Consider the following dataset, with (only) ten points

x=c(.4,.55,.65,.9,.1,.35,.5,.15,.2,.85)
y=c(.85,.95,.8,.87,.5,.55,.5,.2,.1,.3)
plot(x,y,pch=19,cex=2)

We want to get – say – two clusters. Or more specifically, two sets of observations, each of them sharing some similarities.

Since the number of observations is rather small, it is actually possible to get an exhaustive list of all partitions, and to minimize some criteria, such as the within variance. Given a vector with clusters, we compute the within variance using

within_var = function(I){
I0=which(I==0)
I1=which(I==1)
xbar0=mean(x[I0])
xbar1=mean(x[I1])
ybar0=mean(y[I0])
ybar1=mean(y[I1])
w=sum(I0)*sum( (x[I0]-xbar0)^2+(y[I0]-ybar0)^2 )+
  sum(I1)*sum( (x[I1]-xbar1)^2+(y[I1]-ybar1)^2 )
return(c(I,w))
}

Then, to compute all possible partitions, use

base2=function(z,n=10){
  Base.b=rep(0,n)
  ndigits=(floor(logb(z, base=2))+1)
  for(i in 1:ndigits){
    Base.b[ n-i+1]=(z%%2)
    z=(z%/%2)}
  return(Base.b)}
L=function(x) within_var(base2(x))
S=sapply(1:(2^10),L)

The cluster indices at the mimimum is here

I=S[1:n,which.min(S[n+1,])]

To visualize those clusters, use

cluster_viz = function(indices){
library(RColorBrewer)
CL2palette=rev(brewer.pal(n = 9, name = "RdYlBu"))
CL2f=CL2palette[c(1,9)]
plot(x,y,pch=19,xlab="",ylab="",xlim=0:1,ylim=0:1,cex=2,col=CL2f[1+I])
CL2c=CL2palette[c(3,7)]
I0=which(indices==0)
I1=which(indices==1)
xbar0=mean(x[I0])
xbar1=mean(x[I1])
ybar0=mean(y[I0])
ybar1=mean(y[I1])
segments(x[I0],y[I0],xbar0,ybar0,col=CL2c[1])
segments(x[I1],y[I1],xbar1,ybar1,col=CL2c[2])
points(xbar0,ybar0,pch=19,cex=1.5,col=CL2c[1])
points(xbar1,ybar1,pch=19,cex=1.5,col=CL2c[2])}

and then, simply

cluster_viz(I)

But that was possible only because is not to large (since the total number of scenarios – with only 2 clusters – is , or if we changes zeros in ones).

Continue reading Visualizing Clusters →