Number theoretic methods in designing experiments

Abstract

The objective of this dissertation is to explore and contrast several methods of generating robust space-filling uniform designs (UDs) and provide tables of UDs for practitioners to use. Four methods of constructing UDs considered in this research are the good lattice point (glp) method, the cyclotomic field (CF or good point (gp)) method, the Halton method and the Hammersley method. UD tables are provided for experimental regions in the s dimensional unit cube C s and sphere B s for s = 2; 3; 4 and for mixture UDs for the q component simplex S q for q = 3; 4; 5: For each of these experimental regions the glp method was found to be the best method for generating UDs. However, the glp method is computationally expensive. To circumvent this problem two methods have been proposed to reduce the computational cost of the glp method. These are the equivalence method and the projection method. The projection method is easier to implement and is the recommended alternative especially for higher dimensions. Two classes of measures to assess uniformity (i.e., space-filling properties of a design) used in this dissertation are new discrepancy measures and distance-based measures. The new discrepancy measures are proposed for all the aforementioned experimental regions. Specifically, the modified star discrepancy (MSTRD) is proposed for a unit cube C s and new discrepancy measures are also proposed for two and three dimensional spherical regions B ² and B ³: Directional systematic discrepancy (DSD) and the non-directional random discrepancy (NDRD) are proposed for the simplex S ³ for a 3 component mixture experiment. However, for mixture experiments with single component constraints (SCCs) the NDRD is the new proposed measure of discrepancy. In higher dimensions, the distance-based criteria are easier to implement and hence are the recommended alterative to discrepancy measures. Generating UDs for mixture experiments with multiple component constraints (MCCs) is complicated and, hence, a modification of the one-pass exchange algorithm of Borkowski and Piepel (2009) that generates UDs for experiments with MCCs is presented. This new method is called the one-step exchange algorithm.