矩阵方程与Cramer法则的推广

本文讨论了矩阵方程AXB=C(A,B可逆)的用行列式表示的求解公式,然后指出它是Cramer法则的重要推广.     

代数学中的同构与同态思想

以线性空间为模型刻划了代数学中的同构与同态思想。     

The structure of norms and relations in patronage systems

Patronage systems are composed of a social relation (between patron and client), the norms associated with this relation (support, defend, not attack), and network-formation rules (governing the creation of patronage ties). Actors in a patronage system face moral dilemmas when they become obligated to fight on both sides of a conflict, and patronage systems are normatively consistent when these dilemmas can never arise. This paper examines the structure of normatively consistent patronage systems, exploring the logical relationships among norms and between norms and network structure. The paper illustrates how sociologically relevant concerns from deontic logic can be incorporated into network analysis using algebraic methods familiar to network analysts.     

Indicator function and complex coding for mixed fractional factorial designs

In a general fractional factorial design, the n levels of a factor are coded by the nth roots of the unity. This device allows a full generalization to mixed-level designs of the theory of the polynomial indicator function which has already been introduced for two-level designs in a joint paper with Fontana. The properties of orthogonal arrays and regular fractions are discussed.     

Algebraic statistics of level codings for fractional factorial designs

We discuss the applications of algebraic statistics to fractional factorial design with special emphasis on the choice of level coding. In particular, we deal with the theory of Bayley's level codings in that framework.     

Estimation of exponential-polynomial distribution by holonomic gradient descent

ABSTRACT

We study the holonomic gradient decent for maximum likelihood estimation of exponential-polynomial distribution, whose density is the exponential function of a polynomial in the random variable. We first consider the case that the support of the distribution is the set of positive reals. We show that the maximum likelihood estimate (MLE) can be easily computed by the holonomic gradient descent, even though the normalizing constant of this family does not have a closed-form expression, and discuss the determination of the degree of the polynomial based on the score test statistic. Then, we present extensions to the whole real line and to the bivariate distribution on the positive orthant.     

Optimal and efficient semi-Latin squares

An (n×n)/k$(n\times n)/k$semi-Latin square is an n×n square array in which nk distinct symbols (representing treatments) are placed in such a way that there are exactly k   symbols in each cell (row–column intersection) and each symbol occurs once in each row and once in each column. Semi-Latin squares form a class of row–column designs generalising Latin squares, and have applications in areas including the design of agricultural experiments, consumer testing, and via their duals, human–machine interaction. In the present paper, new theoretical and computational methods are developed to determine optimal or efficient (n×n)/k$(n\times n)/k$ semi-Latin squares for values of n and k for which such semi-Latin squares were previously unknown. The concept of subsquares of uniform semi-Latin squares is studied, new applications of the DESIGN package for GAP are developed, and exact algebraic computational techniques for comparing efficiency measures of binary equireplicate block designs are described. Applications include the complete enumeration of the (4×4)/k$(4\times 4)/k$ semi-Latin squares for k=2,…,10$k=2,\dots ,10$, and the determination of those that are A-, D-, and E-optimal, the construction of efficient (6×6)/k$(6\times 6)/k$ semi-Latin squares for k=4,5,6$k=4,5,6$, and counterexamples to a long-standing conjecture of R.A. Bailey and to a similar conjecture of D. Bedford and R.M. Whitaker.     

Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation

The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but in fact it extends back at least into the 19th century. Moreover, it remains an active area of research today. In this paper we give an overview of this history focussing on the development of log-linear models and their estimation via the method of maximum likelihood. Roy played a crucial role in this development with two papers co-authored with his students, Mitra and Marvin Kastenbaum, at roughly the mid-point temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for log-linear models. Understanding the problem of non-existence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem.     

Fractional Factorial Designs for Model-Based Evaluation of Customer Preferences

In this article, we explore the connection between Conjoint Analysis (CA) and a recent theory for minimum size orthogonal fractional factorial design generation (Fontana, 2013 Fontana , R. ( 2013 ). Algebraic generation of minimum size orthogonal fractional factorial designs: an approach based on integer linear programming . Computat. Statist. 28 : 241 – 253 .[Crossref], [Web of Science ®] , [Google Scholar]).

We show how searching for a minimum size OFFD that satisfies a set of constraints, expressed in terms of orthogonality between simple and interaction effects, is equivalent to solving an integer linear programming problem. It is worth noting that the methodology puts no restriction either on the number of levels of each factor or on the orthogonality constraints and so it can be applied to a very wide range of designs, including mixed orthogonal arrays. An algorithm, that has been implemented in SAS/IML, is briefly described.

The use of this algorithm during the design stage of a generic CA is shown in two applications.