Sunday, September 18, 2005

Orthogonal Arrays

Orthogonal Arrays: "Orthogonal arrays are beautiful and useful. They are essential in statistics and they are used in computer science and cryptography. In statistics they are primarily used in designing experiments, which simply means that they are immensely important in all areas of human investigation: for example in medicine, agriculture and manufacturing.

Your automobile lasts longer today because of orthogonal arrays ['The new mantra: MVT', Forbes, Mar. 11, 1996, pp. 114-118.]

The mathematical theory is extremely beautiful: orthogonal arrays are related to combinatorics, finite fields, geometry and error-correcting codes. The definition of an orthogonal array is simple and natural, and we know many elegant constructions - yet there are at least as many unsolved problems.

Here is an example of an orthogonal array of strength 2:

0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 0 0 0
0 1 1 1 0 1 1 0 1 0 0
0 0 1 1 1 0 1 1 0 1 0
0 0 0 1 1 1 0 1 1 0 1
1 0 0 0 1 1 1 0 1 1 0
0 1 0 0 0 1 1 1 0 1 1
1 0 1 0 0 0 1 1 1 0 1
1 1 0 1 0 0 0 1 1 1 0
0 1 1 0 1 0 0 0 1 1 1
1 0 1 1 0 1 0 0 0 1 1
1 1 0 1 1 0 1 0 0 0 1

Pick any two columns, say the first and the last:

0 0
1 0
0 0
0 0
0 1
1 0
0 1
1 1
1 0
0 1
1 1
1 1

Each of the four possible rows we might see there,
0 0, 0 1, 1 0, 1 1,

does appear, and they all appear the same number of times (three times, in fact). That's the property that makes it an orthogonal array.

Only 0's and 1's appear in that array, but for use in statistics

0 or 1

in the first column might be replaced by

'butter' or 'margarine' ,

and in the second column by

'sugar' or 'no sugar' ,

and so on. Or

'slow cooling' or 'fast cooling',
'catalyst' or 'no catalyst',

etc., depending on the application.

Since only 0's and "
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