On third order rotatability

Abstract

Bi-axial rotations, finite matrix groups, information surfaces, invariant subspaces of symmetric matrices, orthogonal groups, permutation groups, response surface designs, tensor representations of classical matrix groups,

On optimal third order rotatable designs

“…However, the associated technicalities are not identical, due to the fact that the moment matrices MK, Ms, Mull of the Kronecker, Schliiflian, or Box-Hunter representations differ. It is precisely because of these technical differences that the eigenvalue decomposition of the Kronecker representation moment matrix MK takes an almost explicit form, as in (3.27) of Draper and Pukelsheim (1994). The same could also be achieved under the Sehl/iflian calculus, but we find the details less transparent.…”

“…The moment matrix of boundary nucleus designs is given by equation (3.27) in Draper and Pukelsheim (1994). From this we can obtain the eigenvalues as functions of the weight c~ and the radius r, for dimension m _> 3, as follows: …”

“…Theorem 4.1 and Corollary 4.2 of Draper and Pukelsheim (1994) show that any design, rotatable or not, can be improved upon by an appropriate boundary nucleus design, with respect to a wide class of optimality criteria. Furthermore, Theorem 4.1 implies that the boundary nucleus designs form a complete class under the Loewner matrix ordering, while Lemma 2 of Heiligers and Schneider (1992) then entails that this is actually a minimal complete class.…”

“…Draper and Pukelsheim (1994) choose a Kronecker representation because rotatability of any order is then easily handled using the methods of multilinear algebra. Although the Kronecker representation is redundant, in that it repeats such terms as mixed products, the redundancy is of the same quality as in a dispersion matrix: We preferably think of the latter as a matrix, albeit symmetric, rather than replacing it through the set of functionally independent terms on and above the diagonal.…”

“…For a detailed discussion of this issue see Section 8.18 in Pukelsheim (1993). Draper and Pukelsheim (1994) discuss the pattern of the moment matrices associated with third order designs under the Kronecker representation. A conclusion emerging from that work is that a complete class of designs that provides third order rotatability consists of the family of boundary nucleus designs, reviewed in Section 2.…”