arXiv subjects

Ruriko Yoshida

Publications and source records attributed to Ruriko Yoshida.

At least 73 records · Page 4Linked to original sources

arXiv2009

The Geometry of the Neighbor-Joining Algorithm for Small Trees

In 2007, Eickmeyer et al. showed that the tree topologies outputted by the Neighbor-Joining (NJ) algorithm and the balanced minimum evolution (BME) method for phylogenetic reconstruction are each determined by a polyhedral subdivision of the space of dissimilarity maps ${\R}^{n \choose 2}$, where $n$ is the number of taxa. In this paper, we will analyze the behavior of the Neighbor-Joining algorithm on five and six taxa and study the geometry and combinatorics of the polyhedral subdivision of the space of dissimilarity maps for six taxa as well as hyperplane representations of each polyhedral subdivision. We also study simulations for one of the questions stated by Eickmeyer et al., that is, the robustness of the NJ algorithm to small perturbations of tree metrics, with tree models which are known to be hard to be reconstructed via the NJ algorithm.

math.CO↗

arXiv2009

Computing holes in semi-groups and its applications to transportation problems

An integer feasibility problem is a fundamental problem in many areas, such as operations research, number theory, and statistics. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup generated by a finite set of vectors in $\Z^d$ and its saturation. In this paper we present an algorithm to compute an explicit description for the set of holes which is the difference of a semi-group $Q$ generated by the vectors and its saturation. We apply our procedure to compute an infinite family of holes for the semi-group of the $3\times 4\times 6$ transportation problem. Furthermore, we give an upper bound for the entries of the holes when the set of holes is finite. Finally, we present an algorithm to find all $Q$-minimal saturation points of $Q$.

math.CO↗

arXiv2009

First steps toward the geometry of cophylogeny

Here we introduce researchers in algebraic biology to the exciting new field of cophylogenetics. Cophylogenetics is the study of concomitantly evolving organisms (or genes), such as host and parasite species. Thus the natural objects of study in cophylogenetics are tuples of related trees, instead of individual trees. We review various research topics in algebraic statistics for phylogenetics, and propose analogs for cophylogenetics. In particular we propose spaces of cophylogenetic trees, cophylogenetic reconstruction, and cophylogenetic invariants. We conclude with open problems.

q-bio.PE↗

arXiv2008

On connectivity of fibers with positive marginals in multiple logistic regression

In this paper we consider exact tests of a multiple logistic regression, where the levels of covariates are equally spaced, via Markov beses. In usual application of multiple logistic regression, the sample size is positive for each combination of levels of the covariates. In this case we do not need a whole Markov basis, which guarantees connectivity of all fibers. We first give an explicit Markov basis for multiple Poisson regression. By the Lawrence lifting of this basis, in the case of bivariate logistic regression, we show a simple subset of the Markov basis which connects all fibers with a positive sample size for each combination of levels of covariates.

math.ST↗

arXiv2008

A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.

math.CO↗

arXiv2008

Parametric k-best alignment

Optimal sequence alignments depend heavily on alignment scoring parameters. Given input sequences, {\em parametric alignment} is the well-studied problem that asks for all possible optimal alignment summaries as parameters vary, as well as the {\em optimality region} of alignment scoring parameters which yield each optimal alignment. But biologically correct alignments might be {\em suboptimal} for all parameter choices. Thus we extend parametric alignment to {\em parametric $k$-best alignment}, which asks for all possible $k$-tuples of $k$-best alignment summaries $(s_1, s_2, ..., s_k)$, as well as the {\em $k$-best optimality region} of scoring parameters which make $s_1, s_2, ..., s_k$ the top $k$ summaries. By exploiting the integer-structure of alignment summaries, we show that, astonishingly, the complexity of parametric $k$-best alignment is only polynomial in $k$. Thus parametric $k$-best alignment is tractable, and can be applied at the whole-genome scale like parametric alignment.

q-bio.PE↗

arXiv2008

A Novel Test for Host-Symbiont Codivergence Indicates Ancient Origin of Fungal Endophytes in Grasses

