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As a basic tool of the experience plans, the Hadamard matrices have allowed to “automate” the planning of the experiments while guaranteeing the quality of the mathematical result.

Introduction

The plans of experiments are all based on the matrixes of Hadamard. These matrices were invented by Sylvester in 18671, which built of any order, a power of 2, thanks to his idea of doubling. Hadamard, a famous French mathematician known for his first-number theorem, built of order 12 and 20 and formulated his famous conjecture in 1893: ” for any n multiple of 4, there is a matrix of Hadamard of order n”.

Jacques Salomon Hadamard (1865-1963)

Property of Hadamard

A Hadamard array of order N is an array with n rows and n columns such as:

  • All its coefficients are worth-1 or + 1
  • For each pair of lines, there are as many concordances as inconsistencies, i.e. columns where the coefficients are equal, and columns where the coefficients are distinct.

In other words, in the following matrix where n = 8, take two lines, and count the columns whose two squares are of the same color, that is, both white or both orange. So we find exactly 4 color matches , and thus automatically 4 color discrepancies .

Hadamard matrix with n = 8

Lines 1 and 2 of the logo

The color matches are in columns 1, 3, 5 and 7, and the discrepancies in the remaining four columns.

Lines 3 and 5 of the logo

Color matches are held here in columns 1, 2, 7 and 8.

Interest of Hadamard

Hadamard arrays have various properties that are widely used in encoding, signal processing… In the case of experience plans, we are interested in 2 properties:

  • They allow to maximize the determinant of a matrix, in other words, the criterion called D-optimality2 (d for determinant). It is described as follows: tXX = NIn. By maximizing the determinant of a matrix, the variance and covariance of the result are minimized. This allows us, in our system of equations to be solved, to minimise the Variance of the effects of each factor and to have a good accuracy in the estimates.
  • They ensure the orthogonality: if we replace the color of the squares with + 1 and-1 or several other values (if we have more than 2 levels per factor), the sum of each column must be equal to 0. So the more orthogonal the matrix, the better the plan will be. If the plane is orthogonal, it is ensured that factor A is taken into account at the same weight as factor B. So there is not one factor that is taken more into consideration than the other.

Source

1 – J. J. Sylvester (1867)-Thoughts on inverse orthogonal matrixes, simultaneous sign-inheritances, and tessellated pavements in two or more colors, with applications to Newton rule, ornamental tile-work, and the theory of numbers

2 – J. J. Droesbeke, J. Fine, G. Soporta (1996) – Experience Plans, application to the company

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