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Mar 5, 2012 9:48 AM
(622 views)

Mathemagical musings

Take a look at this matrix of symbols.

++++++++++++++++++++++++++++++++

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

++−−++−−++−−++−−++−−++−−++−−++−−

+−−++−−++−−++−−++−−++−−++−−++−−+

++++−−−−++++−−−−++++−−−−++++−−−−

+−+−−+−++−+−−+−++−+−−+−++−+−−+−+

++−−−−++++−−−−++++−−−−++++−−−−++

+−−+−++−+−−+−++−+−−+−++−+−−+−++−

++++++++−−−−−−−−++++++++−−−−−−−−

+−+−+−+−−+−+−+−++−+−+−+−−+−+−+−+

++−−++−−−−++−−++++−−++−−−−++−−++

+−−++−−+−++−−++−+−−++−−+−++−−++−

++++−−−−−−−−++++++++−−−−−−−−++++

+−+−−+−+−+−++−+−+−+−−+−+−+−++−+−

++−−−−++−−++++−−++−−−−++−−++++−−

+−−+−++−−++−+−−++−−+−++−−++−+−−+

++++++++++++++++−−−−−−−−−−−−−−−−

+−+−+−+−+−+−+−+−−+−+−+−+−+−+−+−+

++−−++−−++−−++−−−−++−−++−−++−−++

+−−++−−++−−++−−+−++−−++−−++−−++−

++++−−−−++++−−−−−−−−++++−−−−++++

+−+−−+−++−+−−+−+−+−++−+−−+−++−+−

++−−−−++++−−−−++−−++++−−−−++++−−

+−−+−++−+−−+−++−−++−+−−+−++−+−−+

++++++++−−−−−−−−−−−−−−−−++++++++

+−+−+−+−−+−+−+−+−+−+−+−++−+−+−+−

++−−++−−−−++−−++−−++−−++++−−++−−

+−−++−−+−++−−++−−++−−++−+−−++−−+

++++−−−−−−−−++++−−−−++++++++−−−−

+−+−−+−+−+−++−+−−+−++−+−+−+−−+−+

++−−−−++−−++++−−−−++++−−++−−−−++

+−−+−++−−++−+−−+−++−+−−++−−+−++−

Beautiful isn’t it? This matrix has several wonderful properties. Ignoring the first column composed of all pluses, each column has the same number of pluses and minuses. Also, each pair of columns has eight replicates of all possible pairs of symbols, (++, +−,−+,−−). That makes this matrix a so-called **strength-two orthogonal array**. Using this matrix as a design for an experiment, you can do a sensitivity analysis of the main effects of 31 factors using only 32 runs.

*What is the probability of generating this matrix by chance?*

Here is one possible answer. There are 32 rows and 32 columns in this matrix. That makes 32x32 = 1024 total symbols. Each symbol can take two values, so the number of possible matrices is 2^{1024}. If one generated a random 32 by 32 matrix of pluses and minuses, then the probability of generating the matrix above is 1/2^{1024} – and that is a very small number. Since this way of generating the matrix by chance is not feasible in our lifetime (or very possibly the life of the universe), we might suppose that the matrix could not be generated by chance.

*Could there be another way?*

We have two symbols, + and −. Consider this rule for combining symbols. If I give you a symbol or matrix of symbols, H, you give me back a matrix that looks like this,

H H

H –H

For example, starting with + and applying our rule we get

++

+−

Applying the rule to the matrix above yields

++++

+−+−

++−−

+−−+

Three more iterations produce our elegant original matrix.

The rule is extremely simple. The probability of generating *the rule* by chance could be as large as ½^{4} = 1/16. So, to generate the original matrix all we need are the two symbols, our rule of combination and five repeats.

*And your point is?*

Our original probability assessment involved the assumption of 1,024 independent chance events. This assumption, this declaration of independence as it were, is not necessary.

The assumption of independence is extremely strong. Use it in statistical, or philosophical, arguments with care!

*Footnote*

The original matrix is called a Hadamard matrix after the 19th century French mathematician of that name. Hadamard also invented the simple rule that allows one to take one Hadamard matrix and turn it into another that has twice as many rows and columns.

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