Results 1  10
of
82
Statistical challenges with high dimensionality: feature selection in knowledge discovery
, 2006
"... ..."
(Show Context)
The identifiability of tree topology for phylogenetic models, including covarion and mixture models
, 2005
"... 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 phylogene ..."
Abstract

Cited by 42 (12 self)
 Add to MetaCart
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.
Toric fiber products
, 2006
"... We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Gröbner bases f ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
(Show Context)
We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Gröbner bases for these ideals. This allows us to unify and generalize some results in algebraic statistics.
Performance of a New Invariants Method on Homogeneous and Nonhomogeneous Quartet Trees
, 2006
"... ..."
Toric geometry of cuts and splits
 MICHIGAN MATH. J
, 2007
"... Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial structure of the graph and the corresponding cut polytope to alg ..."
Abstract

Cited by 20 (2 self)
 Add to MetaCart
(Show Context)
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal. Cut ideals generalize toric ideals arising in phylogenetics and the study of contingency tables.
Algebraic Statistical Models
 Statistica Sinica
, 2007
"... Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice propert ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an 'algebraic exponential family'. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models.
ON THE IDEALS OF EQUIVARIANT TREE MODELS
, 2008
"... We introduce equivariant tree models in algebraic statistics, which unify and generalize existing tree models such as the general Markov model, the strand symmetric model, and group based models such as the JukesCantor and Shimura models. We focus on the ideals of such models. We show how the ide ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
We introduce equivariant tree models in algebraic statistics, which unify and generalize existing tree models such as the general Markov model, the strand symmetric model, and group based models such as the JukesCantor and Shimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices. The main novelty is that our results yield generators of the full ideal rather than an ideal which only defines the model settheoretically.