arXiv subjects

Frederick A. Matsen

Publications and source records attributed to Frederick A. Matsen.

15 recordsLinked to original sources

arXiv2012

Minimizing the average distance to a closest leaf in a phylogenetic tree

When performing an analysis on a collection of molecular sequences, it can be convenient to reduce the number of sequences under consideration while maintaining some characteristic of a larger collection of sequences. For example, one may wish to select a subset of high-quality sequences that represent the diversity of a larger collection of sequences. One may also wish to specialize a large database of characterized "reference sequences" to a smaller subset that is as close as possible on average to a collection of "query sequences" of interest. Such a representative subset can be useful whenever one wishes to find a set of reference sequences that is appropriate to use for comparative analysis of environmentally-derived sequences, such as for selecting "reference tree" sequences for phylogenetic placement of metagenomic reads. In this paper we formalize these problems in terms of the minimization of the Average Distance to the Closest Leaf (ADCL) and investigate algorithms to perform the relevant minimization. We show that the greedy algorithm is not effective, show that a variant of the Partitioning Among Medoids (PAM) heuristic gets stuck in local minima, and develop an exact dynamic programming approach. Using this exact program we note that the performance of PAM appears to be good for simulated trees, and is faster than the exact algorithm for small trees. On the other hand, the exact program gives solutions for all numbers of leaves less than or equal to the given desired number of leaves, while PAM only gives a solution for the pre-specified number of leaves. Via application to real data, we show that the ADCL criterion chooses chimeric sequences less often than random subsets, while the maximization of phylogenetic diversity chooses them more often than random. These algorithms have been implemented in publicly available software.

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arXiv2012

A format for phylogenetic placements

We have developed a unified format for phylogenetic placements, that is, mappings of environmental sequence data (e.g. short reads) into a phylogenetic tree. We are motivated to do so by the growing number of tools for computing and post-processing phylogenetic placements, and the lack of an established standard for storing them. The format is lightweight, versatile, extensible, and is based on the JSON format which can be parsed by most modern programming languages. Our format is already implemented in several tools for computing and post-processing parsimony- and likelihood-based phylogenetic placements, and has worked well in practice. We believe that establishing a standard format for analyzing read placements at this early stage will lead to a more efficient development of powerful and portable post-analysis tools for the growing applications of phylogenetic placement.

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arXiv2011

Reconciling taxonomy and phylogenetic inference: formalism and algorithms for describing discord and inferring taxonomic roots

Although taxonomy is often used informally to evaluate the results of phylogenetic inference and find the root of phylogenetic trees, algorithmic methods to do so are lacking. In this paper we formalize these procedures and develop algorithms to solve the relevant problems. In particular, we introduce a new algorithm that solves a "subcoloring" problem for expressing the difference between the taxonomy and phylogeny at a given rank. This algorithm improves upon the current best algorithm in terms of asymptotic complexity for the parameter regime of interest; we also describe a branch-and-bound algorithm that saves orders of magnitude in computation on real data sets. We also develop a formalism and an algorithm for rooting phylogenetic trees according to a taxonomy. All of these algorithms are implemented in freely-available software.

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arXiv2011

Edge principal components and squash clustering: using the special structure of phylogenetic placement data for sample comparison

Principal components (PCA) and hierarchical clustering are two of the most heavily used techniques for analyzing the differences between nucleic acid sequence samples sampled from a given environment. However, a classical application of these techniques to distances computed between samples can lack transparency because there is no ready interpretation of the axes of classical PCA plots, and it is difficult to assign any clear intuitive meaning to either the internal nodes or the edge lengths of trees produced by distance-based hierarchical clustering methods such as UPGMA. We show that more interesting and interpretable results are produced by two new methods that leverage the special structure of phylogenetic placement data. Edge principal components analysis enables the detection of important differences between samples that contain closely related taxa. Each principal component axis is simply a collection of signed weights on the edges of the phylogenetic tree, and these weights are easily visualized by a suitable thickening and coloring of the edges. Squash clustering outputs a (rooted) clustering tree in which each internal node corresponds to an appropriate "average" of the original samples at the leaves below the node. Moreover, the length of an edge is a suitably defined distance between the averaged samples associated with the two incident nodes, rather than the less interpretable average of distances produced by UPGMA. We present these methods and illustrate their use with data from the microbiome of the human vagina.

