THE EMPIRICAL DISCOVERY OF PHYLOGENETIC INVARIANTS

An invariant PHI of a tree T under a k-state Markov model, where the t ime parameter is identified with the edges of T, allows us to recogniz e whether data on N observed species can be associated with the N term inal vertices of T in the sense of having been generated on T rather t han on any other tree with N terminals. The invariance is with respect to the (time) lengths associated with the edges of the tree. We propo se a general method of finding invariants of a parametrized functional form. It involves calculating the probability f of all k(N) data poss ibilities for each of m edge-length configurations of T, then solving for the parameters using the m equations of form PHI(f) = 0. We apply this to the case of quadratic invariants for unrooted binary trees wit h four terminals, for all k, using the Jukes-Cantor type of Markov mat rix. We report partial results on finding the smallest algebraically i ndependent set of invariants.