併殺を考慮したマルコフ連鎖に基づく投手評価指標とそ Titleの 1997 年度日本プロ野球シーズンでの考察 ( 最適化のための連続と離散数理 ) Author(s) 穴太, 克則 Citation 数理解析研究所講究録 (1999), 1114: 114-125 Issue Date 1999-11 URL http://hdlhandlenet/2433/63391 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
1114 1999 114-125 114 1997 $\text{ ^{ } }\dot{\text{ }}$ KATSUNORI ANO, NANZAN UNIVERSITY, DERA MDERA, MDERA 1, (, ERA(Earned Run Average) ),, ERA 1,,,, ERA ERA, 9 Defensive Earned Run Average(DERA) DERA, 1, DERA ERA, DERA,,, (1), (2), 1,,,,,,, DERA (Modified Defensive Earned Run Average, MDERA $[] 4h4\mathrm{a}\mathfrak{B}$ Typaeet
115 ), ERA,DERA,MDERA, 1997 DERA, Cover and Keilers(1977) 9 Offensive Earned Run Average(OERA) OERA, (1998) (1993) OERA $\mathrm{n}_{\mathrm{u}\mathrm{e}\mathrm{m}}\mathrm{a}\mathrm{n}(1977)$,, D Espo and Leflcowitz(1977), Bellman(1977) MDERA, 9 DERA, ERA DERA, MDERA,, 2 MDERA 3 1997,, 4 $R^{2}$, ERA, DERA MDERA, MDERA Mathematica 301,,, $5\mathrm{J}$ SPSS 7 $\mathrm{t}$ 2 MDERA, $S$ $S=\{0,1, \cdots, 24\}$, $0$, 1 $\cdots$ 1 2,, 24, (1)
$\cdot$, 116 (2), (3) (4), (5),, (6), $-$ $\text{ ^{ } }$ ( \rightarrow $2arrow$ - \vdash $arrow$ ( $5arrow$ 18) - $arrow$ ( $6arrow$ 20) 17) - $arrow$ ( $8arrow$ 23) - ( $arrow$ $10arrow$ $-$ $arrow$ ( $13arrow$ - $arrow$ ( $14arrow$ - $arrow$ ( $16arrow$ $0$ ) $0$ ) $0$ ) $0$ ) Po=Pr( $+$ ) $= \frac{}(\text{ ^{ } ^{ } })}{\text{ }$ -ph=pr( ) $=\text{ },\text{ _{}S\text{ } _{ }}+\text{ }$ PB=Pr( ) $= \frac{ }{\text{ }+-\S \text{ }}$ pl=pr( ) $= \frac{ }{\text{ _{}+}\text{ ^{ }}}$ p2=pr( q ) $= \frac{ _{}-}^{-\text{ }}*\mathrm{j}\text{ }{\text{ }\mathrm{f}\mathrm{t}\text{ }+\text{ }}$ $Ps$ $= \frac{ _{}-}^{-}-\text{ }t\mathrm{t}\ddot{\text{ }}{\text{ }*\mathrm{r}\text{ }+\text{ ^{ }}}$ =Pr( ) p4=pr( ) $= \frac{}n\text{ }\mathrm{r}\text{ }{\text{ }+\text{ }}$ 6 2,5,6,8,10,13,14,16,
$T=24171615141312110\mathfrak{g}11\cdot\cdot$ 117 1 $i$ $j$ 1 p( ) $P=(P_{ij})=$ $p(j i),$ $i,j=0,1,$ $\cdots,24$ $P= _{T}^{1}$ $Q0$, $T,$ $Q$ $,$ $Q=2417169\cdot\cdot[_{Q_{0}}81,Q_{11}Q_{0}$ $Q_{11}Q_{12}Q0$ $Q_{1}1Q12]Q_{H}$, $Q_{0}=0$ (8 $\mathrm{x}8$ ), $Q_{11}=$ $Q_{12}=-[_{-}^{0}p_{0}+p_{H}000000p00000000$ $\prime p0+ph\mathrm{o}000000$ $p0+ph000\mathrm{o}\mathrm{o}00$ $p00000_{0}00$ $p0000\mathrm{o}_{0}\mathrm{o}0$ $p_{0}+0^{p_{h}}000000$ $p_{0}000]0000$,
$arrow$ 118 $Q_{H}=[^{p}0_{H}000000$ $p_{h}0000000$ $00000000$ $p_{h}0000000$ $00000000$ $00000000^{\cdot}p_{H}0000000$ $000000]00^{\cdot}$, 12( ) 20( ),,,, $p(20 12)=p0+PH$ $i$ $R(i)$, $R==241617981\cdot\cdot\cdot$ $[_{R_{1}}^{R_{1}}R_{1}]$,, $R_{1}=$, 10( - ), 1, 1, 2, 10 $R(10)=2p_{4}+p_{3}+p_{2}$
