Title: Circuit bases in optimal experimental design and randomization
Authors: Henry Wynn - London School of Economics (United Kingdom) [presenting]
Fabio Rapallo - University of Genova (Italy)
Roberto Fontana - Politecnico di Torino (Italy)
Abstract: Circuits are a construction used extensively in integer programming, numerical linear algebra and the application of toric ideals in algebraic statistics. Recent work is brought together. Circuits have been previously used to construct small sample optimally robust factorial designs and to construct randomisation schemes to de-bias the analysis of factorial experiments. The Binet-Cauchy theorem has been used alongside the circuit theory to construct robust designs. A subset of binary (0-1) circuits turned out to provide the set of generators, under set partition theory, for all valid randomization schemes. In both cases, the circuits are for the kernel $K$ of the usual $X$-matrix of the model/design pair, namely the residual space. At least one connection is that the extent of the available randomisation schemes depends on the extent of symmetry in the design, which is also a feature of optimal design. Heuristically: good designs lead to versatility in randomization. The complexity of the circuit structure even for small problems requires computer algebra. The package 4ti2 is suggested. The methods are applied to some standard and new examples, revealing deep combinatorial structures.