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 WPGMA method.

Algorithm

The UPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise similarity matrix (or a dissimilarity matrix). At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters A and B is taken to be the average of all distances between pairs of objects "x" in A and "y" in B, that is, the mean distance between elements of each cluster:

${1 \over {|\mathcal{A}|\cdot|\mathcal{B}|}}\sum_{x \in \mathcal{A}}\sum_{ y \in \mathcal{B}} d(x,y)$

The UPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption - that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal. When the tips are molecular data (i.e., DNA, RNA and protein), the ultrametricity assumption is called the molecular clock.

Uses

• In ecology, 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).
• In bioinformatics, UPGMA is used for the creation of phenetic trees (phenograms). UPGMA was initially designed for use in protein electrophoresis studies, but is currently most often used to produce guide trees for more sophisticated algorithms. This algorithm is for example used in sequence 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.
• In phylogenetics, UPGMA assumes a constant rate of evolution (molecular clock hypothesis), and is not a well-regarded method for inferring relationships unless this assumption has been tested and justified for the data set being used.

Time complexity

A trivial implementation of the algorithm to construct the UPGMA tree has $O(n^3)$ time complexity, and using a heap for each cluster to keep its distances from other cluster reduces its time to $O(n^2 \log n)$. Fionn Murtagh presented some other approaches for special cases, a $O(k3^kn^2)$ time algorithm by Day and Edelsbrunner for k-dimensional data that is optimal $O(n^2)$ for constant k, and another $O(n^2)$ algorithm for restricted inputs, when "the anglomerative strategy satisfies the reducibility property."

See also

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

External links

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