R exercise1

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
# 其中scale函数提供数据中心化功能，所谓数据的中心化是指数据集中的各项数据减去数据集的均值，这个函数还提供数据的标准化功能，所谓数据的标准化是指中心化之后的数据在除以数据集的标准差，即数据集中的各项数据减去数据集的均值再除以数据集的标准差。见http://it.zhans.org/10/1834.htm。
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)))
# kmeas聚类满足如下条件
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))
)