\name{plot1DProjection}
\alias{plot1DProjection}
\title{PLOT A PAIR OF CLUSTERS AND THEIR DENSITY ESTIMATES, WHICH ARE PROJECTED  ALONG A SPECIFIED 1-D PROJECTION DIRECTION}
\description{
Plot a pair of clusters and their density estimates, which are projected along a specified 1-D projection direction. 
}
\usage{
plot1DProjection(y1, y2, projDir, 
  sepValMethod=c("normal", "quantile"), bw="nrd0", 
  xlim=NULL, ylim=NULL, 
  xlab="1-D projected clusters", ylab="density estimates", 
  title="1-D Projected Clusters and their density estimates",
  font=2, font.lab=2, cex=1.2, cex.lab=1.2, cex.main=1.5,
  lwd=4, lty1=1, lty2=2, pch1=18, pch2=19, col1=2, col2=4, 
  type="l", alpha=0.05, eps=1.0e-10, quiet=TRUE)
}
\arguments{
  \item{y1}{
Data matrix of cluster 1. Rows correspond to observations. Columns correspond to variables.
  }
  \item{y2}{
Data matrix of cluster 2. Rows correspond to observations. Columns correspond to variables.
  }
  \item{projDir}{
1-D projection direction along which two clusters will be projected.
  }
  \item{sepValMethod}{
Method to calculate separation index for a pair of clusters projected onto a 
1-D space. \code{sepValMethod="quantile"} indicates the quantile version of
separation index will be used: $sepVal=(L_2-U_1)/(U_2-L_1)$ where $L_i$ and 
$U_i$, $i=1, 2$, are the lower and upper \code{alpha/2} sample percentiles 
of projected cluster $i$. \code{sepValMethod="normal"} indicates the 
normal version of separation index will be used: 
$sepVal=[(\bar{x}_2-\bar{x}_1)-z_{\alpha/2}(s_1+s_2)]/
[(\bar{x}_2-\bar{x}_1)+z_{\alpha/2}(s_1+s_2)]$, 
where $\bar{x}_i$ and $s_i$ are the sample mean and standard deviation 
of projected cluster $i$.
  }
  \item{bw}{
The smoothing bandwidth to be used by the function \code{density}.
  }
  \item{xlim}{
Range of X axis.
  }
  \item{ylim}{
Range of Y axis.
  }
  \item{xlab}{
X axis label.
  }
  \item{ylab}{
Y axis label.
  }
  \item{title}{
Title of the plot.
  }
  \item{font}{
An integer which specifies which font to use for text (see \code{par}).
  }
  \item{font.lab}{
The font to be used for x and y labels (see \code{par}).
  }
  \item{cex}{
A numerical value giving the amount by which plotting text
and symbols should be scaled relative to the default (see \code{par}).
  }
  \item{cex.lab}{
The magnification to be used for x and y labels relative
to the current setting of 'cex' (see \code{par}).
  }
  \item{cex.main}{
The magnification to be used for main titles relative
to the current setting of 'cex' (see \code{par}).
  }
  \item{lwd}{
The line width, a \_positive\_ number, defaulting to '1' (see \code{par}).
  }
  \item{lty1}{
  Line type for cluster 1 (see \code{par}).
  }
  \item{lty2}{
  Line type for cluster 2 (see \code{par}).
  }
  \item{pch1}{
Either an integer specifying a symbol or a single character
to be used as the default in plotting points for cluster 1 (see \code{points}).
  }
  \item{pch2}{
Either an integer specifying a symbol or a single character
to be used as the default in plotting points for cluster 2 (see \code{points}).
  }
  \item{col1}{
Color to indicates cluster 1.
  }
  \item{col2}{
Color to indicates cluster 2.
  }
  \item{type}{
What type of plot should be drawn (see \code{plot}).
  }
  \item{alpha}{
Tuning parameter reflecting the percentage in the two
tails of a projected cluster that might be outlying.
  }
  \item{eps}{
A small positive number to check if a quantitiy \eqn{q} is equal to zero.  
If \eqn{|q|<}\code{eps}, then we regard \eqn{q} as equal to zero.  
\code{eps} is used to check the denominator in the formula of the separation 
index is equal to zero. Zero-value denominator indicates two clusters are 
totally overlapped. Hence the separation index is set to be $-1$.
The default value of \code{eps} is \eqn{1.0e-10}.
  }
  \item{quiet}{
A flag to switch on/off the outputs of intermediate results and/or possible warning messages. The default value is \code{TRUE}.
  }
}
\details{
The ticks along X axis indicates the positions of points of the projected 
two clusters. The positions of $L_i$ and $U_i$, $i=1, 2$, are also indicated 
on X axis, where $L_i$ and $U_i$ are the lower and upper $\alpha/2$ sample 
percentiles of cluster $i$ if \code{sepValMethod="quantile"}. 
If \code{sepValMethod="normal"},
$L_i=\bar{x}_i-z_{\alpha/2}s_i$, where $\bar{x}_i$ and $s_i$ are the 
sample mean and standard deviation of cluster $i$, and $z_{\alpha/2}$ 
is the upper $\alpha/2$ percentile of standard normal distribution.
}
\value{
  \item{sepVal}{
    value of the separation index for the projected two clusters along
    the projection direction \code{projDir}.
  }
  \item{projDir}{
    projection direction. To make sure the projected cluster 1 is on the 
    left-hand side of the projected cluster 2, the input \code{projDir}
    might be changed to \code{-projDir}.
  }
}
\references{
  Qiu, W.-L. and Joe, H. (2006)
  Separation Index and Partial Membership for Clustering.
  \emph{Computational Statistics and Data Analysis}, \bold{50}, 585--603.
}
\author{
Weiliang Qiu \email{stwxq@channing.harvard.edu}\cr
Harry Joe \email{harry@stat.ubc.ca}
}
\seealso{
  \code{\link{plot2DProjection}}
  \code{\link{viewClusters}}
}
\examples{
n1<-50
mu1<-c(0,0)
Sigma1<-matrix(c(2,1,1,5),2,2)
n2<-100
mu2<-c(10,0)
Sigma2<-matrix(c(5,-1,-1,2),2,2)
projDir<-c(1, 0)

library(MASS)
set.seed(1234)
y1<-mvrnorm(n1, mu1, Sigma1)
y2<-mvrnorm(n2, mu2, Sigma2)
y<-rbind(y1, y2)
cl<-rep(1:2, c(n1, n2))

b<-getSepProjData(y, cl, iniProjDirMethod="SL", projDirMethod="newton")
# projection direction for clusters 1 and 2
projDir<-b$projDirArray[1,2,]

plot1DProjection(y1, y2, projDir)

}
\keyword{cluster}