Evaluation Model for Batter based on

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1 Evaluation Model for Batter based on Markov Chain Katsunori Ano Faculty of Business, Nanzan University Abstract We propose an improved Offensive Earned-Run Average(OERA) value which is based on a stationary Markov Chain and is a well-known index to evaluate a contribution of batter in baseball game However an original OERA value doesn t consider steal s effect It seems to be unrealistic Proposed value in this paper includes the effect Some-examples are demonstrated 1,,,,,, \iota / Cover and Keilers[1] OERA(Offensive Earned-Run Average) $-\tau \text{ }\triangleright$, OERA 1 \iota / OERA 2 95 \nearrow o 25, 86 )$\ddagger$ -p\check, 49 OERA $97469(342$, ), 6 ( 302, 31, 98 ) OERA ,, j OERA,, $\text{ }\triangleright$ OERA (TOERA ),, OERA TOERA TOERA, (49, 9 ), (6, 6 ), TOERA,,

2 46 2 TOERA $\mathfrak{l}\mathrm{h}\llcorner$ OERA, - 9,,,,,,, , 5,, 6 7 1, 2 $\text{ }<><><><><>$ $\mathrm{o}$ $S$ $0$ $S=\{0,1, \ldots, 24\}$,, $-$ 1, 1 2,, 24

3 $\cross$ \frac{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ ^{}>}\mathrm{i}\mathrm{t}\text{ }+\mathfrak{w}\text{ }}$ \frac{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\mathfrak{w}\text{ }\exists\sigma*\text{ }}$ $\cross$ \frac{}-\bigoplus_{-\subseteq \mathrm{i}\rightarrow\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }}{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\text{ }}$ $\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}^{-}\text{ _{}\mathit{0})_{-\text{ }\ovalbox{\tt\small REJECT} \text{ }\mathrm{t}\text{ }k\text{ _{}-\text{ }^{}-}}{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }$ REJECT} \mathrm{t}\text{ }\not\subset)---\text{ }\mathrm{t}fi\text{ }{-\text{ }-*\mathrm{t}\not\subset\rangle \text{ _{}\mathit{0})}\underline{=}\text{ }\acute{\overline{\tau}}\tau \text{ }}$ REJECT} \mathrm{t}\text{ }X\iota X\text{ }{--\subseteq \mathrm{i}\fallingdotseq\ovalbox{\tt\small REJECT}-\mathbb{E}\Xi \mathrm{t}\text{ ^{}-}\overline{-}3\acute{r}\overline{\mathrm{t}}\text{ }}$ 47 $p_{0}$ $=$ Pr( ) $=$ $p_{b}$ $=$ P7FI ) $=$ $p_{1}$ $=$ Pr( $p_{2}$ $=$ Pr( ) $=\overline{\text{ }*\mathrm{t}\text{ }+\text{ }F\mathrm{f}\text{ }}$ ) $= }} }$ $\text{ }$ $\frac{ $p_{3}$ $=$ Pr( ) $= $\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ _{}-\text{ }^{}-}$ $\text{ }$ $= \frac{}-\text{ }-\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\mathrm{h}\text{ }\mathfrak{x}i^{\backslash }\mathrm{r}\text{ }}$ $p_{4}$ $=$ Pr( ) $\frac{}-\text{ }-\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }l;\text{ }---\text{ }{-\leq 1-\mathrm{a}\mathrm{e}\doteqdot\ovalbox{\tt\small REJECT} \mathrm{t}\text{ ^{}\underline{=}}\text{ }\uparrow\urcorner-\text{ }},$ $p_{5}$ $=$ $P_{r}$ ( $ $ $= CH ) $\frac{ }\underline{-}\text{ }\ovalbox{\tt\small REJECT} \text{ }\mathrm{t}\text{ }k\text{ }*\text{ }{-\text{ }-\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }}$ $\frac{}-\text{ }-\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\text{ }}$ $p_{6}$ $=$ Pr( ) $=,$ $\frac{}-=\mathrm{i}=-\mathrm{a}\mathrm{e}\ovalbox{\tt\small $p_{7}$ $=$ $P_{r}$ ( $ $ $= \frac{}\underline{=}\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\mathrm{h}\text{ }}$ $\frac{}--\text{ }-\ovalbox{\tt\small ) REJECT} \mathrm{t}\text{ }k\text{ }\sigma)\text{ }\ovalbox{\tt\small REJECT}\hslash \mathfrak{u}}{--=\mathrm{i}\cong-\mathrm{a}\mathrm{e}\neq \mathrm{t}\text{ _{}(\urcorner}^{\prime-\text{ }}$ $p_{8}$ $=$ $\overline{p_{r}^{p}-}$ $= \frac{}--\text{ }-\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\mathrm{h}\text{ }}$ $\frac{ }\overline{\underline{-}}\text{ }\ovalbox{\tt\small REJECT} \text{ }\mathrm{t}\not\subset)\text{ }*\text{ }{\underline{=}\text{ }\neq \mathrm{t}\text{ }}$ ( ) $= \frac{}--\text{ }-\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }{\text{ }\ovalbox{\tt\small REJECT} \mathrm{t}\text{ }+\text{ }\ddagger*\text{ }}$ $p_{9}$ $=$ Pr(H ) $\frac{}\underline{=}\text{ }\ovalbox{\tt\small $p_{10}$ $=$ Pr( ) $=$ $P=(P_{ij})=p(j i)$, $i,$ $j=0,1,$ $24$ $\cdots,$ $P$ $=$, $T,$ $Q$

