Chapter 4 Unsupervised Learning

4.1 Introduction

In unsupervised learning (UML), no labels are provided, and the learning algorithm focuses solely on detecting structure in unlabelled input data. One generally differentiates between

  • Clustering, where the goal is to find homogeneous subgroups within the data; the grouping is based on distance between observations.

  • Dimensionality reduction, where the goal is to identify patterns in the features of the data. Dimensionality reduction is often used to facilitate visualisation of the data, as well as a pre-processing method before supervised learning.

UML presents specific challenges and benefits:

  • there is no single goal in UML
  • there is generally much more unlabelled data available than labelled data.

4.2 k-means clustering

The k-means clustering algorithms aims at partitioning n observations into a fixed number of k clusters. The algorithm will find homogeneous clusters.

In R, we use

stats::kmeans(x, centers = 3, nstart = 10)

where

  • x is a numeric data matrix
  • centers is the pre-defined number of clusters
  • the k-means algorithm has a random component and can be repeated nstart times to improve the returned model

Challenge:

  • To learn about k-means, let’s use the iris dataset with the sepal and petal length variables only (to facilitate visualisation). Create such a data matrix and name it x
  • Run the k-means algorithm on the newly generated data x, save the results in a new variable cl, and explore its output when printed.
  • The actual results of the algorithms, i.e. the cluster membership can be accessed in the clusters element of the clustering result output. Use it to colour the inferred clusters to generate a figure like that shown below.
k-means algorithm on sepal and petal lengths

Figure 4.1: k-means algorithm on sepal and petal lengths

i <- grep("Length", names(iris))
x <- iris[, i]
cl <- kmeans(x, 3, nstart = 10)
plot(x, col = cl$cluster)

4.2.1 How does k-means work

Initialisation: randomly assign class membership

set.seed(12)
init <- sample(3, nrow(x), replace = TRUE)
plot(x, col = init)
k-means random intialisation

Figure 4.2: k-means random intialisation

Iteration:

  1. Calculate the centre of each subgroup as the average position of all observations is that subgroup.
  2. Each observation is then assigned to the group of its nearest centre.

It’s also possible to stop the algorithm after a certain number of iterations, or once the centres move less than a certain distance.

par(mfrow = c(1, 2))
plot(x, col = init)
centres <- sapply(1:3, function(i) colMeans(x[init == i, ], ))
centres <- t(centres)
points(centres[, 1], centres[, 2], pch = 19, col = 1:3)

tmp <- dist(rbind(centres, x))
tmp <- as.matrix(tmp)[, 1:3]

ki <- apply(tmp, 1, which.min)
ki <- ki[-(1:3)]

plot(x, col = ki)
points(centres[, 1], centres[, 2], pch = 19, col = 1:3)
k-means iteration: calculate centers (left) and assign new cluster membership (right)

Figure 4.3: k-means iteration: calculate centers (left) and assign new cluster membership (right)

Termination: Repeat iteration until no point changes its cluster membership.

k-means convergence (credit Wikipedia)

k-means convergence (credit Wikipedia)

4.2.2 Model selection

Due to the random initialisation, one can obtain different clustering results. When k-means is run multiple times, the best outcome, i.e. the one that generates the smallest total within cluster sum of squares (SS), is selected. The total within SS is calculated as:

For each cluster results:

  • for each observation, determine the squared euclidean distance from observation to centre of cluster
  • sum all distances

Note that this is a local minimum; there is no guarantee to obtain a global minimum.

Challenge:

Repeat k-means on our x data multiple times, setting the number of iterations to 1 or greater and check whether you repeatedly obtain the same results. Try the same with random data of identical dimensions.

