A Description of Experimental Design on the Basis of an Orthonormal System

The Fourier series representation of a function is a classic representation which is widely used to approximate real functions (Stein & Shakarchi, 2003). In digital signal processing (Oppenheim & Schafer, 1975), the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f . This theorem can also be applied to functions over finite domains (Stankovic & Astola, 2007; Takimoto & Maruoka, 1997). Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set I is referred to as “bandlimited to I”. Ukita et al. obtained a sampling theorem for bandlimited functions over Boolean (Ukita et al., 2003) and GF(q)n domains (Ukita et al., 2010a), where q is a prime power and GF(q) is Galois field of order q. The sampling theorem can be applied in various fields as well as in digital signal processing, and one of the fields is the experimental design. In most areas of scientific research, experimentation is a major tool for acquiring new knowledge or a better understanding of the target phenomenon. Experiments usually aim to study how changes in various factors affect the response variable of interest (Cochran & Cox, 1992; Toutenburg & Shalabh, 2009). Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters and is not expressed in terms of an orthonormal system, a considerable amount of time is often necessary to implement the procedure for estimating the effects. In this chapter, we propose that the model of experimental design be expressed as an orthonormal system, and show that the model contains no redundant parameters. Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a system allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. In addition, the effect of each factor can be easily obtained from the Fourier coefficients (Ukita & Matsushima, 2011). Therefore, it is possible to implement easily the estimation procedures as well as to understand how each factor affects the response variable in a model based on an orthonormal system. Moreover, the analysis of variance can also be performed in a model based on an orthonormal system (Ukita et al., 2010b). Hence, 18


Introduction
The Fourier series representation of a function is a classic representation which is widely used to approximate real functions (Stein & Shakarchi, 2003). In digital signal processing (Oppenheim & Schafer, 1975), the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f . This theorem can also be applied to functions over finite domains (Stankovic & Astola, 2007;Takimoto & Maruoka, 1997). Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set I is referred to as "bandlimited to I". Ukita et al. obtained a sampling theorem for bandlimited functions over Boolean (Ukita et al., 2003) and GF(q) n domains (Ukita et al., 2010a), where q is a prime power and GF(q) is Galois field of order q. The sampling theorem can be applied in various fields as well as in digital signal processing, and one of the fields is the experimental design. In most areas of scientific research, experimentation is a major tool for acquiring new knowledge or a better understanding of the target phenomenon. Experiments usually aim to study how changes in various factors affect the response variable of interest (Cochran & Cox, 1992;Toutenburg & Shalabh, 2009). Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters and is not expressed in terms of an orthonormal system, a considerable amount of time is often necessary to implement the procedure for estimating the effects. In this chapter, we propose that the model of experimental design be expressed as an orthonormal system, and show that the model contains no redundant parameters. Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a system allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. In addition, the effect of each factor can be easily obtained from the Fourier coefficients (Ukita & Matsushima, 2011). Therefore, it is possible to implement easily the estimation procedures as well as to understand how each factor affects the response variable in a model based on an orthonormal system. Moreover, the analysis of variance can also be performed in a model based on an orthonormal system (Ukita et al., 2010b). Hence, 18 www.intechopen.com it is clear that two main procedures in the experimental design, that is, the estimation of the effects and the analysis of variance can be executed in a description of experimental design on the basis of an orthonormal system. This chapter is organized as follows. In Section 2, we give preliminaries that are necessary for this study. In Section 3, we provide an introduction to experimental design and describe the characteristic of the previous model in .experimental design. In Section 4, we propose the new model of experimental design on the basis of an orthonormal system and clarify the characteristic of the model. Finally, Section 5 concludes this chapter.

Fourier analysis on finite Abelian groups
Here, we provide a brief explanation of Fourier analysis on finite Abelian groups. Characters are important in the context of finite Fourier series.

Characters
Let G be a finite Abelian group (with additive notation), and let S 1 be the unit circle in the complex plane. A character on G is a complex-valued function X : G → S 1 that satisfies the condition In other words, a character is a homomorphism from G to the circle group.

Fourier transform
Let G i , i = 1, 2, . . . , n, be Abelian groups of respective orders |G i | = g i , i = 1, 2, . . . , n, g 1 ≤ g 2 ≤···≤g n , and let Since the character group of G is isomorphic to G, we can index the characters by the elements of G, that is, {X a (x)|a ∈ G} are the characters of G. Note that X 0 (x) is the principal character and identically equal to 1. The characters {X a (x)|a ∈ G} form an orthonormal system: . Any function f : G → C, where C is the field of complex numbers, can be uniquely expressed as a linear combination of the following characters: where the complex number is the a-th Fourier coefficient of f .

