arXiv subjects

John A. Rhodes

Publications and source records attributed to John A. Rhodes.

At least 37 records · Page 2Linked to original sources

arXiv2018

Species tree inference from genomic sequences using the log-det distance

The log-det distance between two aligned DNA sequences was introduced as a tool for statistically consistent inference of a gene tree under simple non-mixture models of sequence evolution. Here we prove that the log-det distance, coupled with a distance-based tree construction method, also permits consistent inference of species trees under mixture models appropriate to aligned genomic-scale sequences data. Data may include sites from many genetic loci, which evolved on different gene trees due to incomplete lineage sorting on an ultrametric species tree, with different time-reversible substitution processes. The simplicity and speed of distance-based inference suggests log-det based methods should serve as benchmarks for judging more elaborate and computationally-intensive species trees inference methods.

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arXiv2017

Split probabilities and species tree inference under the multispecies coalescent model

Using topological summaries of gene trees as a basis for species tree inference is a promising approach to obtain acceptable speed on genomic-scale datasets, and to avoid some undesirable modeling assumptions. Here we study the probabilities of splits on gene trees under the multispecies coalescent model, and how their features might inform species tree inference. After investigating the behavior of split consensus methods, we investigate split invariants --- that is, polynomial relationships between split probabilities. These invariants are then used to show that, even though a split is an unrooted notion, split probabilities retain enough information to identify the rooted species tree topology for trees of more than 5 taxa, with one possible 6-taxon exception.

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arXiv2016

Split scores: a tool to quantify phylogenetic signal in genome-scale data

Detecting variation in the evolutionary process along chromosomes is increasingly important as whole-genome data becomes more widely available. For example, factors such as incomplete lineage sorting, horizontal gene transfer, and chromosomal inversion are expected to result in changes in the underlying gene trees along a chromosome, while changes in selective pressure and mutational rates for different genomic regions may lead to shifts in the underlying mutational process. We propose the split score as a general method for quantifying support for a particular phylogenetic relationship within a genomic data set. Because the split score is based on algebraic properties of a matrix of site pattern frequencies, it can be rapidly computed, even for data sets that are large in the number of taxa and/or in the length of the alignment, providing an advantage over other methods (e.g., maximum likelihood) that are often used to assess such support. Using simulation we explore the properties of the split score, including its dependence on sequence length, branch length, size of a split and its ability to detect true splits in the underlying tree. Using a sliding window analysis, we show that split scores can be used to detect changes in the underlying evolutionary process for genome-scale data from primates, mosquitoes, and viruses in a computationally efficient manner. Computation of the split score has been implemented in the software package SplitSup.

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arXiv2016

Phylogenetic trees and Euclidean embeddings

It was recently observed by de Vienne et al. that a simple square root transformation of distances between taxa on a phylogenetic tree allowed for an embedding of the taxa into Euclidean space. While the justification for this was based on a diffusion model of continuous character evolution along the tree, here we give a direct and elementary explanation for it that provides substantial additional insight. We use this embedding to reinterpret the differences between the NJ and BIONJ tree building algorithms, providing one illustration of how this embedding reflects tree structures in data.

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arXiv2016

Species tree inference from gene splits by Unrooted STAR methods

The $\text{NJ}_{st}$ method was proposed by Liu and Yu to infer a species tree topology from unrooted topological gene trees. While its statistical consistency under the multispecies coalescent model was established only for a 4-taxon tree, simulations demonstrated its good performance on gene trees inferred from sequences for many taxa. Here we prove the statistical consistency of the method for an arbitrarily large species tree. Our approach connects $\text{NJ}_{st}$ to a generalization of the STAR method of Liu, Pearl and Edwards, and a previous theoretical analysis of it. We further show $\text{NJ}_{st}$ utilizes only the distribution of splits in the gene trees, and not their individual topologies. Finally, we discuss how multiple samples per taxon per gene should be handled for statistical consistency.

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arXiv2016

Statistically-Consistent k-mer Methods for Phylogenetic Tree Reconstruction

Frequencies of $k$-mers in sequences are sometimes used as a basis for inferring phylogenetic trees without first obtaining a multiple sequence alignment. We show that a standard approach of using the squared-Euclidean distance between $k$-mer vectors to approximate a tree metric can be statistically inconsistent. To remedy this, we derive model-based distance corrections for orthologous sequences without gaps, which lead to consistent tree inference. The identifiability of model parameters from $k$-mer frequencies is also studied. Finally, we report simulations showing the corrected distance out-performs many other $k$-mer methods, even when sequences are generated with an insertion and deletion process. These results have implications for multiple sequence alignment as well, since $k$-mer methods are usually the first step in constructing a guide tree for such algorithms.

