Construction Of Latin Hypercube Designs With Complex Structures

Experimental design has been playing an important role in agriculture, industry and sciences. Traditional experiments are always time-consuming, expensive and even can be ruinous. Following the rapid development of computing power, computer experiments are developed, and becoming more and more indispensable. Compared with traditional experiments, computer experiments have no random errors, that is to say, the same inputs yields equal responses. Therefore, the three criteria-replication, randomization and blocking in traditional experiments (Wu and Hamada (2009)) are no need in the design and analysis of computer experiments. In recent years, computer experiments got great development in both design construction and data analysis. For the design construction, space-filling designs are most popular, including Latin hypercube designs (LHDs, Mckay, Conover and Beckman (1979)), uniform designs (Fang, Li and Sudjianto (2006)) and lots of their variants. For the data analysis, after Sacks, Welch, Mitchell and Wynn (1989) introduced the kriging method into computer experiments, this method was widely used and got great improvements. In this dissertation, we investigate some frontier issues, including the construction of LHDs for computer experiments with different levels of accuracy, with both qualitative and quantitative factors and with branching and nested factors.The main task of computer experiments is to simulate complicated system, such as the whether variations, the track of the guided missile, etc., and learn the law of the nature, then make predictions. For a complicated system, we usually approxi-mate it by a meta-model, and build a simple surrogate model for the meta-model after simulating the system. Although the meta-models are simple approximation to the real systems, they may still need a lot of time for one result, for example, a com-puter code using the finite element method or the finite difference method. Luckily, scientists found that such complicated programs can be run at different levels of accuracy, such as high-accuracy and low-accuracy, and proposed methods for de-sign construction and data analysis. A high-accuracy experimental run takes much more time but can obtain a much more accurate result than a low-accuracy one. We can carry out more low-accuracy experimental runs and build a low-accuracy model, then adjust this model by executing a small number of high-accuracy ex-perimental runs at part of the design points that have been run at low-accuracy. The corresponding literature about the data analysis include Kennedy and O’Hagan (2000), Qian, Seepersad, Roshan, Allen and Wu (2006) and Qian and Wu (2008). For the collection of data, nested space-filling designs are prevalent, see Qian, Tang and Wu (2009), Qian, Ai and Wu (2009), Qian (2009), Haaland and Qian (2010), Sun, Yin and Liu (2013), Sun, Liu and Qian (2013), Yang, Liu and Lin (2014), etc. Particularly, Qian (2009) constructed the nested Latin hypercube design (NLHD), which means a large LHD contains a small LHD as a subset for each an NLHD. For instance, to run a computer experiment with k precision, we can follow Qian (2009) to generate an NLHD with k layers Dl,...,Dk, which should satisfy the following conditions:(1) nested structure:D1(?)…(?)Dk;(2) maintain LHDs:each Di should still be an LHD.In addition, the early literature about computer experiments only consider quan-titative factors (Santner, Williams and Notz (2003), Fang, Li and Sudjianto (2006)). However, there can exist qualitative factors. For example, Schmidt, Cruz and Iyen-gar (2005) investigated a thermal diffusion facility, in which the places of the facility and the vent of the hot air are qualitative factors. Han, Santner, Notz and Bartel (2009) studied a knee prothesis model, where the manner of baring the stress is a qualitative factor. Therefore, studying the design and analysis of the computer experiments with both qualitative and quantitative factors is important. The cor-responding literature about the design construction include Qian and Wu (2009), Qian (2012), Yang, Lin, Qian and Lin (2013), Huang, Yang and Liu (2013), Yang, Chen, Lin and Liu (2013), etc.; and those on the data analysis include Qian, Wu and Wu (2008), Han, Santner, Notz and Bartel (2009), Han, Santner, Notz and Long (2009), Zhou, Qian and Zhou (2011), etc. Specially, Qian (2012) proposed the sliced Latin hypercube design (SLHD), which is a special kind of LHD, and can be divided into slices each of which is still a small LHD and corresponds to a level combination of qualitative factors involved in the experiment. Just like the ordinary LHDs, SLHDs proposed by Qian (2012) have one-dimensional projection uniformi-ty, but cannot guarantee the uniformity in the whole experimental region. In this dissertation, we will optimize SLHDs using the Threshold-accepting (TA) algorithm (Dueck and Scheuer (1990)) based on the uniformity measure of the centered L2(CL2) discrepancy (Hickernell (1998)). In addition, since each slice of an SLHD is also an LHD, its uniformity in the