arXiv subjects

Seth Sullivant

Publications and source records attributed to Seth Sullivant.

At least 37 records · Page 2Linked to original sources

arXiv2019

Exchangeable and Sampling Consistent Distributions on Rooted Binary Trees

We introduce a notion of finite sampling consistency for phylogenetic trees and show that the set of finitely sampling consistent and exchangeable distributions on n leaf phylogenetic trees is a polytope. We use this polytope to show that the set of all exchangeable and infinite sampling consistent distributions on 4 leaf phylogenetic trees is exactly Aldous' beta-splitting model and give a description of some of the vertices for the polytope of distributions on 5 leaves. We also introduce a new semialgebraic set of exchangeable and sampling consistent models we call the multinomial model and use it to characterize the set of exchangeable and sampling consistent distributions.

math.CO↗

arXiv2018

Bounds on the expected size of the maximum agreement subtree for a given tree shape

We show that the expected size of the maximum agreement subtree of two $n$-leaf trees, uniformly random among all trees with the shape, is $Θ(\sqrt{n})$. To derive the lower bound, we prove a global structural result on a decomposition of rooted binary trees into subgroups of leaves called blobs. To obtain the upper bound, we generalize a first moment argument for random tree distributions that are exchangeable and not necessarily sampling consistent.

math.CO↗

arXiv2017

On Mixing Behavior of a Family of Random Walks Determined by a Linear Recurrence

We study random walks on the integers mod $G_n$ that are determined by an integer sequence $\{ G_n \}_{n \geq 1}$ generated by a linear recurrence relation. Fourier analysis provides explicit formulas to compute the eigenvalues of the transition matrices and we use this to bound the mixing time of the random walks.

math.PR↗

arXiv2017

Algebraic Aspects of Conditional Independence and Graphical Models

This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces binomial ideals and some ideas from real algebraic geometry. When random variables are discrete or Gaussian, tools from computational algebraic geometry can be used to understand implications between conditional independence statements. This is accomplished by computing primary decompositions of conditional independence ideals. As examples the chapter presents in detail the graphical model of a four cycle and the intersection axiom, a certain implication of conditional independence statements. Another important problem in the area is to determine all constraints on a graphical model, for example, equations determined by trek separation. The full set of equality constraints can be determined by computing the model's vanishing ideal. The chapter illustrates these techniques and ideas with examples from the literature and provides references for further reading.

math.ST↗

arXiv2017

Identifiability of phylogenetic parameters from k-mer data under the coalescent

Distances between sequences based on their $k$-mer frequency counts can be used to reconstruct phylogenies without first computing a sequence alignment. Past work has shown that effective use of k-mer methods depends on 1) model-based corrections to distances based on $k$-mers and 2) breaking long sequences into blocks to obtain repeated trials from the sequence-generating process. Good performance of such methods is based on having many high-quality blocks with many homologous sites, which can be problematic to guarantee a priori. Nature provides natural blocks of sequences into homologous regions---namely, the genes. However, directly using past work in this setting is problematic because of possible discordance between different gene trees and the underlying species tree. Using the multispecies coalescent model as a basis, we derive model-based moment formulas that involve the divergence times and the coalescent parameters. From this setting, we prove identifiability results for the tree and branch length parameters under the Jukes-Cantor model of sequence mutations.

q-bio.PE↗

arXiv2017

Strongly robust toric ideals in codimension 2

A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set. For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials. We characterize the codimension 2 strongly robust toric ideals by their Gale diagrams. This gives a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.

math.AC↗

arXiv2016

Matrix Schubert varieties and Gaussian conditional independence models

Matrix Schubert varieties are certain varieties in the affine space of square matrices which are determined by specifying rank conditions on submatrices. We study these varieties for generic matrices, symmetric matrices, and upper triangular matrices in view of two applications to algebraic statistics: we observe that special conditional independence models for Gaussian random variables are intersections of matrix Schubert varieties in the symmetric case. Consequently, we obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.

