project520.Rnw

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\begin{document}
<<setup, include=FALSE, cache=FALSE>>=
library(knitr)
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opts_chunk$set(fig.path='figure/', cache.path='cache/', fig.align='center', out.width='8cm', out.height='6cm', echo=FALSE, message=FALSE)
@


\title{A Bayesian Model for Evaluating MLB Hitting Performance}


\author{Carson Sievert}

\maketitle

\section{Introduction}

Statistical methods have been used to understand and analyze the game
of baseball for many years. An obvious and popular application is
modeling and predicting player performance. \citet{bayesball} proposes
a bayesian model for predicting hitting performance at the season
level. This model was shown to be effective at prediction - especially
for players with little experience at the Major League level thanks
to a shrinking towards the population mean component. In the discussion
of that paper, reviewers call for a model at the game (rather than)
level. The reasoning is to account for well-known factors on performance
such as a ``park effects'' and ``home field effects''. This paper
explores a step in that direction by modeling home run performance
at the game level and accounting for any ``home field effect''.


\section{Data Collection}

The data used for this project was taken from the Major League Baseball
Advanced Media (MLBAM) website \url{http://gd2.mlb.com/components/game/mlb/}
using the \texttt{R} package \texttt{pitchRx} \citet{pitchRx}.
The number of home runs and number of atbats in every game over the
2012 season was collected for the five players with the highest home
run total in 2012.


\section{Model Formulation}

Since home runs are such a rare event, we may want some type of zero-inflated
model if we want to model the number of home runs in a particular
game. For that reason, consider a latent Bernoulli random variable
$X_{ij}$ that will govern whether player $i$ in game $j$ has ``potential''
to get at least one success in $n_{ij}$ atbats. The Bernoulli parameter
$\theta_{ij}$ could thought of as a players ``potential'' for a
particular game.

\[
f(X_{ij}=x_{ij}|\theta_{ij})=\theta_{ij}^{x_{ij}}(1-\theta_{ij})^{(1-x_{ij})},0<\theta_{ij}<1
\]


where 
\[
x_{ij}=\begin{cases}
0 & \mbox{no "potential success" }\\
1 & \mbox{"potential success" }
\end{cases}
\]
 

To model the actual number of successes for player $i$ in game $j$,
we use a Binomial random variable $Y_{ij}$ where the number of trials
$n_{ij}$ is considered fixed. In some sense, the success probability
$\psi_{ij}$ could thought of as a player's ``ability'' to hit a
home run in a particular game. To get the marginal distribution for
$Y_{ij}$, we could ``integrate out'' the latent $X_{ij}$. For
example,

\begin{align*}
P(Y_{ij}=0)= & P(Y_{ij}=0|X_{ij}=0)P(X_{ij}=0)+P(Y_{ij}=0|X_{ij}=1)P(X_{ij}=1)=(1-\theta_{ij})+\theta_{ij}(1-\psi_{ij})^{n_{ij}}\\
P(Y_{ij}=1)= & P(Y_{ij}=1|X_{ij}=1)P(X_{ij}=1)=n_{ij}\psi_{ij}(1-\psi_{ij})^{n_{ij}-1}\theta_{ij}\\
P(Y_{ij}=2)= & P(Y_{ij}=2|X_{ij}=1)P(X_{ij}=1)=\binom{{n_{ij}}}{2}\psi_{ij}^{2}(1-\psi_{ij})^{n_{ij}-2}\theta_{ij}\\
\vdots
\end{align*}
\begin{align*}
\end{align*}


We can describe this distribution in general by:
\[
\]


\[
P(Y_{ij}=y_{ij})=\begin{cases}
(1-\theta_{ij})+\theta_{ij}(1-\psi_{ij})^{n_{ij}} & \mbox{if }y_{ij}=0\\
\binom{{n_{ij}}}{y_{ij}}\psi_{ij}^{y_{ij}}(1-\psi_{ij})^{(n_{ij}-y_{ij})}\theta_{ij} & \mbox{if }y_{ij}\in\{{1,2,\dots,n_{ij}}\}
\end{cases},0<\psi_{ij}<1
\]


