混合實驗是在一包含 k 個非負的成分 x_i,i=1,2,….k 的
(k-1)維之單純型機率空間 S^{k-1}上所設計的實驗，並且以一單純的限制式 Σx_i=1 為實驗的條件。在本論文中，我們研究具有對稱之實驗區域的混合實驗條件的線性與二次對數對比模型的最適設計。在以往的文獻中，對於高維度線性模型的最適設計問題，常見的作法是先在Kiefer 次序(Kiefer ordering) 之下建構一個完備集合(essentially complete class)，再從此集合中求出最適設計。

基於此作法，在第一部份，
我們提出具有對稱之實驗區域的線性對數對比模型的完備集合，利用前述的集合我們可以很容易得到
Φ_p-最適設計(包括 D-與 A-最適設計)。另外，我們也探討 k=3
與 k=4 的線性對數對比混合實驗模型下之正合 D-最適設計問題。Chan (1988)
曾證明 k=3 的線性對數對比模型的近似 D-最適設計(approximate
D-optimal design) 的支撐點是實驗區域的極端點(extreme point) S_1
或S_2，且均勻分散權重於這些支撐點上之設計，或是前面所提近似
D-最適設計的凸組合(convex combination)。 當樣本數N=3p+q,1 ≤q≤2，我們證明正合D-最適設計是先放置相同之權重n/N, 0≤n≤p
於 S_1 (或 S_2)，而剩餘的權重 (N-3n)/N 均勻分配於
S_2 (或 S_1)。 對於 k=4 的線性對數對比模型中，當樣本數 N=6p+q,1 ≤q≤5， 正合 D-最適設計(exact D-optimal
design)的支撐點和近似
D-最適設計一樣，且應盡量均勻分散樣本點於這些支撐點上。
在第二部份，對於具有混合實驗條件的二次對數對比模型，我們也用類似的作法求得Φ_p -最適設計與區分線性與二次模型的
D_s-最適設計。其中 D_s-最適設計是在實驗區域的中心點放置 1/(2^{k-1})
的權重，而將
1-1/(2^{k-1})的權重均勻分配在實驗區域的端點。並且對於二次對數對比模型的
D-最適設計與 Aitchison and Bacon-Shone (1984) 所提出之一設計討論
D_s-效率。

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&#8804;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 &#8804;q&#8804;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&#8804;n&#8804;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 &#8804;q&#8804;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&#8804;1 are
obtained.

1. Introduction 1
2. Preliminaries 9
2.1. R-invariant designs in log contrast models 9
2.2. Invariant symmetric block matrices 13
2.3. Preliminary for exact D-optimal design 16
3. Optimal designs for linear log contrast model 19
3.1. Introduction 19
3.2. -optimal designs 21
3.2.1. Essentially complete class 21
3.2.2. -optimal designs 31
3.3. Exact D-optimal designs 36
3.3.1. Three ingredients 36
3.3.2. Four ingredients 42
3.4. Conclusion 45
3.5. Appendix 46
3.5.1. Proof of Lemma 3.2.1. 46
3.5.2. Proof of Theorem 3.2.2. 47
3.5.3. Proof of Theorem 3.3.3. 50

4. Optimal designs for quadratic log contrast model 53
4.1. Introduction 53
4.2. D_s-optimal designs 55
4.2.1. Invariance of designs 55
4.2.2. Quadratic D_s-optimal designs 61
4.2.3 D_s-efficiency of the design in an example 63
4.3. -optimal designs 65
4.3.1. Essentially complete class 66
4.3.2. Numerical -optimal designs 71
4.4. Conclusion 76
4.5. Appendix 77
4.5.1. Proof of Theorem 4.2.2. 77
References 79

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