GROSTAT-VI

GROSTAT VI and International School on Algebraic Statistics Menton - February 17-20, 2003
Abstracts
SHAO-WEI CHENG Geometric Isomorphism and Minimum Aberration Factorial designs have broad applications in agriculture, engineering, and scientific studies. In constructing and studying properties of factorial designs, traditional design theory treats all factors as nominal. However, this is not appropriate for experiments that involve quantitative factors. For designs with quantitative factors, level permutation of one or more factors in a design matrix could result in different geometric structures, and thus different design properties. In this paper, indicator functions are introduced to represent factorial designs. A polynomial form of indicator functions is used to characterize the geometric structure of those designs. Geometric isomorphism is defined for classifying designs with quantitative factors. Based on indicator functions, a new aberration criteria is proposed and some minimum aberration designs are presented.
ALESSANDRO DI BUCCHIANICO Umbral calculus and approximation operators (Joint work with Maria Craciun) slides We show how to use umbral calculus to compute the action of binomial approximation operators of Popoviciu type, or equivalently to compute moments of finite support discrete probability distributions. The umbral calculus is used to convert the problem to computations with linear operators acting on polynomials, in particular operators involving divided differences and generalized differentiation operators. Examples of explicit calculations will be presented. We will also briefly touch on the multivariate case and indicate a link with work by Wynn, Pistone and Riccomagno on Groebner bases and divided difference formulas.
IAN DINWOODIE Maximum Likelihood, Polynomial Systems, and Numerical Solutions An iterative method based on a fixed-point property is proposed for solving a polynomial system for maximum likelihood estimators in a model of network reliability with spatial dependence. The method is shown to converge at a geometric rate under natural conditions on data.
LUIS GARCIA Algebraic Geometry of Bayesian Networks (Joint work with Bernd Sturmfels and Mike Stillman) slides The main focus of the talk will be the algebraic classification, in terms of primary decomposition, of the independence ideals of Bayesian networks on at most five vertices The set of probability distributions that satisfy a conditional independece statement is the zero set of certain polynomials and can hence be studied using methods from algebraic geometry. We call such a set an independence variety. I will focus on a particular probabilistic model based on directed acyclic graphs known in Artificial Intelligence as Bayesian Networks. These models are the method of choice for uncertain reasoning in AI and expert systems. In this talk, I will present algebraic versions of important theorems like the Factorization theorem for Bayesian networks.
RAYMOND HEMMECKE Algebraic Algorithms for enumeration of multi-index contingency tables slides In this talk we present novel polyhedral-algebraic algorithms to count exactly the number of contingency tables with fixed given margins. The key tools necessary are Barvinok's cone decomposition of polytopes and the symbolic algebra of formal power series.
SERKAN HOSTEN Toric Initial Ideals and Disclosure Limitation slides One of the fundamental problems in disclosure limitation is to estimate lower and upper bounds for cell entries in a multidimensional contingency table with given marginals. One approach is to model this problem as an integer program: minimize (or maximize) the cell entry x over all tables with integer entries and fixed marginals. Because general integer programs are NP-complete, one looks at the linear programming relaxation where the requirement that the entries of the table be integer is dropped. However, this approach does not yield the correct answer in general. In this talk, I will present a result that says that the set of all possible marginals can be partitioned into finitely many subsets where for every marginal in the same subset the gap between the linear and integer programming solutions is constant. This partition comes from the standard pair decomposition of the associated toric initial ideal. I will also give bounds on this gap as well as show that the maximum gap is always attained.
CHRISTOS KOUKOUVINOS A comparison between Grobner bases approach and hidden projection properties in factorial designs (joint work with H. Evangelaras) Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. Plackett and Burman designs have traditionally been used for this purpose. Since these designs are only main effects plans and since the number of runs are greater than the number of active factors (main effects), there are plenty degrees of freedom unused for identifying and estimating interactions of factors. Computational Algebraic Geometry can be used to solve identi?ability problems in design of experiments in Statistics. The key idea is to view the design as a solution set of a system of non-linear polynomial equations. Then, the theory of Grobner bases allows one to identify the whole set of estimable effects (main or interactions) of the factors of the design. On the other hand, the hidden projection property approach, as introduced by Wang and Wu (1995), that deals with the same identi?cation problem, gives us a measure of how efficient our identi?cation of active effects is. In this paper we discuss the advantages and disadvantages of both methods in certain two level (fractional) factorial designs that arise from Plackett and Burman designs.
FERO MATUS On the conditional independence structures of Gaussian vectors ps file
DAVID MOND Stochastic Matrices, sandwiched simplices and the topology of the space of explanations: This is an account of joint work with Jim Smith and Duco van Straten. Given a Bayes net consisting of two observed discrete random variables and and one unobserved, we consider the space of possible distributions of the unobserved variable. We show that this space can be geometrically modelled as the space of simplices sandwiched between two nested convex polyhedra, and use this geometrical model to gain some insight into its structure and topology. The talk will give an overview of our joint paper, together with motivation for studying the topology of such spaces. It will not assume prior knowledge of topology.
MAN NGUYEN VAN MINH Finding balanced fractions in experimental designs dvi file
EVA RICCOMAGNO Algebraic Causality, the story so far (Joint work with J. Q. Smith and D. Mond) slides We discuss the possibility to use polynomial algebra for the generalisation of the notion of causality as atomic intervention as developed by Pearl (2001) to non DAG models. Results so far obtained are briefly discussed and ideas for future research are outlined.
MARIA-PIERA ROGANTIN Indicator polynomial functions for multilevel factorial designs (Joint work with Giovanni Pistone) slides The description of fractional factorial designs using polynomial representations of their indicator functions has been introduced for binary designs in Fontana, Pistone, Rogantin (2000), and generalized to replicates by Ye (2003). See also Tang, Deng (1999) for a related approach. Here we represent the levels of a factor as the elements of the multiplicative group of complex roots of unity, generalizing all the properties already known for binary designs to the case where the numbers of levels are prime numbers. In particular we read informations about the orthogonality between simple or interaction terms and the regularity from indicator function.
KENNY YE Indicator functions and their application in factorial designs Factorial designs are popular in industrial and scientific experiments. Each factorial designs can be represented by indicator functions. Polynomial forms of the indicator functions are closed related to many design criteria, in particular, aberration criteria. This relation can be applied to extend many works on regular factorial design to non-regular factorial designs. In this talk, i will discuss its applications in studying foldover design and blocked designs.
RURIKO YOSHIDA LattE: Software for effective lattice point Enumeration (with R. Hemmecke) slides We are presenting polyhedral-algebraic algorithms to count the number of contingency tables with fixed given marginals. We will describe our program LattE, the first implementation which applies Barvinok's cone decomposition and the symbolic algebra of power series to the algorithms. Barvinok's cone decomposition is the algorithm that can decompose any cone into unimodular cones. If we fix the dimension, Barvinok's cone decomposition runs in polynomial time. After we decompose each vertex cone of a given polytope into unimodular cones, we can write the formal power series and count the number of lattice points inside the given polytope. This decomposition can be used to write Hilbert-function like generating functions counting lattice points at dilations of the polytope (Ehrhart polynomials).

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