FX Jollois

Here are the packages we use in this work.

library(reshape2)
library(ggplot2)
library(FactoMineR)
library(NbClust)

Here is the previous code (see here for details) that we need in this part.

# Importation
pen.tra = read.table("donnees/pendigits.tra", sep = ",")
pen.tes = read.table("donnees/pendigits.tes", sep = ",")
pen = rbind(pen.tra, pen.tes)
names(pen) = c(paste0(c("X", "Y"), rep(1:8, each = 2)), "digit")
pen$digit = factor(pen$digit)

# Drawn function
drawn <- function(v, n = NULL, point = FALSE, add = FALSE, color = "black") {
    # Transformation of the data.frame if needed
    if (is.data.frame(v))
        v = unlist(v)
    # extract x and y coordinates
    x = v[seq(1, 15, by = 2)]
    y = v[seq(2, 16, by = 2)]
    # optimize space into graphics in reducing margin (sse ?par for more information)
    opar = par(mar = c(0, 0, 2, 0) + .1)
    if (!add) { # Create a graphic
        plot(x, y, 
             # Specify limits is a way to have always the same frame for plotting lines
             xlim = c(-5, 105), ylim = c(-5, 105),
             # Do not show axes
             axes = FALSE,
             # If point is TRUE, we add a space (with pch = " ") at each point
             # If not, draw a line
             type = ifelse(point, "b", "l"), 
             pch = " ", 
             # Specify color (black by default)
             col = color,
             # Add a title (NULL by default)
             main = n)
        if (point) text(x, y, 1:8)
    } else { # Add line to the plot
        # lines() add lines to an existing plot (the last produce)
        lines(x, y, 
              # same comment than before
              xlim = c(-5, 105), ylim = c(-5, 105),
              type = ifelse(point, "b", "l"), 
              pch = " ", 
              col = color)
    }
    par(opar)
}

# PCA
res = PCA(pen, quali.sup = 17, graph = FALSE)
res2 = data.frame(res$ind$coord, digit = pen$digit)

Application on pendigits

5 digit

In our case, we have to perform a direct clustering for each digit. Here, we focus on the 5 digit. First, we compute \(k\)-means with 2 clusters, as we notice from the PCA.

km5 = kmeans(subset(pen, digit == 5, -digit), 2)
plot(Dim.2 ~ Dim.1, subset(res2, digit == 5, -digit), 
     col = rainbow(2)[km5$cluster], pch = 19)

We now use the NbClust() function to test all criteria, in order to get a good number of clusters.

dig = 5
km5.nbcl = NbClust(subset(pen, digit == dig, -digit), method = "kmeans", 
                   index = "hartigan") # for all indices, use "all"

We can now use the result of the automatic choice made by NbClust() to represent this 4 clusters. We do not use ggplot() to plot the factorial map, to be able to have in one graphic the mean digit for the clusters, and the projections for the points.

km5.sub = data.frame(subset(pen, digit == dig, -digit),
                     cluster = factor(km5.nbcl$Best.partition))
km5.pca = data.frame(subset(res2, digit == dig, -digit),
                     cluster = factor(km5.nbcl$Best.partition))
km5.mean = aggregate(. ~ cluster, km5.sub, mean)
par(mfcol = c(2, km5.nbcl$Best.nc[1]), mar = c(0, 0, 0, 0) + .1)
for (k in 1:km5.nbcl$Best.nc[1]) {
    drawn(km5.mean[k,-1], point = TRUE)
    plot(Dim.2 ~ Dim.1, subset(km5.pca, cluster == k),
         xlim = range(res2$Dim.1),
         ylim = range(res2$Dim.2),
         col = rainbow(km5.nbcl$Best.nc[1])[k], pch = 19, 
         axes = FALSE, xlab = "", ylab = "", frame.plot = TRUE)
}

For all digits

We can apply the preceeding code to all digit, to get an estimation of the number of clusters for all of them. We transform it to directly get the part of the results we want.

km.nbcl = lapply(0:9, function(dig) {
    sub = subset(pen, digit == dig, -digit)
    nbcl = NbClust(sub, 
                   method = "kmeans", index = "hartigan")
    sub$cluster = factor(nbcl$Best.partition)
    res = list(nc = nbcl$Best.nc[1],
               partition = nbcl$Best.partition,
               sub = sub,
               pca = data.frame(subset(res2, digit == dig, -digit),
                                cluster = factor(nbcl$Best.partition)),
               mean = aggregate(. ~ cluster, sub, mean))
})

Now, we can represent all this results.

for (dig in 0:9) {
    par(mfcol = c(2, km.nbcl[[dig+1]]$nc), mar = c(0, 0, 0, 0) + .1)
    for (k in 1:km.nbcl[[dig+1]]$nc) {
        drawn(km.nbcl[[dig+1]]$mean[k,-1], point = TRUE)
        plot(Dim.2 ~ Dim.1, subset(km.nbcl[[dig+1]]$pca, cluster == k),
             xlim = range(res2$Dim.1),
             ylim = range(res2$Dim.2),
             col = rainbow(km.nbcl[[dig+1]]$nc)[k], pch = 19, 
             axes = FALSE, xlab = "", ylab = "", frame.plot = TRUE)
    }
}

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