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arXiv2015

There are no caterpillars in a wicked forest

Species trees represent the historical divergences of populations or species, while gene trees trace the ancestry of individual gene copies sampled within those populations. In cases involving rapid speciation, gene trees with topologies that differ from that of the species tree can be most probable under the standard multispecies coalescent model, making species tree inference more difficult. Such anomalous gene trees are not well understood except for some small cases. In this work, we establish one constraint that applies to trees of any size: gene trees with "caterpillar" topologies cannot be anomalous. The proof of this involves a new combinatorial object, called a population history, which keeps track of the number of coalescent events in each ancestral population.

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arXiv2014

Parameter identifiability of discrete Bayesian networks with hidden variables

Identifiability of parameters is an essential property for a statistical model to be useful in most settings. However, establishing parameter identifiability for Bayesian networks with hidden variables remains challenging. In the context of finite state spaces, we give algebraic arguments establishing identifiability of some special models on small DAGs. We also establish that, for fixed state spaces, generic identifiability of parameters depends only on the Markov equivalence class of the DAG. To illustrate the use of these results, we investigate identifiability for all binary Bayesian networks with up to five variables, one of which is hidden and parental to all observable ones. Surprisingly, some of these models have parameterizations that are generically 4-to-one, and not 2-to-one as label swapping of the hidden states would suggest. This leads to interesting difficulties in interpreting causal effects.

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arXiv2013

Tensors of Nonnegative Rank Two

A nonnegative tensor has nonnegative rank at most 2 if and only if it is supermodular and has flattening rank at most 2. We prove this result, then explore the semialgebraic geometry of the general Markov model on phylogenetic trees with binary states, and comment on possible extensions to tensors of higher rank.

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arXiv2012

A semialgebraic description of the general Markov model on phylogenetic trees

Many of the stochastic models used in inference of phylogenetic trees from biological sequence data have polynomial parameterization maps. The image of such a map --- the collection of joint distributions for a model --- forms the model space. Since the parameterization is polynomial, the Zariski closure of the model space is an algebraic variety which is typically much larger than the model space, but has been usefully studied with algebraic methods. Of ultimate interest, however, is not the full variety, but only the model space. Here we develop complete semialgebraic descriptions of the model space arising from the k-state general Markov model on a tree, with slightly restricted parameters. Our approach depends upon both recently-formulated analogs of Cayley's hyperdeterminant, and the construction of certain quadratic forms from the joint distribution whose positive (semi-)definiteness encodes information about parameter values. We additionally investigate the use of Sturm sequences for obtaining similar results.

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arXiv2012

Tensor Rank, Invariants, Inequalities, and Applications

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$ tensors of rank $n$ over $\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic description of the tensors of rank $n$ and multilinear rank $(n,n,n)$. The polynomials we construct coincide with Cayley's hyperdeterminant in the case $n=2$, and thus generalize it. Though our construction is direct and explicit, we also recast our functions in the language of representation theory for additional insights. We give three applications in different directions: First, we develop basic topological understanding of how the real tensors of complex rank $n$ and multilinear rank $(n,n,n)$ form a collection of path-connected subsets, one of which contains tensors of real rank $n$. Second, we use the invariants to develop a semialgebraic description of the set of probability distributions that can arise from a simple stochastic model with a hidden variable, a model that is important in phylogenetics and other fields. Third, we construct simple examples of tensors of rank $2n-1$ which lie in the closure of those of rank $n$.

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arXiv2012

When Do Phylogenetic Mixture Models Mimic Other Phylogenetic Models?

Phylogenetic mixture models, in which the sites in sequences undergo different substitution processes along the same or different trees, allow the description of heterogeneous evolutionary processes. As data sets consisting of longer sequences become available, it is important to understand such models, for both theoretical insights and use in statistical analyses. Some recent articles have highlighted disturbing "mimicking" behavior in which a distribution from a mixture model is identical to one arising on a different tree or trees. Other works have indicated such problems are unlikely to occur in practice, as they require very special parameter choices. After surveying some of these works on mixture models, we give several new results. In general, if the number of components in a generating mixture is not too large and we disallow zero or infinite branch lengths, then it cannot mimic the behavior of a non-mixture on a different tree. On the other hand, if the mixture model is locally over-parameterized, it is possible for a phylogenetic mixture model to mimic distributions of another tree model. Though theoretical questions remain, these sorts of results can serve as a guide to when the use of mixture models in either ML or Bayesian frameworks is likely to lead to statistically consistent inference, and when mimicking due to heterogeneity should be considered a realistic possibility.

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arXiv2012

Species tree inference by the STAR method, and generalizations

The multispecies coalescent model describes the generation of gene trees from a rooted metric species tree, and thus provides a framework for the inference of species trees from sampled gene trees. We prove that the STAR method of Liu et al., and generalizations of it, are statistically consistent methods of topological species tree inference under this model. We discuss the impact of gene tree sampling schemes for species tree inference using generalized STAR methods, and reinterpret the original STAR as a consensus method based on clades.

