Hadamard matrix designs

Preparation conditions have been varied following an experimental design (Hadamard matrix) as shown in Table 1. 

Statistical optimization of a controlled release formulation obtained by a double-compression process Application of a Hadamard matrix and a factorial design... 

According to the Hadamard matrix, a 22 factorial design was built. The complete linear models were fitted by regression for each response, reflecting the compression behaviour and dissolution kinetics. 

The theory and application of a Hadamard matrix were discussed by Ozil and Rochat [18]. This design is an optimum strategy leading to good accuracy in the main effects... 

From a minimum number of experiments, the Hadamard matrix gives the possibility of estimating the mean effects of four parameters. Among them, the particle size range had the most important effect in the release of diclofenac sodium. By interpreting data, a factorial design including only two parameters was applied from which an optimum formulation was found. 

What happens if n is a multiple of 8 Cheng (1995) showed that, if the widely held conjecture is true that a Hadamard matrix exists for every order that is a multiple of 4, then for every n that is a multiple of 8, one can always construct an OA(n, 2, 2) that has defining words of length three or four. Such designs are not of projectivity 3, a fact also pointed out by Box and Tyssedal (1996), and do not have the desirable hidden projection properties mentioned above. However, this does not mean that, when n is a multiple of 8, there are no OA( ,. 2)s with good... 

This method can be illustrated by taking the Plackett-Burman design for 11 factors in 12 runs, which is based on a 12 x 12 Hadamard matrix and is shown in Table 2. Selecting the rows corresponding to + in Factor 11 and dropping this factor gives the supersaturated design for 10 factors in 6 runs shown in Table 1. 

Various other approaches to constructing nearly /i(.v2)-optimal designs have been suggested. One is the construction of Wu (1993), described in Section 2.2, based on adding interaction columns to a Hadamard matrix which, when it does not produce / (.v2)-opt.imal designs, produces designs with reasonably high E(s2)-efficiencies. 

Deng, L. Y., Lin, D. K. J., and Wang, J. N. (1994). Supersaturated design using Hadamard matrix. IBM Research Report RC19470, IBM Watson Research Center, New York. 

In this chapter, two other principles for the construction of screening designs are discussed design by a Hadamard matrix and D-optimal design. Through such designs the inconveniences mentioned above can be overcome. However, other... 

The construction of a design by a Hadamard matrix is simple. Plackett and Burman have determined how the first row in the design matrix should be constructed so that the remaining rows can be obtained by cyclic permutations of the first row. An example to show this is given below. For a given n there can be many different n xn matrices which are Hadamard matrices. The Plackett-Burman matrices are by no means the unique solutions. 

Hadatnard matrices make the best chemical balance weighing designs, i.e., the best matrices with entries +1, 0, and -1. If there are n unknowns and a Hadamard matrix of order n is used, the mean squared error in each unknown is reduced by a factor of n or in other words the signal-to-noise ratio is increased by a factor of VIT... 

Furthermore it can be shown that Hadamard matrices are also D-optimal (cf. ref. 11, Section 3.2.5 of ref. 2). Equations 11 and 12 describe a weighing design based on a Hadamard matrix of order 4. 

A weighing experimental design containing only elements equal to I is called a Hadamard design if the resulting information matrix X X built from the N points is such that ... 

Strictly speaking, it is not the experimental design that is a Hadamard design but the model matrix X. But this notation is now common and we will use it. In order to build this type of experimental design, it is advised to use the mode of generation proposed by Plackett and Burman (6), which we review here Knowing the number of factors k, we determine the minimum number of experiments needed to study a polynomial model of degree 1 = it + I. and we seek the... 

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