Betti Numbers, Models and Experimental Designs
Henry P. Wynn1, Hugo Maruri-Aguillar2, Eduardo Saenz de Cabezon3
1Statistics, London School of Economics, London, United Kingdom; 2Mathematics, Queen Mary University of London, United Kingdom; 3Matematicas y Computacion, Universidad de la Rioja, Logrono, Rioja, Spain

Polynomial regression models with a hierarchical structure give rise to a monomial ideal generatied by all monomial terms not in the model. The complexity of this ideal gives information about the model. Roughly speaking if the model is simple in having only a few interactions terms the ideal is complex. In particular the Betti numbers (simple, graded and multi-graded) can be studied. In ideal theory an important theorem due to Bigatti (1993) gives a construction when the Betti numbers are maximal: the lex-segment method. The problem addressed here is to determine which statistical models give this case, with intuition saying that such models are “flat” in the sense of not having large discepancies in the degree of their terms. In turns out that this intuition is partly confirmed. There is a class of models called “flat” models where the monomial terms in the models and those outside are separated by a hyperplane in the space of non-negative integer vectors (being the monomial exponents). Such models are closely related to the “flat”, the convex hull of vectors describing the average degree of the model. Indeed, corner models models give the lower vertices. It is shown (i) that every lex-segment construction gives a lower boundary point of the (sub) state polytope of models with the same Hilbert function (ii) that there are situations where the lex-segment construction gives models corresponding to verticesof the (global) state-polytope. Given an experimental design a natural class of models can be found via the algebraic (G-basis) method. Linking this with the above theory we can exhibit Betti numbers and maximal Betti numbers for actual designs. A thorough analysis is carried out for some regular and irregular two-level design including the Plackett Burman design PB8 and PB12.

Keywords: Betti numbers; Polynomial regression; Experimental design; Interactions

Biography: Henry Wynn is Professor Emeritus at the London School of Economics (LSE). Following an MA in mathematics at the University of Oxford and a PhD at Imperial College London, in 1970. He joined Imperial College in 1972 as a Lecturer and later became Reader. He took the Chair in Statistics at City University in 1985 and then was appointed Professor of Industrial Statistics at the University of Warwick and finally a Chair in Statistics at LSE in 2003. He is a Fellow of the IMS, holds the Guy Medal in Silver of the Royal Statístical Society and is an Honorary Fellow of the Institute of Actuaries. He has published widely in statistics, mathematics, optimisation and engineering journals, has a number of monographs and edited volumes to his name and has directed well-funded research centres. Current main interests are: experimental design, algebraic statistics, risk theory and robust engineering design.