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Hierarchical data analysis is crucial in various fields for making discoveries. The linear mixed model is often used for training hierarchical data, but its parameter estimation is computationally expensive, especially with big data. Subsampling techniques have been developed to address this challenge. However, most existing subsampling methods assume homogeneous data and do not consider the possible heterogeneity in hierarchical data. To address this limitation, we develop a new approach called group-orthogonal subsampling (GOSS) for selecting informative subsets of hierarchical data that may exhibit heterogeneity. GOSS selects subdata with balanced data size among groups and combinatorial orthogonality within each group, resulting in subdata that are D- and A-optimal for building linear mixed models. Estimators of parameters trained on GOSS subdata are consistent and asymptotically normal. GOSS is shown to be numerically appealing via simulations and a real data application. Theoretical proofs, R codes, and supplementary numerical results are accessible online as . [ABSTRACT FROM AUTHOR]

Strong orthogonal arrays are suitable designs for computer experiments because of stratification in low‐dimensional projections. However, strong orthogonal arrays may be very expensive for a moderate number of factors. In this article, we develop a method for constructing space‐filling designs with more economical run sizes. These designs are not only column‐orthogonal but also enjoy a large proportion of low‐dimensional stratification properties that strong orthogonal arrays ought to have. Moreover, a class of proposed designs can be 3‐orthogonal. In addition, some theoretical results on regular fractional factorial designs are obtained as a by‐product. [ABSTRACT FROM AUTHOR]

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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]

Space-filling designs are widely used in both computer and physical experiments. Column-orthogonality, maximin distance, and projection uniformity are three basic and popular space-filling criteria proposed from different perspectives, but their relationships have been rarely investigated. We show that the average squared correlation metric is a function of the pairwise L2-distances between the rows only. We further explore the connection between uniform projection designs and maximin L1-distance designs. Based on these connections, we develop new lower and upper bounds for column-orthogonality and projection uniformity from the perspective of distance between design points. These results not only provide new theoretical justifications for each criterion but also help in finding better space-filling designs under multiple criteria. for this article are available online. [ABSTRACT FROM AUTHOR]