Results 1  10
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11
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 183 (15 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
Variation of Cost Functions in Integer Programming
 MATHEMATICAL PROGRAMMING
, 1994
"... We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classe ..."
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Cited by 42 (8 self)
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We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Gröbner bases associated with A. The union of the reduced Gröbner bases as c varies (called the universal Gröbner basis) consists precisely of the edge directions of St(A). We present geometric algorithms for computing St(A), the Graver basis [Gra], and the universal Gröbner basis.
Gröbner Bases And Triangulations Of The Second Hypersimplex
, 1994
"... The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex \Delta(2; n). We present a quadratic Gröbner basis for the associated toric ideal I(Kn ). The simplices in the resulting triangulation of \Delta(2; n) have unit volume, and they are indexed by subg ..."
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Cited by 18 (2 self)
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The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex \Delta(2; n). We present a quadratic Gröbner basis for the associated toric ideal I(Kn ). The simplices in the resulting triangulation of \Delta(2; n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding of Kn . For n 6 the number of distinct initial ideals of I(Kn ) exceeds the number of regular triangulations of \Delta(2; n); more precisely, the secondary polytope of \Delta(2; n) equals the state polytope of I(Kn ) for n 5 but not for n 6. We also construct a nonregular triangulation of \Delta(2; n) for n 9. We determine an explicit universal Gröbner basis of I(Kn ) for n 8. Potential applications in combinatorial optimization and random generation of graphs are indicated.
A Primal AllInteger Algorithm Based on Irreducible Solutions
, 2001
"... This paper introduces an exact primal augmentation algorithm for solving general linear integer programs. The algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities. We prove that various versions ..."
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Cited by 11 (5 self)
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This paper introduces an exact primal augmentation algorithm for solving general linear integer programs. The algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities. We prove that various versions of our algorithm are finite. It is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively. An implementation of the presented algorithms is given. Computational results for a number of hard 0/1 integer programs from the MIPLIB demonstrate the practical power of the method.
Model Fitting and Testing for NonGaussian Data with Large Data Sets
, 1996
"... We consider the application of the smoothing spline to the generalized linear model in large data set situations. First we derive a Generalized Approximate Cross Validation function (GACV ), which is an approximate leaveoutone cross validation function used to choose smoothing parameters. In order ..."
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Cited by 5 (2 self)
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We consider the application of the smoothing spline to the generalized linear model in large data set situations. First we derive a Generalized Approximate Cross Validation function (GACV ), which is an approximate leaveoutone cross validation function used to choose smoothing parameters. In order to apply the GACV function to a large data set situation, we propose a corresponding randomized version of it. To reduce the computational intensity of calculating the smoothing spline estimate, we suggest an approximate solution and a clustering method to choose a subset of the basis functions. Combining randomized GACV with this approximate solution, we apply it to binary response data from the Wisconsin Epidemiological Study of Diabetic Retinopathy in order to establish the accuracy of the model when applied to a large data set. iii Contents Acknowledgements i Abstract ii 1 Introduction 1 1.1 Smoothing Spline for Generalized Linear Model : : : : : : : : : : : : : 2 1.2 The Problem : :...
Lattice Points, Contingency Tables, and Sampling
 In Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization
, 2004
"... Markov chains and sequential importance sampling (SIS) are described as two leading sampling methods for Monte Carlo computations in exact conditional inference on discrete data in contingency tables. Examples are explained from genotype data analysis, graphical models, and logistic regression. ..."
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Cited by 5 (2 self)
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Markov chains and sequential importance sampling (SIS) are described as two leading sampling methods for Monte Carlo computations in exact conditional inference on discrete data in contingency tables. Examples are explained from genotype data analysis, graphical models, and logistic regression.
Test Sets of the Knapsack Problem and Simultaneous Diophantine Approximation
, 1997
"... This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by ..."
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Cited by 2 (1 self)
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This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the corresponding cone the integer analogue of Caratheodory's theorem applies when the numbers are divisible. We show that the elements of the minimal Hilbert basis of the dual cone of all minimal integral solutions of a linear diophantine equation yield best approximations of a rational vector "from above". A recursive algorithm for computing this Hilbert basis is discussed. We also outline an algorithm for determining a Hilbert basis of a family of cones associated with the knapsack problem.
Testing the Generalized Linear Model Null Hypothesis versus `Smooth' Alternatives
, 1995
"... We consider y i ; i = 1; :::n independent observations from an exponential family with canonical parameter j(x i ), where the predictor variable x is in some index set and j is a `smooth' function of x. The usual GLIM models suppose that j has a parametric form j(x) = P p =1 fi OE (x) where the O ..."
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Cited by 2 (1 self)
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We consider y i ; i = 1; :::n independent observations from an exponential family with canonical parameter j(x i ), where the predictor variable x is in some index set and j is a `smooth' function of x. The usual GLIM models suppose that j has a parametric form j(x) = P p =1 fi OE (x) where the OE are given. This paper is concerned with testing the hypothesis that j is in the span of a given (low dimensional) set of OE versus general `smooth' alternatives. In the Gaussian case, studied by Cox, Koh, Wahba and Yandell(1988), test statistics are available whose distributions are independent of the nuisance fi , whereas in general this is not the case. We propose a symmetrized KullbackLeibler (SKL) distance test statistic, based on comparing a smoothing spline (penalized likelihood) fit and a GLIM fit, for testing the hypothesis j `parametric' vs j `smooth', in the nonGaussian situation. The spline fit uses a smoothing parameter obtained from the data via either the unbiased risk ...
Hilbert bases of cones related to simultaneous Diophantine approximations and linear Diophantine equations
, 1997
"... This paper investigates properties of the minimal integral solutions of a linear diophantine equation. We present best possible inequalities that must be satisfied by these elements which improves on former results. We also show that the elements of the minimal Hilbert basis of the dual cone of all ..."
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Cited by 2 (1 self)
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This paper investigates properties of the minimal integral solutions of a linear diophantine equation. We present best possible inequalities that must be satisfied by these elements which improves on former results. We also show that the elements of the minimal Hilbert basis of the dual cone of all minimal integral solutions of a linear diophantine equation yield best approximations of a rational vector "from above". Relations between these cones are applied to the knapsack problem.