UPGMA

Being able to assign branch lengths to a given tree, as we have demonstrated, we need to minimize SSQ(T) over the possible tree topologies. The UPGMA, or Unweighted Pair Group Method with Arithmetic mean [13], is a heuristic algorithm that usually generates satisfactory results. Basically, the algorithm iteratively joins the two nearest clusters (or groups of species), until one cluster is left.

UPGMA algorithm:
Let d be the distance function between species, we define the distance Di,j between two clusters of species C_i and C_j the following:

$$D_{i,j} = \frac{1}{n_i + n_j} \sum_{p\in C_i} \sum_{q\in C_j}d(p,q)$$

where n_i = |C_i| and n_j = |C_j|

• Initialization:

  1.

  Initialize n clusters with the given species, one species per cluster.

  2.

  Set the size of each cluster to 1: $n_i \leftarrow 1$.

  3.

  In the output tree T, assign a leaf for each species.

• Iteration:

  1.

    Find the i and j that have the smallest distance D_ij.

  2.

  Create a new cluster - (ij), which has n_(ij) = n_i + n_j members.

  3.

  Connect i and j on the tree to a new node, which corresponds to the new cluster (ij), and give the two branches connecting i and j to (ij) length $\frac{D_{i,j}}{2}$ each.

  4.

  Compute the distance from the new cluster to all other clusters (except for i and j, which are no longer relevant) as a weighted average of the distances from its components:
  
  

  $$D_{(ij),k} = (\frac{n_i}{n_i + n_j}) D_{i,k} + (\frac{n_j}{n_i + n_j}) D_{j,k}$$

  
  

  5.

  Delete the columns and rows in D that correspond to clusters i and j, and add a column and row for cluster (ij), with D_(ij),k computed as above.

  6.

  Return to 1 until there is only one cluster left.

Complexity: The time and space complexity of UPGMA is O(n²), since there are n-1 iterations, with O(n) work in each one.

A clocklike, or ultrametric, tree is a rooted tree, in which the total branch length from the root to any leaf is equal. In other words, there is a ``molecular clock'' that ticks in a constant pace (i.e., the mutation rate is identical for all species), and all the observed species are at an equal number of ticks from the root (see also page ). If the solution to the least squares problem is 0, and there is a molecular clock (i.e., the solution is a clocklike tree), then UPGMA is guaranteed to return the optimal solution. Actually, UPGMA implicitly assumes the existence of an ultrametric tree, which explains why the new node, (ij), is the mean of the two nodes that were joined to create it, as shown in figure 8.8. It is therefore not surprising that for substantially non-clocklike trees, the algorithm might give seriously misleading results.

  

Figure 8.8: A clocklike tree, showing the clustering (ab) of the two nodes a and b by UPGMA and by the Neighbor-Joining algorithm.

Another assumption that UPGMA does is additivity: In the "real" tree, distances between species are the sum of distances along the path between the corresponding leaves.

There are two corollaries of additivity that the next algorithm will use

• For every three nodes i,j,k connected through an internal node m with the distances: d(i,m)=a, d(j,m)=b, d(k,m)=c then dm,k=1/2(di,j+dj,k-di,j).
• For every four nodes i,j,k,l connected through an two internal nodes m,n where m is connected with i,k and n, and nis connected with j,l and m then di,k+dj,l <= di,j+dk,l = di,l+dk,j (see figure 8.9).

  

Figure 8.9: di,j+dk,l <= di,j+dk,l = di,j+dk,l.