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42
Phylogenetic Tree Construction Using Markov Chain Monte Carlo
, 1999
"... We describe a Bayesian method based on Markov chain simulation to study the phylogenetic relationship in a group of DNA sequences. Under simple models of mutational events, our method produces a Markov chain whose stationary distribution is the conditional distribution of the phylogeny given the obs ..."
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Cited by 58 (0 self)
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We describe a Bayesian method based on Markov chain simulation to study the phylogenetic relationship in a group of DNA sequences. Under simple models of mutational events, our method produces a Markov chain whose stationary distribution is the conditional distribution of the phylogeny given the observed sequences. Our algorithm strikes a reasonable balance between the desire to move globally through the space of phylogenies and the need to make computationally feasible moves in areas of high probability. Since phylogenetic information is described by a tree, we have created new diagnostics to handle this type of data structure. An important byproduct of the Markov chain Monte Carlo phylogeny building technique is that it provides estimates and corresponding measures of variability for any aspect of the phylogeny under study.
Toric ideals of phylogenetic invariants
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 2005
"... Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have tra ..."
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Cited by 52 (13 self)
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Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine minimal generators and Gröbner bases for these toric ideals. For the JukesCantor and Kimura models on a binary tree, our Gröbner basis consists of quadrics, cubics and quartics.
Phylogenetic ideals and varieties for the general Markov model
 Math. Biosciences
"... Abstract. The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statisti ..."
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Cited by 42 (5 self)
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Abstract. The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statistics. For an arbitrary trivalent tree, we determine the full ideal of invariants for the 2state model, establishing a conjecture of PachterSturmfels. For the κstate model, we reduce the problem of determining a defining set of polynomials to that of determining a defining set for a 3leaved tree. Along the way, we prove several new cases of a conjecture of GarciaStillmanSturmfels on certain statistical models on star trees, and reduce their conjecture to a family of subcases. 1.
The identifiability of tree topology for phylogenetic models, including covarion and mixture models, arXive qbio.PE/0511009
"... Abstract. For a model of molecular evolution to be useful for phylogenetic inference, the topology of evolutionary trees must be identifiable. That is, from a joint distribution the model predicts, it must be possible to recover the tree parameter. We establish tree identifiability for a number of p ..."
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Cited by 23 (7 self)
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Abstract. For a model of molecular evolution to be useful for phylogenetic inference, the topology of evolutionary trees must be identifiable. That is, from a joint distribution the model predicts, it must be possible to recover the tree parameter. We establish tree identifiability for a number of phylogenetic models, including a covarion model and a variety of mixture models with a limited number of classes. The proof is based on the introduction of a more general model, allowing more states at internal nodes of the tree than at leaves, and the study of the algebraic variety formed by the joint distributions to which it gives rise. Tree identifiability is first established for this general model through the use of certain phylogenetic invariants. 1.
On The Computational Complexity of Inferring Evolutionary Trees
, 1993
"... The process of reconstructing evolutionary trees can be viewed formally as an optimization problem. Recently, decision problems associated with the most commonly used approaches to reconstructing such trees have been shown to be NPcomplete [Day87, DJS86, DS86, DS87, GF82, Kri88, KM86]. In this t ..."
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Cited by 19 (5 self)
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The process of reconstructing evolutionary trees can be viewed formally as an optimization problem. Recently, decision problems associated with the most commonly used approaches to reconstructing such trees have been shown to be NPcomplete [Day87, DJS86, DS86, DS87, GF82, Kri88, KM86]. In this thesis, a framework is established that incorporates all such problems studied to date. Within this framework, the NPcompleteness results for decision problems are extended by applying theorems from [CT91, Gas86, GKR92, JVV86, KST89, Kre88, Sel91] to derive bounds on the computational complexity of several functions associated with each of these problems, namely ffl evaluation functions, which return the cost of the optimal tree(s), ffl solution functions, which return an optimal tree, ffl spanning functions, which return the number of optimal trees, ffl enumeration functions, which systematically enumerate all optimal trees, and ffl randomselection functions, which return a random...
Probability Models for Genome Rearrangement and Linear Invariants for Phylogenetic Inference
 In Proc. of COCOON
, 1999
"... We review the combinatorial optimization problems in calculating edit distances between genomes and phylogenetic inference based on minimizing gene order changes. With a view to avoiding the computational cost and the "long branches attract" artifact of some treebuilding methods, we explore the pro ..."
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Cited by 18 (3 self)
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We review the combinatorial optimization problems in calculating edit distances between genomes and phylogenetic inference based on minimizing gene order changes. With a view to avoiding the computational cost and the "long branches attract" artifact of some treebuilding methods, we explore the probabilization of genome rearrangment models prior to developing a methodology based on branchlength invariants. We characterize probabilistically the evolution of the structure of the gene adjacency set for inversions on unsigned circular genomes and, using a nontrivial recurrence relation, inversions on signed genomes. Concepts from the theory of invariants developed for the phylogenetics of homologous gene sequences can be used to derive a complete set of linear invariants for unsigned inversions, as well as for a mixed rearrangement model for signed genomes, though not for pure transposition nor pure signed inversion models. The invariants are based on an extended JukesCantor semigroup....
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Performance of a New Invariants Method on Homogeneous and Nonhomogeneous Quartet Trees
, 2006
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Using invariants for phylogenetic tree construction,” in Emerging Applications of Algebraic Geometry
, 2008
"... Abstract. Phylogenetic invariants are certain polynomials in the joint probability distribution of a Markov model on a phylogenetic tree. Such polynomials are of theoretical interest in the field of algebraic statistics and they are also of practical interest—they can be used to construct phylogenet ..."
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Cited by 9 (0 self)
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Abstract. Phylogenetic invariants are certain polynomials in the joint probability distribution of a Markov model on a phylogenetic tree. Such polynomials are of theoretical interest in the field of algebraic statistics and they are also of practical interest—they can be used to construct phylogenetic trees. This paper is a selfcontained introduction to the algebraic, statistical, and computational challenges involved in the practical use of phylogenetic invariants. We survey the relevant literature and provide some partial answers and many open problems.