  The method of flexi-Weighted Least Squares on evolutionary trees uses simple
polynomial or exponential functions of the evolutionary distance in place of
model-based variances. This has the advantage that unexpected deviations from
additivity can be modeled in a more flexible way. At present, only polynomial
weights have been used. However, a general family of exponential weights is
desirable to compare with polynomial weights and to potentially exploit recent
insights into fast least squares edge length estimation on trees. Here describe
families of weights that are multiplicative on trees, along with measures of
fit of data to tree. It is shown that polynomial, but also multiplicative
weights can approximate model-based variance of evolutionary distances well.
Both models are fitted to evolutionary data from yeast genomes and while the
polynomial weights model fits better, the exponential weights model can fit a
lot better than ordinary least squares. Iterated least squares is evaluated and
is seen to converge quickly and with minimal change in the fit statistics when
the data are in the range expected for the useful evolutionary distances and
simple Markov models of character change. In summary, both polynomial and
exponential weighted least squares work well and justify further investment
into developing the fastest possible algorithms for evaluating evolutionary
trees.
