Hierarchical clustering in a nutshell
Agglomerative hierarchical clustering first assigns each observation
(1D point in our case) to a singleton cluster. Thus, we start off with
as many clusters as there are observations. Then, in each step of the
algorithm, the closest two clusters get merged. In order to decide,
which clusters are closest, we need a way to measure either a
distance, a dissimilarity or a similarity between clusters.
Below, for clarity, we will drop saying a distance, or a
dissimilarity, or a similarity and will refer to a
distance in this broader colloquial sense, in which for instance a
triangle inequality may not hold.
Linkage function concept
Please note, that we start off with a measure of distance, but it
works for observations only. So we need to generalize this initial
measure to work for clusters, too. It is easy for singleton clusters -
we simply say, that a distance (or a dissimilarity, or a similarity)
between two singleton clusters is the same as between the two
observations involved.
But in order to say what is a distance between more complex clusters,
we need a concept of a linkage function. This concept
constitutes a link between the distance for clusters and the distance
between observations, hence its name.
For instance, we could say that a distance between two clusters \(A\) and \(B\) is the same as the minimal distance
between any observation \(a \in A\) and
any observation \(b \in B\). This would
be called a single linkage in hierarchical clustering
terminology.
Sometimes, instead of defining a cluster-wise distance in terms of
distances between the clustered observations, it would be easier to
build a distance concept for any two clusters (that are present in a
current step of our hierarchical clustering procedure) inductively upon
the distance concept as it got defined for smaller clusters. Observe,
that we have it defined for singleton clusters already. Then, we could
say, for instance, that after \(A\) and
\(B\) got merged (denoted \(A \cup B\)) the distance between \(A \cup B\) and any other cluster \(C\) is the arithmetic average between two
distances: the one between \(A\) and
\(C\) and the one between \(B\) and \(C\). This would be called a
mcquitty or WPGMA linkage clustering.
Merging height
Now, that we understand a concept of a linkage function, the concept
of closest clusters becomes clear, too. After the closest clusters get
merged, the height of this merging is defined as their
cluster-wise distance. Obviously not only the choice of the closest
clusters, but also the merging height, they both depend on a choice of a
linkage function.
hclust1d supports a comprehensive list of choices of a
linkage function, matching all possible choices in
stats::hclust with an addition of a
true_median linkage.
Linkage functions list
Below find a complete list of all linkage functions supported by
hclust1d. We also state what is a distance of the newly
merged cluster \(A \cup B\) and some
other cluster \(C\). Sometimes it can
be done in terms of the observations involved, and sometimes an
inductive definition is easier. Below, the distance function is denoted
\(d(\cdot, \cdot)\) and it works for
observations, and with a slight abuse of notation for clusters or for
arbitrary points; the number of observations in a cluster \(X\) is denoted \(\left | X \right |\).
complete: a distance between two clusters is the
maximum distance between all pairs of observations in the involved
clusters, formally \(d(A \cup B, C) = \max_{x
\in A \cup B,\,y \in C}d(x,y)\).
single: a distance between two clusters is the
minimum distance between all pairs of observations in the involved
clusters, formally \(d(A \cup B, C) = \min_{x
\in A \cup B,\,y \in C}d(x,y)\).
average, called also UPGMA (Unweighted Pair Group
Method with Arithmetic mean): a distance between two clusters is the
(arithmetic) average distance between all pairs of observations in the
involved clusters, formally \(d(A \cup B, C) =
\frac{\sum_{x \in A \cup B} \sum_{y \in C} d(x,y)}{\left | A \cup B
\right | \cdot \left | C \right |}\).
centroid, called also UPGMC (Unweighted Pair Group
Method with Centroid average): \(d(A \cup B,
C) = \left \| \frac{ \sum_{x \in A \cup B} x }{\left | A \cup B \right
|} - \frac{ \sum_{y \in C} y }{\left | C \right |} \right \|
^2\). Observe, that centroid linkage reports height as
the squared euclidean distance between clusters’
centroids. With \(d(\cdot, \cdot)\)
being an euclidean distance, this can be rewritten as \(d(A \cup B, C) = d \left( \frac{ \sum_{x \in A
\cup B} x}{\left | A \cup B \right |}, \frac{ \sum_{y \in C} y}{\left |
C \right |} \right ) ^2\).