Now we consider the parameters $\theta_{ij}$ and $\psi_{ij}$ to
be unknown quantities and can be described through the deterministic
relationships: 

\[
\log(\frac{\theta_{ij}}{1-\theta_{ij}})=\beta_{i}+\alpha_{i}H_{ij}
\]


\[
\log(\frac{\psi_{ij}}{1-\psi_{ij}})=\delta_{i}+\lambda_{i}H_{ij}
\]


where $H_{ij}$ is an indicator variable that is 1 if game $j$ was
a ``home'' game for player $i$ and 0 otherwise. This leaves use
to choose distributions for the (independent) set of random quantities
$\beta_{i},\delta_{i},\alpha_{i},\lambda_{i}:$

\begin{align*}
 & \beta_{i}\sim N(\beta_{0},\sigma_{\beta}^{2})\\
 & \alpha_{i}\sim N(\alpha_{0},\sigma_{\alpha}^{2})\\
 & \delta_{i}\sim N(\delta_{0},\sigma_{\delta}^{2})\\
 & \lambda_{i}\sim N(\lambda_{0},\sigma_{\lambda}^{2})
\end{align*}


where $\beta_{0},\sigma_{\beta},\alpha_{0},\sigma_{\alpha},\delta_{0},\sigma_{\delta},\lambda_{0},$
and $\sigma_{\lambda}$ are all ``known'' quantities that are chosen
such that we have diffuse priors that reflects a lack of prior knowledge.
In particular, the mean and standard deviation was set to 0 and 5,
respectively.


\section{Fitting the model}

This section addresses how samples from the joint posterior distribution
were obtained. Full conditional posterior distributions were derived
for all quantities involved (besides the observed data). A Gibbs sampling
algorithm was employed on these full conditionals to obtain draws
from the joint distribution. 

\[
p(\beta_{i},\delta_{i},\alpha_{i},\lambda_{i}|\textbf{{\ensuremath{y_{i}}}})\propto p(\textbf{{\ensuremath{y_{i}}}}|\beta_{i},\delta_{i},\alpha_{i},\lambda_{i})p(\beta_{i})p(\delta_{i})p(\alpha_{i})p(\lambda_{i})
\]


The notation $p(x|\cdot)$ is now used to represent the conditional
density of $X$ given all other quantities. Note that $y_{i}$ is
used to represent a vector whose length is equal to the number of
games played by player $i$.


\subsection{Full conditionals}
\begin{enumerate}
\item The full conditional densities of $\beta_{i}$ for $i=1,\dots,5$
are:
\[
p(\beta_{i}|\cdot)\propto p(y_{i}|\cdot)p(\beta_{i})\propto p(y_{i}|\cdot)exp\left\{ \frac{{-(\beta_{i}-\beta_{0})^{2}}}{2\sigma_{\beta}^{2}}\right\} 
\]
In order to sample from this form, an \emph{adaptive} Metropolis-Hastings
algorithm is used with a Gaussian proposal $N(\beta_{i}^{(k)},(\tau_{\beta}^{2})^{(k)})$
where $\beta_{i}^{(k)}$is the simulated value from the $k^{th}$
iteration. The adaptive piece helps to obtain a reasonable acceptance
rate by increasing or decreasing $(\tau_{\beta}^{2})^{(k+1)}$ based
on whether or not $k^{th}$ proposal was accepted.
\item The full conditional densities of $\delta_{i}$ for $i=1,\dots,5$
are:
\[
p(\delta_{i}|\cdot)\propto p(y_{i}|\cdot)p(\delta_{i})\propto p(y_{i}|\cdot)exp\left\{ \frac{{-(\delta_{i}-\delta_{0})^{2}}}{2\sigma_{\delta}^{2}}\right\} 
\]
In order to sample from this form, an \emph{adaptive} Metropolis-Hastings
algorithm is used with a Gaussian proposal $N(\delta_{i}^{(k)},(\tau_{\delta}^{2})^{(k)})$
where $\delta_{i}^{(k)}$is the simulated value from the $k^{th}$
iteration. The adaptive piece helps to obtain a reasonable acceptance
rate by increasing or decreasing $(\tau_{\delta}^{2})^{(k+1)}$ based
on whether or not $k^{th}$ proposal was accepted.
\item The full conditional densities of $\alpha_{i}$ for $i=1,\dots,5$
are:
\[
p(\alpha_{i}|\cdot)\propto p(y_{i}|\cdot)p(\alpha_{i})\propto p(y_{i}|\cdot)exp\left\{ \frac{{-(\alpha_{i}-\alpha_{0})^{2}}}{2\sigma_{\alpha}^{2}}\right\} 
\]
In order to sample from this form, an \emph{adaptive} Metropolis-Hastings
algorithm is used with a Gaussian proposal $N(\alpha_{i}^{(k)},(\tau_{\alpha}^{2})^{(k)})$
where $\alpha_{i}^{(k)}$is the simulated value from the $k^{th}$
iteration. The adaptive piece helps to obtain a reasonable acceptance
rate by increasing or decreasing $(\tau_{\alpha}^{2})^{(k+1)}$ based
on whether or not $k^{th}$ proposal was accepted.
\item The full conditional densities of $\lambda_{i}$ for $i=1,\dots,5$
are:
\[
p(\lambda_{i}|\cdot)\propto p(y_{i}|\cdot)p(\lambda_{i})\propto p(y_{i}|\cdot)exp\left\{ \frac{{-(\lambda_{i}-\lambda_{0})^{2}}}{2\sigma_{\lambda}^{2}}\right\} 
\]
In order to sample from this form, an \emph{adaptive} Metropolis-Hastings
algorithm is used with a Gaussian proposal $N(\lambda_{i}^{(k)},(\tau_{\lambda}^{2})^{(k)})$
where $\lambda_{i}^{(k)}$is the simulated value from the $k^{th}$
iteration. The adaptive piece helps to obtain a reasonable acceptance
rate by increasing or decreasing $(\tau_{\lambda}^{2})^{(k+1)}$ based
on whether or not $k^{th}$ proposal was accepted.
\end{enumerate}

