Complete-linkage clustering

Algorithms

Naive scheme

The following algorithm is an agglomerative scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The ${\displaystyle N\times N}$ proximity matrix D contains all distances d(i,j). The clusterings are assigned sequence numbers 0,1,......, (n − 1) and L(k) is the level of the kth clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters (r) and (s) is denoted d[(r),(s)].

The complete linkage clustering algorithm consists of the following steps:

1. Begin with the disjoint clustering having level ${\displaystyle L(0)=0}$ and sequence number ${\displaystyle m=0}$.
2. Find the most similar pair of clusters in the current clustering, say pair ${\displaystyle (r),(s)}$, according to ${\displaystyle d[(r),(s)]=maxd[(i),(j)]}$where the maximum is over all pairs of clusters in the current clustering.
3. Increment the sequence number: ${\displaystyle m=m+1}$. Merge clusters ${\displaystyle (r)}$ and ${\displaystyle (s)}$ into a single cluster to form the next clustering ${\displaystyle m}$. Set the level of this clustering to ${\displaystyle L(m)=d[(r),(s)]}$
4. Update the proximity matrix, ${\displaystyle D}$, by deleting the rows and columns corresponding to clusters ${\displaystyle (r)}$ and ${\displaystyle (s)}$ and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted ${\displaystyle (r,s)}$, and an old cluster ${\displaystyle (k)}$ is defined as ${\displaystyle d[(r,s),(k)]=max\{d[(k),(r)],d[(k),(s)]\}}$.
5. If all objects are in one cluster, stop. Else, go to step 2.

Optimally efficient scheme

The algorithm explained above is easy to understand but of complexity ${\displaystyle O(n^3)}$. In May 1976, D. Defays proposed an optimally efficient algorithm of only complexity ${\displaystyle O(n^2)}$ known as CLINK (published 1977) inspired by the similar algorithm SLINK for single-linkage clustering.

Working example

The working example is based on a JC69 genetic distance matrix computed from the 5S ribosomal RNA sequence alignment of five bacteria: Bacillus subtilis (${\displaystyle a}$), Bacillus stearothermophilus (${\displaystyle b}$), Lactobacillus viridescens (${\displaystyle c}$), Acholeplasma modicum (${\displaystyle d}$), and Micrococcus luteus (${\displaystyle e}$).

First step

• First clustering

Let us assume that we have five elements ${\displaystyle (a,b,c,d,e)}$ and the following matrix ${\displaystyle D_1}$ of pairwise distances between them:

In this example, ${\displaystyle D_1(a,b)=17}$ is the smallest value of ${\displaystyle D_1}$, so we join elements ${\displaystyle a}$ and ${\displaystyle b}$.

• First branch length estimation

Let ${\displaystyle u}$ denote the node to which ${\displaystyle a}$ and ${\displaystyle b}$ are now connected. Setting ${\displaystyle \delta (a,u)=\delta (b,u)=D_1(a,b)/2}$ ensures that elements ${\displaystyle a}$ and ${\displaystyle b}$ are equidistant from ${\displaystyle u}$. This corresponds to the expectation of the ultrametricity hypothesis. The branches joining ${\displaystyle a}$ and ${\displaystyle b}$ to ${\displaystyle u}$ then have lengths ${\displaystyle \delta (a,u)=\delta (b,u)=17/2=8.5}$ (see the final dendrogram)

• First distance matrix update

We then proceed to update the initial proximity matrix ${\displaystyle D_1}$ into a new proximity matrix ${\displaystyle D_2}$ (see below), reduced in size by one row and one column because of the clustering of ${\displaystyle a}$ with ${\displaystyle b}$. Bold values in ${\displaystyle D_2}$ correspond to the new distances, calculated by retaining the maximum distance between each element of the first cluster ${\displaystyle (a,b)}$ and each of the remaining elements:

${\displaystyle D_2((a,b),c)=max(D_1(a,c),D_1(b,c))=max(21,30)=30}$

${\displaystyle D_2((a,b),d)=max(D_1(a,d),D_1(b,d))=max(31,34)=34}$

${\displaystyle D_2((a,b),e)=max(D_1(a,e),D_1(b,e))=max(23,21)=23}$

Italicized values in ${\displaystyle D_2}$ are not affected by the matrix update as they correspond to distances between elements not involved in the first cluster.

