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% File src/library/stats/man/kmeans.Rd
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% File src/library/stats/man/kmeans.Rd
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% Part of the R package, https://www.R-project.org
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% Part of the R package, https://www.R-project.org
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% Copyright 1995-2023 R Core Team
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% Copyright 1995-2025 R Core Team
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% Distributed under GPL 2 or later
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% Distributed under GPL 2 or later
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\name{kmeans}
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\name{kmeans}
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\alias{kmeans}
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\alias{kmeans}
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\alias{print.kmeans}
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\alias{print.kmeans}
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  which aims to partition the points into \eqn{k} groups such that the
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  which aims to partition the points into \eqn{k} groups such that the
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  sum of squares from points to the assigned cluster centres is minimized.
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  sum of squares from points to the assigned cluster centres is minimized.
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  At the minimum, all cluster centres are at the mean of their \I{Voronoi}
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  At the minimum, all cluster centres are at the mean of their \I{Voronoi}
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  sets (the set of data points which are nearest to the cluster centre).
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  sets (the set of data points which are nearest to the cluster centre).
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  The algorithm of \bibcite{Hartigan and Wong (1979)} is used by
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  The algorithm of \bibcitet{R:Hartigan+Wong:1979} is used by
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  default.  Note that some authors use \eqn{k}-means to refer to a
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  default.  Note that some authors use \eqn{k}-means to refer to a
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  specific algorithm rather than the general method: most commonly the
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  specific algorithm rather than the general method: most commonly the
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  algorithm given by \bibcite{MacQueen (1967)} but sometimes that given
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  algorithm given by \bibcitet{R:MacQueen:1967} but sometimes that given
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  by \bibcite{Lloyd (1957)} and \bibcite{Forgy (1965)}.
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  by \bibcitet{R:Lloyd:1957, R:Lloyd:1982} and \bibcitet{R:Forgy:1965}.
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  The \I{Hartigan}--\I{Wong} algorithm generally does a better job than
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  The \I{Hartigan}--\I{Wong} algorithm generally does a better job than
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  either of those, but trying several random starts (\code{nstart}\eqn{>
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  either of those, but trying several random starts (\code{nstart}\eqn{>
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  1}) is often recommended.  In rare cases, when some of the points
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  1}) is often recommended.  In rare cases, when some of the points
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  (rows of \code{x}) are extremely close, the algorithm may not converge
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  (rows of \code{x}) are extremely close, the algorithm may not converge
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  in the \dQuote{Quick-Transfer} stage, signalling a warning (and
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  in the \dQuote{Quick-Transfer} stage, signalling a warning (and
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  \item{iter}{The number of (outer) iterations.}
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  \item{iter}{The number of (outer) iterations.}
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  \item{ifault}{integer: indicator of a possible algorithm problem
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  \item{ifault}{integer: indicator of a possible algorithm problem
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    -- for experts.}
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    -- for experts.}
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}
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}
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\references{
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\references{
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  Forgy, E. W. (1965).
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  Cluster analysis of multivariate data: efficiency vs interpretability
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  of classifications.
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  \emph{Biometrics}, \bold{21}, 768--769.
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  Hartigan, J. A. and Wong, M. A. (1979).
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  Algorithm AS 136: A K-means clustering algorithm.
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  \emph{Applied Statistics}, \bold{28}, 100--108.
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  \doi{10.2307/2346830}.
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  Lloyd, S. P. (1957, 1982).
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  Least squares quantization in PCM.
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  Technical Note, Bell Laboratories.
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  Published in 1982 in \emph{IEEE Transactions on Information Theory},
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  \bold{28}, 128--137.
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  MacQueen, J. (1967).
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  Some methods for classification and analysis of multivariate
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  observations.
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  \bibshow{*}
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  In \emph{Proceedings of the Fifth Berkeley Symposium on  Mathematical
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    Statistics and Probability},
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  eds L. M. Le Cam & J. Neyman,
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  \bold{1}, pp.\sspace{}281--297.
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  Berkeley, CA: University of California Press.
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}
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}
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\examples{
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\examples{
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require(graphics)
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require(graphics)
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# a 2-dimensional example
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# a 2-dimensional example