cl1 <- kmeans(x, centers = 3, nstart = 10)
cl2 <- kmeans(x, centers = 3, nstart = 10)
table(cl1$cluster, cl2$cluster)
##    
##      1  2  3
##   1  0 41  0
##   2 51  0  0
##   3  0  0 58
cl1 <- kmeans(x, centers = 3, nstart = 1)
cl2 <- kmeans(x, centers = 3, nstart = 1)
table(cl1$cluster, cl2$cluster)
##    
##      1  2  3
##   1 41  0  0
##   2  0  0 58
##   3  0 51  0
set.seed(42)
xr <- matrix(rnorm(prod(dim(x))), ncol = ncol(x))
cl1 <- kmeans(xr, centers = 3, nstart = 1)
cl2 <- kmeans(xr, centers = 3, nstart = 1)
table(cl1$cluster, cl2$cluster)
##    
##      1  2  3
##   1  0 52  0
##   2  0  0 46
##   3 52  0  0
diffres <- cl1$cluster != cl2$cluster
par(mfrow = c(1, 2))
plot(xr, col = cl1$cluster, pch = ifelse(diffres, 19, 1))
plot(xr, col = cl2$cluster, pch = ifelse(diffres, 19, 1))
Different k-means results on the same (random) data

Figure 4.4: Different k-means results on the same (random) data

4.2.3 How to determine the number of clusters

  1. Run k-means with k=1, k=2, …, k=n
  2. Record total within SS for each value of k.
  3. Choose k at the elbow position, as illustrated below.

Challenge

Calculate the total within sum of squares for k from 1 to 5 for our x test data, and reproduce the figure above.

ks <- 1:5
tot_within_ss <- sapply(ks, function(k) {
    cl <- kmeans(x, k, nstart = 10)
    cl$tot.withinss
})
plot(ks, tot_within_ss, type = "b")

4.3 Hierarchical clustering

4.3.1 How does hierarchical clustering work

Initialisation: Starts by assigning each of the n points its own cluster

Iteration

  1. Find the two nearest clusters, and join them together, leading to n-1 clusters
  2. Continue the cluster merging process until all are grouped into a single cluster

Termination: All observations are grouped within a single cluster.

Hierarchical clustering: initialisation (left) and colour-coded results after iteration (right).

Figure 4.5: Hierarchical clustering: initialisation (left) and colour-coded results after iteration (right).

The results of hierarchical clustering are typically visualised along a dendrogram, where the distance between the clusters is proportional to the branch lengths.

Visualisation of the hierarchical clustering results on a dendrogram

Figure 4.6: Visualisation of the hierarchical clustering results on a dendrogram

In R:

  • Calculate the distance using dist, typically the Euclidean distance.
  • Hierarchical clustering on this distance matrix using hclust

Challenge

Apply hierarchical clustering on the iris data and generate a dendrogram using the dedicated plot method.

d <- dist(iris[, 1:4])
hcl <- hclust(d)
hcl
## 
## Call:
## hclust(d = d)
## 
## Cluster method   : complete 
## Distance         : euclidean 
## Number of objects: 150
plot(hcl)

4.3.2 Defining clusters

After producing the hierarchical clustering result, we need to cut the tree (dendrogram) at a specific height to defined the clusters. For example, on our test dataset above, we could decide to cut it at a distance around 1.5, with would produce 2 clusters.

Cutting the dendrogram at height 1.5.

Figure 4.7: Cutting the dendrogram at height 1.5.

In R we can us the cutree function to

  • cut the tree at a specific height: cutree(hcl, h = 1.5)
  • cut the tree to get a certain number of clusters: cutree(hcl, k = 2)

Challenge

  • Cut the iris hierarchical clustering result at a height to obtain 3 clusters by setting h.
  • Cut the iris hierarchical clustering result at a height to obtain 3 clusters by setting directly k, and verify that both provide the same results.
plot(hcl)
abline(h = 3.9, col = "red")

cutree(hcl, k = 3)
##   [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
##  [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 2 3 2 3 2 3 3 3 3 2 3 2 3 3 2 3
##  [71] 2 3 2 2 2 2 2 2 2 3 3 3 3 2 3 2 2 2 3 3 3 2 3 3 3 3 3 2 3 3
##  [ reached getOption("max.print") -- omitted 50 entries ]
cutree(hcl, h = 3.9)
##   [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
##  [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 2 3 2 3 2 3 3 3 3 2 3 2 3 3 2 3
##  [71] 2 3 2 2 2 2 2 2 2 3 3 3 3 2 3 2 2 2 3 3 3 2 3 3 3 3 3 2 3 3
##  [ reached getOption("max.print") -- omitted 50 entries ]
identical(cutree(hcl, k = 3), cutree(hcl, h = 3.9))
## [1] TRUE

Challenge

Using the same value k = 3, verify if k-means and hierarchical clustering produce the same results on the iris data.

