The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the `equal input model'. This model generalizes the `Felsenstein 1981' model (and thereby the Jukes--Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a `random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees -- the so called `model invariants'), on any number $n$ of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of $n=4$ leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake in 1987.
In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join varieties. Determining properties of these secant and join varieties can aid both in model selection and establishing parameter identifiability. Here we explore the first natural geometric property of these varieties: their dimension. The expected projective dimension of the join variety of a set of varieties is one more than the sum of their dimensions. A join …
The Fair Proportion of a species in a phylogenetic tree is a very simple measure that has been used to assess its value relative to the overall phylogenetic diversity represented by the tree. It has recently been proved by Fuchs and Jin to be equal to the Shapley Value of the coallitional game that sends each subset of species to its rooted Phylogenetic Diversity in the tree. We prove in this paper that this result extends to the natural translations of the Fair Proportion and the rooted Phylogenetic Diversity to rooted phylogenetic networks. We also generalize to rooted phylogenetic networks …
The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of a tree shape (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short elementary proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted t…
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters ar…
Phylogenetic diversity indices provide a formal way to apportion 'evolutionary heritage' across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called 'evolutionary distinctiveness' and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this paper, we investigate the extent to which FP and ES can differ, characterise tree shapes on which the indices are identical, and study the equivalence of FP and SV and its implications in more detail. We also define and investigate analogues of these ind…
Phylogenetic diversity indices such as the Fair Proportion (FP) index are frequently discussed as prioritization criteria in biodiversity conservation. They rank species according to their contribution to overall diversity by taking into account the unique and shared evolutionary history of each species as indicated by its placement in an underlying phylogenetic tree. Traditionally, phylogenetic trees were inferred from single genes and the resulting gene trees were assumed to be a valid estimate for the species tree, i.e., the "true" evolutionary history of the species under consideration. Ho…
Measures of phylogenetic balance, such as the Colless and Sackin indices, play an important role in phylogenetics. Unfortunately, these indices are specifically designed for phylogenetic trees, and do not extend naturally to phylogenetic networks (which are increasingly used to describe reticulate evolution). This led us to consider a lesser-known balance index, whose definition is based on a probabilistic interpretation that is equally applicable to trees and to networks. This index, known as the $B_2$ index, was first proposed by Shao and Sokal in 1990. Surprisingly, it does not seem to have…
In phylogenetics it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction of computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most re…
An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set $X$ of species from a collection of trees, each having leaf-set some subset of $X$. In the 1980's characterizations, certain inference rules were given for when a collection of 4-leaved trees, one for each 4-element subset of $X$, can all be simultaneously displayed by a single supertree with leaf-set $X$. Recently, it has become of interest to extend such results to phylogenetic …
Phylogenetic networks extend phylogenetic trees to model non-vertical inheritance, by which a lineage inherits material from multiple parents. The computational complexity of estimating phylogenetic networks from genome-wide data with likelihood-based methods limits the size of networks that can be handled. Methods based on pairwise distances could offer faster alternatives. We study here the information that average pairwise distances contain on the underlying phylogenetic network, by characterizing local and global features that can or cannot be identified. For general networks, we clarify t…
The Bayesian method is noted to produce spuriously high posterior probabilities for phylogenetic trees in analysis of large datasets, but the precise reasons for this over-confidence are unknown. In general, the performance of Bayesian selection of misspecified models is poorly understood, even though this is of great scientific interest since models are never true in real data analysis. Here we characterize the asymptotic behavior of Bayesian model selection and show that when the competing models are equally wrong, Bayesian model selection exhibits surprising and polarized behaviors in large…
In Francis and Steel (2015), it was shown that there exists non-trivial networks on $4$ leaves upon which the distance metric affords a metric on a tree which is not the base tree of the network. In this paper we extend this result in two directions. We show that for any tree $T$ there exists a family of non-trivial HGT networks $N$ for which the distance metric $d_N$ affords a metric on $T$. We additionally provide a class of networks on any number of leaves upon which the distance metric affords a metric on a tree which is not the base tree of the network. The family of networks are all "flo…
Phylogenetic trees in genetics and biology in general are all binary. We make an attempt to answer one fundamental question: Is such binary branching from the coarsest to the finest scales sustained by data? We convert this question into an equivalent one: where is the structural information of tree in a data matrix? Results from this conceptual as well as computing issue afford us to conclude a negative answer: Each branch being split into two at each inter-node of tree from the top to bottom levels is a man-made structure. The data-driven computing paradigm Data Mechanics is employed here to…
