Creates stratum orthogonal arrays (also known as strong orthogonal arrays).
This package constructs arrays in \(s^{el}\) levels from orthogonal arrays in s levels.
These are all based on equations of the type
$$D = s^{el-1} A_1 + ... + s A_{el-1} + A_{el},$$
or for \(s^2\) levels, $$D = s A + B$$
and for \(s^3\) levels, $$D = s^2 A + s B + C.$$
The constructions differ in how they obtain the ingredient matrices, and what properties can be guaranteed for the resulting D.
Where a construction function guarantees orthogonal columns for all matrices D it produces, its name starts with a OSOA, otherwise with SOA.
If optimization is requested (default TRUE), space filling properties of D are improved using a level permutation algorithm
by Weng (2014). This algorithm is applied for improving the phi_p
criterion, which is often a reasonable surrogate for increasing the minimum distance.
Groemping (2023a) describes the constructions by He and Tang (2013, function SOAs),
Liu and Liu (2015, function OSOAs_LiuLiu), He, Cheng and Tang (2018, function SOAs2plus_regular),
Zhou and Tang (2019), Shi and Tang (2020, function SOAs_8level) and Li, Liu and Yang (2021) in unified notation.
The constructions by Zhou and Tang (2019) and Li et al. (2021) are very close to each other and are both implemented
in the three functions OSOAs, OSOAs_hadamard and OSOAs_regular.
Within the package, available SOA constructions for specific situations can be queried using
the guide functions guide_SOAs and guide_SOAs_from_OA.
Besides the construction functions, properties of the resulting array D can be checked using the aforementioned function
phi_p as well as check functions ocheck, ocheck3 for orthogonality and
soacheck2D, soacheck3D for (O)SOA stratification properties, and
Spattern for the space-filling pattern proposed by Tian and Xu (2022); the implementation
of the latter will presumably become more important than the
2D and 3D check functions eventually.
There is one further construction, maximin distance level expansion (XiaoXuMDLE, MDLEs),
that does not yield stratum (aka strong) orthogonal arrays and is available for comparison only (Xiao and Xu 2018).
Groemping, U. (2022). Implementation of the stratification pattern by Tian and Xu via power coding. Report 2022/03, Reports in Mathematics, Physics and Chemistry, Berliner Hochschule fuer Technik. http://www1.bht-berlin.de/FB_II/reports/Report-2022-003.pdf
Groemping, U. (2023a). A unifying implementation of stratum (aka strong) orthogonal arrays. Computational Statistics and Data Analysis 183, 1-28. doi:10.1016/j.csda.2023.107739
Groemping, U. (2023b). Implementating the stratification pattern for space-filling, with dimension by weight tables. Report 2023/01, Reports in Mathematics, Physics and Chemistry, Berliner Hochschule fuer Technik. http://www1.bht-berlin.de/FB_II/reports/Report-2023-001.pdf
He, Y., Cheng, C.S. and Tang, B. (2018). Strong orthogonal arrays of strength two plus. The Annals of Statistics 46, 457-468. doi:10.1214/17-AOS1555
He, Y. and Tang, B. (2013). Strong orthogonal arrays and associated Latin hypercubes for computer experiments. Biometrika 100, 254-260. doi:10.1093/biomet/ass065
Li, W., Liu, M.-Q. and Yang, J.-F. (2021). Construction of column-orthogonal strong orthogonal arrays. Statistical Papers doi:10.1007/s00362-021-01249-w .
Liu, H. and Liu, M.-Q. (2015). Column-orthogonal strong orthogonal arrays and sliced strong orthogonal arrays. Statistica Sinica 25, 1713-1734. doi:10.5705/ss.2014.106
Shi, L. and Tang, B. (2020). Construction results for strong orthogonal arrays of strength three. Bernoulli 26, 418-431. doi:10.3150/19-BEJ1130
Tian, Y. and Xu, H. (2022). A minimum aberration-type criterion for selecting space-filling designs. Biometrika 109, 489-501. doi:10.1093/biomet/asab021
Weng, J. (2014). Maximin Strong Orthognal Arrays. Master's thesis at Simon Fraser University under supervision of Boxin Tang and Jiguo Cao. https://summit.sfu.ca/item/14433
Xiao, Q. and Xu, H. (2018). Construction of Maximin Distance Designs via Level Permutation and Expansion. Statistica Sinica 28, 1395-1414. doi:10.5705/ss.202016.0423
Zhou, Y.D. and Tang, B. (2019). Column-orthogonal strong orthogonal arrays of strength two plus and three minus. Biometrika 106, 997-1004. doi:10.1093/biomet/asz043
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