median, called also WPGMC (Weighted Pair Group
Method with Centroid average): \(d(A \cup B,
C) = d(m_{A \cup B}, m_C)^2\) with \(m_X\) equal to the observation in cases of
\(X\) being a singleton, and \(m_{A \cup B} = \frac{1}{2}\left (m_A + m_B \right
)\) for merged clusters \(A\)
and \(B\). Observe, that
median linkage reports height as the squared
distance, similarly to centroid linkage. Observe also, that
this definition is in odds with the Wikipedia
hierarchical clustering page, and although it is compatible with
stats::hclust, this behavior is not well documented in
stats::hclust, either.
mcquitty, called also WPGMA (Weighted Pair Group
Method with Arithmetic mean): \(d(A \cup B, C)
= \frac{d(A, C) + d(B, C)}{2}\)
ward.D, called also MISSQ (Minimum Increase of Sum
of SQuares): \(d(A \cup B, C) = 2 \cdot
\frac{\left | A \cup B \right | \cdot \left | C \right |}{\left | A \cup
B \cup C \right |} \cdot \left \| \frac{ \sum_{x \in A \cup B} x}{\left
| A \cup B \right |} - \frac{ \sum_{y \in C} y}{\left | C \right
|} \right \| ^2\). Observe, that ward.D linkage reports
height as the squared euclidean distance between
clusters’ centroids weighted with a harmonic mean of relevant clusters’
sizes. This definition is in odds with what one can read in the Wikipedia
hierarchical clustering page. With \(d(\cdot, \cdot)\) being an euclidean
distance, this can be rewritten as \(d(A \cup
B, C) = 2 \cdot \frac{\left | A \cup B \right | \cdot \left | C \right
|}{\left | A \cup B \cup C \right |} \cdot d \left( \frac{ \sum_{x \in A
\cup B} x}{\left | A \cup B \right |}, \frac{ \sum_{y \in C} y}{\left |
C \right |} \right ) ^2\).
ward.D2: added to stats::hclust in
R >3.0.3 versions to implement the original
Ward’s linkage function [Murtagh and Legendre, 2014] which is
not implemented with ward.D. The reported height
in ward.D2 is the square root of the height in
ward.D, i.e. \(d(A \cup B, C) =
\sqrt{ 2 \cdot \frac{\left | A \cup B \right | \cdot \left | C \right
|}{\left | A \cup B \cup C \right |} } \cdot \left \| \frac{ \sum_{x \in
A \cup B} x}{\left | A \cup B \right |} - \frac{ \sum_{y \in C}
y}{\left | C \right |} \right \|\). So ward.D2 linkage
reports height as the unsquared euclidean distance
between clusters’ centroids weighted with a square root of harmonic mean
of relevant clusters’ sizes. With \(d(\cdot,
\cdot)\) being an euclidean distance, this can be rewritten as
\(d(A \cup B, C) = \sqrt{2 \cdot \frac{\left |
A \cup B \right | \cdot \left | C \right |}{\left | A \cup B \cup C
\right |} } \cdot d \left( \frac{ \sum_{x \in A \cup B} x }{\left | A
\cup B \right |}, \frac{ \sum_{y \in C} y }{\left | C \right |} \right
)\).
true_median: \(d(A \cup B,
C) = d(m_{A \cup B}, m_C)\) with \(m_X\) being the median of observations in
\(X\) (specifically, the middle-valued
observation in case of \(\left | X \right
|\) odd, and the arithmetic mean of the two middle-valued
observations in case of \(\left | X \right
|\) even). Note also, that a concept of a median makes sense only
for 1D points.
Computational complexity of 1D hierarchical clustering
Hierarchical clustering in the 1D setting has time complexity of
\(\mathcal{O}(n\log n)\) time
regardless of the linkage function used.
Compatibility with stats::hclust was high in the
priority list and thus for 1D data it is simply a matter of a
plug-and-play replacement of stats::hclust calls to be able
to use the advantage of our fast implementation of this asymptotically
much more efficient algorithm. The how-to is covered in detail in our replacing
stats::hclust vignette.
OK, so let’s get started
To load the package into memory execute the following line:
We will work with random normally distributed data points in this
vignette.
Working with hclust1d is very simple an requires only
passing the data points vector and optionally a linkage method to
hclust1d function (complete linkage is used as a default,
if the linkage method is not provided). The simplest example of a
complete linkage clustering:
result <- hclust1d(points)
The hclust1d function returns an object of the same type
that is returned from stats::hclust (a list object with S3
class , to be specific).