4 $ \cdot\cdot$ $p_{b}+p_{2}p_{2}p_{2}p_{2}0000$ $p_{3}+p_{5}p_{3}+p_{5}p_{3}+p_{5}p\mathrm{s}+p_{5}p_{5}p_{5}p_{5}p_{5}$ $p_{6}+p_{8}p_{6}+p_{8}p_{6}+p_{8}p_{6}+p_{8}p_{6}+p_{8}p_{6}+p_{8}p_{6}+p_{8}p_{6}+p_{8}$ $p_{0} $ $p_{0} $ $ ^{\cdot}\cdot\cdot$ $p_{b}p_{b}000000$ $p_{1}p_{0}p_{1}p_{1}p_{1}000$ $T=$, $Q=$, $Q_{11}$ $=$ $[p_{9}+p_{10}p_{9}+p_{10}p_{9}+p_{10}p_{9}+p_{10}p_{9}+p_{10}p_{9}+p_{10}p_{9}+p_{10}p_{9}+p_{10}$ $p_{b}p_{2}p_{2}p_{2}p_{2}000$ $p_{3}p_{3}p_{3}p_{3}0000$ $p_{b}p_{b}p_{b}p_{b}0000]$, $Q_{12}$ $=$ $[p_{0}+p_{1}+p_{4}+p_{7}p_{1}+p_{4}+p_{7}p_{1}+p_{4}+p_{7}p_{1}+p_{4}+p_{7}p_{4}+p_{7}p_{4}+p_{7}p_{4}+p_{7}p_{4}+p_{7} $ $p_{0} $ $p_{0} $ $p_{0} $ $p_{0} ]$, $Q_{13}=Q_{21}=Q_{31}=Q_{32}=0$ $Q_{11}=Q_{22}=Q_{33}$, $Q_{12}=Q_{23}$ (8 $\mathrm{x}8$ ),,, 1( ) 4( ),, 1 4 $p(4 1)=p_{6}+p_{8}$ TOERA, $i$ $E(i)$ 1 1 $E(1)$, $E(1)$,

5 $\text{ }\triangleright E$ 49 9 TOERA, TOERA (11) TOERA $=$ $9E(1)$, $i$ $E(i)$ $j$ $R(j, i)$ $\text{ }\mathrm{t}$, First Step Analysis (,[5] ), $E(i)= \sum_{j=1}^{24}p(j i)\{r(j, i)+e(j)\}$, $i=1,$ $\cdots,$ $24$ $i$ $\Sigma_{j=1}^{24}p(j i)r(j, i)$, $R(i)$ $R(i)$ $= \sum_{j=1}^{24}p(j i)r(j, i),$ $i=1,$ $\cdots$,24, $R,$ $E$ $R=$, $E=$,, (12) $E$ $=$ $QE+R$, E (12) $E$ $=$ $(I-Q)^{-1}R$ 1 1 $E(1)$, TOERA (12) $Q$,, $(I-Q)^{-1}$ \nabla $(I -Q)^{-1}=I+Q^{2}+Q^{3}+\cdots$ 1 $i$ $i,j$,, $j$ $j$ $R$ li $i$, 1 $R(i)$, 1 $R$ $=$ 24

6 ,,\nu 50 $R_{1}$ $=$ $R_{2}=$, $R_{3}$ $=$ $R_{1}+$, 2( 1 ),,, 2,,,,,,, 1, 2 $R(2)=2(p_{9}+p_{10})+p_{4}+p_{5}+p_{6}+p_{7}+p_{8}$ 3,96 *y o $J -$ \check 30, 2 60,, 2, TOERA o, 7 \check -- $\cdot$ Table1, Table2 * OERA TOERA

7 Table 1 $96*\cdot J-pP\text{ }$ 30 OERA TOERA 51

8 52 $\mathrm{t}\circ \mathrm{h}1_{\circ} ) 06_{J\backslash }\cdot 1\mathrm{l}$ \eta 19$\mathrm{q}\cap$ OERA TOERA {

9 $\text{ }\triangleright$ 53 $ \text{ }\triangleright$ 1 $\overline{\tau} =$, TOERA OERA, 2 TOERA 9, (103399), (92045), ( )(94960), 3 OERA ( ) 06599, TOERA OERA TOERA, (35,3 ), ( ) (50,9 ),, 09, 06 4 $/\mathrm{t}\mathrm{o}\mathrm{e}\mathrm{r}\mathrm{a}$ TOERA,, $-$ $-$, -t \check TOERA } OERA, TOERA [1] Cover, TM, and Keilers, CW, An Offensive Earned-Run for Baseball, Operations Research, Vol25(1977), $\wedge^{\backslash ^{\backslash }}$ $-\text{ }\triangleright$ [2], 1996 $-$,, 1997 [3],,,, 1993 [4],,,,, 1988, [5] Taylor, HM, and Karlin, S, An Introduction to Stochastic Modeling, Revised Edition, Academic Press, San Diego, 1994

Baseball and Statistics: Rethinking Slugging Percentage. Tanner Mortensen. December 5, 2013

Baseball and Statistics: Rethinking Slugging Percentage Tanner Mortensen December 5, 23 Abstract In baseball, slugging percentage represents power, speed, and the ability to generate hits. This statistic