Orthogonal array
In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a "table" (array) whose entries come from a fixed finite set of symbols (for example, {1,2,...,v}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times.
In the example on the right,[1] the rows restricted to the first three columns contain the 8 possible ordered triples consisting of 0's and 1's, each appearing once.
(Subarrays of t columns may have repeated rows, as in the OA(18, 7, 3, 2) example pictured in this section.)
An orthogonal array is linear if X is a finite field Fq of order q (q a prime power) and the rows of the array form a subspace of the vector space (Fq)k.[2] The right-hand example in the introduction is linear over the field F2.
This is effective when the parameters all have specific numerical values, but less so when a class of orthogonal arrays is intended.
For example, when indicating the class of arrays having strength t = 2 and index λ=1, the notation OA(N, k, v, 2) is insufficient to determine λ by itself.
While notations that explicitly include the parameter λ do not have this problem, they cannot easily be extended to denote mixed-level arrays.
Hedayat, Sloane and Stufken[6] recommend it as standard, but list eight alternatives found in the literature, and there are others.
Any OA(N, k, v, k) would be considered trivial since such arrays are easily constructed by simply listing all the k-tuples of the v-set λ times.
Let A be a strength 2, index 1 orthogonal array on an n-set of elements, identified with the set of natural numbers {1,...,n}.
[10] Orthogonal arrays provide a uniform way to describe these diverse objects which are of interest in the statistical design of experiments.
In fact, no row, column or file (the cells of a particular position in the different layers) need be a permutation of the n symbols.
A set of k − 3 mutually orthogonal Latin cubes of order n is equivalent to an OA(n3, k, n, 2).
An m-dimensional Latin hypercube of order n of the rth class is an n × n × ... ×n m-dimensional matrix having nr distinct elements, each repeated nm − r times, and such that each element occurs exactly n m − r − 1 times in each of its m sets of n parallel (m − 1)-dimensional linear subspaces (or "layers").
The possibility of non-simple arrays arose naturally when making treatment combinations the rows of a matrix.
Hedayat, Sloane and Stufken[17] credit K. Bush[18] with the term "orthogonal array".
[19] To proceed in one direction, let H be a Hadamard matrix of order 4m in standardized form (first row and column entries are all +1).
Delete the first row and take the transpose to obtain the desired orthogonal array.
[24] In a full factorial experiment all combinations of levels of the factors need to be tested.
An experimental run is a row of the orthogonal array, that is, a specific combination of factor levels.
This article incorporates public domain material from the National Institute of Standards and Technology
