Perform k-means clustering on a data matrix.

```
kmeans(x, centers, iter.max = 10, nstart = 1,
algorithm = c("Hartigan-Wong", "Lloyd", "Forgy",
"MacQueen"), trace=FALSE)
# S3 method for kmeans
fitted(object, method = c("centers", "classes"), ...)
```

x

numeric matrix of data, or an object that can be coerced to such a matrix (such as a numeric vector or a data frame with all numeric columns).

centers

either the number of clusters, say \(k\), or a set of
initial (distinct) cluster centres. If a number, a random set of
(distinct) rows in `x`

is chosen as the initial centres.

iter.max

the maximum number of iterations allowed.

nstart

if `centers`

is a number, how many random sets
should be chosen?

algorithm

character: may be abbreviated. Note that
`"Lloyd"`

and `"Forgy"`

are alternative names for one
algorithm.

object

an R object of class `"kmeans"`

, typically the
result `ob`

of `ob <- kmeans(..)`

.

method

character: may be abbreviated. `"centers"`

causes
`fitted`

to return cluster centers (one for each input point) and
`"classes"`

causes `fitted`

to return a vector of class
assignments.

trace

logical or integer number, currently only used in the
default method (`"Hartigan-Wong"`

): if positive (or true),
tracing information on the progress of the algorithm is
produced. Higher values may produce more tracing information.

…

not used.

`kmeans`

returns an object of class `"kmeans"`

which has a
`print`

and a `fitted`

method. It is a list with at least
the following components:

A vector of integers (from `1:k`

) indicating the cluster to
which each point is allocated.

A matrix of cluster centres.

The total sum of squares.

Vector of within-cluster sum of squares, one component per cluster.

Total within-cluster sum of squares,
i.e.`sum(withinss)`

.

The between-cluster sum of squares,
i.e.`totss-tot.withinss`

.

The number of points in each cluster.

The number of (outer) iterations.

integer: indicator of a possible algorithm problem -- for experts.

The data given by `x`

are clustered by the \(k\)-means method,
which aims to partition the points into \(k\) groups such that the
sum of squares from points to the assigned cluster centres is minimized.
At the minimum, all cluster centres are at the mean of their Voronoi
sets (the set of data points which are nearest to the cluster centre).

The algorithm of Hartigan and Wong (1979) is used by default. Note
that some authors use \(k\)-means to refer to a specific algorithm
rather than the general method: most commonly the algorithm given by
MacQueen (1967) but sometimes that given by Lloyd (1957) and Forgy
(1965). The Hartigan--Wong algorithm generally does a better job than
either of those, but trying several random starts (`nstart`

\(>
1\)) is often recommended. In rare cases, when some of the points
(rows of `x`

) are extremely close, the algorithm may not converge
in the “Quick-Transfer” stage, signalling a warning (and
returning `ifault = 4`

). Slight
rounding of the data may be advisable in that case.

For ease of programmatic exploration, \(k=1\) is allowed, notably
returning the center and `withinss`

.

Except for the Lloyd--Forgy method, \(k\) clusters will always be returned if a number is specified. If an initial matrix of centres is supplied, it is possible that no point will be closest to one or more centres, which is currently an error for the Hartigan--Wong method.

Forgy, E. W. (1965) Cluster analysis of multivariate data:
efficiency vs interpretability of classifications.
*Biometrics* **21**, 768--769.

Hartigan, J. A. and Wong, M. A. (1979).
A K-means clustering algorithm.
*Applied Statistics* **28**, 100--108.

Lloyd, S. P. (1957, 1982) Least squares quantization in PCM.
Technical Note, Bell Laboratories. Published in 1982 in
*IEEE Transactions on Information Theory* **28**, 128--137.

MacQueen, J. (1967) Some methods for classification and analysis of
multivariate observations. In *Proceedings of the Fifth Berkeley
Symposium on Mathematical Statistics and Probability*,
eds L. M. Le Cam & J. Neyman,
**1**, pp.281--297. Berkeley, CA: University of California Press.

require(graphics) # a 2-dimensional example x <- rbind(matrix(rnorm(100, sd = 0.3), ncol = 2), matrix(rnorm(100, mean = 1, sd = 0.3), ncol = 2)) colnames(x) <- c("x", "y") (cl <- kmeans(x, 2)) plot(x, col = cl$cluster) points(cl$centers, col = 1:2, pch = 8, cex = 2) # sum of squares ss <- function(x) sum(scale(x, scale = FALSE)^2) ## cluster centers "fitted" to each obs.: fitted.x <- fitted(cl); head(fitted.x) resid.x <- x - fitted(cl) ## Equalities : ---------------------------------- cbind(cl[c("betweenss", "tot.withinss", "totss")], # the same two columns c(ss(fitted.x), ss(resid.x), ss(x))) stopifnot(all.equal(cl$ totss, ss(x)), all.equal(cl$ tot.withinss, ss(resid.x)), ## these three are the same: all.equal(cl$ betweenss, ss(fitted.x)), all.equal(cl$ betweenss, cl$totss - cl$tot.withinss), ## and hence also all.equal(ss(x), ss(fitted.x) + ss(resid.x)) ) kmeans(x,1)$withinss # trivial one-cluster, (its W.SS == ss(x)) ## random starts do help here with too many clusters ## (and are often recommended anyway!): (cl <- kmeans(x, 5, nstart = 25)) plot(x, col = cl$cluster) points(cl$centers, col = 1:5, pch = 8)