Algebraic geometry of Poisson regression

Abstract

Abstract. Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semi-algebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry.

On the Feasibility of Semi-algebraic Sets in Poisson Regression

“…For brevity we omit any details of statistical theory and focus on mathematical and computational problems. The interested reader should consult [12] and its references. We also stick to that paper's notation.…”

“…This case is particularly well-behaved, well-studied, and relevant for practitioners. It was investigated in depth in [7,8,9,12]. The pairwise interaction model arises for d = 2, where…”

“…A special design, relevant for practitioners and studied in [12], is the corner design w * k,d . It is the saturated design with equal weights w x = 1/p for all x ∈ {0, 1} k with |x| 1 ≤ d. For example, for k = 3 rules and interaction order d = 2 the regression function is f (…”

“…For example, the inequalities corresponding to the corner design are the topic of [12]. It can be seen that there always exist parameters β 1 , .…”

“…In practice, this means that appropriate parameters have to be guessed a priori, or fixed by other means. In [12] the authors approached this problem from a global perspective. They study the regions of optimality of fixed designs and demonstrate that these are often defined by semi-algebraic constraints.…”