Normal Binary Hierarchical Models

Unimodular binary hierarchical models

“…See [12], Theorem 19.2 for the equivalence of (1) and (2). We can see that condition (3) of Definition 1.1 is equivalent to condition (2) because the circuits can be computed via a determinantal formula using Cramer's rule [14, p. 35].…”

“…The first is that the set of matroids underlying the design matrices of hierarchical models is not closed under the operations of direct sum, two sum, three sum, deletion, nor contraction. The second is that normality, unlike (total) unimodularity, is not a matroidal property and we are interested in extending our methods to the normal classification problem [2].…”

“…However, selecting sufficiently large random subsets of the columns produced Graver basis elements of the desired form. Computations are available on our website [3]. For the infinite family ∂∆ n ⊔ {v} where n ≥ 1, examples of a non-squarefree Graver basis element appear in [15].…”

“…Let C be the simplicial complex on ground set [3] = {1, 2, 3} with facet(C) = {{1}, {2, 3}}. The linear transformation π C,d maps a three-way tensor u = (u ijk :…”

“…By Table 1 in [9], (1) is not unimodular. Computing a Graver basis with 4ti2 [1] shows that (2) and (4) are not unimodular. Computations are available on our website [3].…”