Space-filling designs for mixture/process variable experiments
Obiri, Moses Yeboah
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The ultimate objective of this dissertation was to present a statistical methodology and an algorithm for generating uniform designs for the combined mixture/process variable experiment. There are many methods available for constructing uniform designs and four of such methods have been used in this study. These are the Good Lattice Point (GLP) method, the cyclotomic field (CF) method, the square root sequence (SRS) method, and the power-of-a-prime (PP) method. A new hybrid algorithm is presented for generating uniform designs for mixture/process variable experiments. The algorithm uses the G function introduced by Fang and Yang (2000), and adopted by Borkowski and Piepel (2009) to map q-1 points from q + k - 1 points generated in the hypercube to the simplex. Two new criteria based on the Euclidean Minimum Spanning Tree (EMST) which are more computationally efficient for assessing uniformity of mixture designs and mixture/process variable designs are presented. The two criteria were found to be interchangeable and the geometric mean of the edge lengths (GMST) criterion is preferred to the average and standard deviation of edge lengths (adMST, sdMST) criterion. The GMST criterion uses only one statistic to quantify the uniformity properties of mixture and mixture/process variable designs. Tables of good uniform designs are provided for mixture experiments in the full simplex (S q) for q = 3; 4; 5 and practical design sizes, 9 _ n _ 30, using the four number theoretic methods in this study. A conditional approach based on the GMST criterion for generating good uniform mixture/process variable designs is also introduced and tables of good uniform designs are given for the combined q-mixture and k process variable experiments for q = 3; 4; 5, k = 1; 2 and practical numbers of runs, 9 _ n _ 30. A new algorithm is provided to augment existing mixture design points with space-filling points including designs with existing clustered design points. In this algorithm, new design points are chosen from a candidate set of points such that the resulting augmented design has good space-filling properties. The SRS method is found to produce the best augmented space-filling mixture designs.