\subsection{Monitoring Convergence}

Three different chains with randomly dispersed starting values were
each run for 10000 iterations with an adaptation period of 1000 and
burnin of 5000. The Gelman and Rubin scale reduction factor for each
parameter was computed and is presented below. Clearly, the factor
value looks good (very close or equal to 1) for all $\alpha_{i}$
in table 1. Similarly, the factor value looks good for all $\beta_{i}$
in table 2. The factor values for some $\delta_{i}$ in table 3 and
$\lambda_{i}$ in table 4 are a little higher than we would like to
see, but they don't cause an overwhelming reason for worry.

<<alphas, results='asis'>>=
library(coda) 
library(xtable)
load("results.rda")
n.iter <- 10000
burnin <- 5000
alphas <- mcmc.list(lapply(res, function(x) mcmc(x$alphas[burnin:n.iter,]))) 
betas <- mcmc.list(lapply(res, function(x) mcmc(x$betas[burnin:n.iter,]))) 
deltas <- mcmc.list(lapply(res, function(x) mcmc(x$deltas[burnin:n.iter,]))) 
lambdas <- mcmc.list(lapply(res, function(x) mcmc(x$lambdas[burnin:n.iter,]))) 
print(xtable(gelman.diag(alphas)[[1]], caption="The potential scale reduction factor for all alpha parameters"), table.placement='H')
@

<<betas, results='asis'>>=
print(xtable(gelman.diag(betas)[[1]], caption="The potential scale reduction factor for all beta parameters"), table.placement='H')
@

<<deltas, results='asis'>>=
print(xtable(gelman.diag(deltas)[[1]], caption="The potential scale reduction factor for all delta parameters"), table.placement='H')
@

<<lambdas, results='asis'>>=
print(xtable(gelman.diag(lambdas)[[1]], caption="The potential scale reduction factor for all lambda parameters"), table.placement='H')
@

To ensure the algorithm has explored the entire sample space and has
also not over represented areas of high probability, we track the
proportion of proposed jumps that are accepted for each parameter
sampled via Metropolis-Hastings in table 5. Note that all of these
proportions are between 0.31 and 0.61; thus, all these rates are inside
the rule of thumb of 0.2 to 0.6.

<<MHrates, results='asis'>>=
lu <- function(x) length(unique(x)) 
ac.alpha <- apply(res[[1]]$alphas, 2, lu)/n.iter 
ac.beta <- apply(res[[1]]$betas, 2, lu)/n.iter 
ac.delta <- apply(res[[1]]$deltas, 2, lu)/n.iter 
ac.lambda <- apply(res[[1]]$lambda, 2, lu)/n.iter
df <- data.frame(alphas=ac.alpha, betas=ac.beta, deltas=ac.delta, lambdas=ac.lambda) 
xtable(df, caption="Acceptance Rates for model parameters sampled via Metropolis-Hastings")
@


\subsection{Is there a home effect?}

It's natural to think that players tend to perform better at their
home stadium. According to the central 95\% credible intervals for
each $\alpha_{i}$ (in table 6), there obviously is not a significant
home field effect on the ``potential'' to hit a home run for any
of these players. Similarly, according to the central 95\% credible
intervals for each $\lambda_{i}$ (in table 7), there obviously is
not a significant home field effect on the ``ability'' to hit a
home run for any of these players.

<<alphaCI, results='asis'>>=
tab <- summary(alphas)[[2]][,c(1,5)]
names(tab) <- c("Lower", "Upper")
print(xtable(tab, caption="Central 95 percent credible intervals for every alpha."), table.placement='H')
@

<<lambdaCI, results='asis'>>=
tab <- summary(lambdas)[[2]][,c(1,5)]
names(tab) <- c("Lower", "Upper")
print(xtable(tab, caption="Central 95 percent credible intervals for every lambda."), table.placement='H')
@

<<getDat, cache=TRUE>>=
dat <- read.csv("data/dat.csv", header=TRUE, stringsAsFactors=FALSE)
N <- length(dat$y) 
n.players <- length(unique(dat$batter_name)) 
ni <- as.numeric(table(dat$batter_name))
@