119 MDERA $i$ $E(i)$ 1 1,1 $E(1)$,E(l), 9 MDERA, MDERA, MDERA $=9E(1)$, $E(i)$ $i$ j $R(j, i)$ $\mathrm{k}\mathrm{e}\mathrm{r}\mathrm{l}\dot{\mathrm{m}}(\backslash 9194)$ First Step Analysis (, Taylor and ), $E(i)= \sum_{j=1}p(j i)\{r(j,i)+e(j)\},$, $i=1,$ $\cdots$ 24, (11) $i$ $\sum_{j=1}^{24}\mathrm{p}(j i)r(i,i)$, }, $R(i)$ $R(i)= \sum_{j=1}p(j i)r(j,i),$ $i=1,$ $\cdots,$ $24$,, $R,E$ $\mathrm{t}$ $R=$, $E=$, (11) $E=R+QE$, $E$, $E=(I-Q)-1R$ (12) $-$ $E(1)$ $E$, 9 3 1997 MDERA 1997, ;Y DERA MDERA,,, 1997, $=7\mathrm{x}$,
: 120 [2], p 1, 712, 364, 7 7 $\mathrm{x}$ $17=119$,,, ), ( $\overline{\sigma}\underline-$ - MDERA MDERA 7 $1\sim 4$ 5\sim 7 3 \sim 11 MDERA $\mathrm{w}$ $\mathrm{h}$, AB:, :, $\mathrm{l}$:, $2\mathrm{B}$ $3\mathrm{B}$, :, :, $\mathrm{h}\mathrm{r}$:, $\mathrm{b}\mathrm{b}$:, $\mathrm{d}\mathrm{p}$:, $\mathrm{h}$ ERA:, ( 1: ) $2\mathrm{B},3\mathrm{B},\mathrm{H}\mathrm{R}$ 2: DERA, MDERA $()$
121 $*\cdot$ $J\mathrm{X}$ 1,, ERA DERA ERA MDERA J\o, DERA $\int\backslash ^{\mathrm{q}}\cdot$ 2, : $\iota y-f,$ ERA MDERA $*\cdot$ ERA 6, 8 ( ) ( ) MD\ ERA $J\mathrm{X}$ 1, 2, ERA 8, 10 ( ), (ff) MDERA 1, 2 ERA, ERA=( x9)/,, 1 MDERA,,,,,,,, 1, MDERA, MDERA, 1997, $\text{ }$ $(\dot{\text{ }})$ ( ), ( ),, (ff) ERA MDERA
122 4 ERA MDERA ERA, ERA DERA MDERA, DERA MDERA,, $\tau$ $\rho$ DERA, MDERA Kendall, Spearman $J\backslash ^{\mathrm{o}}\cdot$ $\rho$, 97 ERA-DERA Spearman 5%, 7 1%,, ERA-MDERA ERA-DERA, ERA $R^{2}$ DERA MDERA 4 8, 8 1% $\mathit{1}\backslash ERA, DERA MDERA,, MDERA $R^{2}$ $R^{2}$,, ERA, DERA MDERA ^{\mathrm{o}}\cdot$
123 DERA, MDERA Mathematica,,, [1],, Working Paper Series, No9804, Center for Management Studies, Nanzan University, 1998 [2], 1997, (,, 1997 [3] R Bellman, Dynamic Programming and Markov Decision Processes, with Application to Baseball, Optimal Strareqies in Sports, North-Holland, New York, 1977, 77-85 [4] T M Cover and C W Keilers, An Offensive Earned-Run Average for Baseball, $\mathrm{i}977$ Operations Research, 25,, 729-740 [5] D A D Espo and B Lefkowitz, The Distribution of Runs in the Game of Baseball, Optimal Strareqies in Spon8, North-Holland, New York, 1977, 55-62 [6],,,, 1993 $\text{ }\mathfrak{f}$ [7],,,, 1988 [8] H M Taylor and S Karlin, An Introduction to Stochastic Modelin$q$, Revised Edition, Academic Press, San Diego, 1994 [9] R E Trueman, Analysis of Baseball as a Markov Process, Optimal Strareqies in Sports, North-Holland, New York, 1977, 68-76 [10], 1997,
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