Least squares inference in phylogeny

The discrepancy between the observed pairwise distances $D\_\{ij\}$

and the distances $T\_\{ij\}$ over a phylogenetic tree (i.e. the sum

of the branch lengths in the path from leaf $i$ to leaf

$j$) is measured by

:$S\; =\; \backslash sum\_\{ij\}\; w\_\{ij\}\; (D\_\{ij\}-T\_\{ij\})^2$

where the weights $w\_\{ij\}$ depend on the least squares method used.

Least squares

distance tree construction aims to find the tree (topology and branch lengths)

with minimal S. This is a non-trivial problem. It involves searching the

discrete space of unrooted binary tree topologies whose size is exponential in

the number of leaves. For n leaves there are

1 • 3 • 5 • ... • (2n-3)

different topologies. Enumerating them is not feasible already for a small

number of leaves. Heuristic search methods are used to find a reasonably

good topology. The evaluation of S for a given topology (which includes the

computation of the branch lengths) is a linear least squares problem.

There are several ways to weight the squared errors

$(D\_\{ij\}-T\_\{ij\})^2$,

depending on the knowledge and assumptions about the variances of the observed

distances. When nothing is known about the errors, or if they are assumed to be

independently distributed and equal for all observed distances, then all the

weights $w\_\{ij\}$ are set to one. This leads to an ordinary least

squares estimate.

In the weighted least squares case the errors are assumed to be independent

(or their correlations are not known). Given independent errors, a particular

weight should ideally be set to the inverse of the variance of the corresponding distance

estimate. Sometimes the variances may not be known, but they

can be modeled as a function of the distance estimates. In the Fitch and

Margoliash method

Fitch WM, Margoliash E. (1967). Construction of phylogenetic trees. Science 155: 279-84.

for instance it is assumed that the variances are proportional to the squared

distances.

The ordinary and weighted least squares methods described above

assume independent distance estimates. If the distances

are derived from genomic data their estimates covary, because evolutionary

events on internal

branches (of the true tree) can push several distances up or down at

the same time. The resulting covariances can be taken into account using the

method of generalized least squares, i.e. minimizing the following quantity

:$\backslash sum\_\{ij,\; kl\}\; w\_\{ij,kl\}\; (D\_\{ij\}-T\_\{ij\})\; (D\_\{kl\}-T\_\{kl\})$

where $w\_\{ij,kl\}$ are the entries of the inverse of the covariance matrix of the distance estimates.

Finding the tree and branch lengths minimizing the least squares residual is an NP-complete problem.William H.E. Day, [http://www.sciencedirect.com/science/article/B6WC7-4GP2507-7/2/8bed127a0bba589021e7cd6e8f0306f6 Computational complexity of inferring phylogenies from dissimilarity matrices], Bulletin of Mathematical Biology, Volume 49, Issue 4, 1987, Pages 461-467, ISSN 0092-8240, {{doi|10.1016/S0092-8240(87)80007-1}}. However, for a given tree, the optimal branch lengths can be determined in $O(n^2)$ time for ordinary least squares, $O(n^3)$ time for weighted least squares, and $O(n^4)$ time for generalised least squares (given the inverse of the covariance matrix).David Bryant, Peter Waddell,

[http://mbe.oxfordjournals.org/content/15/10/1346.full.pdf Rapid Evaluation of Least-Squares and Minimum-Evolution Criteria on Phylogenetic Trees]{{dead link|date=May 2021|bot=medic}}{{cbignore|bot=medic}}, Mol Biol Evol (1998) 15(10): 1346

• [http://evolution.genetics.washington.edu/phylip.html PHYLIP], a freely distributed phylogenetic analysis package containing an implementation of the weighted least squares method
• [http://paup.csit.fsu.edu/ PAUP], a similar package available for purchase
• [https://web.archive.org/web/20070228025332/http://www.cbrg.ethz.ch/darwin/index Darwin], a programming environment with a library of functions for statistics, numerics, sequence and phylogenetic analysis

{{Reflist}}

{{Phylogenetics}}

Category:Computational phylogenetics