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Pages
 Title
 Maximum Likelihood for Matrices with Rank Constraints
 Creator
 Hauenstein, Jonathan, Rodriguez, Jose Israel, Sturmfels, Bernd
 Date
 2014, 20140430
 Description

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded...
Show moreMaximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.
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 Journal of Algebraic Statistics
 Title
 Generic Identification of BinaryValued Hidden Markov Processes
 Creator
 Schönhuth, Alexander
 Date
 2014, 20140430
 Description

The generic identification problem is to decide whether a stochastic process (X_t) is a hidden Markov process and if yes to infer its...
Show moreThe generic identification problem is to decide whether a stochastic process (X_t) is a hidden Markov process and if yes to infer its parameters for all but a subset of parametrizations that form a lowerdimensional subvariety in parameter space. Partial answers so far available depend on extra assumptions on the processes, which are usually centered around stationarity. Here we present a general solution for binaryvalued hidden Markov processes. Our approach is rooted in algebraic statistics hence it is geometric in nature. We find that the algebraic varieties associated with the probability distributions of binaryvalued hidden Markov processes are zero sets of determinantal equations which draws a connection to wellstudied objects from algebra. As a consequence, our solution allows for algorithmic implementation based on elementary (linear) algebraic routines.
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 Journal of Algebraic Statistics
 Title
 Estimation for DyadicDependent Exponential Random Graph Models
 Creator
 Yang, Xiaolin, Rinaldo, Alessandro, Fienberg, Stephen E.
 Date
 2014, 20140430
 Description

Graphs are the primary mathematical representation for networks, with nodes or vertices corresponding to units (e.g., individuals) and edges...
Show moreGraphs are the primary mathematical representation for networks, with nodes or vertices corresponding to units (e.g., individuals) and edges corresponding to relationships. Exponential Random Graph Models (ERGMs) are widely used for describing network data because of their simple structure as an exponential function of a sum of parameters multiplied by their corresponding sufficient statistics. As with other exponential family settings the key computational difficulty is determining the normalizing constant for the likelihood function, a quantity that depends only on the data. In ERGMs for network data, the normalizing constant in the model often makes the parameter estimation intractable for large graphs, when the model involves dependence among dyads in the graph. One way to deal with this problem is to approximate the likelihood function by something tractable, e.g., by using the method of pseudolikelihood estimation suggested in the early literature. In this paper, we describe the family of ERGMs and explain the increasing complexity that arises from imposing different edge dependence and homogeneous parameter assumptions. We then compare maximum likelihood (ML) and maximum pseudolikelihood (MPL) estimation schemes with respect to existence and related degeneracy properties for ERGMs involving dependencies among dyads.
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 Journal of Algebraic Statistics
 Title
 A Family of Quasisymmetry Models
 Creator
 Kateri, Maria, Mohammadi, Fatemeh, Sturmfels, Bernd
 Date
 2015, 20150611
 Description

We present a oneparameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its...
Show moreWe present a oneparameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue. Algebraically, this corresponds to deformations of toric ideals associated with graphs. Our discussion of the statistical issues centers around maximum likelihood estimation.
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 Journal of Algebraic Statistics
 Title
 On the Connectivity of Fiber Graphs
 Creator
 Hemmecke, Raymond, Windisch, Tobias
 Date
 2015, 20150611
 Description

We consider the connectivity of fiber graphs with respect to Gröbner basis and Graver basis moves. First, we present a sequence of fiber...
Show moreWe consider the connectivity of fiber graphs with respect to Gröbner basis and Graver basis moves. First, we present a sequence of fiber graphs using moves from a Gröbner basis and prove that their edgeconnectivity is lowest possible and can have an arbitrarily large distance from the minimal degree. We then show that graphtheoretic properties of fiber graphs do not depend on the size of the righthand side. This provides a counterexample to a conjecture of Engström on the nodeconnectivity of fiber graphs. Our main result shows that the edgeconnectivity in all fiber graphs of this counterexample is best possible if we use moves from Graver basis instead.
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 Journal of Algebraic Statistics
 Title
 The precision space of interpolatory cubature formulæ
 Creator
 Fassino, Claudia, Riccomagno, Eva
 Date
 2015, 20150611
 Description

Methods from Commutative Algebra and Numerical Analysis are combined to address a problem common to many disciplines: the estimation of the...
Show moreMethods from Commutative Algebra and Numerical Analysis are combined to address a problem common to many disciplines: the estimation of the expected value of a polynomial of a random vector using a linear combination of a finite number of its values. In this work we remark on the error estimation in cubature formulæ for polynomial functions and introduce the notion of a precision space for a cubature rule.
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 Journal of Algebraic Statistics
 Title
 The degeneration of the Grassmannian into a toric variety and the calculation of the eigenspaces of a torus action
 Creator
 Witaszek, Jakub
 Date
 2015, 20150611
 Description

