Talk:Orthogonal array

Notation; mixed-level arrays

There are two problems with this article.

1. It ignores so-called mixed-level orthogonal arrays, where different factors can have different numbers of symbols. These arise naturally as fractional factorial designs.
2. The notation t-(v,k,λ) is problematic. It is very unusual for orthogonal arrays, and it can’t be used for mixed-level arrays. (The more common notation is OA(N,k,v,t) where N is the number of runs, and this notation is easily generalized to denote mixed-level arrays.)

More problematic, the same t-notation is used for combinatorial designs known as t-designs, where the parameters k and λ have very different meanings from those used here. Such designs are unrelated to orthogonal arrays. This is very confusing.

It would not be hard to fix the first problem, by simply adding a section on mixed-level arrays and making small modifications/additions elsewhere. The second issue is trickier. One option is to mention the OA(N,k,v,t) notation as an alternative at the beginning, so that it can be used for mixed-level arrays when suitably modified. But this means that the article would contain two distinct systems of notation.

A cleaner alternative is simply to replace all instances of the t-notation with the OA notation. This would be time-consuming but not difficult, but I don’t know what the editorial policy is on this. (I assume the original author is Douglas Stinson, whose book uses the t-notation for orthogonal arrays. That book is cited numerous times in the article.)

It would be worth having a brief section on notation, since there are yet other notations for orthogonal arrays. For example, the notation L_N is used in many industrial applications, probably due to Taguchi. There are others.

I would be happy to make all the modifications I've suggested.

(I note, incidentally, that the Combinatorial_design article doesn’t seem to define t-designs, although it indicates when such designs are quasi-symmetric.) Johsebb (talk) 18:14, 16 February 2023 (UTC)

Mixed-level arrays and other additions/corrections

I've made the following changes:

Added mixed-level arrays in several sections (and inserted the term “fixed-level” as needed)

Added an example of strength 3 to intro (which also illustrates the concept of a linear orthogonal array that is mentioned in the Definition section)

Added clarifications:

In the definition of a simple array, noted that subarrays may have repeated rows

Changed the first sentence on Hadamard matrices to “if and only if”

Added notes to History section.

Corrected minor points:

In the intro:

The set of symbols is not “typically” {1, …, n}.

Changed n to v for consistency with later sections.

In the Factorial Design subsection:

Corrected the description of “levels” (need not be “integral”, and need not be “high, low or intermediate”). This is more consistent with the Street and Street citation.

Orthogonal arrays are used to design fractional factorial experiments, and need not be of strength 2 as originally stated.

Added a remark about resolution.

Eliminated this: “but to minimize confounding influences the levels should be varied within any experimental run.” I believe it makes no sense.

Added references and citations as needed.

In all, I modified these sections/subsections:

Introduction

Definition

History

Hadamard Matrices

Factorial designs

Quality control

References

Johsebb (talk) 20:43, 1 June 2023 (UTC)

Minor changes

In the Definition section: I inserted "v-set" parenthetically in the definition, with a link to Finite set. (The term v-set is used later with no explanation.) I slightly reworded a couple of references to displayed tables so that they would work better (I think) when viewed on mobile devices.

I inserted a sentence in the Trivial example subsection that I think clarifies the original statement. Added a link to trivial (mathematics), and made a small grammar correction. Johsebb (talk) 16:59, 6 June 2023 (UTC)

Notational changes and minor edits

I have changed the t-notation (t-(v,k,λ)) to OA-notation (OA(N,k,v,t)) throughout, as I indicated in a previous comment. This required some rewording in the "Definition" section, including notation for mixed-level arrays.

I have inserted a "Terminology and notation" section. This includes the recommendation for OA-notation in the text by Hedayat et al; I am not one of the authors. (The reference at the end of the section will lead the reader to a source of the t-notation. It is worth noting that the name of the file imported in the "Definition" section uses the OA-notation, although the original caption did not.)

Aside from this, I combined small paragraphs in the "Latin cubes" and "Hadamard" subsections, along with some minor rewording. Johsebb (talk) 18:23, 26 June 2023 (UTC)

References