Theory and Construction of Some Space-filing Designs

某些空間填充設計的理論與構造

Student thesis: Doctoral Thesis

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Detail(s)

Awarding Institution
Supervisors/Advisors
  • Matthias Hwai-yong TAN (Supervisor)
  • Min-Qian Liu (External person) (Supervisor)
  • Minqian LIU (External person) (External Supervisor)
Award date17 Jun 2021

Abstract

Computer experiments are becoming more commonly used in science and engineering compared to physical experiments. This is primarily because the actual physical systems are too time-consuming and expensive, and some systems are destructive or even unable to implement. Space-filling designs, which aim to spread the design points over the entire design space as uniformly as possible, are most widely used for computer experiments. Latin hypercube design (LHD) and uniform design are two classes of most popular space-filling designs. LHDs satisfy one-dimensional space-filling property, while they cannot guarantee good space-filling properties in two or higher-dimensional projections. Therefore, constructing good LHDs is an important topic for computer experiments.

Generally, we adopt some optimality criteria, model-based criteria or model-independent criteria, for improving the performance of LHDs. The entropy criterion is a widely employed model-based criterion. For Gaussian process (GP) models, which are popular for modeling computer experiments data sets, maximizing the entropy is equivalent to maximizing the log determinant of the covariance matrix. Aside from the maximum entropy criterion, the integrated mean squared error (IMSE), the maximum prediction variance (MPV) and the average prediction variance (APV) criteria are some commonly used model-based criteria. Alternatively, the maximin distance, the minimax distance, the orthogonality and the uniformity criteria are widely used model-independent criteria. Additionally, many efforts have been made to construct designs based on two or more optimality criteria.

By the effect sparsity principle, only a few of factors are expected to be active in a computer experiment. Existing designs (such as maximin distance designs and uniform designs) may have bad low-dimensional projections, which is always undesirable. Therefore, space-filling designs with good projection properties become more popular in practice recently. In addition, for computer experiments, designs with many levels are desirable, but it is not essential to make the run size equal to the number of levels. In recent years, U-type designs with flexible run sizes and numbers of factor levels have received more attention in computer experiments.

The main challenge of this dissertation is to construct designs with good space-filling properties for computer experiments. In this dissertation, we aim to address the above related design issues by proposing new optimality criteria and new construction methods. Existing projection designs always attempt to achieve good space-filling properties in all projections in terms of some model-independent criteria. While for GP models, model-based design criteria are more appropriate. Then, we propose some new optimality design criteria for GP models and employ input transformations to generate such good projection designs. Additionally, for model-independent design criteria such as the orthogonality and the maximin distance criteria, existing construction methods always have restrictions on the run sizes and numbers of factor levels. We propose some new construction methods to construct good space-filling designs which cannot be obtained by existing methods.

What follows is the organization of this dissertation.

Chapter 1 introduces the research progress related to space-filling designs and the preliminary knowledge of computer experiments and commonly used criteria that will be used in the following chapters.

In Chapter 2, we employ the entropy criterion averaged over a set of projections, called expected entropy criterion (EEC), to generate projection designs. We also demonstrate that transformation of each column of an LHD based on a monotonic function, which preserves the Latin hypercube structure, can substantially improve the EEC. Existing projection designs (e.g., maximum projection designs) attempt to achieve good space-filling properties in all projections in terms of distance, uniformity, or orthogonality. However, as fitting a GP model is often an important goal, a model-based criterion such as the entropy criterion is more appropriate in many cases. Two types of input transformations are considered: a quantile function of a symmetric Beta distribution with distribution parameter chosen to optimize the EEC, and a nonparametric transformation corresponding to the quantile function of a symmetric density chosen to optimize the EEC. Modification of the transformation method to exploit factor importance ranking information is also considered. Numerical studies show that the proposed transformations of the LHD are computationally efficient and effective methods for building robust maximum EEC designs. These designs are shown to give projections with markedly higher entropies and lower MPV's at a cost of small increases in APV's compared to some state-of-the-art space-filling designs over wide ranges of covariance parameter values.

Chapter 3 proposes a new rotation method for constructing orthogonal Latin hypercube designs (OLHDs) and nearly OLHDs with flexible run sizes that cannot be obtained by existing methods. Orthogonality is a desirable property for LHDs, as it allows the estimates of the main effects in linear models to be uncorrelated with each other, and is a stepping stone to the space-filling property for fitting GP models. Among the available methods for constructing OLHDs, the rotation method is particularly attractive due to its theoretical elegance as well as its contribution to space-filling properties in low dimensional projections. Furthermore, the resulting OLHDs are improved in terms of the maximin distance criterion and the alias matrices and a new kind of orthogonal designs are constructed. Theoretical properties as well as construction algorithms are provided.

Chapter 4 employs incomplete block designs to propose a systematic combinatorial construction method for U-type designs, which can generate designs with various run sizes and numbers of factor levels. While designs with many levels are desirable for computer experiments, sometimes it is not necessary to make the number of levels for each factor equal the run size as in LHDs. The quality of these newly constructed designs is guaranteed by small designs. By using different small designs and block designs, the proposed method produces various types of U-type designs in terms of the maximin distance and the orthogonality criteria. Theoretical properties as well as the construction algorithm are provided.

Chapter 5 concludes this dissertation with some remarks.

    Research areas

  • Computer experiment, Entropy criterion, Factor sparsity, Factorial design, Incomplete block design, Input transformation, Latin hypercube design, Maximin distance, Orthogonality, Projection, Space-filling design