Significant phylogenetic codivergence between plant or animal hosts ($H$) and their symbionts or parasites ($P$) indicate the importance of their interactions on evolutionary time scales. However, valid and realistic methods to test for codivergence are not fully developed. One of the systems where possible codivergence has been of interest involves the large subfamily of temperate grasses (Pooideae) and their endophytic fungi (epichloae). These widespread symbioses often help protect host plants from herbivory and stresses, and affect species diversity and food web structures. Here we introduce the MRCALink (most-recent-common-ancestor link) method and use it to investigate the possibility of grass-epichloë codivergence. MRCALink applied to ultrametric $H$ and $P$ trees identifies all corresponding nodes for pairwise comparisons of MRCA ages. The result is compared to the space of random $H$ and $P$ tree pairs estimated by a Monte Carlo method. Compared to tree reconciliation the method is less dependent on tree topologies (which often can be misleading), and it crucially improves on phylogeny-independent methods such as {\tt ParaFit} or the Mantel test by eliminating an extreme (but previously unrecognized) distortion of node-pair sampling. Analysis of 26 grass species-epichloë species symbioses did not reject random association of $H$ and $P$ MRCA ages. However, when five obvious host jumps were removed the analysis significantly rejected random association and supported grass-endophyte codivergence. Interestingly, early cladogenesis events in the Pooideae corresponded to early cladogenesis events in epichloae, suggesting concomitant origins of this grass subfamily and its remarkable group of symbionts. We also applied our method to the well-known gopher-louse data set.

q-bio.PE↗

arXiv2008

A Markov Basis for Conditional Test of Common Diagonal Effect in Quasi-Independence Model for Square Contingency Tables

In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger(or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets.

stat.ME↗

arXiv2008

A generalization of the integer linear infeasibility problem

Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is closely related to non-squarefree elements of Markov bases for sampling contingency tables with given marginals. To study a family of systems with no integer solution, we focus on a commutative semigroup generated by a finite subset of $\Z^d$ and its saturation. An element in the difference of the semigroup and its saturation is called a ``hole''. We show the necessary and sufficient conditions for the finiteness of the set of holes. Also we define fundamental holes and saturation points of a commutative semigroup. Then, we show the simultaneous finiteness of the set of holes, the set of non-saturation points, and the set of generators for saturation points. We apply our results to some three- and four-way contingency tables. Then we will discuss the time complexities of our algorithms.

math.ST↗

arXiv2007

On the optimality of the neighbor-joining algorithm

The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy algorithm for finding the balanced minimum evolution (BME) tree associated to a dissimilarity map. From this point of view, NJ is ``optimal'' when the algorithm outputs the tree which minimizes the balanced minimum evolution criterion. We use the fact that the NJ tree topology and the BME tree topology are determined by polyhedral subdivisions of the spaces of dissimilarity maps ${\R}_{+}^{n \choose 2}$ to study the optimality of the neighbor-joining algorithm. In particular, we investigate and compare the polyhedral subdivisions for $n \leq 8$. A key requirement is the measurement of volumes of spherical polytopes in high dimension, which we obtain using a combination of Monte Carlo methods and polyhedral algorithms. We show that highly unrelated trees can be co-optimal in BME reconstruction, and that NJ regions are not convex. We obtain the $l_2$ radius for neighbor-joining for $n=5$ and we conjecture that the ability of the neighbor-joining algorithm to recover the BME tree depends on the diameter of the BME tree.

q-bio.QM↗

arXiv2007

Markov bases for two-way subtable sum problems

It has been well-known that for two-way contingency tables with fixed row sums and column sums the set of square-free moves of degree two forms a Markov basis. However when we impose an additional constraint that the sum of a subtable is also fixed, then these moves do not necessarily form a Markov basis. Thus, in this paper, we show a necessary and sufficient condition on a subtable so that the set of square-free moves of degree two forms a Markov basis.

math.CO↗

arXiv2007

Saturation points on faces of a rational polyhedral cone

Different commutative semigroups may have a common saturation. We consider distinguishing semigroups with a common saturation based on their ``sparsity''. We propose to qualitatively describe sparsity of a semigroup by considering which faces of the corresponding rational polyhedral cone have saturation points. For a commutative semigroup we give a necessary and sufficient condition for determining which faces have saturation points. We also show that we can construct a commutative semigroup with arbitrary consistent patterns of faces with saturation points.

