WPGMA (Weighted Pair Group Method with Arithmetic Mean) is a simple agglomerative (bottom-up)
hierarchical clustering method, generally attributed to
Sokal
Sokal ( uk, Сокаль, Romanization of Ukrainian, romanized: ''Sokal'') is a city located on the Bug River in Chervonohrad Raion, Lviv Oblast of western Ukraine. It hosts the administration of Sokal urban hromada, one of the hromadas of Ukrain ...
and
Michener.
The WPGMA method is similar to its ''unweighted'' variant, the
UPGMA
UPGMA (unweighted pair group method with arithmetic mean) is a simple agglomerative (bottom-up) hierarchical clustering method. The method is generally attributed to Sokal and Michener.
The UPGMA method is similar to its ''weighted'' variant, the ...
method.
Algorithm
The WPGMA algorithm constructs a rooted tree (
dendrogram
A dendrogram is a diagram representing a tree. This diagrammatic representation is frequently used in different contexts:
* in hierarchical clustering, it illustrates the arrangement of the clusters produced by the corresponding analyses.
...
) that reflects the structure present in a pairwise
distance matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the ''dist ...
(or a
similarity matrix
In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such meas ...
). 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 average distances between members of
and
and
and
:
The WPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption: it produces an
ultrametric
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
tree in which the distances from the root to every branch tip are equal. This
ultrametricity
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
assumption is called the
molecular clock
The molecular clock is a figurative term for a technique that uses the mutation rate of biomolecules to deduce the time in prehistory when two or more life forms diverged. The biomolecular data used for such calculations are usually nucleoti ...
when the tips involve
DNA,
RNA
Ribonucleic acid (RNA) is a polymeric molecule essential in various biological roles in coding, decoding, regulation and expression of genes. RNA and deoxyribonucleic acid ( DNA) are nucleic acids. Along with lipids, proteins, and carbohydra ...
and
protein
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...
data.
Working example
This working example is based on a
JC69 genetic distance matrix computed from the
5S ribosomal RNA sequence alignment of five bacteria: ''
Bacillus subtilis
''Bacillus subtilis'', known also as the hay bacillus or grass bacillus, is a Gram-positive, catalase-positive bacterium, found in soil and the gastrointestinal tract of ruminants, humans and marine sponges. As a member of the genus ''Bacillus ...
'' (
), ''
Bacillus stearothermophilus'' (
), ''
Lactobacillus
''Lactobacillus'' is a genus of Gram-positive, aerotolerant anaerobes or microaerophilic, rod-shaped, non-spore-forming bacteria. Until 2020, the genus ''Lactobacillus'' comprised over 260 phylogenetically, ecologically, and metabolically diver ...
viridescens'' (
), ''
Acholeplasma modicum'' (
), and ''
Micrococcus luteus
''Micrococcus luteus'' is a Gram-positive, to Gram-variable, nonmotile, coccus, tetrad-arranging, pigmented, saprotrophic bacterium that belongs to the family Micrococcaceae. It is urease and catalase positive. An obligate aerobe, ''M. luteus' ...
'' (
).
First step
* First clustering
Let us assume that we have five elements
and the following matrix
of pairwise distances between them :
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
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
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 each element of 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
:
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
:
Of note, this average calculation of the new distance does not account for the larger size of the
cluster (two elements) with respect to
(one element). Similarly:
The averaging procedure therefore gives differential weight to the initial distances of matrix
. This is the reason why the method is ''weighted'', not with respect to the mathematical procedure but with respect to the initial distances.
Third step
* Third clustering
We again reiterate the three previous steps, starting from the updated distance matrix
.
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:
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
![WPGMA Dendrogram 5S data](https://upload.wikimedia.org/wikipedia/commons/3/39/WPGMA_Dendrogram_5S_data.svg)
The dendrogram is now complete. It is ultrametric because all tips (
to
) are equidistant from
:
The dendrogram is therefore rooted by
, its deepest node.
Comparison with other linkages
Alternative linkage schemes include
single linkage clustering
In statistics, single-linkage clustering is one of several methods of hierarchical clustering. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of ...
,
complete linkage clustering
Complete-linkage clustering is one of several methods of agglomerative hierarchical clustering. At the beginning of the process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters until all ...
, and
UPGMA average linkage clustering. Implementing a different linkage is simply a matter of using a different formula to calculate inter-cluster distances during the distance matrix update steps of the above algorithm. Complete linkage clustering avoids a drawback of the alternative single linkage clustering 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.
See also
*
Neighbor-joining
In bioinformatics, neighbor joining is a bottom-up (agglomerative) clustering method for the creation of phylogenetic trees, created by Naruya Saitou and Masatoshi Nei in 1987. Usually based on DNA or protein sequence data, the algorithm requi ...
*
Molecular clock
The molecular clock is a figurative term for a technique that uses the mutation rate of biomolecules to deduce the time in prehistory when two or more life forms diverged. The biomolecular data used for such calculations are usually nucleoti ...
*
Cluster analysis
Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of ...
*
Single-linkage clustering
In statistics, single-linkage clustering is one of several methods of hierarchical clustering. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of el ...
*
Complete-linkage clustering
Complete-linkage clustering is one of several methods of agglomerative hierarchical clustering. At the beginning of the process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters until all ...
*
Hierarchical clustering
References
{{Phylogenetics
Bioinformatics algorithms
Computational phylogenetics
Cluster analysis algorithms