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 combinatorial, rather than a probability-theoretic, formulation. We introduce vector versions of approximate entropy to quantify the degrees of irregularity of planar (and higher dimensional) arrangements. Selection rules, applied to the elements of irregular permutations, define randomization in strictly combinatorial terms. These concepts are developed in the context of Latin square arrangements and valid randomization of them. Conflicts and tradeoffs between the objectives of irregular arrangements and valid randomization are highlighted. Extensions to broad classes of designs, and a diverse range of scientific applications are indicated, including lattice-based models in physics and signal detection in seismology and physiology.