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Latin Squares were first investigated by Euler in 1782, as a sort of mathematical curiosity, such as magic squares. Their first application as experimental designs method was found in agriculture. With this method it is possible to estimate only the main effects.
A Latin Square of order n is an n by n matrix with entries from some alphabet of size n (i.e. a list of n symbols) such that the entries in each row and the entries in each column are distinct. The name is explained by the fact that originally the list of symbols was taken from the Latin alphabet. Here n is the number of levels of each variable.
For example a Latin Square for a 3-variables of 4-levels problem is the following one:
this matrix represents 4x4 = 16 designs: the levels of the first variable are represented by the row indices {1,2,3,4}, the levels of the second variable are represented by the column indices {1,2,3,4}, and finally the levels of the third variable are represented by the entries in the matrix {a,b,c,d}.The number of generated designs is n2, where n is the number of levels. The number of levels is the same for all the variables. The Latin Square experiments result balanced. Since the maximum number of levels is limited to 20, the maximum number of generated designs is limited to 400. This algorithm can be used only with at least three variables.
With Nv input variables, the number of Latin squares generated by the algorithm is NLS = Nv - 2. So if the number of variables is equal to four, two Latin squares are used. In this case a nice feature can be used: Graeco-Latin squares. Two Latin squares of order n are said to be orthogonal if the n2 pairs formed by superimposing the two matrices are all distinct. Such orthogonal squares are called Graeco-Latin squares (or Euler squares), because originally the lists of symbols were taken from the Latin and the Greek alphabets.
An example of Graeco-Latin squares for a 4-variables of 3-levels problem is:
the levels of the first variable are represented by the row indices, the levels of the second variable are represented by the column indices, the levels of the third variable are represented by the entries in the first matrix (Latin letters), and the levels of the fourth variable are represented by the entries in the second matrix (Greek letters).Within modeFRONTIERTM the Graeco-Latin squares feature can be used by selecting the Orthogonality Criterion, as explained below.
A better description of the algorithm is available in the paper Latin Square DOE.
Several parameters can be defined for this algorithm:
Number of Levels an integer number between 2 and 20 (the real minimum is automatically fixed according to the number of input variables). If n is the number of levels, the number of generated designs is n2.
Criterion two options exist:
Reduce Correlation The configuration with the minimum variables correlation is chosen. This method can be applied only if the number of input variables is greater than three. If the Orthogonality Criterion cannot be applied, then the Reduce Correlation method is automatically selected
Orthogonality The two generated Latin Squares are orthogonal, i.e. they are Graeco-Latin squares. This method can be applied only if the number of input variables is equal to four, and if the number of levels is 3, 4, or 5.
Number of Iterations a positive integer number between 1 and the maximum number of iterations (i.e. 9999). This parameter is used if the above Reduce Correlation method is performed.
Random Generator Seed a positive integer number, used for sequence repeatability. If two Latin Square DOEs are created with the same seed, they will generate and return identical sequences of numbers. If the seed value is 0, the sequence is automatically seeded with a clock value.
[1] Fisher, R. and Yates, F. 1934, The 6 x 6 Latin Squares, Proceedings of the Cambridge Philosophical Society, 30, 492-507
[2] Bose, R. and Manvel, B. 1984, Introduction to Combinatorial Theory, John Wiley & Sons, 135-149 (Ch. 7)
[3] McKay, B. and Rogoyski, E. 1995. Latin Squares of Order 10, Electronic Journal of Combinatorics, 2(3), 1-4
[4] Jacobson, M. and Matthews, P. 1996, Generating Uniformly Distributed Latin Squares, Journal of Combinatorial Designs, 4(6), 405-437
[5] Eric W. Weisstein. "Latin Square." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LatinSquare.html
