How to cite

Grömping U. A new kid on the block: The stratification pattern for space-filling, with dimension by weight tables. Qual Reliab Engng Int. 2024; 1–21. https://doi.org/10.1002/qre.3627

Abstract

Computer experiments in several or even many quantitative experimental variables require experimental designs that use many levels for each variable, in order to provide enough information for identifying the functional form from the data. The “curse of dimensionality” implies that an increasing number of experimental variables dramatically increases the size of the experimental space. Filling that experimental space is the most important property of experimental designs for computer experiments, and there are many criteria that assess aspects of space-filling. This paper explains a newly-introduced criterion, the stratification pattern for space-filling that comes with supplementary dimension by weight tables. This pattern is applicable for designs whose number of levels, i.e., number of different values for each experimental variable, is a power of an integer, e.g., 2³ or 3⁵ or 2¹⁰. Such designs are also known as Generalized Stratified Orthogonal Arrays (GSOAs; some authors use “Strong” instead of “Stratified” for expanding the S); in the special case for which there are as many different levels as there are experimental runs, a GSOA is a Latin Hypercube Design (LHD).

This paper explains the ideas behind the proposed stratification pattern and the related ranking criteria. A practical example in 64 runs with 9 experimental variables in 64 levels each, as well as several toy examples, aid the illustration. The stratification pattern can be calculated using the R package SOAs, which does not only provide the pattern itself but also provides more detail in a dimension by weight table, in the spirit of the refinement by Shi and Xu.

The recent addition to the toolbox for assessing space-filling behaviour supplements many existing metrics. Typically, the proposal of a new metric is accompanied by a demonstration of its benefits. This is also the case for the stratification pattern and its refinements presented in this paper: their success was demonstrated in small simulation studies by the author pairs Tian and Xu as well as Shi and Xu. In this paper, a small simulation study compares the design from the practical example (which is a GSOA from a construction by Ye from a time before GSOAs were invented), with several further GSOAs of the same general layout (64 runs in 9 experimental variables) regarding their performance on some metrics as well as their performance in predicting a well-known benchmark function, the so-called borehole function. The partly surprising results of this very small study point to a need for research into the use of space-filling metrics for selecting designs for practical applications.