UPGMA (unweighted pair group method with arithmetic mean) is a simple agglomerative (bottom-up)hierarchical clustering method. It also has a weighted variant,WPGMA, and they are generally attributed toSokal andMichener.^[1]

Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved. Thus the simple averaging in WPGMA produces a weighted result and the proportional averaging in UPGMA produces an unweighted result (see the working example).^[2]

The UPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwisesimilarity matrix (or adissimilarity matrix).At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters${\displaystyle \mathcal{A}}$ and${\displaystyle \mathcal{B}}$, each of size (i.e.,cardinality)${\displaystyle |\mathcal{A}|}$ and${\displaystyle |\mathcal{B}|}$, is taken to be the average of all distances${\displaystyle d(x,y)}$ between pairs of objects${\displaystyle x}$ in${\displaystyle \mathcal{A}}$ and${\displaystyle y}$ in${\displaystyle \mathcal{B}}$, that is, the mean distance between elements of each cluster:

${\displaystyle \frac{1}{|\mathcal{A}|\cdot |\mathcal{B}|}\sum _{x\in \mathcal{A}}\sum _{y\in \mathcal{B}}d(x,y)}$

In other words, at each clustering step, the updated distance between the joined clusters${\displaystyle \mathcal{A}\cup \mathcal{B}}$ and a new cluster${\displaystyle X}$ is given by the proportional averaging of the${\displaystyle d_{\mathcal{A},X}}$ and${\displaystyle d_{\mathcal{B},X}}$ distances:

${\displaystyle d_{(\mathcal{A}\cup \mathcal{B}),X}=\frac{|\mathcal{A}|\cdot d_{\mathcal{A},X}+|\mathcal{B}|\cdot d_{\mathcal{B},X}}{|\mathcal{A}|+|\mathcal{B}|}}$

The UPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption - that is, it assumes anultrametric tree in which the distances from the root to every branch tip are equal. When the tips are molecular data (i.e.,DNA,RNA andprotein) sampled at the same time, theultrametricity assumption becomes equivalent to assuming amolecular clock.

This working example is based on aJC69 genetic distance matrix computed from the5S ribosomal RNA sequence alignment of five bacteria:Bacillus subtilis (${\displaystyle a}$),Bacillus stearothermophilus (${\displaystyle b}$),Lactobacillus viridescens (${\displaystyle c}$),Acholeplasma modicum (${\displaystyle d}$), andMicrococcus luteus (${\displaystyle e}$).^[3][4]

• 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 theultrametricity 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 distance matrix${\displaystyle D_1}$ into a new distance 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 byaveraging distances between each element of the first cluster${\displaystyle (a,b)}$ and each of the remaining elements:

${\displaystyle D_2((a,b),c)=(D_1(a,c)\times 1+D_1(b,c)\times 1)/(1+1)=(21+30)/2=25.5}$

${\displaystyle D_2((a,b),d)=(D_1(a,d)+D_1(b,d))/2=(31+34)/2=32.5}$

${\displaystyle D_2((a,b),e)=(D_1(a,e)+D_1(b,e))/2=(23+21)/2=22}$

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 clustering

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

Here,${\displaystyle D_2((a,b),e)=22}$ is the smallest value of${\displaystyle D_2}$, so we join cluster${\displaystyle (a,b)}$ and 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 length:${\displaystyle \delta (a,v)=\delta (b,v)=\delta (e,v)=22/2=11}$

We deduce the missing branch length:${\displaystyle \delta (u,v)=\delta (e,v)-\delta (a,u)=\delta (e,v)-\delta (b,u)=11-8.5=2.5}$ (see the final dendrogram)

• Second distance matrix update

We then proceed to update${\displaystyle D_2}$ 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${\displaystyle e}$. Bold values in${\displaystyle D_3}$ correspond to the new distances, calculated byproportional averaging:

${\displaystyle D_3(((a,b),e),c)=(D_2((a,b),c)\times 2+D_2(e,c)\times 1)/(2+1)=(25.5\times 2+39\times 1)/3=30}$

