Design for Computer Experiments: Constructions and Properties

Abstract

Computer experiments have become an indispensable tool in modern scientific research, providing a flexible, cost-effective, and safe alternative to physical experimentation. They allow researchers to investigate complex systems, assess multiple factors simultaneously, and explore scenarios that may be impractical or unsafe to replicate in real life. In contrast, physical experiments often require substantial time and financial resources and can be limited by practical or ethical restrictions. The efficiency, repeatability, and broad applicability of computer experiments make them an increasingly attractive choice in fields such as engineering, applied sciences, and data analysis. Within this context, we develop new design constructions that advance the field of computer experiments. This thesis is guided by three research objectives: (1) to propose novel Orthogonal Designs for Computer Experiments (ODCEs), (2) to develop new Sliced Orthogonal Designs for Computer Experiments (SODCEs), and (3) to compare these newly constructed designs with existing methods in the literature. Together, these objectives capture the novelty of our contributions to the design of computer experiments. The first chapter introduces the fundamental concepts of the Design of Experiments (DoE), defining key terms such as factors, levels, treatments, and responses, and explaining the core principles, including randomization, replication, and blocking. The chapter highlights the distinction between physical and computer experiments, underlining the advantages of the latter in terms of cost, flexibility, and scalability. The scope and significance of the research are also described, outlining how this work contributes to the creation of more effective and practical designs for computer experiments. The second chapter reviews the background and development of various experimental designs. It begins with traditional approaches such as one-factor-at-a-time (OF AT ), factorial, fractional factorial, Latin square, and Graeco–Latin square designs. The focus then shifts to designs for computer experiments, particularly Latin hypercube designs (LHDs) and sliced Latin hypercube designs (SLHDs), discussing their construction methods, strengths, and limitations. The chapter also examines other designs that have been adapted or specifically developed to enhance the accuracy and practicality of computer-based studies. The third chapter lays the technical foundation for the proposed constructions. It discusses the polynomial regression models and their relationship to design matrices, then explores mathematical tools such as autocorrelation functions (periodic and non-periodic), circulant matrices, and structured arrays such as Goethals-Seidel and Kharaghani arrays. A major emphasis is placed on sequence families with zero autocorrelation, T-sequences, Base sequences, Golay sequences, and disjoint amicable sequences, which are central to generating efficient and highly structured experimental designs. The fold-over technique is also discussed as a means of enhancing symmetry and orthogonality in the resulting designs. In the fourth chapter, we develop our new design, called "Orthogonal Designs for Computer Experiments (ODCEs). This part of the research has been published in Applied Numerical Mathematics(see Appendix A). The method combines T-sequences(and other sequences converted to T-sequences) with circulant matrices, Goethals–Seidel arrays, and the fold-over technique to create designs with strong structural properties. The resulting ODCEs achieve full orthogonality not only between main effects and second-order terms but also between odd- and even-order interaction effects. This improves clarity and reduces confounding in model estimation. The approach supports a wide range of run sizes and factor combinations, while avoiding exhaustive search procedures and the reliance on rare sequence types. The designs are easy to generate, efficient to use, and adaptable to various experimental needs. The proposed ODCE is also evaluated and compared with selected existing designs from the literature to highlight its performance and advantages. In the fifth chapter, we contribute to the field of computer experiments by developing the first infinite families of Sliced Orthogonal Designs for Computer Experiments (SODCEs). Our newly constructed designs are presented here for the first time. These designs extend and generalize the concept of Sliced Latin Hypercube Designs (SLHDs). This new design ensures that orthogonality is maintained throughout the full design and within every slice. Each slice is a smaller ODCE, making the designs suitable for step-by-step experimentation and multi-stage studies. The construction uses T-sequences, Golay sequences, and disjoint amicable sequences, allowing flexible parameter choices and broad applicability in scientific and engineering studies. The proposed SODCE is evaluated and compared with existing sliced designs from the literature to demonstrate its effectiveness and practical benefits. The final chapter provides a concluding review of the thesis. It summarizes the work presented in each chapter and highlights the main contributions of the new constructions ODCE and SODCE. It shows their practical strengths and applicability in computer experiments. It also outlines some future work in the areas covered in this thesis.