Polynomials for classification trees and applications

This paper relates computational commutative algebra to tree classification with binary covariates. With a single classification variable, properties of uniqueness of a tree polynomial are established. In a binary multivariate context, it is shown how trees for many response variables can be made into a single ideal of polynomials for computations. Finally, a new sequential algorithm is proposed for uniform conditional sampling. The algorithm combines the lexicographic Groebner basis with importance sampling and it can be used for conditional comparisons of regulatory network maps. The binary state space leads to an explicit form for the design ideal, which leads to useful radical and extension properties that play a role in the algorithms.     

Quantifying the effectiveness of VMI and integrated inventory management in a supply chain with uncertain lead-times and uncertain demands

This article considers the inventory management problem in a supply chain with uncertain replenishment lead-times and uncertain demands. The optimal integrated inventory management (IIM) policy is developed using stochastic dynamic programming theory. The IIM policy is contrasted with two pull-type vendor-managed inventory policies (VMI-1 and VMI-2) and a traditional retailer-managed inventory policy (RMI). Computational results show that in such stochastic supply chains, IIM performs about 23, 15, and 3% better than the optimised RMI, VMI-1 and VMI-2 policies, respectively, while two VMI policies are about 8 and 20% better than the best RMI. The basestock-based VMI-2 is a very good form of VMI. The ANOVA analysis reveals that the replenishment lead-times have the largest effect on the relative performance between IIM and other policies. Numerical examples demonstrated that the IIM policy has good structural properties and can be characterised by a set of switching curves.     

Transform methods for the hypergeometric distribution

Two new methods for computing with hypergeometric distributions on lattice points are presented. One uses Fourier analysis, and the other uses Gröbner bases in the Weyl algebra. Both are very general and apply to log-linear models that are graphical or non-graphical.     

Algebraic Methods for Polynomial Statistical Models

We describe applications of computational algebra to statistical problems of parameter identifiability, sufficiency, and estimation. The methods work for a family of statistical models that includes Poisson and binomial examples in network tomography.     

Sequential importance sampling of binary sequences

Two sequential methods are described for sampling constrained binary sequences from partial solutions. The backward method computes elimination ideals over finite fields and constructs partial solutions that extend. The forward method uses numerical global optimization to determine which partial solutions extend. The methods are applied to restricted orderings, binary dynamics, and random graphs.