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```
#' Demonstration of the k-Means clustering algorithm
#'
#' This function provides a demo of the k-Means cluster algorithm for data
#' containing only two variables (columns).
#'
#' The k-Means cluster algorithm may be regarded as a series of iterations of:
#' finding cluster centers, computing distances between sample points, and
#' redefining cluster membership.
#'
#' The data given by \code{x} is clustered by the \eqn{k}-means method, which
#' aims to partition the points into \eqn{k} groups such that the sum of squares
#' from points to the assigned cluster centers 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).
#' @param x A numercal matrix or an object that can be coerced to such a matrix
#' (such as a numeric vector or a data frame with all numeric columns)
#' containing \emph{only} 2 columns.
#' @param centers Either the number of clusters or a set of initial (distinct)
#' cluster centres. If a number, a random set of (distinct) rows in \code{x}
#' is chosen as the initial centres.
#' @param pch,col Symbols and colors for different clusters; the length of these
#' two arguments should be equal to the number of clusters, or they will be
#' recycled.
#' @param hints Two text strings indicating the steps of k-means clustering:
#' move the center or find the cluster membership?
#' @return A list with components \item{cluster }{A vector of integers
#' indicating the cluster to which each point is allocated.} \item{centers }{A
#' matrix of cluster centers.}
#' @note This function is only for demonstration purpose. For practical
#' applications please refer to \code{\link{kmeans}}.
#'
#' Note that \code{ani.options('nmax')} is defined as the maximum number of
#' iterations in such a sense: an iteration includes the process of computing
#' distances, redefining membership and finding centers. Thus there should be
#' \code{2 * ani.options('nmax')} animation frames in the output if the other
#' condition for stopping the iteration has not yet been met (i.e. the cluster
#' membership will not change any longer).
#' @author Yihui Xie
#' @references Examples at \url{https://yihui.org/animation/example/kmeans-ani/}
#' @seealso \code{\link{kmeans}}
#' @export
kmeans.ani = function(
x = cbind(X1 = runif(50), X2 = runif(50)), centers = 3, hints = c('Move centers!', 'Find cluster?'),
pch = 1:3, col = 1:3
) {
x = as.matrix(x)
ocluster = sample(centers, nrow(x), replace = TRUE)
if (length(centers) == 1) centers = x[sample(nrow(x), centers), ] else
centers = as.matrix(centers)
numcent = nrow(centers)
dst = matrix(nrow = nrow(x), ncol = numcent)
j = 1
pch = rep(pch, length = numcent)
col = rep(col, length = numcent)
for (j in 1:ani.options('nmax')) {
dev.hold()
plot(x, pch = pch[ocluster], col = col[ocluster], panel.first = grid())
mtext(hints[1], 4)
points(centers, pch = pch[1:numcent], cex = 3, lwd = 2, col = col[1:numcent])
ani.pause()
for (i in 1:numcent) {
dst[, i] = sqrt(apply((t(t(x) - unlist(centers[i, ])))^2, 1, sum))
}
ncluster = apply(dst, 1, which.min)
plot(x, type = 'n')
mtext(hints[2], 4)
grid()
ocenters = centers
for (i in 1:numcent) {
xx = subset(x, ncluster == i)
polygon(xx[chull(xx), ], density = 10, col = col[i], lty = 2)
points(xx, pch = pch[i], col = col[i])
centers[i, ] = apply(xx, 2, mean)
}
points(ocenters, cex = 3, col = col[1:numcent], pch = pch[1:numcent], lwd = 2)
ani.pause()
if (all(centers == ocenters)) break
ocluster = ncluster
}
invisible(list(cluster = ncluster, centers = centers))
}
```

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