in-IOC-Protocol-Order.pdf. referred in 2016/6/15. 23

9 Rating: Dataset 4 (just before Rio WOQT, Men) FRA 044 1 BRA 044 2 GER 018 3 USA 0.2764 4 POL 0.2542 5 SRB 0.2176 6 ITA 0.2152 7 IRI 0.1871 8 RUS 0.1734 9 ARG 0.1621 10 JPN 0.1013 11 BEL 0.0976 12 TUR 0.0871 13 MNE 0.0662 14 KOR 0.0632 15 BUL 0.0624 16 CZE 0.0612 17 CAN 0.0549 18 AUS 0.0549 19 NED 0.0503 20 CUB 0.0145 21 EGY 0.0084 22 ESP 0.0081 23 CHN 0.0060 24 GRE -0.0200 25 SVK -0.0472 26 ALG -0.0631 27 FIN -0.0699 28 POR -0.1014 29 TUN -0.1132 30 VEN -0.1180 31 CHI -0.1581 32 PUR -0.1585 33 KAZ -0.1740 34 MEX -0.1874 35 CMR -0.2662 36 COL -0.2954 37 CGO -238 38 COD -898 39 NGR -468 40 24

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