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URN etd-0205109-154131
Author Miao-kuan Huang
Author's Email Address huangmk@math.nsysu.edu.tw
Statistics This thesis had been viewed 5066 times. Download 1274 times.
Department Applied Mathematics
Year 2008
Semester 1
Degree Ph.D.
Type of Document
Language English
Title Optimal Designs for Log Contrast Models in Experiments with Mixtures
Date of Defense 2009-01-10
Page Count 96
Keyword
  • Equivalence theorem
  • Geometric-arithmetic means inequality
  • Invariant symmetric block matrices
  • Kiefer ordering
  • Lagrange interpolation polynomial
  • Loewner ordering
  • Abstract A mixture experiment is an
    experiment in which the k ingredients are nonnegative and subject
    to the simplex restriction Σx_i=1 on the
    (k-1)-dimensional probability simplex S^{k-1}. This dissertation
    discusses optimal designs for linear and
    quadratic log contrast models for experiments with
    mixtures suggested by Aitchison and Bacon-Shone (1984),
    where the experimental domain is restricted further as in Chan (1992).
    In this study, firstly, an essentially complete
    class of designs under the Kiefer ordering for linear log contrast
    models with mixture experiments is presented. Based on the
    completeness result, Φ_p-optimal designs for all p, -∞<p≤1 including D- and A-optimal are obtained, where
    the eigenvalues of the design moment matrix are used. By using the
    approach presented here, we gain insight on how these
    Φ_p-optimal designs behave.
    Following that, the exact N-point D-optimal designs for
    linear log contrast models with three and four ingredients are
      further investigated.
     The results show that for k=3 and N=3p+q ,1 ≤q≤2, there is an exact
    N-point D-optimal design supported at the points of S_1 (S_2)
    with equal weight n/N, 0≤n≤p , and puts the remaining
    weight (N-3n)/N uniformly on the points of S_2 (S_1) as evenly as
    possible, where S_1 and S_2 are sets of the supports of the
    approximate D-optimal designs. When k=4 and N=6p+q , 1 ≤q≤5, an exact N-point design which distributes the weights as
    evenly as possible among the supports of the approximate D-optimal
    design is proved to be exact D-optimal.
    Thirdly, the approximate D_s-optimal designs for
    discriminating between linear and
    quadratic log contrast models for experiments with
    mixtures are derived.
    It is shown that for a symmetric subspace of the finite
    dimensional simplex, there is a D_s-optimal design with the nice structure that
    puts a weight 1/(2^{k-1}) on the centroid of this subspace and the remaining weight is
    uniformly distributed on the vertices of the experimental domain.
    Moreover, the D_s-efficiency of the D-optimal design for
    quadratic model and the design given by Aitchison and Bacon-Shone
    (1984) are also discussed
    Finally, we show that an essentially complete class of designs under
    the Kiefer ordering for the quadratic log contrast model is the set
    of all designs in the boundary of T or origin of T
    . Based on the completeness result, numerical
    Φ_p -optimal designs for some  p, -∞<p≤1 are
    obtained.
    Advisory Committee
  • none - chair
  • none - co-chair
  • none - co-chair
  • none - co-chair
  • none - co-chair
  • Mong-Na Lo Huang - advisor
  • Files
  • etd-0205109-154131.pdf
  • indicate accessible in a year
    Date of Submission 2009-02-05

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