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arXiv2011

The phylogenetic Kantorovich-Rubinstein metric for environmental sequence samples

Using modern technology, it is now common to survey microbial communities by sequencing DNA or RNA extracted in bulk from a given environment. Comparative methods are needed that indicate the extent to which two communities differ given data sets of this type. UniFrac, a method built around a somewhat ad hoc phylogenetics-based distance between two communities, is one of the most commonly used tools for these analyses. We provide a foundation for such methods by establishing that if one equates a metagenomic sample with its empirical distribution on a reference phylogenetic tree, then the weighted UniFrac distance between two samples is just the classical Kantorovich-Rubinstein (KR) distance between the corresponding empirical distributions. We demonstrate that this KR distance and extensions of it that arise from incorporating uncertainty in the location of sample points can be written as a readily computable integral over the tree, we develop $L^p$ Zolotarev-type generalizations of the metric, and we show how the p-value of the resulting natural permutation test of the null hypothesis "no difference between the two communities" can be approximated using a functional of a Gaussian process indexed by the tree. We relate the $L^2$ case to an ANOVA-type decomposition and find that the distribution of its associated Gaussian functional is that of a computable linear combination of independent $χ_1^2$ random variables.

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arXiv2010

Polyhedral geometry of Phylogenetic Rogue Taxa

It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this "rogue taxon" effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a "rogue taxon" such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the optimal tree for the original data. Second, we prove a theorem which restricts the topology of BME-optimal trees for data sets of this type, thus showing that a rogue taxon cannot have an arbitrary effect on the optimal tree. Third, we construct polyhedral cones computationally which give complete answers for BME rogue taxon behavior when our original data fits a tree on four, five, and six taxa. We use these cones to derive sufficient conditions for rogue taxon behavior for four taxa, and to understand the frequency of the rogue taxon effect via simulation.

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arXiv2009

constNJ: an algorithm to reconstruct sets of phylogenetic trees satisfying pairwise topological constraints

This paper introduces constNJ, the first algorithm for phylogenetic reconstruction of sets of trees with constrained pairwise rooted subtree-prune regraft (rSPR) distance. We are motivated by the problem of constructing sets of trees which must fit into a recombination, hybridization, or similar network. Rather than first finding a set of trees which are optimal according to a phylogenetic criterion (e.g. likelihood or parsimony) and then attempting to fit them into a network, constNJ estimates the trees while enforcing specified rSPR distance constraints. The primary input for constNJ is a collection of distance matrices derived from sequence blocks which are assumed to have evolved in a tree-like manner, such as blocks of an alignment which do not contain any recombination breakpoints. The other input is a set of rSPR constraints for any set of pairs of trees. ConstNJ is consistent and a strict generalization of the neighbor-joining algorithm; it uses the new notion of "maximum agreement partitions" to assure that the resulting trees satisfy the given rSPR distance constraints.

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arXiv2008

Fourier transform inequalities for phylogenetic trees

Phylogenetic invariants are not the only constraints on site-pattern frequency vectors for phylogenetic trees. A mutation matrix, by its definition, is the exponential of a matrix with non-negative off-diagonal entries; this positivity requirement implies non-trivial constraints on the site-pattern frequency vectors. We call these additional constraints ``edge-parameter inequalities.'' In this paper, we first motivate the edge-parameter inequalities by considering a pathological site-pattern frequency vector corresponding to a quartet tree with a negative internal edge. This site-pattern frequency vector nevertheless satisfies all of the constraints described up to now in the literature. We next describe two complete sets of edge-parameter inequalities for the group-based models; these constraints are square-free monomial inequalities in the Fourier transformed coordinates. These inequalities, along with the phylogenetic invariants, form a complete description of the set of site-pattern frequency vectors corresponding to \emph{bona fide} trees. Said in mathematical language, this paper explicitly presents two finite lists of inequalities in Fourier coordinates of the form ``monomial $\leq 1$,'' each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.

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arXiv2008

To what extent does genealogical ancestry imply genetic ancestry?

