Search results

Title: The geometry of Sloppiness
Creator: Dufresne, Emilie, Harrington , Heather A, Raman, Dhruva V
Date: 2018, 2018-09-24
Description: The use of mathematical models in the sciences often requires the estimation of unknown parameter values from data. Sloppiness provides...
Show moreThe use of mathematical models in the sciences often requires the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. In this paper, we develop a precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold', in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric. This opens up the possibility of alternative quantification of sloppiness, beyond the standard use of the Fisher Information Matrix, which assumes that parameter space is equipped with the usual Euclidean and the measurement error is infinitesimal. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models.
Show less
Collection: Journal of Algebraic Statistics

Title: Mixture models for rating data: the method of moments via Groebner bases
Creator: Iannario, Maria, Simone, Rosaria
Date: 2017, 2017-12-26
Description: A recent thread of research in ordinal data analysis involves a class of mixture models that designs the responses as the combination of the...
Show moreA recent thread of research in ordinal data analysis involves a class of mixture models that designs the responses as the combination of the two main aspects driving the decision pro- cess: a feeling and an uncertainty components. This novel paradigm has been proven flexible to account also for overdispersion. In this context, Groebner bases are exploited to estimate model parameters by implementing the method of moments. In order to strengthen the validity of the moment procedure so derived, alternatives parameter estimates are tested by means of a simulation experiment. Results show that the moment estimators are satisfactory per se, and that they significantly reduce the bias and perform more efficiently than others when they are set as starting values for the Expectation-Maximization algorithm.
Show less
Collection: Journal of Algebraic Statistics

Title: Ideal-Theoretic Strategies for Asymptotic Approximation of Marginal Likelihood Integrals
Creator: Lin, Shaowei
Date: 2017, 2017-02-08
Description: The accurate asymptotic evaluation of marginal likelihood integrals is a fundamental problem in Bayesian statistics. Following the approach...
Show moreThe accurate asymptotic evaluation of marginal likelihood integrals is a fundamental problem in Bayesian statistics. Following the approach introduced by Watanabe, we translate this into a problem of computational algebraic geometry, namely, to determine the real log canonical threshold of a polynomial ideal, and we present effective methods for solving this problem. Our results are based on resolution of singularities. They apply to parametric models where the Kullback-Leibler distance is upper and lower bounded by scalar multiples of some sum of squared real analytic functions. Such models include finite state discrete models.
Show less
Collection: Journal of Algebraic Statistics

Title: Unimodular hierarchical models and their Graver bases
Creator: Bernstein, Daniel Irving, O'Neill, Christopher
Date: 2017, 2017-12-26
Description: Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding...
Show moreGiven a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted simplicial complexes that give rise to unimodular vector configurations. We also provide a combinatorial characterization of their Graver bases.
Show less
Collection: Journal of Algebraic Statistics

Title: Mixtures and products in two graphical models
Creator: Seigal,Anna, Montufar, Guido
Date: 2018, 2018-09-24
Description: We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called...
Show moreWe compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.
Show less
Collection: Journal of Algebraic Statistics

Title: Markov bases for two-way change-point models of ladder determinantal tables
Creator: Aoki, Satoshi, Hibi, Takayuki
Date: 2017, 2017-02-08
Description: To evaluate the goodness-of-fit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method...
Show moreTo evaluate the goodness-of-fit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method is one of the effective approaches. For this purpose, a Markov basis plays an important role because it guarantees the connectivity of the chain, which is needed for unbiasedness of the estimation, and therefore is investigated in various settings such as incomplete tables or subtable sum constraints. In this paper, we consider the two-way change-point model for the ladder determinantal table, which is an extension of these two previous works, i.e., works on incomplete tables by Aoki and Takemura (2005, J. Stat. Comput. Simulat.) and subtable some constraints by Hara, Takemura and Yoshida (2010, J. Pure Appl. Algebra). Our main result is based on the theory of Gr ?obner basis for the distributive lattice. We give a numerical example for actual data.
Show less
Collection: Journal of Algebraic Statistics

Title: Cubature Rules and Expected Value of Some Complex Functions, Special Volume in honor of memory of S.E.Fienberg
Creator: Fassino, Claudia, Riccomagno, Eva, Rogantin, Maria Piera
Date: 2019, 2019-04-12
Description: The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in...
Show moreThe expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for finite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments is determined.
Show less
Collection: Journal of Algebraic Statistics

