First and second order rotatability of experimental designs,moment matrices,and information surfaces |
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Authors: | N R Draper N Gaffke F Pukelsheim |
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Institution: | 1. Department of Statistics, University of Wisconsin-Madison, 53706, Madison, WI, USA 2. Institut für Mathematik, Universit?t Augsburg, Universit?tsstra?e 8, 8900, Augsburg, FRG
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Abstract: | Summary We place the well-known notion of rotatable experimental designs into the more general context of invariant design problems.
Rotatability is studied as it pertains to the experimental designs themselves, as well as to moment matrices, or to information
surfaces. The distinct aspects become visible even in the case of first order rotatability. The case of second order rotatability
then is conceptually similar, but technically more involved. Our main result is that second order rotatability may be characterized
through a finite subset of the orthogonal group, generated by sign changes, permutations, and a single reflection. This is
a great reduction compared to the usual definition of rotatability which refers to the full orthogonal group. Our analysis
is based on representing the second order terms in the regression function by a Kronecker power. We show that it is essentially
the same as using the Schl?flian powers, or the usual minimal set of second order monomials, but it allows a more transparent
calculus. |
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Keywords: | Finite groups of matrices invariant subspaces of symmetric matrices measures of rotatability orthodiagonal reflection group orthogonal group permutation group quadratic subspaces of symmetric matrices response surface designs sign change group |
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