Recent statistical and computational analyses have shown that a genealogical most recent common ancestor (MRCA) may have lived in the recent past. However, coalescent-based approaches show that genetic most recent common ancestors for a given non-recombining locus are typically much more ancient. It is not immediately clear how these two perspectives interact. This paper investigates relationships between the number of descendant alleles of an ancestor allele and the number of genealogical descendants of the individual who possessed that allele for a simple diploid genetic model extending the genealogical model of Joseph Chang.

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arXiv2007

Mixed-up trees: the structure of phylogenetic mixtures

In this paper we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how ``common'' non-identifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of rates-across-sites models. Finally, we answer a question raised in a previous paper concerning ``mixed branch repulsion'' on trees larger than quartet trees under the CFN model.

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arXiv2007

Phylogenetic mixtures on a single tree can mimic a tree of another topology

Phylogenetic mixtures model the inhomogeneous molecular evolution commonly observed in data. The performance of phylogenetic reconstruction methods where the underlying data is generated by a mixture model has stimulated considerable recent debate. Much of the controversy stems from simulations of mixture model data on a given tree topology for which reconstruction algorithms output a tree of a different topology; these findings were held up to show the shortcomings of particular tree reconstruction methods. In so doing, the underlying assumption was that mixture model data on one topology can be distinguished from data evolved on an unmixed tree of another topology given enough data and the ``correct'' method. Here we show that this assumption can be false. For biologists our results imply that, for example, the combined data from two genes whose phylogenetic trees differ only in terms of branch lengths can perfectly fit a tree of a different topology.

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arXiv2006

The Bayesian `star paradox' persists for long finite sequences

The `star paradox' in phylogenetics is the tendency for a particular resolved tree to be sometimes strongly supported even when the data is generated by an unresolved (`star') tree. There have been contrary claims as to whether this phenomenon persists when very long sequences are considered. This note settles one aspect of this debate by proving mathematically that there is always a chance that a resolved tree could be strongly supported, even as the length of the sequences becomes very large.

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arXiv2006

Optimization over a class of tree shape statistics

Tree shape statistics quantify some aspect of the shape of a phylogenetic tree. They are commonly used to compare reconstructed trees to evolutionary models and to find evidence of tree reconstruction bias. Historically, to find a useful tree shape statistic, formulas have been invented by hand and then evaluated for utility. This article presents the first method which is capable of optimizing over a class of tree shape statistics, called Binary Recursive Tree Shape Statistics (BRTSS). After defining the BRTSS class, a set of algebraic expressions is defined which can be used in the recursions. The tree shape statistics definable using these expressions in the BRTSS is very general, and includes many of the statistics with which phylogenetic researchers are already familiar. We then present a practical genetic algorithm which is capable of performing optimization over BRTSS given any objective function. The chapter concludes with a successful application of the methods to find a new statistic which indicates a significant difference between two distributions on trees which were previously postulated to have similar properties.

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arXiv2006

Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials

There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such representations involve a specific labeling of the vertices or at least the leaves, and so it is natural to attempt to identify trees by some feature of the associated matrices that is invariant under relabeling. An obvious candidate is the spectrum of eigenvalues (or, equivalently, the characteristic polynomial). We show for any of these choices of matrix that the fraction of binary trees with a unique spectrum goes to zero as the number of leaves goes to infinity. We investigate the rate of convergence of the above fraction to zero using numerical methods. For the adjacency and Laplacian matrices, we show that that the {\em a priori} more informative immanantal polynomials have no greater power to distinguish between trees.

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arXiv2005

A geometric approach to tree shape statistics

This article presents a new way to understand the descriptive ability of tree shape statistics. Where before tree shape statistics were chosen by their ability to distinguish between macroevolutionary models, the ``resolution'' presented in this paper quantifies the ability of a statistic to differentiate between similar and different trees. We term this a ``geometric'' approach to differentiate it from the model-based approach previously explored. A distinct advantage of this perspective is that it allows evaluation of multiple tree shape statistics describing different aspects of tree shape. After developing the methodology, it is applied here to make specific recommendations for a suite of three statistics which will hopefully prove useful in applications. The article ends with an application of the tree shape statistics to clarify the impact of omission of taxa on tree shape.

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