By Rahul Mukerjee, C.F. J. Wu
The final two decades have witnessed an important development of curiosity in optimum factorial designs, less than attainable version uncertainty, through the minimal aberration and comparable standards. This e-book supplies, for the 1st time in publication shape, a complete and updated account of this contemporary idea. Many significant periods of designs are lined within the ebook. whereas protecting a excessive point of mathematical rigor, it additionally presents vast layout tables for examine and useful reasons. except being helpful to researchers and practitioners, the e-book can shape the middle of a graduate point path in experimental layout.
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Paperback: 246 pages
writer: Cambridge collage Press (September 19, 2005)
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Extra info for A Modern Theory of Factorial Designs
24) Consider now the pencils b∗ = (0, . . , 0, −bf +1 , . . , −bp , 0, . . 25) where 0 appears in the ﬁrst f and the last n − p positions of b∗ . 25), (b − b∗ ) = bdef ∈ R(B), so that by deﬁnition, the pencils b and b∗ are aliased with each other. 25), the number of nonzero entries in b∗ is p − f , which is at most t, since p ≤ f + t. Consequently, (ii) above is violated. Thus every deﬁning pencil must have at least f + t + 1 nonzero entries, which proves the “only if” part. 3, the behavior of an sn−k design depends on the numbers of nonzero entries in the deﬁning pencils and, in particular, on the minimum of these numbers.
Continuing the previous interpretation, the minimum distance of a linear code C is mathematically equivalent to the resolution of the corresponding sn−k design. 5. The concept of minimum distance plays an equally important role in coding theory, since it determines the capability of error correction of a code. 3). 2) because pencils with proportional entries are identical. Recall that the MA criterion was deﬁned on the basis of the wordlength pattern and that its importance was justiﬁed by the eﬀect hierarchy principle.
17), the result is evident. 1. 1, are called corresponding contrasts. The lemma shows that corresponding contrasts belonging to pencils that are aliased with each other cannot be distinguished on the basis of the design d(B). In this sense, such pencils are said to be confounded with each other. 1, does not belong to R(B) and B has full row rank, it is clear that the (k + 1) × n matrix [b, B ] also has full row rank. 16) has cardinality sn−k−1 for each j. 18), L(B) (or, equivalently, L∗ (B)) itself is a contrast involving the treatment combinations included in d(B).
A Modern Theory of Factorial Designs by Rahul Mukerjee, C.F. J. Wu