Irregular arrays and randomization
- *Office of Population Research, Princeton University, Princeton, NJ 08544; and ‡990 Moose Hill Road, Guilford, CT 06437
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Contributed by Burton H. Singer
Abstract
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.
Footnotes
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↵ † To whom reprint requests should be addressed. e-mail: singer{at}opr.princeton.edu.
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↵ § In the core text, we exclude blocks that exceed the boundaries from all calculations, except as noted; a wrap-around version of ApEn will be developed and considerably used in the upcoming paper, “A Recipe for Randomness.”
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↵ ¶ We use conventional lattice array indexing in our (incremental) vector notation.
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‖ Recall that mutually orthogonal Latin squares of order 6 do not exist.
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↵ ** A strict KV Latin square satisfies the definition given in Example 2. A Latin square is called weak KV when all cells with the same symbol can be traversed by knight’s moves without visiting cells with other symbols, where rows and columns are both considered to form endless cyclic sequences with first row (or column) following the last, i.e., in a wraparound or toroidal manner.
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↵ ‡‡ A full exposition by the authors, “Irregular Permutations,” will be published elsewhere.
- ABBREVIATIONS:
- ApEn,
- approximate entropy;
- KV,
- Knut-Vik;
- CMOS,
- complete mutually orthogonal system;
- FY,
- Fisher and Yates
- Copyright © 1998, The National Academy of Sciences
