\documentclass[a4paper]{article} 
\begin{document} 
\title{Statistics II: Histograms in R}
\author{Richard Gill\thanks{{\tt http://www.math.leidenuniv.nl/$\sim$gill/teaching/statistics}}}
\date{\today}
\maketitle

\noindent These scripts and notes illustrate the histogram as a density estimator. First we make some data. And before that, I set the random seed, so that the results will be perfectly reproducible.
%
<<>>= 
set.seed(11091951)
x<-rgamma(100,10)
@ 
%
Now we show the plain vanilla R histogram.
%
%
\begin{center} 
<<fig=TRUE,echo=TRUE>>= 
hist(x)
@ 
\end{center} 
%

\newpage\noindent
The area under the graph was equal to the number of observations. Using the option prob=TRUE we make the area under the graph equal to one. On this graph I will superimpose a plot of the true density of the data. I'll also fix the axes to get a nicer plot. The bin-width is determined by Sturges' method, the default, it doesn't hurt to make that explicit.
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
hist(x,prob=T,xlim=c(0,30),ylim=c(0,0.15),breaks="Sturges")
xd<-seq(from=0,to=30,length=1000)
yd<-dgamma(xd,shape=10)
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
Now let's look at the results of the other bin-width algorithms. First, Scott:
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
hist(x,prob=T,xlim=c(0,30),ylim=c(0,0.15),breaks="Scott")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
And Freedman-Diaconis:
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
hist(x,prob=T,xlim=c(0,30),ylim=c(0,0.15),breaks="FD")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
Not much to see, right? But I'll draw these all again, but now with an outlier added to the dataset. First, plain vanilla (Sturges):
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
hist(c(x,50),prob=T,xlim=c(0,50),ylim=c(0,0.15),breaks="Sturges")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
Then Scott:
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
hist(c(x,50),prob=T,xlim=c(0,50),ylim=c(0,0.15),breaks="Scott")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
And Freedman-Diaconis
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
hist(c(x,50),prob=T,xlim=c(0,50),ylim=c(0,0.15),breaks="Freedman-Diaconis")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
Venables and Ripley think they have a much better histogram. Let's take a look.
Scott:
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
library(MASS)
truehist(c(x,50),prob=T,xlim=c(0,50),ylim=c(0,0.15),nbins="Scott")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%

\newpage\noindent
And Freedman-Diaconis
%
\begin{center}
<<fig=TRUE,echo=TRUE>>= 
truehist(c(x,50),prob=T,xlim=c(0,50),ylim=c(0,0.15),nbins="Freedman-Diaconis")
lines(xd,yd)
abline(h=0)
@ 
\end{center} 
%
\bigskip
\noindent{\bf Exercise}. Figure out how to draw a frequency polygon (join the mid-points of histogram bars by straight lines...). Theoretically you can use a larger bin-width, at least, if the true density is twice differentiable. 

\end{document} 





help(hist)
help(rgamma)
x<-rgamma(100,10)
hist(x)
hist(x,prob=T)
y<-seq(from=0,to=30,length=1000)
lines(y,dgamma(y,shape=10))
hist(x,prob=T,breaks="Scott")
lines(y,dgamma(y,shape=10))
hist(x,prob=T,breaks="FD")
lines(y,dgamma(y,shape=10))
hist(c(x,50),prob=T,breaks="FD")
lines(y,dgamma(y,shape=10))
hist(c(x,50),prob=T,breaks="FD",ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))
hist(c(x,50),prob=T,breaks="Scott",ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))
hist(c(x,50),prob=T,breaks="Sturges",ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))
library(MASS)
truehist(c(x,50),prob=T,ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))
truehist(c(x,50),prob=T,nbins="FD",ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))
truehist(c(x,50),prob=T,nbins="Scott",ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))
truehist(c(x,50),prob=T,nbins="FD",ylim=c(0,0.15))
lines(y,dgamma(y,shape=10))