## multi-dimensional scaling example
## a.k.a. principal coordinates analysis

## use 'UScitiesD' data
## matrix of pairwise distances between cities 
## (as the crow flies, in miles)
View(as.matrix(UScitiesD))

## do MDS to find two features
mds <- cmdscale(UScitiesD, k=2)

## plot the 
x <- -mds[, 1] # reflect so West is left
y <- -mds[, 2] # reflect so North is at the top

## note asp = 1, to ensure Euclidean distances are represented correctly
plot(x, y, type = "n", xlab = "", ylab = "",
     asp = 1, axes=FALSE,
     main = "MDS of US Cities")
axis(1, labels=FALSE); mtext('MDS Component 1', 1, 2)
axis(2, labels=FALSE); mtext('MDS Component 2', 2, 2)
points(x, y, pch=20)
text(x, y, rownames(loc), cex = 0.6,
     xpd='NA', adj=c(0.5, -0.75))


library('ElemStatLearn')
## zipcode training data
## first column gives digit
## remaining columns (2:257) give 16x16 pixels columnwise
data(zip.train)
dim(zip.train)

## plot the first training example
image(z=matrix(zip.train[1,-1],16,16)[,16:1],
      col=gray.colors(12,1,0), axes=FALSE)
axis(1); axis(2);

## plot first ten sets of each digit
plot_digits <- function(dat) {
  ## 10x10 figure, no spacing between panels, 1in outer margin
  par(mfrow=c(10,10), mar=rep(0,4),omi=rep(1,4))
  for(i in 1:10) {
    ## match next 0:9 digits
    idx <- match(0:9, dat[,1])
    ## plot each using 'image' function
    for(i in idx)
      image(z=matrix(dat[i,-1],16,16)[,16:1],
            col=gray.colors(12,1,0), 
            axes=FALSE)
    ## remove digits that were plotted
    dat <- dat[-idx,]
  }
}
plot_digits(zip.train)

## use MDS to represent digits in 2 dimensions instead of 256
mds <- cmdscale(dist(zip.train[1:100,-1]), k=2)

plot(mds[,1], mds[,2], type = "n", xlab = "", ylab = "",
     axes=FALSE, main = "MDS of Zipcodes")
axis(1, labels=FALSE); mtext('MDS Component 1', 1, 2)
axis(2, labels=FALSE); mtext('MDS Component 2', 2, 2)
points(mds[,1], mds[,2], pch=as.character(zip.train[,1]))