<<zeros-and-range, cache=TRUE>>=
t <- exp(as.matrix(betas)) 
theta <- t/(1+t)
p <- exp(as.matrix(lambdas)) 
psi <- p/(1+p)
betaz <- apply(as.matrix(betas), 1, function(x) rep(x, ni)) 
alphaz <- apply(as.matrix(alphas), 1, function(x) rep(x, ni)) 
t <- exp(betaz + alphaz*dat$home) 
rm(betaz) 
rm(alphaz) 
thetaz <- t/(1+t) 
rm(t)
lambdaz <- apply(as.matrix(lambdas), 1, function(x) rep(x, ni)) 
deltaz <- apply(as.matrix(deltas), 1, function(x) rep(x, ni)) 
p <- exp(deltaz + lambdaz*dat$home) 
rm(lambdaz) 
rm(deltaz) 
psiz <- p/(1+p) 
rm(p)

n.zeros <- NULL 
ranges <- NULL 
for (m in 1:1000) {      
	sim.x <- rbinom(N, size=1, prob=theta[m,])  
	sim.y <- rbinom(N, size=dat$n, prob=psi[m,])   
	sim.y[sim.x == 0] <- 0   
	n.zeros <- c(n.zeros, sum(sim.y == 0))   
	ranges <- c(ranges, max(sim.y) - min(sim.y)) 
}

real.n.zero <- sum(dat$y == 0) 
p1 <- sum(n.zeros > real.n.zero)/length(n.zeros) 
real.range <- max(dat$y)-min(dat$y) 
p2 <- sum(ranges > real.range)/length(ranges)
@

Perhaps more interesting is ``home run potential''. Since Edwin
Encarnacion is the player with the highest posterior mean for $\alpha_{i}$
among all the players, he was chosen to see whether playing at home
has much of any difference in this probability. In figure 1, the empirical
distribution functions clearly show that playing at home has a very
small positive impact on this probability. 

<<ecdf-edwin, fig.cap="Empirical Distribution Function for Edwin Encarnacion's potential to hit home runs away (black) versus home (red).", out.width"\\textwith">>=
edwina <- ecdf(theta[,4]) 
plot(edwina, main="", ylab="Empirical Distribution Function",
      xlab="Home run potential") 
edwinh <- ecdf(thetaz[ni[4]+1,]) 
lines(edwinh, col=2) 
legend(0, 1, c("Home", "Away"), col=2:1, lty=1)
@

In a similar manner, we can address the impact of playing at home
on the probability of actually hitting a home run given that a player
has ``potential''. Since Josh Hamilton is the player with the highest
posterior mean for $\lambda_{i}$ among all the players, he was chosen
to see whether playing at home has much of any difference in this
probability. In figure 2, the empirical distribution functions clearly
show that playing at home has a very large positive impact on this
probability. Based on figure 2, we would conclude that the probability
Josh Hamilton has a probability of hitting a home run (given potential)
is less than 0.2 is 0.9 at home but 0.2 away.

<<ecdf-josh, fig.cap="Empirical Distribution Function for Josh Hamilton's home run hitting ability given potential in a game with 4 atbats. The function for a home game is shown in red while away is in black.">>=
josha <- ecdf(4 *psi[,2]) 
plot(josha, main="", ylab="Empirical Distribution Function",       
xlab="Home runs given potential in a game with 4 atbats") 
joshh <- ecdf(4 * psiz[ni[2]+5,]) 
lines(joshh, col=2) 
legend(3, 0.4, c("Home", "Away"), col=2:1, lty=1)
@


\section{Model Assessment}

To assess the model we address the adequacy of the ``zero inflated-binomial''
model for describing the data with respect to the number of games
with no home runs and the range in the number of home runs per game.
Using the procedure outlined on page 311 of the course notes, we obtain
a posterior predictive p-value of \Sexpr{round(p1, 2)} for the number
of zeros and \Sexpr{round(p2, 2)} for the range. The first p-value
suggests that this model is adequate for describing the number of
games without a home run. The second p-value suggests that this model
lacks an ability to describe the spread in the number of home runs
per game. Figure 3 shows the two different posterior predictive distributions
and the corresponding actual values (vertical dashed line) computed
from the observed data.

<<plots, fig.show='hold', fig.cap="Posterior predictive distributions.">>=
hist(n.zeros, main="Number of games without a home run.") 
real.n.zero <- sum(dat$y == 0) 
abline(v=real.n.zero, lty=2) 
hist(ranges, main="Range of home runs per game.") 
real.range <- max(dat$y)-min(dat$y) 
abline(v=real.range, lty=2)
@


\section{Conclusion}

The ``zero inflated-binomial model'' for home run hitting at the
game level allows for inference related to the impact that playing
at home has on both a player's ``potential'' to hit home runs and
``ability'' to hit home runs. As it turns out, there seems to be
a much higher positive impact on the ``ability'' to hit a home run
when playing at home (at least for Josh Hamilton).

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\end{document}