# PLOT A PAIR OF CLUSTERS ALONG A 2-D PROJECTION SPACE Plot a pair of clusters along a 2-D projection space. ```r 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 - `y1`: Data matrix of cluster 1. Rows correspond to observations. Columns correspond to variables. - `y2`: Data matrix of cluster 2. Rows correspond to observations. Columns correspond to variables. - `projDir`: 1-D projection direction along which two clusters will be projected. - `sepValMethod`: Method to calculate separation index for a pair of clusters projected onto a 1-D space. `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 `alpha/2` sample percentiles of projected cluster $i$. `sepValMethod="normal"` indicates the normal version of separation index will be used: c("$sepVal=[(xbar_2-xbar_1)-z_{\\alpha/2}(s_1+s_2)]/\n$", "$[(xbar_2-xbar_1)+z_{\\alpha/2}(s_1+s_2)]$"), where $xbar_i$ and $s_i$ are the sample mean and standard deviation of projected cluster $i$. - `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). `iniProjDirMethod`=SL indicates the initial projection direction is the sample version of the SL's projection direction (Su and Liu, 1993) $\left(\boldsymbol{\Sigma}_1+\boldsymbol{\Sigma}_2\right)^{-1}\left(\boldsymbol{\mu}_2-\boldsymbol{\mu}_1\right)$ `iniProjDirMethod`=naive indicates the initial projection direction is $\boldsymbol{\mu}_2-\boldsymbol{\mu}_1$ - `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). `projDirMethod`=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 fixedpoint method could be used. The 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). - `xlim`: Range of X axis. - `ylim`: Range of Y axis. - `xlab`: X axis label. - `ylab`: Y axis label. - `title`: Title of the plot. - `font`: An integer which specifies which font to use for text (see `par`). - `font.lab`: The font to be used for x and y labels (see `par`). - `cex`: A numerical value giving the amount by which plotting text and symbols should be scaled relative to the default (see `par`). - `cex.lab`: The magnification to be used for x and y labels relative to the current setting of 'cex' (see `par`). - `cex.main`: The magnification to be used for main titles relative to the current setting of 'cex' (see `par`). - `lwd`: The line width, a positive number, defaulting to '1' (see `par`). - `lty1`: Line type for cluster 1 (see `par`). - `lty2`: Line type for cluster 2 (see `par`). - `pch1`: Either an integer specifying a symbol or a single character to be used as the default in plotting points for cluster 1 (see `points`). - `pch2`: Either an integer specifying a symbol or a single character to be used as the default in plotting points for cluster 2 (see `points`). - `col1`: Color to indicates cluster 1. - `col2`: Color to indicates cluster 2. - `alpha`: Tuning parameter reflecting the percentage in the two tails of a projected cluster that might be outlying. - `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. - `eps`: A small positive number to check if a quantitiy $q$ is equal to zero. If $|q|<$`eps`, then we regard $q$ as equal to zero. `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 `eps` is $1.0e-10$. - `quiet`: A flag to switch on/off the outputs of intermediate results and/or possible warning messages. The default value is `TRUE`. ## Details To get the second projection direction, we first construct an orthogonal matrix with first column `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 `projDir2` for the rotated data points in the remaining dimensions. Finally, we rotate the vector `projDir3=(0, projDir2)` back to the original space. The vector `projDir3` is the second projection direction. 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 `sepValMethod="quantile"`. If `sepValMethod="normal"`, $L_i=xbar_i-z_{\alpha/2}s_i$, where $xbar_i$ 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. ## Returns - **sepValx**: value of the separation index for the projected two clusters along the 1st projection direction. - **sepValy**: value of the separation index for the projected two clusters along the 2nd projection direction. - **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. **Journal of Classification**, 23 (2), 315-334. Qiu, W.-L. and Joe, H. (2006b) Separation Index and Partial Membership for Clustering. **Computational Statistics and Data Analysis**, 50 , 585--603. ## Author(s) Weiliang Qiu weiliang.qiu@gmail.com Harry Joe harry@stat.ubc.ca ## See Also `plot1DProjection` `viewClusters` ## Examples ```r 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 = y, cl = cl, iniProjDirMethod = "SL", projDirMethod = "newton") # projection direction for clusters 1 and 2 projDir <- b$projDirArray[1,2,] par(mfrow = c(2,1)) plot1DProjection( y1 = y1, y2 = y2, projDir = projDir) plot2DProjection( y1 = y1, y2 = y2, projDir = projDir) ```

plot2DProjection function