WPGMA
WPGMA (Weighted Pair Group Method with Arithmetic Mean) is a simple agglomerative (bottom-up) hierarchical clustering method, generally attributed to Sokal and Michener.[1]
The WPGMA method is similar to its unweighted variant, the UPGMA method.
Algorithm
The WPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise distance matrix (or a similarity matrix). At each step, the nearest two clusters, say and
, are combined into a higher-level cluster
. Then, its distance to another cluster
is simply the arithmetic mean of the distances between
and members of
:
The WPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption: it produces an ultrametric tree in which the distances from the root to every branch tip are equal. This ultrametricity assumption is called the molecular clock when the tips involve DNA, RNA and protein data.
Working example
First step
- First clustering
Let us assume that we have five elements and the following matrix
of pairwise distances between them :
a | b | c | d | e | |
---|---|---|---|---|---|
a | 0 | 17 | 21 | 31 | 23 |
b | 17 | 0 | 30 | 34 | 21 |
c | 21 | 30 | 0 | 28 | 39 |
d | 31 | 34 | 28 | 0 | 43 |
e | 23 | 21 | 39 | 43 | 0 |
In this example, is the smallest value of
, so we join elements
and
.
- First branch length estimation
Let denote the node to which
and
are now connected. Setting
ensures that elements
and
are equidistant from
. This corresponds to the expectation of the ultrametricity hypothesis.
The branches joining
and
to
then have lengths
(see the final dendrogram)
- First distance matrix update
We then proceed to update the initial distance matrix into a new distance matrix
(see below), reduced in size by one row and one column because of the clustering of
with
.
Bold values in
correspond to the new distances, calculated by averaging distances between the first cluster
and each of the remaining elements:
Italicized values in 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 :
(a,b) | c | d | e | |
---|---|---|---|---|
(a,b) | 0 | 25.5 | 32.5 | 22 |
c | 25.5 | 0 | 28 | 39 |
d | 32.5 | 28 | 0 | 43 |
e | 22 | 39 | 43 | 0 |
Here, is the smallest value of
, so we join cluster
and element
.
- Second branch length estimation
Let denote the node to which
and
are now connected. Because of the ultrametricity constraint, the branches joining
or
to
, and
to
are equal and have the following length:
We deduce the missing branch length:
(see the final dendrogram)
- Second distance matrix update
We then proceed to update the matrix into a new distance matrix
(see below), reduced in size by one row and one column because of the clustering of
with
:
Third step
- Third clustering
We again reiterate the three previous steps, starting from the updated distance matrix .
((a,b),e) | c | d | |
---|---|---|---|
((a,b),e) | 0 | 32.25 | 37.75 |
c | 32.25 | 0 | 28 |
d | 37.75 | 28 | 0 |
Here, is the smallest value of
, so we join elements
and
.
- Third branch length estimation
Let denote the node to which
and
are now connected.
The branches joining
and
to
then have lengths
(see the final dendrogram)
- Third distance matrix update
There is a single entry to update:
Final step
The final matrix is:
((a,b),e) | (c,d) | |
---|---|---|
((a,b),e) | 0 | 35 |
(c,d) | 35 | 0 |
So we join clusters and
.
Let denote the (root) node to which
and
are now connected.
The branches joining
and
to
then have lengths:
We deduce the two remaining branch lengths:
The WPGMA dendrogram

The dendrogram is now complete. It is ultrametric because all tips ( to
) are equidistant from
:
The dendrogram is therefore rooted by , its deepest node.
See also
- Neighbor-joining
- Molecular clock
- Cluster analysis
- Single-linkage clustering
- Complete-linkage clustering
- Hierarchical clustering
References
- ↑ Sokal R and Michener C (1958). "A statistical method for evaluating systematic relationships". University of Kansas Science Bulletin 38: 1409–1438.
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