silhouette: Compute or Extract Silhouette Information from Clustering

Description

Compute silhouette information according to a given clustering in $k$ clusters.

Usage

silhouette(x, …)
# S3 method for default
silhouette  (x, dist, dmatrix, …)
# S3 method for partition
silhouette(x, …)
# S3 method for clara
silhouette(x, full = FALSE, …)
sortSilhouette(object, …)
# S3 method for silhouette
summary(object, FUN = mean, …)
# S3 method for silhouette
plot(x, nmax.lab = 40, max.strlen = 5,
     main = NULL, sub = NULL, xlab = expression("Silhouette width "* s[i]),
     col = "gray",  do.col.sort = length(col) > 1, border = 0,
     cex.names = par("cex.axis"), do.n.k = TRUE, do.clus.stat = TRUE, …)

Arguments

an object of appropriate class; for the default method an integer vector with $k$ different integer cluster codes or a list with such an x$clustering component. Note that silhouette statistics are only defined if $2 \le k \le n-1$.

dist

a dissimilarity object inheriting from class dist or coercible to one. If not specified, dmatrix must be.

dmatrix

a symmetric dissimilarity matrix ($n \times n$), specified instead of dist, which can be more efficient.

full

logical specifying if a full silhouette should be computed for clara object. Note that this requires $O(n^2)$ memory, since the full dissimilarity (see daisy) is needed internally.

object

an object of class silhouette.

…

further arguments passed to and from methods.

FUN

function used to summarize silhouette widths.

nmax.lab

integer indicating the number of labels which is considered too large for single-name labeling the silhouette plot.

max.strlen

positive integer giving the length to which strings are truncated in silhouette plot labeling.

main, sub, xlab

arguments to title; have a sensible non-NULL default here.

col, border, cex.names

arguments passed barplot(); note that the default used to be col = heat.colors(n), border = par("fg") instead. col can also be a color vector of length $k$ for clusterwise coloring, see also do.col.sort:

do.col.sort

logical indicating if the colors col should be sorted “along” the silhouette; this is useful for casewise or clusterwise coloring.

do.n.k

logical indicating if $n$ and $k$ “title text” should be written.

do.clus.stat

logical indicating if cluster size and averages should be written right to the silhouettes.

Value

silhouette() returns an object, sil, of class silhouette which is an $n \times 3$ matrix with attributes. For each observation i, sil[i,] contains the cluster to which i belongs as well as the neighbor cluster of i (the cluster, not containing i, for which the average dissimilarity between its observations and i is minimal), and the silhouette width $s(i)$ of the observation. The colnames correspondingly are c("cluster", "neighbor", "sil_width").

summary(sil) returns an object of class summary.silhouette, a list with components

si.summary:: numerical summary of the individual silhouette widths $s(i)$.
clus.avg.widths:: numeric (rank 1) array of clusterwise means of silhouette widths where mean = FUN is used.
avg.width:: the total mean FUN(s) where s are the individual silhouette widths.
clus.sizes:: table of the $k$ cluster sizes.
call:: if available, the call creating sil.
Ordered:: logical identical to attr(sil, "Ordered"), see below.

sortSilhouette(sil) orders the rows of sil as in the silhouette plot, by cluster (increasingly) and decreasing silhouette width $s(i)$. attr(sil, "Ordered") is a logical indicating if sil is ordered as by sortSilhouette(). In that case, rownames(sil) will contain case labels or numbers, and attr(sil, "iOrd") the ordering index vector.

Details

For each observation i, the silhouette width $s(i)$ is defined as follows: Put a(i) = average dissimilarity between i and all other points of the cluster to which i belongs (if i is the only observation in its cluster, $s(i) := 0$ without further calculations). For all other clusters C, put $d(i,C)$ = average dissimilarity of i to all observations of C. The smallest of these $d(i,C)$ is $b(i) := \min_C d(i,C)$, and can be seen as the dissimilarity between i and its “neighbor” cluster, i.e., the nearest one to which it does not belong. Finally, $$s(i) := \frac{b(i) - a(i) }{max(a(i), b(i))}.$$

silhouette.default() is now based on C code donated by Romain Francois (the R version being still available as cluster:::silhouette.default.R).

Observations with a large $s(i)$ (almost 1) are very well clustered, a small $s(i)$ (around 0) means that the observation lies between two clusters, and observations with a negative $s(i)$ are probably placed in the wrong cluster.

References

Rousseeuw, P.J. (1987) Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math., 20, 53--65.

chapter 2 of Kaufman and Rousseeuw (1990), see the references in plot.agnes.

Examples

Run this code

# NOT RUN {
data(ruspini)
pr4 <- pam(ruspini, 4)
str(si <- silhouette(pr4))
(ssi <- summary(si))
plot(si) # silhouette plot
plot(si, col = c("red", "green", "blue", "purple"))# with cluster-wise coloring

si2 <- silhouette(pr4$clustering, dist(ruspini, "canberra"))
summary(si2) # has small values: "canberra"'s fault
plot(si2, nmax= 80, cex.names=0.6)

op <- par(mfrow= c(3,2), oma= c(0,0, 3, 0),
          mgp= c(1.6,.8,0), mar= .1+c(4,2,2,2))
for(k in 2:6)
   plot(silhouette(pam(ruspini, k=k)), main = paste("k = ",k), do.n.k=FALSE)
mtext("PAM(Ruspini) as in Kaufman & Rousseeuw, p.101",
      outer = TRUE, font = par("font.main"), cex = par("cex.main")); frame()

## the same with cluster-wise colours:
c6 <- c("tomato", "forest green", "dark blue", "purple2", "goldenrod4", "gray20")
for(k in 2:6)
   plot(silhouette(pam(ruspini, k=k)), main = paste("k = ",k), do.n.k=FALSE,
        col = c6[1:k])
par(op)

## clara(): standard silhouette is just for the best random subset
data(xclara)
set.seed(7)
str(xc1k <- xclara[ sample(nrow(xclara), size = 1000) ,]) # rownames == indices
cl3 <- clara(xc1k, 3)
plot(silhouette(cl3))# only of the "best" subset of 46
## The full silhouette: internally needs large (36 MB) dist object:
sf <- silhouette(cl3, full = TRUE) ## this is the same as
s.full <- silhouette(cl3$clustering, daisy(xc1k))
stopifnot(all.equal(sf, s.full, check.attributes = FALSE, tolerance = 0))
## color dependent on original "3 groups of each 1000": % __FIXME ??__
plot(sf, col = 2+ as.integer(names(cl3$clustering) ) %/% 1000,
     main ="plot(silhouette(clara(.), full = TRUE))")

## Silhouette for a hierarchical clustering:
ar <- agnes(ruspini)
si3 <- silhouette(cutree(ar, k = 5), # k = 4 gave the same as pam() above
    	           daisy(ruspini))
plot(si3, nmax = 80, cex.names = 0.5)
## 2 groups: Agnes() wasn't too good:
si4 <- silhouette(cutree(ar, k = 2), daisy(ruspini))
plot(si4, nmax = 80, cex.names = 0.5)
# }

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