Second step

• Second clustering

We now reiterate the three previous steps, starting from the new distance matrix ${\displaystyle D_2}$ :

Here, ${\displaystyle D_2((a,b),e)=23}$ is the lowest value of ${\displaystyle D_2}$, so we join cluster ${\displaystyle (a,b)}$ with element ${\displaystyle e}$.

• Second branch length estimation

Let ${\displaystyle v}$ denote the node to which ${\displaystyle (a,b)}$ and ${\displaystyle e}$ are now connected. Because of the ultrametricity constraint, the branches joining ${\displaystyle a}$ or ${\displaystyle b}$ to ${\displaystyle v}$, and ${\displaystyle e}$ to ${\displaystyle v}$, are equal and have the following total length: ${\displaystyle \delta (a,v)=\delta (b,v)=\delta (e,v)=23/2=11.5}$

We deduce the missing branch length: ${\displaystyle \delta (u,v)=\delta (e,v)-\delta (a,u)=\delta (e,v)-\delta (b,u)=11.5-8.5=3}$ (see the final dendrogram)

• Second distance matrix update

We then proceed to update the ${\displaystyle D_2}$ matrix into a new distance matrix ${\displaystyle D_3}$ (see below), reduced in size by one row and one column because of the clustering of ${\displaystyle (a,b)}$ with ${\displaystyle e}$ :

${\displaystyle D_3(((a,b),e),c)=max(D_2((a,b),c),D_2(e,c))=max(30,39)=39}$

${\displaystyle D_3(((a,b),e),d)=max(D_2((a,b),d),D_2(e,d))=max(34,43)=43}$

Third step

• Third clustering

We again reiterate the three previous steps, starting from the updated distance matrix ${\displaystyle D_3}$.

Here, ${\displaystyle D_3(c,d)=28}$ is the smallest value of ${\displaystyle D_3}$, so we join elements ${\displaystyle c}$ and ${\displaystyle d}$.

• Third branch length estimation

Let ${\displaystyle w}$ denote the node to which ${\displaystyle c}$ and ${\displaystyle d}$ are now connected. The branches joining ${\displaystyle c}$ and ${\displaystyle d}$ to ${\displaystyle w}$ then have lengths ${\displaystyle \delta (c,w)=\delta (d,w)=28/2=14}$ (see the final dendrogram)

• Third distance matrix update

There is a single entry to update: ${\displaystyle D_4((c,d),((a,b),e))=max(D_3(c,((a,b),e)),D_3(d,((a,b),e)))=max(39,43)=43}$

Final step

The final ${\displaystyle D_4}$ matrix is:

So we join clusters ${\displaystyle ((a,b),e)}$ and ${\displaystyle (c,d)}$.

Let ${\displaystyle r}$ denote the (root) node to which ${\displaystyle ((a,b),e)}$ and ${\displaystyle (c,d)}$ are now connected. The branches joining ${\displaystyle ((a,b),e)}$ and ${\displaystyle (c,d)}$ to ${\displaystyle r}$ then have lengths:

${\displaystyle \delta (((a,b),e),r)=\delta ((c,d),r)=43/2=21.5}$

We deduce the two remaining branch lengths:

${\displaystyle \delta (v,r)=\delta (((a,b),e),r)-\delta (e,v)=21.5-11.5=10}$

${\displaystyle \delta (w,r)=\delta ((c,d),r)-\delta (c,w)=21.5-14=7.5}$

The complete-linkage dendrogram

The dendrogram is now complete. It is ultrametric because all tips (${\displaystyle a}$ to ${\displaystyle e}$) are equidistant from ${\displaystyle r}$ :

${\displaystyle \delta (a,r)=\delta (b,r)=\delta (e,r)=\delta (c,r)=\delta (d,r)=21.5}$

The dendrogram is therefore rooted by ${\displaystyle r}$, its deepest node.

Comparison with other linkages

Alternative linkage schemes include single linkage clustering and average linkage clustering - implementing a different linkage in the naive algorithm is simply a matter of using a different formula to calculate inter-cluster distances in the initial computation of the proximity matrix and in step 4 of the above algorithm. An optimally efficient algorithm is however not available for arbitrary linkages. The formula that should be adjusted has been highlighted using bold text.

Complete linkage clustering avoids a drawback of the alternative single linkage method - the so-called chaining phenomenon, where clusters formed via single linkage clustering may be forced together due to single elements being close to each other, even though many of the elements in each cluster may be very distant to each other. Complete linkage tends to find compact clusters of approximately equal diameters.