math.AG↗

arXiv2016

Algebraic Tools for the Analysis of State Space Models

We present algebraic techniques to analyze state space models in the areas of structural identifiability, observability, and indistinguishability. While the emphasis is on surveying existing algebraic tools for studying ODE systems, we also present a variety of new results. In particular: on structural identifiability, we present a method using linear algebra to find identifiable functions of the parameters of a model for unidentifiable models. On observability, we present techniques using Gröbner bases and algebraic matroids to test algebraic observability of state space models. On indistinguishability, we present a sufficient condition for distinguishability using computational algebra and demonstrate testing indistinguishability.

math.OC↗

arXiv2016

Unimodular Binary Hierarchical Models

Associated to each simplicial complex is a binary hierarchical model. We classify the simplicial complexes that yield unimodular binary hierarchical models. Our main theorem provides both a construction of all unimodular binary hierarchical models, together with a characterization in terms of excluded minors, where our definition of a minor allows the taking of links and induced complexes. A key tool in the proof is the lemma that the class of unimodular binary hierarchical models is closed under the Alexander duality operation on simplicial complexes.

math.CO↗

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.

q-bio.PE↗

arXiv2016

Normal Binary Hierarchical Models

Each simplicial complex and integer vector yields a vector configuration whose combinatorial properties are important for the analysis of contingency tables. We study the normality of these vector configurations including a description of operations on simplicial complexes that preserve normality, constructions of families of minimally nonnormal complexes, and computations classifying all of the normal complexes on up to six vertices. We repeat this analysis for compressed vector configurations, classifying all of the compressed complexes on up to six vertices.

math.CO↗

arXiv2015

The Maximum Likelihood Threshold of a Graph

The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $n-1$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold.

math.CO↗

arXiv2015

Bounds on the Expected Size of the Maximum Agreement Subtree

We prove polynomial upper and lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and Yule-Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.

q-bio.PE↗

arXiv2015

Lifting Markov Bases and Higher Codimension Toric Fiber Products

We study how to lift Markov bases and Gröbner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.

math.AC↗

arXiv2014

Identifiability results for several classes of linear compartment models

Identifiability concerns finding which unknown parameters of a model can be estimated from given input-output data. If some subset of the parameters of a model cannot be determined given input-output data, then we say the model is unidentifiable. In past work we identified a class of models, that we call identifiable cycle models, which are not identifiable but have the simplest possible identifiable functions (so-called monomial cycles). Here we show how to modify identifiable cycle models by adding inputs, adding outputs, or removing leaks, in such a way that we obtain an identifiable model. We also prove a constructive result on how to combine identifiable models, each corresponding to strongly connected graphs, into a larger identifiable model. We apply these theoretical results to several real-world biological models from physiology, cell biology, and ecology.

math.AG↗

arXiv2014

Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model

The strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the double-stranded structure of DNA. We show that the set of known phylogenetic invariants for the general strand symmetric model of the three leaf claw tree entirely defines the ideal. This knowledge allows one to determine the vanishing ideal of the general strand symmetric model of any trivalent tree. Our proof of the main result is computational. We use the fact that the Zariski closure of the strand symmetric model is the secant variety of a toric variety to compute the dimension of the variety. We then show that the known equations generate a prime ideal of the correct dimension using elimination theory.

q-bio.PE↗

arXiv2014

Sequential Importance Sampling for Two-dimensional Ising Models

In recent years, sequential importance sampling (SIS) has been well developed for sampling contingency tables with linear constraints. In this paper, we apply SIS procedure to 2-dimensional Ising models, which give observations of 0-1 tables and include both linear and quadratic constraints. We show how to compute bounds for specific cells by solving linear programming (LP) problems over cut polytopes to reduce rejections. The computational results, which includes both simulations and real data analysis, suggest that our method performs very well for sparse tables and when the 1's are spread out: the computational times are short, the acceptance rates are high, and if proper tests are used then in most cases our conclusions are theoretically reasonable.

stat.CO↗

arXiv2014

Identifiability of 3-Class Jukes-Cantor Mixtures

We prove identifiability of the tree parameters of the 3-class Jukes-Cantor mixture model. The proof uses ideas from algebraic statistics, in particular: finding phylogenetic invariants that separate the varieties associated to different triples of trees; computing dimensions of the resulting phylogenetic varieties; and using the disentangling number to reduce to trees with a small number of leaves. Symbolic computation also plays a key role in handling the many different cases and finding relevant phylogenetic invariants.

q-bio.PE↗