Analysis of evolution: evolutionary rates, independence and treeness.

Populations that separate according to a given pattern of fissions (a “tree” of descent) and evolve independently after fissions have a patterned variance—covariance matrix which reflects the history of fissions. Spectral analysis of the matrix can help reconstruct the pattern; multivariate analysis techniques can be used to test if an observed matrix is compatible with a given tree of descent, and with the assumption of constant or of variable evolutionary rates. We call these tests of “treeness” and discuss evolutionary implications, giving also an example on human evolution. The methods can be used more generally to test if a loss of information is involved in using a tree as a formal representation of any particular set of data.