This makes it straightforward to further work with the clustering
results. E.g. we can plot it (observe that plot.hclust gets
called internally below):
We can also generate clustering for the named 1D points:
names(points) <- paste("point", 0:9, sep = "-")
result <- hclust1d(points)
plot(result)
We can print the clustering result:
print(result)
#>
#> Call:
#> hclust1d(x = points)
#>
#> Cluster method : complete
#> Distance : euclidean
#> Number of objects: 10
Or we can convert the clustering result to a dendrogram and
further work with it (observe that plot.dendrogram gets
called internally below):
dendrogram <- as.dendrogram(result)
plot(dendrogram, type = "triangle", ylab = "Height")
Or we can use any other specialized packages, like
ggdendro with ggplot2 or ape
packages, to further visualize and work with the result.
Should I provide points or rather a distance matrix to
hclust1d?
In a default statistical package in R,
stats::hclust requires that the dist structure
is provided for clustering. In fact, stats::hclust cannot
be executed on raw \(\mathbb{R}^d\)
points.
However, hclust1d is more flexible in this regard. It
accepts both \(\mathbb{R}^d\) points as
input (with \(d=1\), because
hclust1d works only in 1D setting) and a dist
S3 class input. Actually, the raw points are recreated from a distance
structure anyway, so raw point input is preferred and works a little bit
faster.
If you want to provide the dist structure, please
remember to change a distance argument to TRUE
in a call to hclust1d. The two examples below return
results that are equivalent (but not equal - a note on that follows
below).
Examples
For diversity, single linkage is used in the two
examples below:
result_points <- hclust1d(points, method = "single")
plot(result_points)
distances <- dist(points)
result_dist <- hclust1d(distances, distance = TRUE, method = "single")
plot(result_dist)
But are the results the same? On a close inspection, you’ll notice
that the resulting clusterings are mirror reflections of each other. It
is perfectly OK, the distance structure specifies the mutual distances,
but not the order or shift of points, so the resulting clusterings are
equivalent up to the order and shift.
What about squared distances?
In a default statistical package in R,
stats::hclust requires that the squared
dist structure is provided for ward.D,
centroid and median linkage functions. It is
also reflected in merging height resulting in those linkages: the height
is returned as the appropriate distance measure,
squared.
Again, hclust1d is more flexible in this regard. It
accepts both squared and unsquared distances (squared or unsquared
dist S3 class input), and it even accepts raw \(\mathbb{R}^d\) points as input (with \(d=1\), because hclust1d works
only in 1D setting). Actually, the raw points are preferred in this
setting, too.
The below points should be observed:
- If you want to provide the unsquared
dist structure, please remember to change a
distance argument to TRUE in a call to
hclust1d.
- If you want to provide the squared
dist structure, please remember to change both
distance and squared arguments to
TRUE in a call to hclust1d.
Examples
result_points <- hclust1d(points, method = "ward.D")
plot(result_points)
distances <- dist(points)
result_dist <- hclust1d(distances, distance = TRUE, method = "ward.D")
plot(result_dist)
squared_distances <- distances ^ 2
result_dist <- hclust1d(squared_distances, distance = TRUE, squared = TRUE, method = "ward.D")
plot(result_dist)
The three examples above return results that are equivalent (up to
the order and shift, because again you will notice, that the first
clustering is a mirror reflection of the second and the third
clusterings).
Please also note, that regardless of input, the height is returned as
the appropriately squared distance measure for
ward.D, centroid and median
linkage functions, so the results remain compatible with
stats::hclust results for the squared dist
input (for those linkages). Please check the result below, for
comparison. It presents the same clustering with the same heights, only
the presented order of points is reorganized.
result_dist_stats_hclust <- stats::hclust(squared_distances, method = "ward.D")
plot(result_dist_stats_hclust)
A note on ward.D2 and ward.D
There is a lot of confusion on ward.D and
ward.D2 linkages on Internet. Maybe not so surprisingly,
the difference is very simple: the returned merging heights in
ward.D are squared, while in ward.D2 they are
not. You can see it by comparing the following output:
distances <- dist(points)
result_ward.D <- hclust1d(distances, distance = TRUE, method = "ward.D")
result_ward.D2 <- hclust1d(distances, distance = TRUE, method = "ward.D2")
print(result_ward.D$height)
#> [1] 0.002748790 0.003426376 0.022595304 0.054504098 0.157314251
#> [6] 0.232947678 0.903435672 3.683388750 12.403250822
print(result_ward.D2$height ^ 2)
#> [1] 0.002748790 0.003426376 0.022595304 0.054504098 0.157314251
#> [6] 0.232947678 0.903435672 3.683388750 12.403250822
Unfortunately, stats::hclust adds another layer of
confusion, by requiring implicitly that the input is provided as
squared distances for ward.D and
unsquared for ward.D2. Let’s try to recreate the
above outputs by using stats::hclust:
distances <- dist(points)
squared_distances <- distances ^ 2
result_ward.D_stats_hclust <- stats::hclust(squared_distances, method = "ward.D")
result_ward.D2_stats_hclust <- stats::hclust(distances, method = "ward.D2")
print(result_ward.D_stats_hclust$height)
#> [1] 0.002748790 0.003426376 0.022595304 0.054504098 0.157314251
#> [6] 0.232947678 0.903435672 3.683388750 12.403250822
print(result_ward.D2_stats_hclust$height ^ 2)
#> [1] 0.002748790 0.003426376 0.022595304 0.054504098 0.157314251
#> [6] 0.232947678 0.903435672 3.683388750 12.403250822
The distinction that stats::hclust makes about its input
(squared or unsquared) is unfortunately implicit, not explicit.