Thanks to this proportional average, the calculation of this new distance accounts for the larger size of the${\displaystyle (a,b)}$ cluster (two elements) with respect to${\displaystyle e}$ (one element). Similarly:

${\displaystyle D_3(((a,b),e),d)=(D_2((a,b),d)\times 2+D_2(e,d)\times 1)/(2+1)=(32.5\times 2+43\times 1)/3=36}$

Proportional averaging therefore gives equal weight to the initial distances of matrix${\displaystyle D_1}$. This is the reason why the method isunweighted, not with respect to the mathematical procedure but with respect to the initial distances.

• 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, keeping in mind that the two elements${\displaystyle c}$ and${\displaystyle d}$ each have a contribution of${\displaystyle 1}$ in theaverage computation:

${\displaystyle D_4((c,d),((a,b),e))=(D_3(c,((a,b),e))\times 1+D_3(d,((a,b),e))\times 1)/(1+1)=(30\times 1+36\times 1)/2=33}$

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:

${\displaystyle \delta (((a,b),e),r)=\delta ((c,d),r)=33/2=16.5}$

We deduce the two remaining branch lengths:

${\displaystyle \delta (v,r)=\delta (((a,b),e),r)-\delta (e,v)=16.5-11=5.5}$

${\displaystyle \delta (w,r)=\delta ((c,d),r)-\delta (c,w)=16.5-14=2.5}$

The dendrogram is now complete.^[5] It is ultrametric because all tips (${\displaystyle a}$ to${\displaystyle e}$) are equidistant from${\displaystyle r}$ :

${\displaystyle \delta (a,r)=\delta (b,r)=\delta (e,r)=\delta (c,r)=\delta (d,r)=16.5}$

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

Alternative linkage schemes includesingle linkage clustering,complete linkage clustering, andWPGMA 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-calledchaining 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.^[6]

Comparison of dendrograms obtained under different clustering methods from the samedistance matrix.

Single-linkage clustering	Complete-linkage clustering	Average linkage clustering:WPGMA	Average linkage clustering: UPGMA.

• Inecology, it is one of the most popular methods for the classification of sampling units (such as vegetation plots) on the basis of their pairwise similarities in relevant descriptor variables (such as species composition).^[7] For example, it has been used to understand the trophic interaction between marine bacteria and protists.^[8]
• Inbioinformatics, UPGMA is used for the creation ofphenetictrees (phenograms). UPGMA was initially designed for use inprotein electrophoresis studies, but is currently most often used to produce guide trees for more sophisticated algorithms. This algorithm is for example used insequence alignment procedures, as it proposes one order in which the sequences will be aligned. Indeed, the guide tree aims at grouping the most similar sequences, regardless of their evolutionary rate or phylogenetic affinities, and that is exactly the goal of UPGMA^[9]
• Inphylogenetics, UPGMA assumes a constant rate of evolution (molecular clock hypothesis) and that all sequences were sampled at the same time, and is not a well-regarded method for inferring relationships unless this assumption has been tested and justified for the data set being used. Notice that even under a 'strict clock', sequences sampled at different times should not lead to an ultrametric tree.

A trivial implementation of the algorithm to construct the UPGMA tree has${\displaystyle O(n^3)}$ time complexity, and using a heap for each cluster to keep its distances from other cluster reduces its time to${\displaystyle O(n^2\log n)}$. Fionn Murtagh presented an${\displaystyle O(n^2)}$ time and space algorithm.^[10]

• Neighbor-joining
• Cluster analysis
• Single-linkage clustering
• Complete-linkage clustering
• Hierarchical clustering
• Models of DNA evolution
• Molecular clock

• UPGMA clustering algorithm implementation in Ruby (AI4R)
• Example calculation of UPGMA using a similarity matrix
• Example calculation of UPGMA using a distance matrix

• Bioinformatics algorithms
• Computational phylogenetics
• Cluster analysis algorithms
• Phylogenetics

• CS1: long volume value
• Articles with short description
• Short description matches Wikidata