\name{plot2DProjection}
\alias{plot2DProjection}
\title{PLOT A PAIR OF CLUSTERS ALONG A 2-D PROJECTION SPACE}
\description{
Plot a pair of clusters along a 2-D projection space. 
}
\usage{
plot2DProjection(y1, y2, projDir, 
  sepValMethod=c("normal", "quantile"), 
  iniProjDirMethod=c("SL", "naive"), 
  projDirMethod=c("newton", "fixedpoint"), 
  xlim=NULL, ylim=NULL, 
  xlab="1st projection direction", 
  ylab="2nd projection direction", 
  title="Scatter plot of 2-D Projected Clusters",
  font=2, font.lab=2, cex=1.2, cex.lab=1, cex.main=1.5,
  lwd=4, lty1=1, lty2=2, pch1=18, pch2=19, col1=2, col2=4, 
  alpha=0.05, ITMAX=20, 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: \eqn{sepVal=(L_2-U_1)/(U_2-L_1)} where \eqn{L_i} and 
\eqn{U_i}, \eqn{i=1, 2}, are the lower and upper \code{alpha/2} sample percentiles 
of projected cluster \eqn{i}. \code{sepValMethod="normal"} indicates the 
normal version of separation index will be used: 
\eqn{sepVal=[(xbar_2-xbar_1)-z_{\alpha/2}(s_1+s_2)]/
[(xbar_2-xbar_1)+z_{\alpha/2}(s_1+s_2)]}, 
where \eqn{xbar_i} and \eqn{s_i} are the sample mean and standard deviation 
of projected cluster \eqn{i}.
  }
  \item{iniProjDirMethod}{
Indicating the method to get initial projection direction when calculating
the separation index between a pair of clusters (c.f. Qiu and Joe,
2006a, 2006b). \cr
     \code{iniProjDirMethod}=\dQuote{SL} indicates the initial projection 
direction is the sample version of the SL's projection direction 
(Su and Liu, 1993)
\eqn{\left(\boldsymbol{\Sigma}_1+\boldsymbol{\Sigma}_2\right)^{-1}\left(\boldsymbol{\mu}_2-\boldsymbol{\mu}_1\right)}\cr
     \code{iniProjDirMethod}=\dQuote{naive} indicates the initial projection 
direction is \eqn{\boldsymbol{\mu}_2-\boldsymbol{\mu}_1}
  }
  \item{projDirMethod}{
Indicating the method to get the optimal projection direction when calculating 
the separation index between a pair of clusters (c.f. Qiu and Joe,
2006a, 2006b). \cr
     \code{projDirMethod}=\dQuote{newton} indicates we use the Newton-Raphson 
method to search the optimal projection direction (c.f. Qiu and Joe, 2006a). 
This requires the assumptions that both covariance matrices of the pair of 
clusters are positive-definite. If this assumption is violated, the 
\dQuote{fixedpoint} method could be used. The \dQuote{fixedpoint} method 
iteratively searches the optimal projection direction based on the first 
derivative of the separation index to the project direction 
(c.f. Qiu and Joe, 2006b).
  }
  \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{alpha}{
Tuning parameter reflecting the percentage in the two
tails of a projected cluster that might be outlying.
  }
  \item{ITMAX}{
Maximum iteration allowed when iteratively calculating the
optimal projection direction.
The actual number of iterations is usually much less than the default value 20.
  }
  \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 \eqn{-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{
To get the second projection direction, we first construct an orthogonal 
matrix with first column \code{projDir}. Then we rotate the data points 
according to this orthogonal matrix. Next, we remove the first dimension 
of the rotated data points, and obtain the optimal projection direction 
\code{projDir2} for the rotated data points in the remaining dimensions. 
Finally, we rotate the vector
\code{projDir3=(0, projDir2)} back to the original space. 
The vector \code{projDir3} is the second projection direction.

The ticks along X axis indicates the positions of points of the projected 
two clusters. The positions of \eqn{L_i} and \eqn{U_i}, \eqn{i=1, 2}, are also indicated 
on X axis, where \eqn{L_i} and \eqn{U_i} are the lower and upper \eqn{\alpha/2} sample 
percentiles of cluster \eqn{i} if \code{sepValMethod="quantile"}. 
If \code{sepValMethod="normal"},
\eqn{L_i=xbar_i-z_{\alpha/2}s_i}, where \eqn{xbar_i} and \eqn{s_i} are the 
sample mean and standard deviation of cluster \eqn{i}, and \eqn{z_{\alpha/2}} 
is the upper \eqn{\alpha/2} percentile of standard normal distribution.
}
\value{
  \item{sepValx}{
    value of the separation index for the projected two clusters along the
    1st projection direction.
  }
  \item{sepValy}{
    value of the separation index for the projected two clusters along the
    2nd projection direction.
  }
  \item{Q2}{
    1st column is the 1st projection direction. 2nd column is the 2nd
    projection direction.
  }
}
\references{
  Qiu, W.-L. and Joe, H. (2006a)
  Generation of Random Clusters with Specified Degree of Separaion.
  \emph{Journal of Classification}, \bold{23}(2), 315-334.

  Qiu, W.-L. and Joe, H. (2006b)
  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{plot1DProjection}}
  \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,]

par(mfrow=c(2,1))
plot1DProjection(y1, y2, projDir)
plot2DProjection(y1, y2, projDir)

}
\keyword{cluster}