SPIn: model selection for phylogenetic mixtures via linear invariants

In phylogenetic inference, an evolutionary model describes the substitution processes along each edge of a phylogenetic tree. Misspecification of the model has important implications for the analysis of phylogenetic data. Conventionally, however, the selection of a suitable evolutionary model is based on heuristics or relies on the choice of an approximate input tree. We introduce a method for model Selection in Phylogenetics based on linear INvariants (SPIn), which uses recent insights on linear invariants to characterize a model of nucleotide evolution for phylogenetic mixtures on any number of components. Linear invariants are constraints among the joint probabilities of the bases in the operational taxonomic units that hold irrespective of the tree topologies appearing in the mixtures. SPIn therefore requires no input tree and is designed to deal with nonhomogeneous phylogenetic data consisting of multiple sequence alignments showing different patterns of evolution, for example, concatenated genes, exons, and/or introns. Here, we report on the results of the proposed method evaluated on multiple sequence alignments simulated under a variety of single-tree and mixture settings for both continuous- and discrete-time models. In the simulations, SPIn successfully recovers the underlying evolutionary model and is shown to perform better than existing approaches.     

Estimation of a covariance matrix with zeros

We consider estimation of the covariance matrix of a multivariaterandom vector under the constraint that certain covariancesare zero. We first present an algorithm, which we call iterativeconditional fitting, for computing the maximum likelihood estimateof the constrained covariance matrix, under the assumption ofmultivariate normality. In contrast to previous approaches,this algorithm has guaranteed convergence properties. Droppingthe assumption of multivariate normality, we show how to estimatethe covariance matrix in an empirical likelihood approach. Theseapproaches are then compared via simulation and on an exampleof gene expression.