Cutting heights
The last topic presented in this introductory vignette is how to get
an actual single clustering from the dendrogram. You’ll notice, that a
dendrogram in hierarchical clustering contains multiple clusterings:
there is a clustering into one cluster only, there is a clustering into
two clusters and ultimately, there is a clustering into as many clusters
as there are observations. The number of actual clusters in a clustering
depends on a height that a dendrogram tree is cut.
We will use a standard stats::cutree function to cut the
results of hclust1d, because those results are fully
compatible with the results of stats::hclust. One can cut a
dendrogram either at a given height, or specifying a desired number of
clusters.
But before we get into cutting, let’s first examine a
complete linkage clustering full dendrogram:
result <- hclust1d(points)
plot(result)
The exact merging height values may not be apparent from the
dendrogram plot. They are as follows:
print(result$height)
#> [1] 0.05242890 0.05853526 0.15944627 0.23346113 0.39662861 0.42454721 0.94765275
#> [8] 1.83833040 3.11484961
The closest points are point-1 and point-5
with a distance of 0.0524289 and thus merged at the first height (valued
0.0524289). So if you cut the dendrogram at 0.0524289 you’ll get \(n-1\) clusters, with \(n\) equal to the number of points, \(n=10\) in this example.
n_minus_one_clusters <- stats::cutree(result, h = result$height[1])
print(n_minus_one_clusters)
#> point-0 point-1 point-2 point-3 point-4 point-5 point-6 point-7 point-8 point-9
#> 1 2 3 4 5 2 6 7 8 9
In this example you can see how the clusters get assigned: each point
in a vector gets assigned a cluster number, with both
point-1 and point-5 being assigned to the same
cluster number 2, and all other points having their own singleton
cluster.
There are \(n-1\) merging heights
for \(n\) points, in our case there are
9 heights. At the last 9-th height, valued 3.1148496, the last two
clusters get merged and we have only one cluster at that height:
one_cluster <- stats::cutree(result, h = result$height[9])
print(one_cluster)
#> point-0 point-1 point-2 point-3 point-4 point-5 point-6 point-7 point-8 point-9
#> 1 1 1 1 1 1 1 1 1 1
Let’s say our goal is to get 3 clusters. We should cut the dendrogram
at any height between the 7-th height 0.9476528 (inclusive) and the 8-th
height 1.8383304 (exclusive). Let’s cut the dendrogram at the height
1.0:
three_clusters <- stats::cutree(result, h = 1.0)
print(three_clusters)
#> point-0 point-1 point-2 point-3 point-4 point-5 point-6 point-7 point-8 point-9
#> 1 2 3 1 2 2 2 2 3 2
Alternatively, one can cut the dendrogram specifying not the cutting
height, but rather explicitly the desired number of clusters. You can
specify the desired number of clusters with a k argument to
cutree:
alt_three_clusters <- stats::cutree(result, k = 3)
print(alt_three_clusters)
#> point-0 point-1 point-2 point-3 point-4 point-5 point-6 point-7 point-8 point-9
#> 1 2 3 1 2 2 2 2 3 2
How to read the clustering resulting from this cut? As we said
earlier, each point gets assigned a cluster number, so in this last
example we get the following clustering (into three clusters):
the first cluster with 2 points: point-0 and
point-3,
the second cluster with 6 points: point-1,
point-4, point-5, point-6,
point-7 and point-9,
the third cluster with 2 points: point-2 and
point-8.
Have a look at the dendrogram plot above to verify that indeed a cut
at the height 1.0, or (equivalently) into 3 clusters, would result in
this clustering structure.
