Lattice points, contingency tables, and sampling

Chen, Yuguo; Dinwoodie, Ian H.; Dobra, Adrian; Huber, Mark

doi:10.1090/conm/374/06899

Cited by 20 publications

(23 citation statements)

References 25 publications

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“…However this constraint may be sometimes too strict in practice. As discussed in [4], the connectivity under the assumption means that, if entries in columns in which response marginals are zeros are allowed to drop down to −1, any two tables with zero response marginals are connected by the set of moves proposed here. Hence it is possible to implement MCMC theoretically.…”

Section: Discussionmentioning

confidence: 99%

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On connectivity of fibers with positive marginals in multiple logistic regression

Hara

Takemura

Yoshida

2010

Journal of Multivariate Analysis

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a b s t r a c tIn this paper we consider exact tests of a multiple logistic regression with categorical covariates via Markov bases. In many applications 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.

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Section: Discussionmentioning

confidence: 99%

“…and let A be defined as in (4). Then the configuration for the bivariate logistic regression model is the Lawrence lifting of…”

Section: Connectivity Of Fibers Of Positive Marginals In Bivariate Lomentioning

confidence: 99%

On connectivity of fibers with positive marginals in multiple logistic regression

Hara

Takemura

Yoshida

2010

Journal of Multivariate Analysis

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“…Such rational function solutions to LDS are of great importance in many applications. For example, they can be used to prove theorems in number theory and combinatorics [4,5,50], compute volumes [16], count integer points in polyhedra [17], to maximize non-linear functions over lattice points in polyhedral [32], to compute Pareto optima in multi-criteria optimization [30], to integrate and sum functions over polyhedra [12], to compute Gröbner bases of toric ideals [29], to perform various operations on rational functions and quasipolynomials [15], and to sample objects from polyhedra [52], from combinatorial families [35] and from statistical distributions [28]. We recommend the textbooks [13,20,31] for an introduction.Note that, while above rfsLDS is stated in terms of pure inequality systems, this restriction is not essential.…”

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confidence: 99%

Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

Breuer

Zafeirakopoulos

2017

Ann. Comb.

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Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.1 arXiv:1501.07773v1 [math.CO] 30 Jan 2015In general, we represent a set S ⊂ Z n 0 of non-negative integer vectors by the multivariate generating functionIt is a well-known fact that when S is the set of solutions to a linear Diophantine system, then φ S is always a rational function ρ S . It is this rational function ρ S that we seek to compute when solving a given linear Diophantine system. We are not going to be interested in the normal form of ρ S , though, since the normal form may be unnecessarily large. Take for example the system x 1 + x 2 = 100. Here, the set of solutions is S = {(100, 0), (99, 1), . . . , (0, 100)} which can be represented by the rational functionNote that the polynomial on the right-hand side is the normal form of this rational function, which has 101 terms. (In fact, the normal form of a rational function representing a finite set will always be a polynomial.) The rational function expression on the left-hand side is not in normal form since the denominator divides the numerator, however it is much shorter, having only 4 terms. We are therefore interested in computing multivariate rational function expressions, which are not uniquely determined, as opposed to the unique rational function in normal form. Given this terminology and notation, we can now state the computational problem of solving linear Diophantine systems that this article is about. Problem 1.1. Rational Function Solution of Linear Diophantine System (rfsLDS)Output: An expression for the rational function ρ ∈ Q(z 1 , . . . , z n ) representing the set of all non-negative integer vectors x ∈ Z n 0 such that Ax b. Such rational function solutions to LDS are of great importance in many applications. For example, they can be used to prove theorems in number theory and combinatorics [4,5,50...

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“…This idea of going outside the original fiber that has only nonnegative tables, has been used in cases where Markov bases are difficult to compute for the original fiber ℱ 0 but easier for an enlarged fiber ℱ 1 by using the Lawrence lifting. For example, Chen et al54 use this technique to compute Markov bases for the Hardy‐Weinberg problem and logistic regression. However, since ℱ 1 is larger than ℱ 0 , the running times of Markov chain can be longer.…”

Section: Computing Markov Basesmentioning

confidence: 99%

Conditional inference given partial information in contingency tables using Markov bases

Karwa

Slavković

2013

WIREs Computational Stats

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In this article, we review a Markov chain Monte Carlo (MCMC) algorithm for performing conditional inference in contingency tables in the presence of partial information using Markov bases, a key tool arising from the area known as algebraic statistics. We review applications of this algorithm to the problems of conditional exact tests, ecological inference, and disclosure limitation and illustrate how these problems fall naturally in the setting of inference with partial information. We also discuss some issues associated with computing Markov bases which are needed as an input to the algorithm. WIREs Comput Stat 2013, 5:207–218. doi: 10.1002/wics.1256 This article is categorized under: Data: Types and Structure > Traditional Statistical Data

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Lattice points, contingency tables, and sampling

Cited by 20 publications

References 25 publications

On connectivity of fibers with positive marginals in multiple logistic regression

On connectivity of fibers with positive marginals in multiple logistic regression

Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

Conditional inference given partial information in contingency tables using Markov bases

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