Which one, if any, is correct?

km <- kmeans(iris[, 1:4], centers = 3, nstart = 10)
hcl <- hclust(dist(iris[, 1:4]))
table(km$cluster, cutree(hcl, k = 3))
##    
##      1  2  3
##   1  0 34 28
##   2  0 38  0
##   3 50  0  0
par(mfrow = c(1, 2))
plot(iris$Petal.Length, iris$Sepal.Length, col = km$cluster, main = "k-means")
plot(iris$Petal.Length, iris$Sepal.Length, col = cutree(hcl, k = 3), main = "Hierarchical clustering")

## Checking with the labels provided with the iris data
table(iris$Species, km$cluster)
##             
##               1  2  3
##   setosa      0  0 50
##   versicolor 48  2  0
##   virginica  14 36  0
table(iris$Species, cutree(hcl, k = 3))
##             
##               1  2  3
##   setosa     50  0  0
##   versicolor  0 23 27
##   virginica   0 49  1

4.4 Pre-processing

Many of the machine learning methods that are regularly used are sensitive to difference scales. This applies to unsupervised methods as well as supervised methods, as we will see in the next chapter.

A typical way to pre-process the data prior to learning is to scale the data, or apply principal component analysis (next section). Scaling assures that all data columns have a mean of 0 and standard deviation of 1.

In R, scaling is done with the scale function.

Challenge

Using the mtcars data as an example, verify that the variables are of different scales, then scale the data. To observe the effect different scales, compare the hierarchical clusters obtained on the original and scaled data.

colMeans(mtcars)
##        mpg        cyl       disp         hp       drat         wt 
##  20.090625   6.187500 230.721875 146.687500   3.596563   3.217250 
##       qsec         vs         am       gear       carb 
##  17.848750   0.437500   0.406250   3.687500   2.812500
hcl1 <- hclust(dist(mtcars))
hcl2 <- hclust(dist(scale(mtcars)))
par(mfrow = c(1, 2))
plot(hcl1, main = "original data")
plot(hcl2, main = "scaled data")

4.5 Principal component analysis (PCA)

Dimensionality reduction techniques are widely used and versatile techniques that can be used to:

  • find structure in features
  • pre-processing for other ML algorithms, and
  • aid in visualisation.

The basic principle of dimensionality reduction techniques is to transform the data into a new space that summarise properties of the whole data set along a reduced number of dimensions. These are then ideal candidates used to visualise the data along these reduced number of informative dimensions.

4.5.1 How does it work

Principal Component Analysis (PCA) is a technique that transforms the original n-dimensional data into a new n-dimensional space.

  • These new dimensions are linear combinations of the original data, i.e. they are composed of proportions of the original variables.
  • Along these new dimensions, called principal components, the data expresses most of its variability along the first PC, then second, …
  • Principal components are orthogonal to each other, i.e. non-correlated.
Original data (left). PC1 will maximise the variability while minimising the residuals (centre). PC2 is orthogonal to PC1 (right).

Figure 4.8: Original data (left). PC1 will maximise the variability while minimising the residuals (centre). PC2 is orthogonal to PC1 (right).

In R, we can use the prcomp function.

Let’s explore PCA on the iris data. While it contains only 4 variables, is already becomes difficult to visualise the 3 groups along all these dimensions.

pairs(iris[, -5], col = iris[, 5], pch = 19)

Let’s use PCA to reduce the dimension.

irispca <- prcomp(iris[, -5])
summary(irispca)
## Importance of components:
##                           PC1     PC2    PC3     PC4
## Standard deviation     2.0563 0.49262 0.2797 0.15439
## Proportion of Variance 0.9246 0.05307 0.0171 0.00521
## Cumulative Proportion  0.9246 0.97769 0.9948 1.00000

A summary of the prcomp output shows that along PC1 along, we are able to retain over 92% of the total variability in the data.

Iris data along PC1.

Figure 4.9: Iris data along PC1.

