Construction Of Latin Hypercube Designs And Analysis Of Supersaturated Designs

Scientific experiment is an important tool to learn about the nature, it has been successfully applied in all areas of human investigation. With the numerous development of scientific technology, the research objects involve more and more factors and the underlying relationships among these factors are becoming more and more complex. The conclusions only based on intuition and experience are far from enough. Such situations lead to the birth of a new scientific curriculum—experimental design. To make a plan of scientific experiment, a series of actions should be included:define the research problem, develop a plan of experiment, design an experiment, implement an experiment and analyze the data. Among the series of actions, the design of experiments and the analysis of data are statistical problems. The art of designing an experiment and the art of analyzing an experiment are closely intertwined. A good experimental design should minimize the number of runs to get as much useful information as possible, that is, the information should be of benefit to the analysis of data. Data collected from different designs based on different criteria need different analysis methods.Design of experiments has been developed more than80years since R. A. Fisher started the pioneering work at Rothamsed farm in Britain, in1930s. The modern discipline in design of experiments was stimulated by the research problems in agri-culture and biology, like the famous Mendel’s pea experiments. The observations in this kind of experiments are subject to randomization, that is, the observations have fluctuations under the identical experiment setting. Experiment implemented in a laboratory, a factory or a farm, called a physical experiment, is always ac- companied with randomization. A variety of techniques, such as randomization, blocking and replication, are proposed to suit physical experiments. Designs, like orthogonal designs, regression designs, block designs and Latin square designs have successful applications in physical experiments. In the latest decades, benefiting from the rapid advancement of computer technology, the computer has become an increasingly popular tool in the research work. In some cases, traditional physical experiment may be very expensive and time consuming to run an experiment. Even more, some physical experiments are prohibitive, for example, if we want to study the damage brought by a hurricane, it is impossible to do a physical experimenta-tion. The computer experiment is a new breakthrough under such conditions. A computer experiment is a number of runs of the code with various inputs to simu-late the system performance. Often, the codes are computationally expensive to run, and a common objective of an experiment is to fit a cheaper predictor of the output to the data, i.e., find a metamodel to approximate the true complex relationship between the output and input variables. Computer experiments are different from physical experiments because the output is deterministic—rerunning the code with the same inputs gives identical experimental output. So computer experiments need new methodologies of design and analysis.Latin hypercube designs (LHDs) have a favorite one-dimensional uniformity which is preferable to the deterministic outputs of computer experiments, and they have been almost exclusively recommended in computer experiments. Since the LHD was proposed in1979, many scholars have done a plenty of work to improve LHDs. Designs are first restricted to the class of LHDs and then a second criterion is applied to this class. For example, there is an effort to find designs which are orthogonal or have two or higher-dimensional uniformity within the class of LHDs. Orthogonal property is desirable because it can ensure the independence of estimates of effects corresponding to the orthogonal columns when the analysis is built on a regression model. Two or higher-dimensional uniformity can decrease the variation of prediction. The above two properties can both help to correctly analyze the col- lected data. The Fourier-polynomial model was first proposed by Butler (2001) as a good approximation not only to the polynomial regression model but also to the spa-tial model. In the initial stage of a scientific experiment, to screen the active effects from a lager number of underlying factors is important under model uncertainty. So a Fourier-polynomial model should be a good choice because of its balance between the polynomial regression model and the spatial model. A Fourier-polynomial model can be rewritten to have a form of the polynomial regression model. So we need to construct LHDs with properties that all linear effects are mutually orthogonal and orthogonal to all second-order effects, i.e., quadratic effects and bilinear inter-actions. Butler (2001) applied the resolution criterion of a factorial design to LHDs and defined LHDs with the above mentioned properties to be resolution IV.It is inevitable that we face computer experiments with both qualitative and quantitative factors. In such an experiment, the whole design is restricted to be an LHD with one or higher-dimensional projective uniformity or orthogonality and each part of the design corresponding to one-level combination of the qualitative factors is also hoped to still have the same properties. Though a computer experiment is economical, it can involve a code that is time-consuming to run. In some finite element models, it is usual for a code to run a few weeks to produce a single response. So there is still a need to reduce the cost. Multi-fidelity computer experiments are good choices for time-consuming experiments. Sliced LHDs can be applied to the above two problems and become a hot research topic in recent years. There are still many issues to be resolved.Once the data is collected, suitable analysis methods should be taken to get useful information. Standard analysis methods, such as the analysis of variance method, the regression analysis, etc., have already been applied in reality and have solved many problems. But for newly appeared designs, these traditional methods cannot produce satisfactory results. Supersaturated designs (SSDs) have attracted many authors’attention for its strong and powerful competition in run-size economy. Many construction methods have been proposed in recent twenty years. However relative to the rapid development of the construction, the analysis of SSDs needs more investigation. Though a few methods for analyzing the data from SSDs have been proposed, none of them seem very convincing. How to analyze and interpret the data from SSDs is a thorny issue, especially the data from SSDs with multiple responses. The forgoing methods concentrate on the situations where only one response is considered, however in practice, SSDs with multiple responses are often encountered. There is a blank in the analysis of SSDs with multiple responses.An outline of the dissertation is given in the following.Chapter1is the introduction, including some background knowledge and definitions that will be used in the following chapters.Chapter2introduces a convenient and flexible algorithm for con-structing orthogonal LHDs which have resolution IV under the Fourier-polynomial model. Most of the resulting designs have different run sizes from that of Butler (2001), and thus are new. In this chapter, we prove that a LHD with resolution IV can study factors with no more than half number of the runs. The newly constructed LHDs are very suitable for factor screening as they require very few experimental runs per factor, and building Fourier-polynomial models in computer experiments as discussed in Butler (2001).Chapter3presents two construction methods for sliced LHDs based on orthogonal arrays. First a new approach to constructing sliced LHDs is pro-vided based on symmetric orthogonal arrays. The resulting sliced LHDs possess a desirable sliced structure and have an attractive low-dimensional uniformity. Mean-while within each slice, it is also a LHD with the same low-dimensional uniformity. Next, new sliced LHDs are constructed via asymmetric orthogonal arrays. The same desirable properties are possessed as those constructed using symmetric or-thogonal arrays, depending on their combinations, the uniformity may be differed. The construction methods are easy to implement, and unlike the existing methods, the resulting designs are very flexible in run sizes and numbers of factors.Chapter4proposes a two-stage variable selection strategy for SSDs with multiple responses. The strategy uses the multivariate partial least squares regression in conjunction with the stepwise regression procedure to select true active effects in SSDs with multiple responses. Compared with the analysis methods for SSDs with one response, the information lying in the matrix of observations for the responses is made full use in the MPLS stage so the method is more rational.Chapter5provides some concluding remarks for the whole disserta-tion.