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 class 'kmeans': fitted((object, method = c("centers", "classes"), ...))
- 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).
- 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
xis chosen as the initial centres.
- the maximum number of iterations allowed.
centersis a number, how many random sets should be chosen?
- character: may be abbreviated. Note that
"Forgy"and alternative names for one algorithm.
- an R object of class
"kmeans", typically the result
ob <- kmeans(..).
- character: may be abbreviated.
fittedto return cluster centers (one for each input point) and
fittedto return a vector of class assignments.
- 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.
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) 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
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.
- 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.,
- The between-cluster sum of squares, i.e.
- The number of points in each cluster.
- The number of (outer) iterations.
- integer: indicator of a possible algorithm problem -- for experts.
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)
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