dvi

Size: px

Start display at page:

Download "dvi"

いぶきかやぬま
5 years ago
Views:

1 MCMC 1. SLG OBP OPS Albert and Benett, OPS Albert and Benett 2003 Albert

2 Albert p 1 (1.1) B(Kη,K(1 η)) pkη 1 (1 p) K(1 η) 1 (K>0, 0 <η<1). B( ) η K K p item response theory; IRT IRT Lord, 1952 IRT, Hambleton et al., 1991; De Ayala, 2008; Baker, 1992 Bock and Aitkin, 1981 MCMC Patz and Junker, 1999a, 1999b, Bilog-MG, 2005; Rizopoulos, 2006; Chalmers, 2012 Jong et al., 2008; Raykov and Calantone, 2014 IRT IRT 1 Albert 2008 p p 3 MCMC

3 y i,j j (j =1,...,n) i (i =1,...,l j) 0 1 l j j x i,j j i x i,j (1, 2,...,m) m 1 j y i,j p i,j p i,j =Pr(y i,j =1) p i,j (2.1) τ(x) (2.2) logit(p i,j) =β 0,j + β 1,jy i 1,j + τ(x i,j). b 1 x =1 b 2 x =2 τ(x) =.. x = m b m b k k =1,...,m 0 b k 0 σ 2 β 0,j μ ξ 2 β 1,j (, ), σ, ξ [0, ), μ (, ) β 1,j β 1,j 90% j h (2.3) exp(β 0,j)/ exp(β 0,h ) 0 0, 1

4 (2.4) exp(b j) (2.5) exp(β 1,j) j P (y i,j = 1) = logit 1 (β 0,j + β 1,jy i 1,j) (2.6) p 0,j = 1 1+exp( β 0,j) (2.7) p 1,j = (2.8) P j = 1 1+exp( (β 0,j + β 1,j)) ( ) 1 p0,j p 0,j 1 p 1,j p 1,j 2 Karlin, 1969 π j (2.9) π jp j = π j π j =(π 0j,π 1j), (2.10) π 0j = 1 p 1j 1+p 0j p 1j, π 1j = p 0j 1+p 0j p 1j π j π 1j j MCMC Bishop, 2006 MCMC IRT MCMC MCMC Fox 2010

5 239 MCMC IRT MCMC IRT Albert 1992 data augmetation 0 1 y i,j z z η i,j = β 0,j +β 1,jy i 1,j +τ(x i,j) 1 z i,j > 0 y i,j =1 z i,j 0 y i,j =0 y i,j =1 0 z i,j y i,j =0 0 Turnbull, 1976 z i,j l n j Φ(0 η i,j, 1) y i,j (1 Φ(0 η i,j, 1)) 1 y i,j (3.1) j=1 i=2 Albert 1992 z z η ij = β 0,j + β 1,jy i 1,j + τ(x i,j) 1 F (z η) 1 η, (3.2) N l j j=1 i=2 F (0 η i,j) y i,j (1 F (0 η i,j)) 1 y i,j, ( m )( m ) n l j π(β 0,β 1,b y)=c φ(b i, 0,σ) φ(β 0,i,μ,ξ) F (0 η ij) y ij {1 F (0 η ij)} 1 y ij (3.3) i=1 i=1 j=1 i=2 φ C Albert IRT β 0,j, β 1,j (j =1,...,75) b k (k =1,...,165) x i,j b k logit 1 (β 0,j + b k ) 1 1 logit 1 (β 0,j + b k ) 0

6 a b b β 0 c β 1 logit 1 (β 0,j + β 1,jy i 1,j + b k ) y i,j y i,j x i,j MCMC a 1 c b 0.98, β , β , 0.94, % α/2 (1 α) 100 (1 α) 90% 90% MCMC 5% 95% 90% 2 90% σ 1.05

7 % μ 0.11, ξ 0.98 σ =1,μ =0,ξ = MCMC Gelman 1996 ˆR Gelman 1996 ˆR ˆR 1.1 b k % b k 2 a 2 b b k b i

8 a b 2 b i a b 3 4 β 0,j β 0,j j β 0,j 90% 2.2 π 1j β 0,j, β 1,j β 0,j

9 β 1,j MCMC β 1,j 90% logit 1 (β 0) β 0 β 0 β 1 β 1 β 0 β 1 π Albert k l c kl k =1, 2, l =1, 2 j c 11 = I{yi 1,j =0,yi,j =0} c12 = i i I{yi 1,j =0,yi,j =1} c21 = i I{yi 1,j =1,yi,j =0} c22 = I{yi 1,j =1,yi,j =1} i

10 β 1 8 β 1 9 I % p

11 % j k j k (5.1) p 0,j,k = logit 1 (β 0,j + b k ). (5.2) p 1,j,k = logit 1 (β 0,j + β 1,j + b k ) j k n 0,j,k h 0,j,k j k n 1,j,k h 1,j,k h 0,j,k /n 0,j,k, h 1,j,k /n 1,j,k p 0,j,k, p 1,j,k h 0,j,k /n 0,j,k h 1,j,k /n 1,j,k p 0,j,k p 1,j,k 3 a 3 b n 0,j,k n 1,j,k n 0,j,k n 1,j,k n 0,j,k n 1,j,k β

