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.