Least Squares Methods

One of the more statistically justified methods to approximate a distance matrix is the least squares approach. Basically, our goal is to find a tree T, whose leaves are the n given species, and that predicts distances d_ij between the species, so that the following expression is minimized:

$$SSQ(T) \equiv \sum_{i=1}^{n} \sum_{j \neq i} w_{ij} (D_{ij} - d_{ij})^2$$ (9.6)

where D_ij is the observed distance between species i and j, and w_ij are given weights. The SSQ is a measure of the discrepancy between the observed distances D_ij and the distances d_ij predicted by T. The weights w_ij are usually all 1, or $w_{ij}=\frac{1}{D_{ij}^2}$.

Problem 9.10   Least Squares Tree.
INPUT: The distance D_ij between species i and j, for each $1 \leq i,j \leq n$, and a corresponding set of weights w_ij.
QUESTION: Find the phylogenetic tree T, with the species as its leaves, that minimizes SSQ(T).

In general, finding the least squares tree is an NP-complete problem  [2]. We will discuss two polynomial heuristics - UPGMA and Neighbor-Joining. We have already studied these algorithms in lecture #5, where we used them to iteratively add one additional string to a growing multiple alignment, thus obtaining a progressive alignment.