math.CO↗

arXiv2007

Partitioning the Sample Space on Five Taxa for the Neighbor Joining Algorithm

In this paper, we will analyze the behavior of the Neighbor Joining algorithm on five taxa and we will show that the partition of the sample (data) space for estimation of a tree topology with five taxa into subspaces, within each of which the Neighbor Joining algorithm returns the same tree topology. A key of our method to partition the sample space is the action of the symmetric group $S_5$ on the set of distance matrices by changing the labels of leaves. The method described in this paper can be generalized to trees with more than five taxa.

math.CO↗

arXiv2006

On the Enumeration of Certain Weighted Graphs

We enumerate weighted graphs with a certain upper bound condition. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that if the given graph is a bipartite graph, then its generating function is of the form $\frac{p(x)}{(1-x)^{m+1}}$, where $m$ is the number of vertices of the graph and $p(x)$ is a polynomial of degree at most $m$.

math.CO↗

arXiv2006

Indispensable monomials of toric ideals and Markov bases

Extending the notion of indispensable binomials of a toric ideal, we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for proving the existence or non-existence of a unique minimal system of binomials generators of a toric ideal. Some examples of indispensable monomials from statistical models for contingency tables are given.

math.ST↗

arXiv2005

Maximum likelihood estimation of phylogenetic tree and substitution rates via generalized neighbor-joining and the EM algorithm

A central task in the study of molecular sequence data from present-day species is the reconstruction of the ancestral relationships. The most established approach to tree reconstruction is the maximum likelihood (ML) method. In this method, evolution is described in terms of a discrete-state continuous-time Markov process on a phylogenetic tree. The substitution rate matrix, that determines the Markov process, can be estimated using the expectation maximization (EM) algorithm. Unfortunately, an exhaustive search for the ML phylogenetic tree is computationally prohibitive for large data sets. In such situations, the neighbor-joining (NJ) method is frequently used because of its computational speed. The NJ method reconstructs trees by clustering neighboring sequences recursively, based on pairwise comparisons between the sequences. The NJ method can be generalized such that reconstruction is based on comparisons of subtrees rather than pairwise distances. In this paper, we present an algorithm for simultaneous substitution rate estimation and phylogenetic tree reconstruction. The algorithm iterates between the EM algorithm for estimating substitution rates and the generalized NJ method for tree reconstruction. Preliminary results of the approach are encouraging.

q-bio.QM↗

arXiv2005

Neighbor joining with phylogenetic diversity estimates

The Neighbor-Joining algorithm is a recursive procedure for reconstructing trees that is based on a transformation of pairwise distances between leaves. We present a generalization of the neighbor-joining transformation, which uses estimates of phylogenetic diversity rather than pairwise distances in the tree. This leads to an improved neighbor-joining algorithm whose total running time is still polynomial in the number of taxa. On simulated data, the method outperforms other distance-based methods. We have implemented neighbor-joining for subtree weights in a program called MJOIN which is freely available under the Gnu Public License at http://bio.math.berkeley.edu/mjoin/ .

q-bio.QM↗

arXiv2004

Barvinok's Rational Functions: Algorithms and Applications to Optimization, Statistics, and Algebra

The main theme of this dissertation is the study of the lattice points in a rational convex polyhedron and their encoding in terms of Barvinok's short rational functions. The first part of this thesis looks into theoretical applications of these rational functions to Optimization, Statistics, and Computational Algebra. The main theorem on Chapter 2 concerns the computation of the \emph{toric ideal} $I_A$ of an integral $n \times d$ matrix $A$. We encode the binomials belonging to the toric ideal $I_A$ associated with $A$ using Barvinok's rational functions. If we fix $d$ and $n$, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal $I_A$, with respect to any term order, in polynomial time. We derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions of the division algorithm. Chapter 3 presents three ways to use Barvinok's rational functions to solve Integer Programs. The second part of the thesis is experimental and consists mainly of the software package {\tt LattE}, the first implementation of Barvinok's algorithm. We report on experiments with families of well-known rational polytopes: multiway contingency tables, knapsack type problems, and rational polygons. We also developed a new algorithm, {\em the homogenized Barvinok's algorithm} to compute the generating function for a rational polytope. We showed that it runs in polynomial time in fixed dimension. With the homogenized Barvinok's algorithm, we obtained new combinatorial formulas: the generating function for the number of $5\times 5$ magic squares and the generating function for the number of $3\times 3 \times 3 \times 3$ magic cubes as rational functions.

math.CO↗