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Computer experiments with both quantitative and qualitative (QQ) inputs are commonly used in science and engineering applications. Constructing desirable emulators for such computer experiments remains a challenging problem. In this article, we propose an easy-to-interpret Gaussian process (EzGP) model for computer experiments to reflect the change of the computer model under the different level combinations of qualitative factors. The proposed modeling strategy, based on an additive Gaussian process, is flexible to address the heterogeneity of computer models involving multiple qualitative factors. We also develop two useful variants of the EzGP model to achieve computational efficiency for data with high dimensionality and large sizes. The merits of these models are illustrated by several numerical examples and a real data application.

Recent researches on designs for computer experiments with both qualitative and quantitative factors have advocated the use of marginally coupled designs. This paper proposes a general method of constructing such designs for which the designs for qualitative factors are multi-level orthogonal arrays and the designs for quantitative factors are Latin hypercubes with desirable space-filling properties. Two cases are introduced for which we can obtain the guaranteed low-dimensional space-filling property for quantitative factors. Theoretical results on the proposed constructions are derived. For practical use, some constructed designs for three-level qualitative factors are tabulated.

Orthogonal array, a classical and effective tool for collecting data, has been flourished with its applications in modern computer experiments and engineering statistics. Driven by the wide use of computer experiments with both qualitative and quantitative factors, multiple computer experiments, multi-fidelity computer experiments, cross-validation and stochastic optimization, orthogonal arrays with certain structures have been introduced. Sliced orthogonal arrays and nested orthogonal arrays are examples of such arrays. This article introduces a flexible, fresh construction method which uses smaller arrays and a special structure. The method uncovers the hidden structure of many existing fixed-level orthogonal arrays of given run sizes, possibly with more columns. It also allows fixed-level orthogonal arrays of nearly strength three to be constructed, which are useful as there are not many construction methods for fixed-level orthogonal arrays of strength three, and also helpful for generating Latin hypercube designs with desirable low-dimensional projections. Theoretical properties of the proposed method are explored. As by-products, several theoretical results on orthogonal arrays are obtained., Comment: The Annals of Statistics (2021)

Computer experiments with both qualitative and quantitative input variables occur frequently in many scientific and engineering applications. How to choose input settings for such experiments is an important issue for accurate statistical analysis, uncertainty quantification and decision making. Sliced Latin hypercube designs are the first systematic approach to address this issue. However, it comes with the increasing cost associated with an increasing large number of level combinations of the qualitative factors. For the reason of run size economy, marginally coupled designs were proposed in which the design for the quantitative factors is a sliced Latin hypercube design with respect to each qualitative factor. The drawback of such designs is that the corresponding data may not be able to capture the effects between any two (and more) qualitative factors and quantitative factors. To balance the run size and design efficiency, we propose a new type of designs, doubly coupled designs, where the design points for the quantitative factors form a sliced Latin hypercube design with respect to the levels of any qualitative factor and with respect to the level combinations of any two qualitative factors, respectively. The proposed designs have the better stratification property between the qualitative and quantitative factors compared with marginally coupled designs. The existence of the proposed designs is established. Several construction methods are introduced, and the properties of the resulting designs are also studied., Comment: Statistica Sinica (2021)

This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred to as space-filling designs.

Regularized linear models, such as Lasso, have attracted great attention in statistical learning and data science. However, there is sporadic work on constructing efficient data collection for regularized linear models. In this work, we propose an experimental design approach, using nearly orthogonal Latin hypercube designs, to enhance the variable selection accuracy of the regularized linear models. Systematic methods for constructing such designs are presented. The effectiveness of the proposed method is illustrated with several examples.

Computer experiments with both qualitative and quantitative factors are widely used in many applications. Motivated by the emerging need of optimal configuration in the high-performance computing (HPC) system, this work proposes a sequential design, denoted as adaptive composite exploitation and exploration (CEE), for optimization of computer experiments with qualitative and quantitative factors. The proposed adaptive CEE method combines the predictive mean and standard deviation based on the additive Gaussian process to achieve a meaningful balance between exploitation and exploration for optimization. Moreover, the adaptiveness of the proposed sequential procedure allows the selection of next design point from the adaptive design region. Theoretical justification of the adaptive design region is provided. The performance of the proposed method is evaluated by several numerical examples in simulations. The case study of HPC performance optimization further elaborates the merits of the proposed method.

Computer simulators are nowadays widely used to understand complex physical systems in many areas such as aerospace, renewable energy, climate modeling, and manufacturing. One fundamental issue in the study of computer simulators is known as experimental design, that is, how to select the input settings where the computer simulator is run and the corresponding response is collected. Extra care should be taken in the selection process because computer simulators can be computationally expensive to run. The selection shall acknowledge and achieve the goal of the analysis. This article focuses on the goal of producing more accurate prediction which is important for risk assessment and decision making. We propose two new methods of design approaches that sequentially select input settings to achieve this goal. The approaches make novel applications of simultaneous and sequential contour estimations. Numerical examples are employed to demonstrate the effectiveness of the proposed approaches., Comment: 24 pages

Design of experiments is a fundamental topic in applied statistics with a long history. Yet its application is often limited by the complexity and costliness of constructing experimental designs, which involve searching a high-dimensional input space and evaluating computationally expensive criterion functions. In this work, we introduce a novel approach to the challenging design problem. We will take a probabilistic view of the problem by representing the optimal design as being one element (or a subset of elements) of a probability space. Given a suitable distribution on this space, a generative point process can be specified from which stochastic design realizations can be drawn. In particular, we describe a scenario where the classical entropy-optimal design for Gaussian Process regression coincides with the mode of a particular point process. We conclude with outlining an algorithm for drawing such design realizations, its extension to sequential designs, and applying the techniques developed to constructing designs for Stochastic Gradient Descent and Gaussian process regression.

We develop a new method for constructing "good" designs for computer experiments. The method derives its power from its basic structure that builds large designs using small designs. We specialize the method for the construction of orthogonal Latin hypercubes and obtain many results along the way. In terms of run sizes, the existence problem of orthogonal Latin hypercubes is completely solved. We also present an explicit result showing how large orthogonal Latin hypercubes can be constructed using small orthogonal Latin hypercubes. Another appealing feature of our method is that it can easily be adapted to construct other designs; we examine how to make use of the method to construct nearly orthogonal and cascading Latin hypercubes., Comment: Published in at http://dx.doi.org/10.1214/09-AOS757 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)