New developments in designs for computer experiments and physical experiments

We develop in this thesis new methodologies for designing both computer experiments and physical experiments. There has been much attention devoted to deterministic computer experiments in the past two decades. Such experiments are performed on computer simulators to study complex physical phenomena that might otherwise be too time-consuming, expensive, or impossible to observe. In this thesis, we develop a new method for constructing “good” designs for computer experiments. The construction uses small designs to construct larger designs with desirable properties. Our method allows orthogonal Latin hypercubes to be constructed for any run size n unequal to 4k+2 and when n=4k+2 we prove that an orthogonal Latin hypercube does not exist. Therefore, in terms of run size, we have completely solved the existence problem for orthogonal Latin hypercubes. Another appealing feature of our method is that it can be adapted easily to construct other designs such as nearly orthogonal Latin hypercubes and cascading Latin hypercubes. In addition to the above, we propose and study two generalizations that allow designs with better projection properties to be constructed. The second part of the thesis is devoted to two-level fractional factorial designs, the most widely used designs in practice. Two research problems are investigated. The first is the algorithmic construction of minimum G and G_2-aberration designs. Our method is applicable for any run size that is a multiple of eight. Results from the application of this algorithm to designs of 24, 32 and 40 runs are obtained and presented. The second problem is to provide a catalogue of “good” two-level folded-over non-orthogonal designs. Such designs are useful in screening experiments. To assess the goodness of designs, we introduce the MDS-resolution and MDS-aberration, based on the notion of minimal dependent sets (MDS). With both criteria, it is possible to systematically compare the statistical properties of designs. Obtaining a catalogue, however, remains challenging because it involves determining whether or not two designs are isomorphic. A fast isomorphism check is developed for this purpose. A catalogue of minimum MDS-aberration designs is obtained for many useful run sizes. An algorithm for obtaining “good” larger designs is discussed.