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Robust A-optimal designs for mixture experiments in Scheffe' modelsChou, Chao-Jin 28 July 2003 (has links)
A mixture experiment is an
experiments in which the q-ingredients are nonnegative
and subject to the simplex restriction on
the (q-1)-dimentional probability simplex. In this
work , we investigate the robust A-optimal designs for mixture
experiments with uncertainty on the linear, quadratic models
considered by Scheffe' (1958). In Chan (2000), a review on the
optimal designs including A-optimal designs are presented for
each of the Scheffe's linear and quadratic models. We will use
these results to find the robust A-optimal design for the linear
and quadratic models under some robust A-criteria. It is shown
with the two types of robust A-criteria defined here, there
exists a convex combination of the individual A-optimal designs
for linear and quadratic models respectively to be robust
A-optimal. In the end, we compare efficiencies of these optimal
designs with respect to different A-criteria.
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Locally D-optimal Designs for Generalized Linear ModelsJanuary 2018 (has links)
abstract: Generalized Linear Models (GLMs) are widely used for modeling responses with non-normal error distributions. When the values of the covariates in such models are controllable, finding an optimal (or at least efficient) design could greatly facilitate the work of collecting and analyzing data. In fact, many theoretical results are obtained on a case-by-case basis, while in other situations, researchers also rely heavily on computational tools for design selection.
Three topics are investigated in this dissertation with each one focusing on one type of GLMs. Topic I considers GLMs with factorial effects and one continuous covariate. Factors can have interactions among each other and there is no restriction on the possible values of the continuous covariate. The locally D-optimal design structures for such models are identified and results for obtaining smaller optimal designs using orthogonal arrays (OAs) are presented. Topic II considers GLMs with multiple covariates under the assumptions that all but one covariate are bounded within specified intervals and interaction effects among those bounded covariates may also exist. An explicit formula for D-optimal designs is derived and OA-based smaller D-optimal designs for models with one or two two-factor interactions are also constructed. Topic III considers multiple-covariate logistic models. All covariates are nonnegative and there is no interaction among them. Two types of D-optimal design structures are identified and their global D-optimality is proved using the celebrated equivalence theorem. / Dissertation/Thesis / Doctoral Dissertation Statistics 2018
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Equivalence Theorems and the Local-Global PropertyBarra, Aleams 01 January 2012 (has links)
In this thesis we revisit some classical results about the MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight-preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study a local-global property that naturally generalizes the MacWilliams equivalence theorems. Making use of F-partitions, we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid.
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D-optimal designs for polynomial regression with weight function exp(alpha x)Wang, Sheng-Shian 25 June 2007 (has links)
Weighted polynomial regression of degree d with weight function Exp(£\ x) on an interval is considered. The D-optimal designs £i_d^* are completely characterized via three differential equations. Some invariant properties of £i_d^* under affine transformation are derived. The design £i_d^* as d goes to 1, is shown to converge weakly to the arcsin distribution. Comparisons of £i_d^* with the arcsin distribution are also made.
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Robust D-optimal designs for mixture experiments in Scheffe modelsHsu, Hsiang-Ling 10 July 2003 (has links)
A mixture experiment is an experiment in which the
q-ingredients {xi,i=1,...,q} are nonnegative and subject to the simplex restriction sum_{i=1}^q x_i=1 on the (q-1)-dimensional probability simplex S^{q-1}. In this work, we investigate the robust D-optimal designs for mixture experiments with consideration on uncertainties in the Scheffe's linear, quadratic and cubic model without 3-way effects. The D-optimal designs for each of the Scheffe's models are used to find the robust D-optimal designs. With uncertianties on the Scheffe's linear and quadratic models, the optimal convex combination of the two model's D-optimal designs can be proved to be a robust D-optimal design. For the case of the Scheffe's linear and cubic model without 3-way effects, we have some numerical results about the robust D-optimal designs, as well as that for Scheffe's linear, quadratic and cubic model without 3-way effects. Ultimately, we discuss the efficiency of a maxmin type criterion D_r under given the robust D-optimal designs for the Scheffe's linear and quadratic models.
