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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]

In many practical experiments, both the level combinations of factors and the addition orders will affect the responses. However, virtually no construction methods have been provided for such experimental designs. This paper focuses on such experiments, introduces a new type of design called the ordering factorial design, and proposes the nominal main effect component-position model and interaction-main effect component-position model. To obtain efficient fractional designs, we provide some deterministic construction methods. The resulting designs are D -optimal, and the run sizes are much smaller than that of the full designs. Moreover, in some cases, some constructed designs are still D -optimal after reducing the number of components and factors. [ABSTRACT FROM AUTHOR]