12 a b 3 n a b

13 247 a b 4 a b a 4 b , b B No.15K21379

14 Albert, J. (1992). Bayesian estimation of normal ogive item response curve using Gibbs sampling, Journal of Educational Statistics, 17(3), Albert, J. (2008). Streaky hitting in baseball, Journal of Quantitative Analysis in Sports, 4(1), DOI: / Albert, J. and Bennett, J. (2003). Curve Ball, Springer-Verlag, New York., (2004). Baker, F. B. (1992). Item Response Theory: Parameter Estimation Techniques, Marcel Dekker, New York. Bilog-MG (2005). Scientific Software International, Bishop, C. (2006). Pattern Recognition and Machine Learning (Information Science and Statistics), Springer, Cambridge.,,,, (2008). Bock, R. D. and Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm, Psychometrika, 46(4), Chalmers. R. (2012). Mirt: A multidimensional item response theory package for the r environment, Journal of Statistical Software, 48(1), De Ayala, R. J. (2008). The Theory and Practice of Item Response Theory, The Guilford Press, New York. Fox, J. P. (2010). Bayesian Item Response Modeling Theory and Applications, Springer-Verlag, New York. Gelman, A. (1996). Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC Interdisciplinary Statistics, London. Hambleton R. K., Swaminathan, H. and Rogers, H. J. (1991). Fundamentals of Item Response Theory, Vol. 2, Sage Publications, New York. Jong, M. G. D., Steenkamp, J. B. E. M., Fox, J. P. and Baumgartner, Hans (2008). Using item response theory to measure extreme response style in marketing research: A global investigation, Journal of Marketing Research, 45(1), Karlin, S. (1969). A First Course in Stochastic Processes, Academic Press, Cambridge., (1974). 3 Lord, F. M. (1952). A theory of test scores, Psychometric Monographs, No. 7, Psychometric Corporation, Richmond. Patz, R. J. and Junker, B. W. (1999a). A straightforward approach to Markov chain Monte Carlo methods for item response models, Journal of Educational and Behavioral Statistics, 24(2), Patz, R. J. and Junker, B. W. (1999b). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, Journal of Educational and Behavioral Statistics, 24(4), Raykov, T. and Calantone, R. J. (2014). The utility of item response modeling in marketing research, Journal of the Academy of Marketing Science, 42(4), Rizopoulos, D. (2006). ltm: An R package for latent variable modeling and item response analysis, Journal of Statistical Software, 17(1), (2005). Turnbull, B. W. (1976). The empirical distribution function with arbitrarily grouped, censored and truncated data, Journal of the Royal Statistical Society, Series B (Methodological), 38,

15 Proceedings of the Institute of Statistical Mathematics Vol. 65, No. 2, (2017) 249 Measurements of Baseball Players Batting Abilities Ko Abe 1, Takenori Sakumura 2 and Toshinari Kamakura 2 1 Graduate School of Science and Engineering, Chuo University 2 Department of Industrial and Systems Engineering, Chuo University Statistics of player performance is an important part of baseball. Many stats have been proposed to measure a batter s performance, including batting average, on-base percentage, and slugging percentage. In the field of baseball analytics, the streakiness of batter s ability is often discussed using a binary sequence of hitting outcomes for a player during a season. Unlike previous studies, which use data from the batter, we take a different approach. To analyze a batter s performance, we simultaneously model the pitcher and batter s ability. To model a batter s streakiness, we employ an extension of a one-parameter logistic item response model. Item response theory (IRT) estimates both the subject s ability and item difficulty. In this study, the ability parameter and item difficulty parameter correspond to the batter s ability and pitcher s ability, respectively. Although simplicity is thought to make the one-parameter logistic model easy to interpret, our model incorporates numerous parameters. However, using the odds ratio allows athletes to be compared. We express streakiness by the interactions of previous at bats and imposing the Markov property on batting data. Specifically, we use MCMC in the Hamiltonian Monte Carlo method (also called the hybrid Monte Carlo method). The computation of Gibbs sampling is complex and time consuming, but the Hamiltonian Monte Carlo method is easily computed once the prior distribution and the likelihood function are defined. Our simulation study shows that the true and estimated values agree well. Additionally, the calculated proportion of times that the credible interval contains the true value is close to the nominal value. To demonstrate the usefulness of our proposed method, we applied it to analyze actual data from Japanese professional baseball. Two-way tables can measure the dependence of the previous success and the current success by the Pearson chi-square statistic and the corresponding p-value of the test of independence. The results provide more information and are consistent with the results of chi-square test. Because comparing streakiness in the hypothesis test is difficult, we ranked streaky players from the credible intervals and the posterior means. IRT requires many subjects to estimate item difficulty parameters. Although we estimated the parameters using fewer batters, the results from our method are similar to those from IRT. Key words: Bayesian hierarchical model, MCMC, sabermetrics, logistic model.

dvi

dvi 2017 65 2 185 200 2017 1 2 2016 12 28 2017 5 17 5 24 PITCHf/x PITCHf/x PITCHf/x MLB 2014 PITCHf/x 1. 1 223 8522 3 14 1 2 223 8522 3 14 1 186 65 2 2017 PITCHf/x 1.1 PITCHf/x PITCHf/x SPORTVISION MLB 30