Clustering Example

Problem: given observations $X_i$ for $i=1,\ldots,n$ group the observations into $k$ populations.

Parallel to discriminant analysis but: no training data.

Here: just show some example analyses:

Cluster iris data: so $k=3$ presumably.

Often: $k$ not known.

Many possible SPlus functions including: agnes, clara, pam, hclust

Example: Cluster the iris data.

Put all 150 observations into 150$\times$4 matrix. (Remove species labels.)

Cluster into 2, 3 ,4 groups using pam:

pamiris2 <- pam(x,2)
pamiris3 <- pam(x,3)
pamiris4 <- pam(x,4)

Output for two clusters:

> pam(x,2)
Call: pam(x = x, k = 2)
Medoids:
     Sepal L. Sepal W. Petal L. Petal W. 
[1,]      5.0      3.4      1.5      0.2
[2,]      6.2      2.8      4.8      1.8
Clustering vector:
  [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 
      2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 [75] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 
      2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2
[112] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 
      2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[149] 2 2

Notice that the algorithm correctly groups together the first 50 observations. The other two species are then lumped together.

> pam(x,3)
Call:
pam(x = x, k = 3)
Medoids:
     Sepal L. Sepal W. Petal L. Petal W. 
[1,]      5.0      3.4      1.5      0.2
[2,]      6.0      2.9      4.5      1.5
[3,]      6.8      3.0      5.5      2.1
Clustering vector:
  [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 2 2 
      2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 [75] 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 
      2 2 2 2 2 2 2 2 3 2 3 3 3 3 2 3 3 3 3
[112] 3 3 2 2 3 3 3 3 2 3 2 3 2 3 3 2 2 3 
      3 3 3 3 2 3 3 3 3 2 3 3 3 2 3 3 3 2 3
[149] 3 2

Notice difficulty with 2 versus 3. Total of two from group 2 clustered into group 3; total of 14 from group 3 clustered into group 2.

Now a method which does not require specification of number of classes but doesn't estimate number of classes either. Hierarchical clustering.

> agnesiris <- agnes(x)
> cutree(agnesiris,k=2)
  [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 
      2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 [38] 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 
      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[112] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[149] 1 1
attr(, "height"):
[1] 0.8964852 0.2645751
> plot(cutree(agnesiris,k=2))
> plot(cutree(agnesiris,k=3))
> plot(cutree(agnesiris,k=4))