Chapter 8 Clustering

The goal of clustering is to discover groups of similar observations among a set of variables \(X_1,\dotsc,X_p\). Clustering is an example of an unsupervised learning method, as we only consider the features \(X_1,\dotsc,X_p\) without an associated response \(Y\).

To illustrate clustering methods, we will use the Auto data in the ISLR R library. The data set contains information on the gas mileage, number of cylinders, displacement, horsepower, weight, acceleration, year, and origin for 392 vehicles.

mpg cylinders displacement horsepower weight acceleration year origin name
18 8 307 130 3504 12.0 70 1 chevrolet chevelle malibu
15 8 350 165 3693 11.5 70 1 buick skylark 320
18 8 318 150 3436 11.0 70 1 plymouth satellite
16 8 304 150 3433 12.0 70 1 amc rebel sst
17 8 302 140 3449 10.5 70 1 ford torino
15 8 429 198 4341 10.0 70 1 ford galaxie 500

8.1 \(k\)-Means

The \(k\)-means clustering algorithm uses the variables \(X_1,\dotsc,X_p\) to partition our observations \(1,\dotsc,n\) into \(k\) non-overlapping groups. The partitioning is done based on the similarity of observations, where similarity is measured using Euclidean distance. Consequently, we will need to rescale our data.

We’ll isolate the first seven variables (mpg, cylinders, displacement, horsepower, weight, acceleration, year) and define them as \(X\).

##       mpg            cylinders       displacement      horsepower    
##  Min.   :-1.8509   Min.   :-1.449   Min.   :-1.208   Min.   :-1.519  
##  1st Qu.:-0.8259   1st Qu.:-0.863   1st Qu.:-0.854   1st Qu.:-0.766  
##  Median :-0.0892   Median :-0.863   Median :-0.415   Median :-0.285  
##  Mean   : 0.0000   Mean   : 0.000   Mean   : 0.000   Mean   : 0.000  
##  3rd Qu.: 0.7116   3rd Qu.: 1.482   3rd Qu.: 0.777   3rd Qu.: 0.559  
##  Max.   : 2.9666   Max.   : 1.482   Max.   : 2.490   Max.   : 3.261  
##      weight        acceleration         year        
##  Min.   :-1.607   Min.   :-2.733   Min.   :-1.6232  
##  1st Qu.:-0.886   1st Qu.:-0.640   1st Qu.:-0.8089  
##  Median :-0.205   Median :-0.015   Median : 0.0055  
##  Mean   : 0.000   Mean   : 0.000   Mean   : 0.0000  
##  3rd Qu.: 0.750   3rd Qu.: 0.538   3rd Qu.: 0.8199  
##  Max.   : 2.546   Max.   : 3.356   Max.   : 1.6343

Let’s start by runing the \(k\)-means algorithm with \(k=2\) and only using the mpg and horsepower variables.

We can see how the algorithm divides the observations into two groups: the red observations have a high mpg but lower horsepower, while the black observations have a low mpg but high horsepower.

Now let’s try using all seven variables to define the clusters.

The plot looks similar: even when we use all variables, the first group of cars (black observations) have a low mpg and high horsepower while the second group (red observations) have a high mpg and low horsepower.

The plot above only shows the clustering solution with respect to two variables (mpg and horsepower). To examine how the clusters are defined over all variables, we can use the pairs( ) function.

Lastly, we can explore clustering solutions for different values of \(k\). For simplicity, we will only examine the clusters for the mpg and horsepower variables.

As \(k\) increases, we get a more granular picture of car segments. However, the problem of interpretation becomes more difficult: What is it that actually differentiates these clusters from each other? We are only plotting the data over two variables, but the other variables also contribute in the determination of cluster assignments.

Moreover, how should we determine an appropriate value of \(k\)? Hierachical clustering provides a partial solution.

8.2 Hierarchical Clustering

Hierarchical clustering addresses the issue of having to choose the number of clusters \(k\), and instead considers a sequence of clusters from \(k = 1,\dotsc,n\). We’ll use the hclust( ) function and dendextend package to fit and plot the output from hierarchical clustering models.

Because we have a few hundred observations, the plot – which called a “dendrogram” – becomes difficult to read and interpret on a small scale (meaning we would need a much larger plotting window!).

Suppose we were interested in a two group clustering solution. Then we can simply color the dendrogram based on the first split.

We can do the same for a larger number of groups, too.

Notice that when interpreting a dendrogram, the analyst must still “choose” \(k\), so the problem still hasn’t really gone away. The only benefit with hierarchical clustering methods is that the analyst can quickly and easily examine the landscape of clustering solutions to understand how the value of \(k\) impacts different clustering solutions.