  Homogeneity across lineages is a common assumption in phylogenetics according
to which nucleotide substitution rates remain constant in time and do not
depend on lineages. This is a simplifying hypothesis which is often adopted to
make the process of sequence evolution more tractable. However, its validity
has been explored and put into question in several papers. On the other hand,
dealing successfully with the general case (heterogeneity across lineages) is
one of the key features of phylogenetic reconstruction methods based on
algebraic tools.
  The goal of this paper is twofold. First, we present a new weighting system
for quartets (ASAQ) based on algebraic and semi-algebraic tools, thus specially
indicated to deal with data evolving under heterogeneus rates. This method
combines the weights two previous methods by means of a test based on the
positivity of the branch length estimated with the paralinear distance. ASAQ is
statistically consistent when applied to GM data, considers rate and base
composition heterogeneity among lineages and does not assume stationarity nor
time-reversibility. Second, we test and compare the performance of several
quartet-based methods for phylogenetic tree reconstruction (namely, Quartet
Puzzling, Weight Optimization and Wilson's method) in combination with ASAQ
weights and other weights based on algebraic and semi-algebraic methods or on
the paralinear distance. These tests are applied to both simulated and real
data and support Weight Optimization with ASAQ weights as a reliable and
successful reconstruction method.
