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Shapley value is originally a concept in econometrics to fairly distribute both gains and costs to players in a coalition game. In the recent decades, its application has been extended to other areas such as marketing, engineering and machine learning. For example, it produces reasonable solutions for problems in sensitivity analysis, local model explanation toward the interpretable machine learning, node importance in social network, attribution models, etc. However, it could be very expensive to compute the Shapley value. Specifically, in a d-player coalition game, calculating a Shapley value requires the evaluation of d ! or 2 d marginal contribution values, depending on whether we are taking the permutation or combination formulation of the Shapley value. Hence, it becomes infeasible to calculate the Shapley value when d is reasonably large. A common remedy is to take a random sample of the permutations to surrogate for the complete list of permutations. We find an advanced sampling scheme can be designed to yield much more accurate estimation of the Shapley value than the simple random sampling (SRS). Our sampling scheme is based on combinatorial structures in the field of design of experiments (DOE), particularly the order-of-addition experimental designs for the study of how the orderings of components would affect the output. We show that the obtained estimates are unbiased, and can sometimes deterministically recover the original Shapley value. Both theoretical and simulations results show that our DOE-based sampling scheme outperforms SRS in terms of estimation accuracy. Surprisingly, it is also slightly faster than SRS. Lastly, real data analysis is conducted for the C. elegans nervous system and the 9/11 terrorist network. for this article are available online. [ABSTRACT FROM AUTHOR]

Maximin distance designs and orthogonal designs are two attractive classes of space-filling designs for computer experiments, but their theoretical constructions are challenging, especially the construction of optimal designs in terms of both the maximin distance and orthogonality criteria. This paper presents a systematic method for constructing orthogonal maximin distance designs with flexible numbers of runs and factors. The method is carried out by rotating the subarrays of a saturated two-level regular design in Yates order or its circular shifting version. The principal objective is to construct high-level designs from two-level designs, and the method is effective because the performance of high-level designs is determined by that of two-level designs under both the maximin distance and orthogonality criteria. The proposed method is also generalized by rotating the subarrays of a saturated two-level nonregular design such that the resulting designs have flexible run sizes. Comparison results reveal that the resulting orthogonal designs are well worthy of recommendation under the maximin distance criterion. An illustrative example is provided to show that the proposed designs have a good two-dimensional stratification property. An application is given to present the effectiveness of the proposed designs in building statistical surrogate models. [ABSTRACT FROM AUTHOR]

Drawing samples from a target distribution is essential for statistical computations when the analytical solution is infeasible. Many existing sampling methods may be easy to fall into the local mode or strongly depend on the proposal distribution when the target distribution is complicated. In this article, the Global Likelihood Sampler (GLS) is proposed to tackle these problems and the GL bootstrap is used to assess the Monte Carlo error. GLS takes the advantage of the randomly shifted low-discrepancy point set to sufficiently explore the structure of the target distribution. It is efficient for multimodal and high-dimensional distributions and easy to implement. It is shown that the empirical cumulative distribution function of the samples uniformly converges to the target distribution under some conditions. The convergence for the approximate sampling distribution of the sample mean based on the GL bootstrap is also obtained. Moreover, numerical experiments and a real application are conducted to show the effectiveness, robustness, and speediness of GLS compared with some common methods. It illustrates that GLS can be a competitive alternative to existing sampling methods. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

A supersaturated design (SSD) is a factorial design whose run size is not enough for estimating all the main effects. Such designs have received much recent interest because of their potential in factor screening experiments. This paper first shows that the design obtained by the Kronecker sum of a balanced design and a generalized Hadamard matrix (i.e., a matrix with both itself and its transpose being difference matrices) has some nice properties. Based on these findings, some new methods for constructing E (f NOD) -optimal SSDs via generalized Hadamard matrices are developed. Meanwhile, the non-orthogonality of the proposed designs is well controlled by the source designs. In addition, some generalized Hadamard matrices with nice properties are constructed for obtaining E (f NOD) -optimal SSDs. The proposed methods are easy to implement and many new SSDs can then be constructed. [ABSTRACT FROM AUTHOR]

The order-of-addition experiment aims at determining the optimal order of adding components such that the response of interest is optimized. Order of addition has been widely involved in many areas, including bio-chemistry, food science, nutritional science, pharmaceutical science, etc. However, such an important study is rather primitive in statistical literature. In this paper, a thorough study on pair-wise ordering designs for order of addition is provided. The recursive relation between two successive full pair-wise ordering designs is developed. Based on this recursive relation, the full pair-wise ordering design can be obtained without evaluating all the orders of components. The value of the D-efficiency for the full pair-wise ordering model is then derived. It provides a benchmark for choosing the fractional pair-wise ordering designs. To overcome the unaffordability of the full pair-wise ordering design, a new class of minimal-point pair-wise ordering designs is proposed. A job scheduling problem as well as simulation studies are conducted to illustrate the performance of the pair-wise ordering designs for determining the optimal orders. It is shown that the proposed designs are very efficient in determining the optimal order of addition. [ABSTRACT FROM AUTHOR]