4.5.2 Visualisation

A biplot features all original points re-mapped (rotated) along the first two PCs as well as the original features as vectors along the same PCs. Feature vectors that are in the same direction in PC space are also correlated in the original data space.

biplot(irispca)

One important piece of information when using PCA is the proportion of variance explained along the PCs, in particular when dealing with high dimensional data, as PC1 and PC2 (that are generally used for visualisation), might only account for an insufficient proportion of variance to be relevant on their own.

In the code chunk below, I extract the standard deviations from the PCA result to calculate the variances, then obtain the percentage of and cumulative variance along the PCs.

var <- irispca$sdev^2
(pve <- var/sum(var))
## [1] 0.924618723 0.053066483 0.017102610 0.005212184
cumsum(pve)
## [1] 0.9246187 0.9776852 0.9947878 1.0000000

Challenge

  • Repeat the PCA analysis on the iris dataset above, reproducing the biplot and preparing a barplot of the percentage of variance explained by each PC.
  • It is often useful to produce custom figures using the data coordinates in PCA space, which can be accessed as x in the prcomp object. Reproduce the PCA plots below, along PC1 and PC2 and PC3 and PC4 respectively.

par(mfrow = c(1, 2))
plot(irispca$x[, 1:2], col = iris$Species)
plot(irispca$x[, 3:4], col = iris$Species)

4.5.3 Data pre-processing

We haven’t looked at other prcomp parameters, other that the first one, x. There are two other ones that are or importance, in particular in the light of the section on pre-processing above, which are center and scale.. The former is set to TRUE by default, while the second one is set the FALSE.

Challenge

Repeat the analysis comparing the need for scaling on the mtcars dataset, but using PCA instead of hierarchical clustering. When comparing the two.

par(mfrow = c(1, 2))
biplot(prcomp(mtcars, scale = FALSE), main = "No scaling")  ## 1
biplot(prcomp(mtcars, scale = TRUE), main = "With scaling") ## 2

Without scaling, disp and hp are the features with the highest loadings along PC1 and 2 (all others are negligible), which are also those with the highest units of measurement. Scaling removes this effect.

4.5.4 Final comments on PCA

Real datasets often come with missing values. In R, these should be encoded using NA. Unfortunately, PCA cannot deal with missing values, and observations containing NA values will be dropped automatically. This is a viable solution only when the proportion of missing values is low.

It is also possible to impute missing values. This is described in greater details in the Data pre-processing section in the supervised machine learning chapter.

Finally, we should be careful when using categorical data in any of the unsupervised methods described above. Categories are generally represented as factors, which are encoded as integer levels, and might give the impression that a distance between levels is a relevant measure (which it is not, unless the factors are ordered). In such situations, categorical data can be dropped, or it is possible to encode categories as binary dummy variables. For example, if we have 3 categories, say A, B and C, we would create two dummy variables to encode the categories as:

x y
A 1 0
B 0 1
C 0 0

so that the distance between each category are approximately equal to 1.

4.6 t-Distributed Stochastic Neighbour Embedding

t-Distributed Stochastic Neighbour Embedding (t-SNE) is a non-linear dimensionality reduction technique, i.e. that different regions of the data space will be subjected to different transformations. t-SNE will compress small distances, thus bringing close neighbours together, and will ignore large distances. It is particularly well suited for very high dimensional data.

In R, we can use the Rtsne function from the Rtsne. Before, we however need to remove any duplicated entries in the dataset.

library("Rtsne")
uiris <- unique(iris[, 1:5])
iristsne <- Rtsne(uiris[, 1:4])
plot(iristsne$Y, col = uiris$Species)

As with PCA, the data can be scaled and centred prior the running t-SNE (see the pca_center and pca_scale arguments). The algorithm is stochastic, and will produce different results at each repetition.

4.6.1 Parameter tuning

t-SNE (as well as many other methods, in particular classification algorithms) has two important parameters that can substantially influence the clustering of the data

  • Perplexity: balances global and local aspects of the data.
  • Iterations: number of iterations before the clustering is stopped.

It is important to adapt these for different data. The figure below shows a 5032 by 20 dataset that represent protein sub-cellular localisation.

Effect of different perplexity and iterations when running t-SNE

Effect of different perplexity and iterations when running t-SNE

As a comparison, below are the same data with PCA (left) and t-SNE (right).

PCA and t-SNE on hyperLOPIT

PCA and t-SNE on hyperLOPIT