UPGMA
   HOME

TheInfoList



OR:

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 to Sokal and Michener. 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'').


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 \mathcal and \mathcal, each of size (''i.e.'',
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
) and , is taken to be the average of all distances d(x,y) between pairs of objects x in \mathcal and y in \mathcal, that is, the mean distance between elements of each cluster: :: \sum_\sum_ d(x,y) In other words, at each clustering step, the updated distance between the joined clusters \mathcal \cup \mathcal and a new cluster X is given by the proportional averaging of the d_ and d_ distances: d_ = \frac 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 Ribonucleic acid (RNA) is a polymeric molecule that is essential for most biological functions, either by performing the function itself (non-coding RNA) or by forming a template for the production of proteins (messenger RNA). RNA and deoxyrib ...
and
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residue (biochemistry), residues. Proteins perform a vast array of functions within organisms, including Enzyme catalysis, catalysing metab ...
) sampled at the same time, the ultrametricity assumption becomes equivalent to assuming a molecular clock.


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 ''Bacill ...
'' (a), '' Bacillus stearothermophilus'' (b), '' Lactobacillus viridescens'' (c), '' Acholeplasma modicum'' (d), and '' Micrococcus luteus'' (e).


First step

* First clustering Let us assume that we have five elements (a,b,c,d,e) and the following matrix D_1 of pairwise distances between them : In this example, D_1 (a,b)=17 is the smallest value of D_1, so we join elements a and b. * First branch length estimation Let u denote the node to which a and b are now connected. Setting \delta(a,u)=\delta(b,u)=D_1(a,b)/2 ensures that elements a and b are equidistant from u. This corresponds to the expectation of the ultrametricity hypothesis. The branches joining a and b to u then have lengths \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 D_1 into a new distance matrix D_2 (see below), reduced in size by one row and one column because of the clustering of a with b. Bold values in D_2 correspond to the new distances, calculated by averaging distances between each element of the first cluster (a,b) and each of the remaining elements: D_2((a,b),c)=(D_1(a,c) \times 1 + D_1(b,c) \times 1)/(1+1)=(21+30)/2=25.5 D_2((a,b),d)=(D_1(a,d) + D_1(b,d))/2=(31+34)/2=32.5 D_2((a,b),e)=(D_1(a,e) + D_1(b,e))/2=(23+21)/2=22 Italicized values in 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 D_2 Here, D_2 ((a,b),e)=22 is the smallest value of D_2, so we join cluster (a,b) and element e. * Second branch length estimation Let v denote the node to which (a,b) and e are now connected. Because of the ultrametricity constraint, the branches joining a or b to v, and e to v are equal and have the following length: \delta(a,v)=\delta(b,v)=\delta(e,v)=22/2=11 We deduce the missing branch length: \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 D_2 into a new distance matrix D_3 (see below), reduced in size by one row and one column because of the clustering of (a,b) with e. Bold values in D_3 correspond to the new distances, calculated by proportional averaging: 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 (a,b) cluster (two elements) with respect to e (one element). Similarly: 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 D_1. This is the reason why the method is ''unweighted'', 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 D_3. Here, D_3 (c,d)=28 is the smallest value of D_3, so we join elements c and d. * Third branch length estimation Let w denote the node to which c and d are now connected. The branches joining c and d to w then have lengths \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 c and d each have a contribution of 1 in the average computation: 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


Final step

The final D_4 matrix is: So we join clusters ((a,b),e) and (c,d). Let r denote the (root) node to which ((a,b),e) and (c,d) are now connected. The branches joining ((a,b),e) and (c,d) to r then have lengths: \delta(((a,b),e),r)=\delta((c,d),r)=33/2=16.5 We deduce the two remaining branch lengths: \delta(v,r)=\delta(((a,b),e),r)-\delta(e,v)=16.5-11=5.5 \delta(w,r)=\delta((c,d),r)-\delta(c,w)=16.5-14=2.5


The UPGMA dendrogram

The dendrogram is now complete. It is ultrametric because all tips (a to e) are equidistant from r : \delta(a,r)=\delta(b,r)=\delta(e,r)=\delta(c,r)=\delta(d,r)=16.5 The dendrogram is therefore rooted by r, its deepest node.


Comparison with other linkages

Alternative linkage schemes include single linkage clustering, complete linkage clustering, and WPGMA 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.


Uses

* In
ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ...
, 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). For example, it has been used to understand the trophic interaction between marine bacteria and protists. * In
bioinformatics Bioinformatics () is an interdisciplinary field of science that develops methods and Bioinformatics software, software tools for understanding biological data, especially when the data sets are large and complex. Bioinformatics uses biology, ...
, 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 In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural biology, structural, or evolutionary relationships between ...
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 In biology, phylogenetics () is the study of the evolutionary history of life using observable characteristics of organisms (or genes), which is known as phylogenetic inference. It infers the relationship among organisms based on empirical dat ...
, 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.


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 an O(n^2) time and space algorithm.


See also

* Neighbor-joining *
Cluster analysis Cluster analysis or clustering is the data analyzing technique in which task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more Similarity measure, similar (in some specific sense defined by the ...
* Single-linkage clustering * Complete-linkage clustering * Hierarchical clustering * Models of DNA evolution * Molecular clock


References


External links


UPGMA clustering algorithm implementation in Ruby (AI4R)Example calculation of UPGMA using a similarity matrixExample calculation of UPGMA using a distance matrix
{{Phylogenetics Bioinformatics algorithms Computational phylogenetics Cluster analysis algorithms Phylogenetics