An efficient variable selection technique for kriging in computer experiments is proposed. Kriging models are popularly used in the analysis of computer experiments. The conventional kriging models, the ordinary kriging, and universal kriging could lead to poor prediction performance because of their prespecified mean functions. Identifying an appropriate mean function for kriging is a critical issue. In this article, we develop a Bayesian variable-selection method for the mean function and the performance of the proposed method can be guaranteed by the convergence property of Gibbs sampler. A real-life application on piston design from the computer experiment literature is used to illustrate its implementation. The usefulness of the proposed method is demonstrated via the practical example and some simulative studies. It is shown that the proposed method compares favorably with the existing methods and performs satisfactorily in terms of several important measurements relevant to variable selection and prediction accuracy. [ABSTRACT FROM AUTHOR]

The uniform design is a kind of important experimental design which has great practical value in production and living. Most existing literatures on this topic focus on the construction of uniform designs on regular regions. However, because of the complexity of practical situations, the irregular design region is more common in real life. In this paper, an algorithm is proposed to the construction of nearly uniform designs on irregular regions. The basic idea is to make use of uniform designs on a larger regular region with the irregular region being a subregion. Some theoretical justifications on the proposed algorithm are provided. Both the comparisons with the existing results and a real-life example show that our proposed algorithm is effective. [ABSTRACT FROM PUBLISHER]

Computer experiments have received a great deal of attention in many fields of science and technology. Most literature assumes that all the input variables are quantitative. However, researchers often encounter computer experiments involving both qualitative and quantitative variables (BQQV). In this article, a new interface on design and analysis for computer experiments with BQQV is proposed. The new designs are one kind of sliced Latin hypercube designs with points clustered in the design region and possess good uniformity for each slice. For computer experiments with BQQV, such designs help to measure the similarities among responses of different level-combinations in the qualitative variables. An adaptive analysis strategy intended for the proposed designs is developed. The proposed strategy allows us to automatically extract information from useful auxiliary responses to increase the precision of prediction for the target response. The interface between the proposed design and the analysis strategy is demonstrated to be effective via simulation and a real-life example from the food engineering literature. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Mixture experiments have attracted increasingly attention due to their great practical value in production and living, while uniform designs over irregular experimental regions have become a hot topic in the area of experimental designs in the past two decades. Noting that the experimental region of a mixture experiment withqcomponents under some constraints is in fact a (q− 1)-dimensional geometry, this article proposes a new method for searching nearly uniform designs for mixture experiments with any complex constraints. Two examples with some tables and figures are given to illustrate this method. [ABSTRACT FROM PUBLISHER]

In computer experiments, space-filling designs with a sliced structure or nested structure have received much recent interest and been studied separately. However, it is likely that designs with both structures are needed in some situations, but there are no suitable designs so far. In this paper, we construct a special class of nested Latin hypercube designs with sliced structures, in such a design, a small sliced Latin hypercube design is nested within a large one. The construction method is easy to implement and the number of factors is flexible. Numerical simulations show the usefulness of the newly proposed designs. [ABSTRACT FROM PUBLISHER]

Fractional factorial split-plot (FFSP) designs have received much attention in recent years. In this article, the matrix representation for FFSP designs with multi-level factors is first developed, which is an extension of the one proposed by Bingham and Sitter (1999b) for the two-level case. Based on this representation, periodicity results of maximum resolution and minimum aberration for such designs are derived. Differences between FFSP designs with multi-level factors and those with two-level factors are highlighted. [ABSTRACT FROM AUTHOR]

Clear effects criterion is one of the most important rules for selecting 2 m - p designs, and it is valuable to know the maximum number of clear two-factor interactions in 2 m - p designs. This article provides lower bounds on the maximum number of clear two-factor interactions for and designs by constructing specific designs, respectively. The lower bounds improve the ones given in Tang et al. (2002) and the construction methods perform quite well. [ABSTRACT FROM AUTHOR]

Abstract The spread of COVID-19 makes it essential to investigate its prevalence. In such investigation research, as far as we know, the widely-used sampling methods didn’t use the information sufficiently about the numbers of the previously diagnosed cases, which provides a priori information about the true numbers of infections. This motivates us to develop a new, two-stage sampling method in this paper, which utilizes the information about the distributions of both population and diagnosed cases, to investigate the prevalence more efficiently. The global likelihood sampling, a robust and efficient sampler to draw samples from any probability density function, is used in our sampling strategy, and thus, our new method can automatically adapt to the complicated distributions of population and diagnosed cases. Moreover, the corresponding estimating method is simple, which facilitates the practical implementation. Some recommendations for practical implementation are given. Finally, several simulations and a practical example verify its efficiency. [ABSTRACT FROM AUTHOR]