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Optimal designs for multivariate calibrations in multiresponse regression modelsGuo, Jia-Ming 21 July 2008 (has links)
Consider a linear regression model with a two-dimensional control vector (x_1, x_2) and an m-dimensional response vector y = (y_1, . . . , y_m). The components of y are correlated with a known covariance matrix. Based on the assumed regression model, there are two problems of interest. The first one is to estimate unknown control vector x_c corresponding to an observed y, where xc will be estimated by the classical estimator. The second one is to obtain a suitable estimation of the control vector x_T corresponding to a given target T = (T_1, . . . , T_m) on the expected responses. Consideration in this work includes the deviation of the expected response E(y_i) from its corresponding target value T_i for each component and defines the optimal control vector x, say x_T , to be the one which minimizes the weighted sum of squares of standardized deviations within the range of x. The objective of this study is to find c-optimal designs for estimating x_c and x_T , which minimize the mean squared error of the estimator of xc and x_T respectively. The comparison of the difference between the optimal calibration design and the optimal design for estimating x_T is provided. The efficiencies of the optimal calibration design relative to the uniform design are also presented, and so are the efficiencies of the optimal design for given target vector relative to the uniform design.
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Locally Optimal Experimental Designs for Mixed Responses ModelsJanuary 2020 (has links)
abstract: Bivariate responses that comprise mixtures of binary and continuous variables are common in medical, engineering, and other scientific fields. There exist many works concerning the analysis of such mixed data. However, the research on optimal designs for this type of experiments is still scarce. The joint mixed responses model that is considered here involves a mixture of ordinary linear models for the continuous response and a generalized linear model for the binary response. Using the complete class approach, tighter upper bounds on the number of support points required for finding locally optimal designs are derived for the mixed responses models studied in this work.
In the first part of this dissertation, a theoretical result was developed to facilitate the search of locally symmetric optimal designs for mixed responses models with one continuous covariate. Then, the study was extended to mixed responses models that include group effects. Two types of mixed responses models with group effects were investigated. The first type includes models having no common parameters across subject group, and the second type of models allows some common parameters (e.g., a common slope) across groups. In addition to complete class results, an efficient algorithm (PSO-FM) was proposed to search for the A- and D-optimal designs. Finally, the first-order mixed responses model is extended to a type of a quadratic mixed responses model with a quadratic polynomial predictor placed in its linear model. / Dissertation/Thesis / Doctoral Dissertation Statistics 2020
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A-optimal designs for weighted polynomial regressionSu, Yang-Chan 05 July 2005 (has links)
This paper is concerned with the problem of constructing
A-optimal design for polynomial regression with analytic weight
function on the interval [m-a,m+a]. It is
shown that the structure of the optimal design depends on a and
weight function only, as a close to 0. Moreover, if the weight
function is an analytic function a, then a scaled version of
optimal support points and weights is analytic functions of a at
$a=0$. We make use of a Taylor expansion which coefficients can be
determined recursively, for calculating the A-optimal designs.
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Ds-optimal designs for weighted polynomial regressionMao, Chiang-Yuan 21 June 2007 (has links)
This paper is devoted to studying the problem of constructing Ds-optimal design for d-th degree polynomial regression with analytic weight function
on the interval [m-a,m+a],m,a in R. It is demonstrated that the structure of the optimal design depends on d, a and weight function only, as a close to 0. Moreover, the Taylor polynomials of the scaled versions of the optimal support points and weights can be computed via a recursive formula.
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Optimal Designs for Log Contrast Models in Experiments with MixturesHuang, Miao-kuan 05 February 2009 (has links)
A mixture experiment is an
experiment in which the k ingredients are nonnegative and subject
to the simplex restriction £Ux_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, £X_p-optimal designs for all p, -¡Û<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
£X_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
£X_p -optimal designs for some p, -¡Û<p≤1 are
obtained.
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