experimental region should also be considered. Hence, we will combine the uniformity of the whole design and the uniformity of the slices, and present a new combined criterion with a weighting parameter to optimize the SLHDs, and get better SLHDs.We also find that there is no sliced structures in the NLHDs constructed by Qian (2009). That is to say, the designs did not consider the case that both the high-accuracy and low-accuracy experiments have qualitative and quantitative factors, which really exists. To accommodate such a case, there should be sliced structures in the NLHDs. Therefore, we will propose nested Latin hypercube designs with sliced structures, and show the superiority of them over the ordinary NLHDs in Qian (2009) through simulations.In computer experiments, we often encounter branching and nested factors. Nested factors depends on the branching factors, and within different levels of the branching factor, there exist different nested factors. For example, Hung, Joseph and Melkote (2009) studied manufacturing of printed circuit board (PCB), in which the surface preparation method is a branching factor with two levels:mechanical scrubbing and chemical treatment. Under the mechanical scrubbing, the pressure is a nested factor, which can only exist within the mechanical scrubbing instead of the chemical treatment. Similarly, the micro-etch rate is another nested factor that only exists under the chemical treatment. To suit for such experiments, Hung, Joseph and Melkote (2009) proposed branching Latin hypercube designs (BLHDs). Howev-er, the designs they constructed did not have sliced structures to accommodate the qualitative branching factor. In addition, the designs cannot be used for the situa-tion where the nested factors are also qualitative. In this dissertation, we will first improve the BLHDs by embedding sliced structures into them, and then propose doubly branching Latin hypercube designs (DBLHDs) when the nested factors are qualitative.Next, let us introduce the contents of each chapter briefly.Chapter1introduces some background, concepts and notation that will be used in this dissertation.Chapter2presents the construction of nested Latin hypercube design-s (NLHDs) with sliced structures. NLHDs were proposed by Qian (2009) for conducting multiple computer experiments with different levels of accuracy, while SLHDs were constructed in Qian (2012) to suit for the computer experiments with both qualitative and quantitative factors. That is to say, they were studied separate-ly. However, such two situations can exist simultaneously in one computer experi-ment. For example, if a computer experiment with both qualitative and quantitative factors needs to be run at different levels of accuracy, then neither the NLHDs nor the SLHDs can suit for such a case. In this chapter, we provide a general method for constructing NLHDs with sliced structures, which are suitable for such a case, and easy to implemented. We compare these three kinds of designs by investigating the model accuracy when they are used for building Gauss kriging models, which are implemented by the toolbox DACE in Matlab. The results show that the new designs outperform the NLHDs and SLHDs. Chapter3discusses the construction of uniform sliced Latin hypercube designs (USLHDs). An SLHD is a special kind of LHD, which can be partitioned into slices each of which is a small LHD. Although the SLHDs maintain the one-dimensional projection uniformity from the ordinary LHDs, they cannot guarantee uniformity in the whole design region. In this chapter, we first optimize SLHDs based on the CL2discrepancy from the whole-design perspective, and obtain uniform SLHDs in terms of the whole-design uniformity. Furthermore, considering each slice is also an LHD, and the uniformity in the design region can not be guaranteed either, we propose a combined criterion by combining the uniformity of both the whole SLHD and its slices with a weighting parameter to optimize the SLHDs. Such a measure takes care of not only the uniformity of the whole design but also of the slices in the design region. In addition, by plotting the "w-trace", we can choose a proper value for the weighting parameter.Chapter4provides the construction of improved branching Latin hy-percube designs (IBLHDs), and doubly branching Latin hypercube de-signs (DBLHDs). In computer experiments, we often encounter the situation where branching and nested factors are involved. Hung, Joseph and Melkote (2009) proposed branching Latin hypercube designs (BLHDs) to suit for such a situation. Note that the branching factors are qualitative, however, there is no sliced structures in a BLHD to accommodate such factors. Hence, we improve the BLHDs (Hung, Joseph and Melkote (2009)) by embedding sliced structures into them. In addition, we also consider the situation where the nested factors are also qualitative. Such a case needs different designs from BLHDs and IBLHDs. In this chapter, we construct DBLHDs which can be used for computer experiments with such a situation.Chapter5concludes the work of this dissertation and provides some discussions.