k-means clustering and Voronoi sets

In the context of https://latex.codecogs.com/gif.latex?k-means, we want to partition the space of our observations into https://latex.codecogs.com/gif.latex?k classes. each observation belongs to the cluster with the nearest mean. Here “nearest” is in the sense of some norm, usually the https://latex.codecogs.com/gif.latex?\ell_2 (Euclidean) norm.

Consider the case where we have 2 classes. The means being respectively the 2 black dots. If we partition based on the nearest mean, with the https://latex.codecogs.com/gif.latex?\ell_2 (Euclidean) norm we get the graph on the left, and with the https://latex.codecogs.com/gif.latex?\ell_1 (Manhattan) norm, the one on the right,

Points in the red region are closer to the mean in the upper part, while points in the blue region are closer to the mean in the lower part. Here, we will always use the standard https://latex.codecogs.com/gif.latex?\ell_2 (Euclidean) norm. Note that the graph above is related to Voronoi diagrams (or Voronoy, from Вороний in Ukrainian, or Вороно́й in Russian) with 2 points, the 2 means.

In order to illustrate the https://latex.codecogs.com/gif.latex?k-means clustering algorithm (here Lloyd’s algorithm) consider the following dataset

set.seed(1)
pts <- cbind(X=rnorm(500,rep(seq(1,9,by=2)/10,100),.022),Y=rnorm(500,.5,.15))
plot(pts)

Here, we have 5 groups.  So let us run a 5-means algorithm here.

  • we draw randomly 5 points in the space (intial values for the means), https://latex.codecogs.com/gif.latex?\boldsymbol{\mu}_1^{(1)},\cdots,\boldsymbol{\mu}_k^{(1)}
  • in the assignment step, we assign each point to the nearest mean

https://latex.codecogs.com/gif.latex?S_i^{(t)}%20=%20\big%20\{%20\boldsymbol{x}_j%20:%20\big%20\|%20\boldsymbol{x}_j%20-%20\boldsymbol{\mu}^{(t)}_i%20\big%20\|^2%20\leq%20\big%20\|%20\boldsymbol{x}_j%20-%20\boldsymbol{\mu}^{(t)}_{i%27}%20\big%20\|^2%20\%20\forall%20i%27\in\{1%20,\cdots,%20k\}%20\big\}

  • in the update step, we compute the new centroids of the clusters

https://latex.codecogs.com/gif.latex?%20%20%20%20\boldsymbol{\mu}^{(t+1)}_i%20=%20\frac{1}{|S^{(t)}_i|}%20\sum_{\boldsymbol{x}_j%20\in%20S^{(t)}_i}%20\boldsymbol{x}_j

To visualize it, see

The code the get the clusters is

kmeans(pts, centers=5, nstart = 1, algorithm = "Lloyd")

Observe that the assignment step is based on computations of Voronoi sets. This can be done in R using

library(tripack)
V <- voronoi.mosaic(means[,1],means[,2])
P <- voronoi.polygons(V)
points(V,pch=19)
plot(V,add=TRUE)

This is what we can visualize below

The code to visualize the https://latex.codecogs.com/gif.latex?k means, and the https://latex.codecogs.com/gif.latex?k clusters (or regions), use

km1 <- kmeans(pts, centers=5, nstart = 1, algorithm = "Lloyd")
library(tripack)
library(RColorBrewer)
CL5 <- brewer.pal(5, "Pastel1")
V <- voronoi.mosaic(km1$centers[,1],km1$centers[,2])
P <- voronoi.polygons(V)
plot(pts,pch=19,xlim=0:1,ylim=0:1,xlab="",ylab="",col=CL5[km1$cluster])
points(km1$centers[,1],km1$centers[,2],pch=3,cex=1.5,lwd=2)
plot(V,add=TRUE)

Here, starting points are draw randomly. If we run it again, we might get

or

On that dataset, it is difficult to get cluster that are the five groups we can actually see. If we use

set.seed(1)
A <- c(rep(.2,100),rep(.2,100),rep(.5,100),rep(.8,100),rep(.8,100))
B <- c(rep(.2,100),rep(.8,100),rep(.5,100),rep(.2,100),rep(.8,100))
pts <- cbind(X=rnorm(500,A,.075),Y=rnorm(500,B,.075))

we usually get something better

Colors are obtained from clusters of the https://latex.codecogs.com/gif.latex?k-means function, but additional lines are obtained using as outputs of Voronoi diagrams functions.


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