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 Title
 On multivariable cumulant polynomial sequences with applications, 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
 Di Nardo, Elvira
 Date
 2016, 20160712
 Description

A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely...
Show moreA new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment sequences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integervalued multivariate random index is considered. Applications are given within parameter estimations, L ?evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper.
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 Journal of Algebraic Statistics
 Title
 Algebraic geometry of Poisson regression, 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
 Kahle, Thomas, Oelbermann, KaiFriederike, Schwabe, Rainer
 Date
 2016, 20160712
 Description

Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate...
Show moreDesigning experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semialgebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry.
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 Journal of Algebraic Statistics
 Title
 A linearalgebraic tool for conditional independence inference
 Creator
 Tanaka, Kentaro, Studeny, Milan, Takemura, Akimichi, Sei, Tomonari
 Date
 2015, 20151109
 Description

In this note, we propose a new linearalgebraic method for the implication problem among conditional independence statements, which is...
Show moreIn this note, we propose a new linearalgebraic method for the implication problem among conditional independence statements, which is inspired by the factorization characterization of conditional independence. First, we give a criterion in the case of a discrete strictly positive density and relate it to an earlier linearalgebraic approach. Then, we extend the method to the case of a discrete density that need not be strictly positive. Finally, we provide a computational result in the case of six variables.
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 Journal of Algebraic Statistics
 Title
 Mode Poset Probability Polytopes, 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
 Montúfar, Guido, Rauh, Johannes
 Date
 2016, 20160712
 Description

A mode of a probability distribution is an elementary event that has more probability mass than each of its direct neighbors, with respect to...
Show moreA mode of a probability distribution is an elementary event that has more probability mass than each of its direct neighbors, with respect to some vicinity structure on the set of elementary events. The mode inequalities cut out a polytope from the simplex of probability distributions. Related to this is the concept of strong modes. A strong mode is an elementary event that has more probability mass than all its direct neighbors together. The set of probability distributions with a given set of strong modes is again a polytope. We study the vertices, the facets, and the volume of such polytopes depending on the sets of (strong) modes and the vicinity structures.
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 Journal of Algebraic Statistics
 Title
 FAST AUTOMATIC BAYESIAN CUBATURE USING MATCHING KERNELS AND DESIGNS
 Creator
 Rathinavel, Jagadeeswaran
 Date
 2019, 20191220
 Publisher
 Chicago
 Description

Automatic cubatures approximate multidimensional integrals to userspecified...
Show moreAutomatic cubatures approximate multidimensional integrals to userspecified error tolerances. In many realworld integration problems, the analytical solution is either unavailable or difficult to compute. To overcome this, one can use numerical algorithms that approximately estimate the value of the integral. For high dimensional integrals, quasiMonte Carlo (QMC) methods are very popular. QMC methods are equalweight quadrature rules where the quadrature points are chosen deterministically, unlike Monte Carlo (MC) methods where the points are chosen randomly. The families of integration lattice nodes and digital nets are the most popular quadrature points used. These methods consider the integrand to be a deterministic function. An alternate approach, called Bayesian cubature, postulates the integrand to be an instance of a Gaussian stochastic process.
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 Title
 Lcumulants, Lcumulant embeddings and algebraic statistics, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
 Creator
 Zwiernik, Piotr, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and
 Description

Focusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to Lcumulants. This generalization...
Show moreFocusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to Lcumulants. This generalization keeps all the desired properties of the classical cumulants like semiinvariance and vanishing for independent blocks of random variables. These properties make Lcumulants useful for the algebraic analysis of statistical models. We illustrate this for general Markov models and hidden Markov processes in the case when the hidden process is binary. The main motivation of this work is to understand cumulantlike coordinates in alge braic statistics and to give a more insightful explanation why tree cumulants give such an elegant description of binary hidden tree models. Moreover, we argue that Lcumulants can be used in the analysis of certain classical algebraic varieties.
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 Journal of Algebraic Statistics
 Title
 Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Richards, Donald, Uhler, Caroline
 Description

We consider the lattice, L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and...
Show moreWe consider the lattice, L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, n(·), on L. We derive from the supermodularity of n(·) some generalized Fr ́echet inequalities comple menting and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from n(·), and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequal ity of Ky Fan to derive a new approach to Fr ́echet inequalities for multidimensional contingency tables.
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 Journal of Algebraic Statistics
 Title
 Maximal Length Projections in Group Algebras with Applications to Linear Rank Tests of Uniformity
 Creator
 Bargagliotti, Anna E., Orrison, Michael
 Description

Let G be a finite group, let CG be the complex group algebra of G, and let p ∈ CG. In this paper, we show how to construct submodules S of CG...
Show moreLet G be a finite group, let CG be the complex group algebra of G, and let p ∈ CG. In this paper, we show how to construct submodules S of CG of a fixed dimension with the property that the orthogonal projection of p onto S has maximal length. We then provide an example of how such submodules for the symmetric group Sn can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of n items.
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 Journal of Algebraic Statistics
 Title
 Connectivity for 3 x 3 x K contingency tables
 Creator
 Sumi, Toshio, 2012
 Description

We consider an exact sequential conditional test for threeway conditional test of no interaction. At each time τ, the test uses as the...
Show moreWe consider an exact sequential conditional test for threeway conditional test of no interaction. At each time τ, the test uses as the conditional inference frame the set F(Hτ) of all tables with the same three twoway marginal tables as the obtained table Hτ . For 3 × 3 × K tables, we propose a method to construct F(Hτ) from F(Hτ−1). This enables us to perform efficiently the sequential exact conditional test. The subset Sτ of F (Hτ ) consisting of s + Hτ − Hτ −1 for s ∈ F(Hτ−1) contains Hτ , where the operations + and − are defined elementwise. Our argument is based on the minimal Markov basis for 3 × 3 × K contingency tables and we give a minimal subset M of some Markov basis which has the property that F (Hτ ) = {s − m  s ∈ Sτ , m ∈ M}.
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 Journal of Algebraic Statistics
 Title
 Matrix Completion for the Independence Model
 Creator
 Kubjas, Kaie, Rosen, Zvi
 Description

