Applied Exercise 10.9

Consider the USArrests data. We will now perform hierarchical clustering on the states.

1. Using hierarchical clustering with complete linkage and Euclidean distance, cluster the states.

set.seed(2)
hc.complete <- hclust(dist(USArrests), method = "complete")
plot(hc.complete)

2. Cut the dendrogram at a height that results in three distinct clusters. Which states belong to which clusters?

cutree(hc.complete, 3)

##        Alabama         Alaska        Arizona       Arkansas     California 
##              1              1              1              2              1 
##       Colorado    Connecticut       Delaware        Florida        Georgia 
##              2              3              1              1              2 
##         Hawaii          Idaho       Illinois        Indiana           Iowa 
##              3              3              1              3              3 
##         Kansas       Kentucky      Louisiana          Maine       Maryland 
##              3              3              1              3              1 
##  Massachusetts       Michigan      Minnesota    Mississippi       Missouri 
##              2              1              3              1              2 
##        Montana       Nebraska         Nevada  New Hampshire     New Jersey 
##              3              3              1              3              2 
##     New Mexico       New York North Carolina   North Dakota           Ohio 
##              1              1              1              3              3 
##       Oklahoma         Oregon   Pennsylvania   Rhode Island South Carolina 
##              2              2              3              2              1 
##   South Dakota      Tennessee          Texas           Utah        Vermont 
##              3              2              2              3              3 
##       Virginia     Washington  West Virginia      Wisconsin        Wyoming 
##              2              2              3              3              2

3. Hierarchically cluster the states using complete linkage and Eu- clidean distance, after scaling the variables to have standard deviation one.

sd.data <- scale(USArrests)
hc.complete.sd <- hclust(dist(sd.data), method = "complete")
plot(hc.complete.sd)

4. What effect does scaling the variables have on the hierarchical clustering obtained? In your opinion, should the variables be scaled before the inter-observation dissimilarities are computed? Provide a justification for your answer.

cutree(hc.complete.sd, 3)

##        Alabama         Alaska        Arizona       Arkansas     California 
##              1              1              2              3              2 
##       Colorado    Connecticut       Delaware        Florida        Georgia 
##              2              3              3              2              1 
##         Hawaii          Idaho       Illinois        Indiana           Iowa 
##              3              3              2              3              3 
##         Kansas       Kentucky      Louisiana          Maine       Maryland 
##              3              3              1              3              2 
##  Massachusetts       Michigan      Minnesota    Mississippi       Missouri 
##              3              2              3              1              3 
##        Montana       Nebraska         Nevada  New Hampshire     New Jersey 
##              3              3              2              3              3 
##     New Mexico       New York North Carolina   North Dakota           Ohio 
##              2              2              1              3              3 
##       Oklahoma         Oregon   Pennsylvania   Rhode Island South Carolina 
##              3              3              3              3              1 
##   South Dakota      Tennessee          Texas           Utah        Vermont 
##              3              1              2              3              3 
##       Virginia     Washington  West Virginia      Wisconsin        Wyoming 
##              3              3              3              3              3

table(cutree(hc.complete, 3), cutree(hc.complete.sd, 3), dnn = c('Not-Scaled', 'Scaled'))

##           Scaled
## Not-Scaled  1  2  3
##          1  6  9  1
##          2  2  2 10
##          3  0  0 20

Scaling the variables affect the clusters obtained although by scaling the data the size of the third cluster increases subastantially making up alomost 1/3 of the data present. Varaibles used in clustering algorithms should always be scaled beforehand. The diffence in scale of the features used could artificially pull a data point towards a particular cluster.