R: Compute or Extract Silhouette Information from Clustering
silhouette
R Documentation
Compute or Extract Silhouette Information from Clustering
Description
Compute silhouette information according to a given clustering in
k clusters.
Usage
silhouette(x, ...)
## Default S3 method:
silhouette(x, dist, dmatrix, ...)
## S3 method for class 'partition'
silhouette(x, ...)
## S3 method for class 'clara'
silhouette(x, full = FALSE, ...)
sortSilhouette(object, ...)
## S3 method for class 'silhouette'
summary(object, FUN = mean, ...)
## S3 method for class '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
x
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 <= k <= 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 * 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.
Details
For each observation i, the silhouette widths(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) := ( 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.
Value
silhouette() returns an object, sil, of class
silhouette which is an [n x 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).
sclus.avg.widths:
numeric (rank 1) array of clusterwise
means of silhouette widths where mean = FUN is used.
savg.width:
the total mean FUN(s) where
s are the individual silhouette widths.
sclus.sizes:
table of the k cluster sizes.
scall:
if available, the call creating sil.
sOrdered:
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 silis
ordered as by sortSilhouette(). In that case,
rownames(sil) will contain case labels or numbers, and attr(sil, "iOrd") the ordering index vector.
Note
While silhouette() is intrinsic to the
partition clusterings, and hence has a (trivial) method
for these, it is straightforward to get silhouettes from hierarchical
clusterings from silhouette.default() with
cutree() and distance as input.
By default, for clara() partitions, the silhouette is
just for the best random subset used. Use full = TRUE
to compute (and later possibly plot) the full silhouette.
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.
See Also
partition.object, plot.partition.
Examples
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) ,])
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$clustering, daisy(xc1k))
if(paste(R.version$major, R.version$minor, sep=".") >= "2.3.0")
stopifnot(all.equal(sf, s.full, check.attributes = FALSE, tolerance = 0))
## color dependent on original "3 groups of each 1000":
plot(sf, col = 2+ as.integer(names(cl3$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)