IRREGULAR ARRAYS AND RANDOMIZATION

Although they lie at the conceptual core of a wide range of scientific questions, the notions of irregular or ''random'' arrangement and the process of randomization itself have never been unambiguously defined . Algorithmic implementation of these concepts requires a combinatoria l, rather than, a probability-theoretic, formulation. We introduce vec tor versions of approximate entropy to quantify the degrees of irregul arity of planar (and higher dimensional) arrangements. Selection rules , applied to the elements of irregular permutations, define randomizat ion in strictly combinatorial terms. These concepts are developed in t he context of Latin square arrangements and valid randomization of the m. Conflicts and tradeoffs between the objectives of irregular arrange ments and valid randomization are highlighted. Extensions to broad cla sses of designs, and a diverse range of scientific applications are in dicated, including lattice-based models in physics and signal detectio n in seismology and physiology.