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arXiv2011

Determining species tree topologies from clade probabilities under the coalescent

One approach to estimating a species tree from a collection of gene trees is to first estimate probabilities of clades from the gene trees, and then to construct the species tree from the estimated clade probabilities. While a greedy consensus algorithm, which consecutively accepts the most probable clades compatible with previously accepted clades, can be used for this second stage, this method is known to be statistically inconsistent under the multispecies coalescent model. This raises the question of whether it is theoretically possible to reconstruct the species tree from known probabilities of clades on gene trees. We investigate clade probabilities arising from the multispecies coalescent model, with an eye toward identifying features of the species tree. Clades on gene trees with probability greater than 1/3 are shown to reflect clades on the species tree, while those with smaller probabilities may not. Linear invariants of clade probabilities are studied both computationally and theoretically, with certain linear invariants giving insight into the clade structure of the species tree. For species trees with generic edge lengths, these invariants can be used to identify the species tree topology. These theoretical results both confirm that clade probabilities contain full information on the species tree topology and suggest future directions of study for developing statistically consistent inference methods from clade frequencies on gene trees.

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arXiv2010

Identifiability of Large Phylogenetic Mixture Models

Phylogenetic mixture models are statistical models of character evolution allowing for heterogeneity. Each of the classes in some unknown partition of the characters may evolve by different processes, or even along different trees. The fundamental question of whether parameters of such a model are identifiable is difficult to address, due to the complexity of the parameterization. We analyze mixture models on large trees, with many mixture components, showing that both numerical and tree parameters are indeed identifiable in these models when all trees are the same. We also explore the extent to which our algebraic techniques can be employed to extend the result to mixtures on different trees.

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arXiv2010

Identifying the Rooted Species Tree from the Distribution of Unrooted Gene Trees under the Coalescent

Gene trees are evolutionary trees representing the ancestry of genes sampled from multiple populations. Species trees represent populations of individuals -- each with many genes -- splitting into new populations or species. The coalescent process, which models ancestry of gene copies within populations, is often used to model the probability distribution of gene trees given a fixed species tree. This multispecies coalescent model provides a framework for phylogeneticists to infer species trees from gene trees using maximum likelihood or Bayesian approaches. Because the coalescent models a branching process over time, all trees are typically assumed to be rooted in this setting. Often, however, gene trees inferred by traditional phylogenetic methods are unrooted. We investigate probabilities of unrooted gene trees under the multispecies coalescent model. We show that when there are 4 species with one gene sampled per species, the distribution of unrooted gene tree topologies identifies the unrooted species tree topology and some, but not all, information in the species tree edges (branch lengths). The location of the root on the species tree is not identifiable in this situation. However, for 5 or more species with one gene sampled per species, we show that the distribution of unrooted gene tree topologies identifies the rooted species tree topology and all its internal branch lengths. The length of any pendent branch leading to a leaf of the species tree is also identifiable for any species from which more than one gene is sampled.

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arXiv2010

Parameter identifiability in a class of random graph mixture models

We prove identifiability of parameters for a broad class of random graph mixture models. These models are characterized by a partition of the set of graph nodes into latent (unobservable) groups. The connectivities between nodes are independent random variables when conditioned on the groups of the nodes being connected. In the binary random graph case, in which edges are either present or absent, these models are known as stochastic blockmodels and have been widely used in the social sciences and, more recently, in biology. Their generalizations to weighted random graphs, either in parametric or non-parametric form, are also of interest in many areas. Despite a broad range of applications, the parameter identifiability issue for such models is involved, and previously has only been touched upon in the literature. We give here a thorough investigation of this problem. Our work also has consequences for parameter estimation. In particular, the estimation procedure proposed by Frank and Harary for binary affiliation models is revisited in this article.

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arXiv2009

Estimating Trees from Filtered Data: Identifiability of Models for Morphological Phylogenetics

As an alternative to parsimony analyses, stochastic models have been proposed (Lewis, 2001), (Nylander, et al., 2004) for morphological characters, so that maximum likelihood or Bayesian analyses may be used for phylogenetic inference. A key feature of these models is that they account for ascertainment bias, in that only varying, or parsimony-informative characters are observed. However, statistical consistency of such model-based inference requires that the model parameters be identifiable from the joint distribution they entail, and this issue has not been addressed. Here we prove that parameters for several such models, with finite state spaces of arbitrary size, are identifiable, provided the tree has at least 8 leaves. If the tree topology is already known, then 7 leaves suffice for identifiability of the numerical parameters. The method of proof involves first inferring a full distribution of both parsimony-informative and non-informative pattern joint probabilities from the parsimony-informative ones, using phylogenetic invariants. The failure of identifiability of the tree parameter for 4-taxon trees is also investigated.

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