Construction of linear invariants in phylogenetic inference
Linear invariants in phylogenetic studies are useful for developing statistical tests of hypotheses on the topology of phylogenetic trees. A linear invariant is defined as a random variable on the set of all nucleotide configurations of an \(n\)-species tree, whose expectation is zero under a semigroup of substitution matrices for the given tree. A new recursive method is presented for constructing linear invariants by expressing their expectation as a compound bilinear form, i.e. a linear form such that all corresponding matrix elements again can be written as bilinear forms. In that way all invariants of an \(n\)-species tree can be derived analytically from those of an \((n-1)\)-species subtree. All linear invariants of trees for 2, 3, 4 and 5 species are derived. The number of linear invariants for a tree is found to increase rapidly with the number of species.
set of all nucleotide configurations of an \(n\)- species tree
scientific article; zbMATH DE number 64803