Title: Strongly Robust Toric Ideals in Codimension 2, Special Volume in honor of memory of S.E.Fienberg
Creator: Sullivant ,Seth
Date: 2019, 2019-04-12
Description: A homogeneous ideal is robust if its universal Gr ?obner basis is also a minimal generating set. For toric ideals, one has the stronger...
Show moreA homogeneous ideal is robust if its universal Gr ?obner 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 Petrovi?, Thoma, and Vladoiu in the case of codimension 2 toric ideals.
Show less
Collection: Journal of Algebraic Statistics

Title: Maximum Likelihood for Matrices with Rank Constraints
Creator: Hauenstein, Jonathan, Rodriguez, Jose Israel, Sturmfels, Bernd
Date: 2014, 2014-04-30
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.
Show less
Collection: Journal of Algebraic Statistics

Title: Generic Identification of Binary-Valued Hidden Markov Processes
Creator: Schönhuth, Alexander
Date: 2014, 2014-04-30
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 lower-dimensional 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 binary-valued 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 binary-valued hidden Markov processes are zero sets of determinantal equations which draws a connection to well-studied objects from algebra. As a consequence, our solution allows for algorithmic implementation based on elementary (linear) algebraic routines.
Show less
Collection: Journal of Algebraic Statistics

Title: A Family of Quasisymmetry Models
Creator: Kateri, Maria, Mohammadi, Fatemeh, Sturmfels, Bernd
Date: 2015, 2015-06-11
Description: We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its...
Show moreWe present a one-parameter 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.
Show less
Collection: Journal of Algebraic Statistics

Title: On the Connectivity of Fiber Graphs
Creator: Hemmecke, Raymond, Windisch, Tobias
Date: 2015, 2015-06-11
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 edge-connectivity is lowest possible and can have an arbitrarily large distance from the minimal degree. We then show that graph-theoretic properties of fiber graphs do not depend on the size of the right-hand side. This provides a counterexample to a conjecture of Engström on the node-connectivity of fiber graphs. Our main result shows that the edge-connectivity in all fiber graphs of this counterexample is best possible if we use moves from Graver basis instead.
Show less
Collection: Journal of Algebraic Statistics

Title: The precision space of interpolatory cubature formulæ
Creator: Fassino, Claudia, Riccomagno, Eva
Date: 2015, 2015-06-11
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.
Show less
Collection: 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, 2015-06-11
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 n-dimensional torus on a Grassmannian of planes in an n-dimensional space.
Show less
Collection: Journal of Algebraic Statistics

Title: Varieties with maximum likelihood degree one
Creator: Huh, June
Date: 2014, 2014-04-30
Description: We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under...
Show moreWe show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal 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 A-discriminantal hypersurfaces to varieties of arbitrary codimension.
Show less
Collection: Journal of Algebraic Statistics

Title: Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model
Creator: Long, Colby, Sullivant, Seth
Date: 2015, 2015-06-11
Description: The strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the double-stranded structure of DNA. We show...
Show moreThe 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.
Show less
Collection: 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, 2019-04-12
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 graph-theoretic 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.
Show less
Collection: 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, 2019-04-12
Description: The Expectation-Maximization (EM) algorithm is routinely used for maximum likelihood estimation in latent class analysis. However, the EM...
Show moreThe Expectation-Maximization (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.
Show less
Collection: Journal of Algebraic Statistics

Title: Exact tests to compare contingency tables under quasi-independence and quasi-symmetry, Special Volume in honor of memory of S.E.Fienberg
Creator: Bocci, Christiano, Rapallo, Fabio
Date: 2019, 2019-04-12
Description: In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model...
Show moreIn this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry 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 real-data examples illustrate the use of tehse models in different fields of applications.
Show less
Collection: Journal of Algebraic Statistics

Title: Detecting epistasis via Markov bases
Creator: Malaspinas, Anna-Sapfo, Uhler, Caroline
Date: 2011, 2011
Description: Rapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on...
Show moreRapid research progress in genotyping techniques have allowed large genome-wide 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 two-stage method for detecting epistasis by combining the traditionally used single-locus 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 two-stage 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 genome-wide data set consisting of 685 dogs and identify epistasis associated with canine hair length for four pairs of single nucleotide polymorphisms (SNPs).
Show less
Collection: Journal of Algebraic Statistics

repository.iit

Search the repository

Pages

Pages

Enabled Filters

Refine Results

Type

Date

Department

Subject

Creator

Part of

Rights