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へいぞうさかど
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1 PITCHf/x PITCHf/x PITCHf/x MLB 2014 PITCHf/x

2 PITCHf/x 1.1 PITCHf/x PITCHf/x SPORTVISION MLB 30 PITCHf/x PITCHf/x 3 x y z x-z 50 x-z pfx Kagan, 2009 (1.1) x t = x 0 + tv 0 + t2 2 a t x (1.2) x t = y z x 0 v 0 a t x t (1.3) x t = x 0 + t v 0 + t 2 2 a x t x x (1.4) 0 = xt z z

3 187 x z MLB Gameday 2014 PITCHf/x MLB PITCHf/x PITCHf/x FF PITCHf/x, 2015 = / (2.1) = + 1

4 PITCHf/x 2 MLB2014 MLB Y i i MLB2014 (2.2) log p i 1 p i = α + β i, Y i Bernoulli(p i) z p ˆβ p

5 % %

6 PITCHf/x 2 4. f Hastie and Tibshirani, 1986 Thin plate regression spline Wood thin plate spline 3 thin plate spline 3 thin plate spline 1 x f (4.1) y i = f(x i)+ɛ i, ɛ i N(0,σ 2 ), q f(x) = β jb j(x) j=1 3 ɛ i b j(x) f β =(β 1,...,β q) T y =(y 1,...,y n) T (4.2) y = Xβ + ɛ

7 191 X i j X (ij) X (ij) = b j(x i) knot q 2 x 1 <x 2 < <x q 2 3 Wood 2006 Gu 2002 b 1(x) =1,b 2(x) =x, b j+2(x) =R(x, x j ) R(x, z) (j =1, 2,...,q 2) (4.3) R(x, z) = [ (z 1/2) 2 1/12 ][ (x 1/2) 2 1/12 ] /4 [ ( x z 1/2) 4 1/2( x z 1/2) 2 +7/240 ] / (4.4) f (x 1)=0,f (x q 2) =0 3 (4.5) V (β) = y Xβ 2 + λ f (x) 2 dx Ω β λ(> 0) Ω f (x) 2 dx λ Ω f f(x) = βjbj(x) β 2 j S (4.6) V (β) = y Xβ 2 + β T Sβ S S (i+2,j+2) = R(x i,x j ) (i, j =1, 2,...,q 2) V (β) β (4.7) ˆβ =(X T X + λs) 1 X T y λ λ λ GCV; Wood, 2008 REML; Wood, thin plate spline x =(x 1,x 2) T (x i,y i),i=1, 2,...,n

8 (4.8) y i = f(x i)+ɛ i f x 1,x 2 2 f J(f) ( ) 2 f J(f) = +2 2 f + 2 f (4.9) dx 1dx x 2 1 x 1 x 2 x η(r) =r 2 log(r)/(8π) n (4.10) f(x) =α 1 + α 2x 1 + α 3x 2 + δ iη( x x i ) Wood, 2006; Green and Silverman, 1994 E E (ij) = η( x i x j ), T i = (1,x 1i,x 2i), T = (T 1,T 2,...,T n) T, α = (α 1,α 2,α 3) T, δ = (δ 1,δ 2,...,δ n) T T T δ = 0 f thin plate spline i=1 (4.11) S(α, δ) = y Eδ T α 2 + λδ T Eδ T T δ = 0 J(f) f δ Thin plate spline 2 (4.12) y g(x) 2 + λj(g) Thin plate regression spline Wood, E k E k. g (4.13) g(μ i)=x iθ + f 1(x 1i)+f 23(x 2i,x 3i)+ R mgcv thin plate regression spline thin plate regression spline 4.2 thin plate regression spline

9 193 PITCHf/x MLB thin plate regression spline p i ( ) pi (4.14) log = α + β 1x i1 + β 2x i2 + + f 1(z i1)+f 23(z i2,z i3)+ 1 p i x i1,x i2,...,z i1,z i2,z i3,... f 1,f 23,... 5 t p p 5

194 65 2 2017 6 2 3 thin plate regression spline Wood, 2006 p p 5% null deviance

10 thin plate regression spline Wood, 2006 p p 5% null deviance deviance AIC AIC AIC AIC AIC AIC 2 2 / 2 y y 5 10 x 5 10

11 % %

12 %

13 197 split-finger fastball ( ) pi (5.1) log = α + β 1x i1 + β 2x i2 + + f 1(z i1)+f 23(z i2,z i3)+ + W iγ 1 p i 4.14 γ γ 2014 W i i k k 1 W i =[0,...,1,...,0] W iγ γ k γ k N(0,σ 2 γ) mgcv gam AIC AIC MLB 2016

14 PITCHf/x

15 199 PITCHf/x 2 5 JSPS Core-to-Core Program Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models, Chapman and Hall, New York. Gu, C. (2013). Smoothing Spline ANOVA Models, Springer, New York. Hastie, T. and Tibshirani, R. (1986). Generalized additive models, Statistical Science, 1, Kagan, D. (2009). The anatomy of a pitch: Doing physics with PITCHf/x data, The Physics Teacher, 42, (2015) (2016). ICT Wood, S. N. (2003). Thin plate regression splines, Journal of the Royal Statistical Society, Series B, 65, Wood, S. N. (2006). Generalized Additive Models: An Introduction with R, Chapman and Hall, New York. Wood, S. N. (2008). Fast stable direct fitting and smoothness selection for generalized additive models, Journal of the Royal Statistical Society, Series B, 70, Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation semi-parametric generalized linear models, Journal of the Royal Statistical Society, Series B, 73, 3 36.

16 200 Proceedings of the Institute of Statistical Mathematics Vol. 65, No. 2, (2017) Factors Affecting Batters Contact with a Four-seam Fastball Daiki Nagata 1 and Mihoko Minami 2 1 Graduate School of Science and Technology, Keio University 2 Department of Mathematics, Keio University In baseball, nobi is a four-seam fastball in which a batter has trouble making contact. Our research aims to understand the origin of nobi. It has been speculated that the velocity a four-seam fastball with nobi does not change much from the time it leaves the pitcher s hand to when it crosses the plate. Our previous analysis of nobi using PITCHf/x, which is a system that measures data such as the coordinates and break of a pitch by tracking the ball s trajectory, revealed the opposite relation. Consequently, we applied a logistic regression model to explain bat contact by the difference in the ball speed after defining the batter s contact with a pitch. A negative relation was obtained. This study focuses on the break of a pitch. We analyzed the relationship between the break of a pitch and contact quantitatively. Additionally, we investigated the break of the ball by a generalized additive model using a multivariate spline smoothing method to evaluate the relationship between the break of the ball and bat contact. Vertical breaks are important. Moreover, adjusting the model to replace pitch quality as a random effect with hitting difficulty by pitcher revealed that in the 2014 MLB (Major League Baseball) season, Uehara was the most difficult pitcher for batters to face. Key words: PITCHf/x data, four-seam fastball, nobi, break of the pitch, generalized additive model, random effect.

dvi

dvi 2017 65 2 217 234 2017 Covariate Balancing Propensity Score 1 2 2017 1 15 4 30 8 28 Covariate Balancing Propensity Score CBPS, Imai and Ratkovic, 2014 1 0 1 2 Covariate Balancing Propensity Score CBPS