Using the method of degenerating a Grassmannian into a toric variety, we calculate formulas for the dimensions of the eigenspaces of the...
Show moreUsing the method of degenerating a Grassmannian into a toric variety, we calculate formulas for the dimensions of the eigenspaces of the action of an ndimensional torus on a Grassmannian of planes in an ndimensional space.
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 Journal of Algebraic Statistics
 Title
 Varieties with maximum likelihood degree one
 Creator
 Huh, June
 Date
 2014, 20140430
 Description

We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced Adiscriminantal varieties under...
Show moreWe show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced Adiscriminantal varieties under monomial maps with finite fibers. The maximum likelihood estimator corresponding to such a variety is Kapranov’s Horn uniformization. This extends Kapranov’s characterization of Adiscriminantal hypersurfaces to varieties of arbitrary codimension.
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 Journal of Algebraic Statistics
 Title
 Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model
 Creator
 Long, Colby, Sullivant, Seth
 Date
 2015, 20150611
 Description

The strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the doublestranded structure of DNA. We show...
Show moreThe strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the doublestranded 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.
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 Journal of Algebraic Statistics
 Title
 On Exchangeability in Network Models, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Lauritzen, Steffen, Rinaldo, Alessandro, Sadeghi, Kayvan
 Date
 2019, 20190412
 Description

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric...
Show moreWe derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graphtheoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size.
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 Journal of Algebraic Statistics
 Title
 Maximum Likelihood Estimation of the Latent Class Model through Model Boundary Decomposition, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Elizabeth S. Allman, Baños Cervantes, Hector, Evans, Robin, Hosten, Serkan, Kubjas, Kaie, Lemke, Daniel, Rhodes, John, Zwiernik, Piotr
 Date
 2019, 20190412
 Description

The ExpectationMaximization (EM) algorithm is routinely used for maximum likelihood estimation in latent class analysis. However, the EM...
Show moreThe ExpectationMaximization (EM) algorithm is routinely used for maximum likelihood estimation in latent class analysis. However, the EM algorithm comes with no global guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata and performance of the EM algorithm. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.
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 Journal of Algebraic Statistics
 Title
 Inference for Ordinal LogLinear Models Based on Algebraic Statistics, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Pham, Thi Mui, Kateri, Maria
 Date
 2019, 20190412
 Description

Tools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit...
Show moreTools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit testing of loglinear models. However, this approach has not been considered so far for association models, which are special loglinear models for tables with ordinal classification variables. The simplest association model for twoway tables, the uniform (U) association model, has just one parameter more than the independence model and is applicable when both classification variables are ordinal. Less parsimonious are the row (R) and column (C) effect association models, appropriate when at least one of the classification variables is ordinal. Association models have been extended for multidimensional contingency tables as well. Here, we adjust algebraic methods for association models analysis and investigate their eligibility, focusing mainly on twoway tables. They are implemented in the statistical software R and illustrated on real data tables. Finally the algebraic model fit and selection procedure is assessed and compared to the asymptotic approach in terms of a simulation study.
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 Journal of Algebraic Statistics
 Title
 Exact tests to compare contingency tables under quasiindependence and quasisymmetry, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Bocci, Christiano, Rapallo, Fabio
 Date
 2019, 20190412
 Description

In this work we define loglinear models to compare several square contingency tables under the quasiindependence or the quasisymmetry model...
Show moreIn this work we define loglinear models to compare several square contingency tables under the quasiindependence or the quasisymmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two realdata examples illustrate the use of tehse models in different fields of applications.
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 Journal of Algebraic Statistics
 Title
 Detecting epistasis via Markov bases
 Creator
 Malaspinas, AnnaSapfo, Uhler, Caroline
 Date
 2011, 2011
 Description

Rapid research progress in genotyping techniques have allowed large genomewide association studies. Existing methods often focus on...
Show moreRapid research progress in genotyping techniques have allowed large genomewide association studies. Existing methods often focus on determining associations between single loci and a specific phenotype. However, a particular phenotype is usually the result of complex relationships between multiple loci and the environment. In this paper, we describe a twostage method for detecting epistasis by combining the traditionally used singlelocus search with a search for multiway interactions. Our method is based on an extended version of Fisher’s exact test. To perform this test, a Markov chain is constructed on the space of multidimensional contingency tables using the elements of a Markov basis as moves. We test our method on simulated data and compare it to a twostage logistic regression method and to a fully Bayesian method, showing that we are able to detect the interacting loci when other methods fail to do so. Finally, we apply our method to a genomewide data set consisting of 685 dogs and identify epistasis associated with canine hair length for four pairs of single nucleotide polymorphisms (SNPs).
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 Journal of Algebraic Statistics
 Title
 Open Problems on Connectivity of Fibers with Positive Margins in Multidimensional Contingency Tables
 Creator
 Yoshida, Ruriko
 Date
 2010, 2010
 Description

DiaconisSturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families...
Show moreDiaconisSturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families, based on the algebraic theory of toric ideals. This algorithm is applied to categorical data analysis through the notion of Markov bases. Initiated with its application to Markov chain Monte Carlo approach for testing statistical fitting of the given model, many researchers have extensively studied the structure of Markov bases for models in computational algebraic statistics. In the Markov chain Monte Carlo approach for testing statistical fitting of the given model, a Markov basis is a set of moves connecting all contingency tables satisfying the given margins. Despite the computational advances, there are applied problems where one may never be able to compute a Markov basis. In general, the number of elements in a minimal Markov basis for a model can be exponentially many. Thus, it is important to compute a reduced number of moves which connect all tables instead of computing a Markov basis. In some cases, such as logistic regression, positive margins are shown to allow a set of Markov connecting moves that are much simpler than the full Markov basis. Such a set is called a Markov subbasis with assumption of positive margins. In this paper we summarize some computations of and open problems on Markov subbases for contingency tables with assumption of positive margins under specific models as well as develop algebraic methods for studying connectivity of Markov moves with margin positivity to develop Markov sampling methods for exact conditional inference in statistical models where the Markov basis is hard to compute.
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 Journal of Algebraic Statistics
 Title
 An Euclidean norm based criterion to assess robots’ 2D pathfollowing performance, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
 Creator
 Saggini, Eleonora, Torrente, MariaLaura
 Date
 2016, 20160712
 Description

A current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper...
Show moreA current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper contributes to this by defining a new criterion for assessing pathfollowing tasks in the planar case, that is, evaluating the performance of robots that are required to follow a desired reference path. Such criterion comes from the study of the local differential geometry of the problem. New conditions for deciding whether or not the zero locus of a given polynomial intersects the neighbourhood of a point are defined. Based on this, new algorithms are presented and tested on both simulated data and experiments conducted at sea employing an Unmanned Surface Vehicle.
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 Journal of Algebraic Statistics
 Title
 Uncovering Proximity of Chromosome Territories using Classical Algebraic Statistics
 Creator
 Arsuaga, Javier, Heskia, Ido, Hosten, Serkan, Maskalevich, Tatiana
 Date
 2015, 20151109
 Description

Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments...
Show moreExchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments rejoin with fragments from other chromosomes or from other regions within the same chromosome. ETCAs are commonly observed in cancer cells and in cells exposed to radiation. The frequency of these chromosome rearrangements is correlated with their spatial proximity, therefore it can be used to infer the three dimensional organization of the genome. Extracting statistical significance of spatial proximity from cancer and radiation data has remained somewhat elusive because of the sparsity of the data. We here propose a new approach to study the three dimensional organization of the genome using algebraic statistics. We test our method on a published data set of irradiated human blood lymphocyte cells. We provide a rigorous method for testing the overall organization of the genome, and in agreement with previous results we find a random relative positioning of chromosomes with the exception of the chromosome pairs {1,22} and {13,14} that have a significantly larger number of ETCAs than the rest of the chromosome pairs suggesting their spatial proximity. We conclude that algebraic methods can successfully be used to analyze genetic data and have potential applications to larger and more complex data sets.
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 Journal of Algebraic Statistics
 Title
 Markov degree of configurations defined by fibers of a configuration
 Creator
 Koyama, Takayuki, Ogawa, Mitsunori, Takemura, Akimichi
 Date
 2015, 20151109
 Description

We consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is...
Show moreWe consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is bounded from above by the Markov complexity of the base configuration. As important examples of base configurations we consider incidence matrices of graphs and study the maximum Markov degree of configurations defined by fibers of the incidence matrices. In particular we give a proof that the Markov degree for twoway transportation polytopes is three.
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 Journal of Algebraic Statistics
 Title
 The maximum likelihood degree of Fermat hypersurfaces
 Creator
 Agostini, Daniele, Alberelli, Davide, Grande, Francesco, Lella, Paolo
 Date
 2015, 20151109
 Description

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in...
Show moreWe study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.
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 Journal of Algebraic Statistics
 Title
 Moment Varieties of Gaussian Mixtures, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
 Creator
 Améndola, Carlos, Faugère, JeanCharles, Sturmfels, Bernd
 Date
 2016, 20160712
 Description

The points of a moment variety are the vectors of all moments up to some order, for a given family of probability distributions. We study the...
Show moreThe points of a moment variety are the vectors of all moments up to some order, for a given family of probability distributions. We study the moment varieties for mixtures of multivariate Gaussians. Following up on Pearson’s classical work from 1894, we apply current tools from computational algebra to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all covariance matrices to zero. We compute the ideals of the 5dimensional moment varieties representing mixtures of two univariate Gaussians, and we offer a comparison to the maximum likelihood approach.
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 Journal of Algebraic Statistics