Construction Of Designs For Complex Computer Experiments And Screening Designs

Experimental design has played an important role in the statistical curriculum, practice and research. Much of our knowledge about products and processes in the engineering and scientific disciplines is derived from experimentation. Design of experiments is a tool to develop an experimentation strategy that maximizes learning using a minimum of required runs. Classical experimental designs, such as factorial designs, orthogonal designs, block designs, Latin square designs and response surface designs, have been studied extensively and have rich theories.This dissertation explores some new topics of experimental designs, including nested orthogonal Latin hypercube designs (LHDs) and sliced orthogonal-maximin LHDs for computer experiments, and the construction of the minimal-point mixed-level screening designs.In the preliminary stage of an experiment, the object is to identify the most important factors contributing to the experimental process. Fractional factorial designs with resolution III or IV are usually used as screening designs. However, there are some drawbacks for identifing the active effects, for example, for any fractional factorial design with resolution III, the main effects of some factors are completely confounded with some two-factor interactions. Then, additional runs are needed to identify the true active effects. How to construct screening designs with good properties is an important issue. For two-level or three-level factors, screening designs have been proposed. However, there are some experiments with both two-level and three-level factors. To our best knowledge, no one has proposed methods to obtain mixed-level screening designs. This dissertation proposes a construction of mixed-level screening designs with the minimum run size and good performance in terms of D-efficiency and variance of estimates.Traditionally, an experiment is implemented in a laboratory, a factory, or an agricultural field. With the rapid development of computer technology, some phys-ical experiments can be simulated by sophisticated computer codes. There are no random errors in a computer experiment. Deterministic response is obtained for a given set of input conditions in a computer experiment. Thus, none of the tradi-tional principles of randomization, blocking and replication are useful in the design and analysis problems associated with computer experiments. LHDs introduced by Mckay, Conover and Beckman (1979) are widely used in computer experiments. However, the original construction of LHDs by randomly choosing a permutation of1,...,n for each factor, where n is the run size, is possible to having potential-ly high correlations among the factors. The presence of high correlations among factors can complicate the subsequent data analysis and make it more difficult to identify the most important factors. It is desirable to include orthogonal variables for a regression model, which can guarantee that the estimates of the coefficients are uncorrelated. For a first-order regression model, the orthogonality can make sure that the estimates of the main effects are uncorrelated. Furthermore, for a first-order regression model with the presence of the second-order effects (quadratic effects and bilinear interactions), it is desirable to use a design which can make sure that the es-timates of all the main effects are uncorrelated with that of the second-order effects. It is an important issue to construct LHDs having the following desirable properties:(a) the estimates of the main effects are uncorrelated with each other;(b) the estimates of all the main effects are uncorrelated with that of the quadratic effects and bilinear interactions.A second-order orthogonal LHD can guarantee the properties (a) and (b). A column centered LHD is second-order orthogonal, if it satisfies that the inner product of any two columns of this LHD is zero and the sum of the elementwise products of any three columns is zero. The detail of this definition will be provided in Chapter1. There are some existing methods to construct LHDs with properties (a) and (b), e.g., Ye (1998), Cioppa and Lucas (2007), Georgiou (2009), Sun, Liu and Lin (2009,2010), Yang and Liu (2012) and Ai, He and Liu (2012).A complicated, expensive computer program, like a finite element analysis model, can be executed at various degrees of fidelity, resulting in a computer experiment with multiple levels of cost and accuracy. It is popular to use both accurate (but slow) computer experiments and less accurate (but fast) computer experiments to study complex physical systems. Observations from such experiments are often used to build a statistical model to predict the response for the most accurate involved (Kennedy and O’Hagan (2000), Qian, Seepersad, Roshan, Allen and Wu (2006) and Qian and Wu (2008)). Efficient data collection is critical to conduct these experiments. Qian, Tang and Wu (2009) proposed a new class of designs called nested space-filling designs for the experiments with high accuracy and low accuracy. Consider a situation involving k computer experiments with responses y1,...,yk respectively, where y1is the most accurate,y2is the second most accurate, and so on. Let D1,..., Dk be the designs for the k experiments. D1,...,Dk usually satisfy the following conditions:(1) nested structure:D1(?)...(?)Dk;(2) space-filling property:each Di is a space-filling design. LHDs achieve uniformity in one dimension. It is a good topic to construct nested LHDs with desirable property, such as orthogonality. Few work has been done to construct nested orthogonal LHDs. Li and Qian (2013) proposed several methods to construct (nearly) column-orthogonal LHDs for two-fidelity computer experiments. This dissertation proposes the construction of nested orthogonal LHDs for multi-fidelity computer experiments. Furthermore, the nested LHDs here is second-order orthogonal. In addition, the variables of a computer experiment are usually assumed to be quantitative. However, it is possible that both qualitative and quantitative vari-ables are involved in a computer experiment. The sliced space-filling design pro-posed by Qian and Wu (2009) is a good choice for experiments with qualitative and quantitative variables. Each slice of the sliced space-filling design is associated with one level-combination of the qualitative factors, and each slice should have space-filling property in low dimensions. Qian (2012) generated sliced LHDs with one-dimensional uniformity. Orthogonality is an important property for space-filling designs. Few work has been done to construct sliced orthogonal LHDs. Yang, Lin, Qian and Lin (2013) proposed several methods to construct sliced orthogonal or second-order orthogonal LHDs. Furthermore, the existing methods only considered orthogonality or uniformity in one-dimension for the sliced LHDs. It is a new top-ic to take into account both of the orthogonality and uniformity in two or more dimensions for the sliced LHDs.Though there are some existing approaches for constructing screening designs, nested space-filling designs and sliced space-filling designs, many new problems need to be resolved. In the following, let us introduce the contents of each chapter briefly.Chapter1introduces some background and provides some concepts, notations and lemmas that will be used in the following chapters.Chapter2presents approaches to constructing nested orthogonal L-HDs. Nested LHDs are proposed for conducting multiple computer experiments with different levels of accuracy. Orthogonality is shown to be an important fea-ture. Little is known about the construction of nested orthogonal LHDs. The existing methods can obtain nested orthogonal LHDs with restricted layers. We, in this chapter, present methods to construct such designs with two or more layers, making use of orthogonal designs. The number of the layers can be any positive integer. The constructed nested LHDs possess the property that the sum of the elementwise products of any three columns is zero, which is shown to be desirable for factor screening. Such designs can guarantee the estimates of all the main effects and that of the quadratic effects and bilinear interactions are uncorrelated. The methods are easy to implement.Chapter3provides the construction of sliced orthogonal-maximin L-HDs. A sliced LHD is a special LHD, which can be divided into slices of smaller LHDs. This type of designs is useful for computer experiments with qualitative and quantitative factors, multiple experiments, data pooling and cross-validation. Orthogonality and uniformity are important properties for LHDs. In this chap-ter, sliced orthogonal-maximin LHDs are constructed using orthogonal designs, Goethals-Seidel arrays and Kharaghani arrays. The resulting designs not only have the second-order orthogonality, but also have a good uniformity measured by the maximin distance criterion.Chapter4introduce a method to construct minimal-point mixed-level screening using conference matrices. Screening designs are frequently used to identify active effects from a large number of factors. Small size designs are preferred when the experiments are costly. Two-level or three-level minimal-point screening designs have been well studied in the literature. However, minimal-point mixed-level designs have not been thoroughly explored. In this chapter, a new class of minimal-point mixed-level designs is constructed using conference matrices. The constructed designs can be used to estimate the main effects and quadratic effects with a good performance of D-efficiency and variance of estimates.Chapter5concludes the work of this dissertation and provides some discussions.