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Suppose there are quite many (hundreds/thousands) variables, $X_1$,...,$X_m$, in a computer model, following some multivariate distribution. How can we get a sample of these variables that has good coverage of the possible values while the sample size is not large saying 100-500?

In addition, if a low-dimensional transformation is applied to these variables saying $Z_1=T_1X$,...,$Z_d=T_dX$, where $X=(X_1,...,X_N)'$ and $d<<m$. If there is a way to sample or make design on the low-dimensional space $(Z_1,...Z_d)$ so that a moderate size of samples on $(Z_1,...Z_d)$ can cover the possible values well. I know Latin Hypercube Design is often used in computer experiment, but here the transformed variables $Z_1$,...$Z_d$ are not independent that the domain of these values are not a hypercube... Would anyone have experience or any thoughts or references on this problem?

Many thanks!

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  • $\begingroup$ If the transformation is PCA, then the $Z$ values will be independent, that might be a solution? $\endgroup$
    – jeff
    Commented Dec 7, 2015 at 22:04
  • $\begingroup$ @HalilPazarlama They are not in a regular shape though... $X$'s are correlated... $\endgroup$
    – Eridk Poliruyt
    Commented Dec 9, 2015 at 16:05

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I suggest a hierarchic serpentine sort of the $Z$'s, followed by a systematic sample. A serpentine sort on the Z's assures that neighboring observations are closer together than they would be with a standard sort. The systematic sample will ensure that the entire sorted list is covered.

A hierarchic serpentine sort was formally defined in Williams, Rick L, and James R Chromy. 1980. SAS sample selection macros. Proceedings of the Fifth Annual SAS Users Group International Conference 5, 392-396. http://www.sascommunity.org/sugi/SUGI80/Sugi-80-71%20Williams%20Chromy.pdf.

A simple three-variable example is shown in Sub-population of large dataset based on multiple criteria.

PROC SURVEYSELECT in SAS can do the sort described by Williams and Chromy and will take the systematic sample. See p. 8484 of the procedure manual chapter (http://www.math.wpi.edu/saspdf/stat/chap63.pdf).

A SAS macro by Barbara Lepidus Carlson and Linda Bandeh does serpentine sorting. I don't have a SAS installation, so haven't tried it, but the logic is explained in some detail. See p.1090 of http://www.sascommunity.org/sugi/SUGI95/Sugi-95-182%20Carlson%20Cox%20Bandeh.pdf.

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