We investigate the problem of completing partial matrices to rankone matrices in the standard simplex ∆mn−1. The motivation for studying this...
Show moreWe investigate the problem of completing partial matrices to rankone matrices in the standard simplex ∆mn−1. The motivation for studying this problem comes from statistics: A lack of eligible completion can provide a falsification test for partial observations to come from the independence model. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists. We also describe the set of valid completions, and we optimize over this set.
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 Journal of Algebraic Statistics
 Title
 One example of general unidentifiable tensors
 Creator
 Chiantini, Luca, Mella, Massimiliano, Ottaviani, Giorgio
 Description

Abstract. Theidentifiabilityofparametersinaprobabilisticmodelisacrucialnotioninstatistical inference. We prove that a general tensor of rank 8...
Show moreAbstract. Theidentifiabilityofparametersinaprobabilisticmodelisacrucialnotioninstatistical inference. We prove that a general tensor of rank 8 in C3 ⊗ C6 ⊗ C6 has at least 6 decompositions as sum of simple tensors, so it is not 8identifiable. This is the highest known example of balanced tensors of dimension 3, which are not kidentifiable, when k is smaller than the generic rank.
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 Journal of Algebraic Statistics
 Title
 Hilbert Polynomial of the Kimura 3Parameter Model, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
 Creator
 Kubjas, Kaie, inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July
 Description

In [2] Buczyn ́ska and Wi ́sniewski showed that the Hilbert polynomial of the algebraic variety associated to the JukesCantor binary model on...
Show moreIn [2] Buczyn ́ska and Wi ́sniewski showed that the Hilbert polynomial of the algebraic variety associated to the JukesCantor binary model on a trivalent tree depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other groupbased models. The JukesCantor binary model has Z2 as the underlying group. We consider the Kimura 3parameter model with Z2 × Z2 as the underlying group. We show that the generalization of the statement about the Hilbert polynomials to the Kimura 3parameter model is not possible as the Hilbert polynomial depends on the shape of a trivalent tree.
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 Journal of Algebraic Statistics
 Title
 AN ACCELERATING COUETTE FLOW IN NEK5000: APPLICATIONS IN OCEANOGRAPHY AND MAGNETOHYDRODYNAMICS
 Creator
 Miksis, Zachary M.
 Date
 2017, 201705
 Description

Nek5000 is a highly scalable spectral element code used in a broad array of problems in computational fluid dynamics. In this thesis, we focus...
Show moreNek5000 is a highly scalable spectral element code used in a broad array of problems in computational fluid dynamics. In this thesis, we focus on applying the code to a model problem of an accelerating Couette flow, or a hydrodynamic flow between two plates, of which the top plate is accelerating and the bottom plate is stationary, and verifying the numerical methods as applied to this problem. We obtain an analytical solution to the hydrodynamic flow problem, and use this to analyze the effects of changing time step length, the size of the computational mesh, and the computational polynomial order on the accuracy and stability of Nek5000. Additionally, we discuss the addition of an applied magnetic field to the hydrodynamic Couette flow, and provide a formulation for an exact solution to this magnetohydrodynamic problem that can be used to further verify Nek5000 in a similar fashion to the hydrodynamic problem.
M.S. In Applied Mathematics, May 2017
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 Title
 RUNTIME FOR PERFORMING EXACT TESTS ON THE PI STATISTICAL MODEL FOR RANDOM GRAPHS
 Creator
 Dillon, Martin
 Date
 2016, 201605
 Description

In statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to...
Show moreIn statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to perform exact testing for the p1 random graph model. By comparing it to the simple switch Markov chain, we prove that it mixes rapidly on many classes of degree sequences, and we discuss why it is sometimes better suited than the simple switch chain, and try to easily introduce the concepts from the general theory along the way.
M.S. in Applied Mathematics, May 2016
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 Title
 A Gröbner Basis Database
 Creator
 Mojsilović, Jelena
 Date
 2022
 Title
 Stephen Fienberg's influence on algebraic statistics, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Petrović, Sonja, Slavkovic, Aleksandra, Yoshida, Ruriko
 Date
 2019, 20190412
 Description

Stephen (Steve) E. Fienberg (19422016) was an eminent statistician, whose impact on research, education and the practice of statistics, and...
Show moreStephen (Steve) E. Fienberg (19422016) was an eminent statistician, whose impact on research, education and the practice of statistics, and many other fields is astonishing in its breadth. He was a visionary when it came to linking many different areas to address real scientific issues. He professed the importance of statistics in many disciplines, but recognized that true interdisciplinary work requires joining of the expertise across different areas, and it is in this spirit that he helped steer algebraic statistics toward becoming a thriving subject. Many of his favorite topics in the area are covered in this special issue. We are grateful to all authors for contributing to this volume to honor him and his influence on the field. During the preparation of this issue, we learned about the tragic killing of his widow, Joyce Fienberg, during the Tree of Life Synagogue massacre in Pittsburgh, PA on October 27, 2018. This issue is dedicated to their memory.
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 Journal of Algebraic Statistics
 Title
 Algorithms for Discrete Data in Statistics and Operations Research
 Creator
 Schwartz, William K.
 Date
 20211119, 202112
 Publisher
 ProQuest, https://www.proquest.com/docview/2622985712
 Description

Sponsorship: The Air Force Office of Scientific Research's grant FA95501410141 supported Prof. Petrović's and my initial work on this project.