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Register a new userOn construction of optimal mixed-level supersaturated designs
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Added by
Fasheng Sun
Publication date
2011
fields
Mathematical Statistics
and research's language is
English
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Supersaturated design (SSD) has received much recent interest because of its potential in factor screening experiments. In this paper, we provide equivalent conditions for two columns to be fully aliased and consequently propose methods for constructing $E(f_{mathrm{NOD}})$- and $chi^2$-optimal mixed-level SSDs without fully aliased columns, via equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs have been obtained. Furthermore, it is proved that the nonorthogonality between columns of the resulting design is well controlled by the source designs. A rather complete list of newly generated optimal mixed-level SSDs are tabulated for practical use.
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We develop $D$-optimal designs for linear models with first-order interactions on a subset of the $2^K$ full factorial design region, when both the number of factors set to the higher level and the number of factors set to the lower level are simultaneously bounded by the same threshold. It turns out that in the case of narrow margins the optimal design is concentrated only on those design points, for which either the threshold is attained or the numbers of high and low levels are as equal as possible. In the case of wider margins the settings are more spread and the resulting optimal designs are as efficient as a full factorial design. These findings also apply to other optimality criteria.
In this paper, we propose two simple yet efficient computational algorithms to obtain approximate optimal designs for multi-dimensional linear regression on a large variety of design spaces. We focus on the two commonly used optimal criteria, $D$- and $A$-optimal criteria. For $D$-optimality, we provide an alternative proof for the monotonic convergence for $D$-optimal criterion and propose an efficient computational algorithm to obtain the approximate $D$-optimal design. We further show that the proposed algorithm converges to the $D$-optimal design, and then prove that the approximate $D$-optimal design converges to the continuous $D$-optimal design under certain conditions. For $A$-optimality, we provide an efficient algorithm to obtain approximate $A$-optimal design and conjecture the monotonicity of the proposed algorithm. Numerical comparisons suggest that the proposed algorithms perform well and they are comparable or superior to some existing algorithms.
In this note we consider the optimal design problem for estimating the slope of a polynomial regression with no intercept at a given point, say z. In contrast to previous work, which considers symmetric design spaces we investigate the model on the interval $[0, a]$ and characterize those values of $z$, where an explicit solution of the optimal design is possible.
A polynomial indicator function of designs is first introduced by Fontana, Pistone and Rogantin (2000) for two-level designs. They give the structure of the indicator function of two-level designs, especially from the viewpoints of the orthogonality of the designs. Based on these structure, they use the indicator functions to classify all the orthogonal fractional factorial designs with given sizes using computational algebraic software. In this paper, generalizing the results on two-level designs, the structure of the indicator functions for multi-level designs is derived. We give a system of algebraic equations for the coefficients of indicator functions of fractional factorial designs with given orthogonality. We also give another representation of the indicator function, a contrast representation, which reflects the size and the orthogonality of the corresponding design directly. The contrast representation is determined by a contrast matrix, and does not depend on the level-coding, which is one of the advantages of it. We use these results to classify orthogonal $2^3times 3$ designs with strength $2$ and orthogonal $2^4times 3$ designs with strength $3$ by computational algebraic software.
The general problem of constructing confidence regions is unsolved in the sense that there is no algorithm that provides such a region with guaranteed coverage for an arbitrary parameter $psiinPsi.$ Moreover, even when such a region exists, it may be absurd in the sense that either the set $Psi$ or the null set $phi$ is reported with positive probability. An approach to the construction of such regions with guaranteed coverage and which avoids absurdity is applied here to several problems that have been discussed in the recent literature and for which some standard approaches produce absurd regions.
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