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Fourier analysis on GF(q) n
Assume that q is a prime power. Let GF(q) be a Galois field of order q which contains a finite number of elements. We also use GF(q) n to denote the set of all n-tuples with entries from GF(q). The elements of GF(q) n are referred to as vectors.
Specifying the group G in Section 2.1.2 to be the support group of GF(q) n and g = q n , the relations (3), (4) and (5) also hold over the GF(q) n domain.

Experimental design
In this section, we provide a short introduction to experimental design.

Model in experimental design
Let F 1 , F 2 ,...,F n denote n factors to be included in an experiment. The levels of each factor can be represented by GF(q), and the combinations of levels can be represented by the n-tuples x =(x 1 , x 2 ,...,x n ) ∈ GF(q) n .
Example 2. Let Machine (F 1 ) and Worker (F 2 ) be factors that might influence the total amount of the product. Assume each factor has two levels. F 1 : new machine (level 0), old machine (level 1). F 2 : skilled worker (level 0), unskilled worker (level 1). For example, x = 01 represents a combination of new machine and unskilled worker. Then, the effect of the machine, averaged over both workers, is referred to as the effect of main factor F 1 . Similarly, the effect of the worker, averaged over both machines, is referred to as the effect of main factor F 2 . The difference between the effect of the machine for an unskilled worker and that for a skilled worker is referred to as the effect of the interaction of F 1 and F 2 .
Let the set A ⊆{0, 1} n represent all factors that might influence the response of an experiment. The Hamming weight w(a) of a vector a =( a 1 , a 2 ,...,a n ) ∈ A is defined as the number of nonzero components. The main factors are represented by It is usually assumed that the set A satisfies the following monotonicity condition (Okuno & Haga, 1969).
Since the set A satisfies (7), A is monotonic.
Let y(x) denote the response of the experiment with level combination x. Assume the model where µ is the general mean, α l (x l ) is the effect of the x l -th level of Factor F l , β l,m (x l , x m ) is the effect of the interaction of the x l -th level of Factor F l and the x m -th level of Factor F m and ǫ x is a random error with a zero mean and a constant variance σ 2 .
Since the model is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, because the constraints are assumed, the model contains redundant parameters.
In experimental design, we are presented with a model of an experiment, which consists of a set A ⊆{0, 1} n . First, we determine a set of level combinations x ∈ X, X ⊆ GF(q) n . The set X is referred to as a design. Then, we perform a set of experiments in accordance to the design X and estimate the effects from the obtained results {(x, y(x))|x ∈ X}. An important standard for evaluating experimental design is the maximum of the variances of the unbiased estimators of effects, as calculated from the results of the conducted experiments. It is known that, for a given number of experiments, this criterion is minimized when using orthogonal design (Takahashi, 1979). Hence, there has been extensive research focusing on orthogonal design (Hedayat et al., 1999;Takahashi, 1979;Ukita et al., 2003;2010a;b;Ukita & Matsushima, 2011).

Orthogonal design
The components of this matrix, h ij ∈ GF(q)( 1 ≤ i ≤ k,1 ≤ j ≤ n), satisfy the following conditions.
1. The set {h ·j |j ∈ v(a ′ + a ′′ )} 1 , where h ·j is the j-th column of H A , is linearly independent over GF(q) for any given a ′ , a ′′ ∈ A.
2. The set {h i· |1 ≤ i ≤ k}, where h i· is the i-th row of H A , is linearly independent over GF(q).
An orthogonal design C ⊥ for main and interactive factors A ⊆{0, 1} n is defined as and |C ⊥ | = q k .
Many algorithms for constructing H A have been proposed (Hedayat et al., 1999;MacWilliams & Sloane, 1977;Takahashi, 1979;Ukita et al., 2003). However, it is still an extremely difficult problem to construct H A when the number of factors n is large and a large number of interactions are included in the model. In this regard, algorithms for the construction of orthogonal design are not presented here since this falls outside the scope of this chapter.
Although there are software packages that can be used to estimate the effects on the basis of (19)- (21), as yet no software can be used for an arbitrary monotonic set A. Therefore, it is often necessary to implement the procedure for estimating the effects, which requires a considerable amount of time.

Analysis of variance
When there are many factors, a comprehensive view of whether an interaction in A can be disregarded is needed. The test procedure involves an analysis of variance. For a detailed explanation of analysis of variance, refer to (Toutenburg & Shalabh, 2009). The statistics needed in analysis of variance are the following. SS Mean is the correction term (the sum of squares due to the mean), SS F l is the sum of squares due to the effect of F l , SS F l ×F m is the sum of squares due to the interaction effect of F l × F m , and SS Error is the sum of squares due to error. These can be computed as follows.
Example 8. Consider the case that a set A is given by (14)

Description of experimental design on the basis of an orthonormal system
In this section, we propose the model of experimental design on the basis of an orthonormal system.

Model on the basis of an orthonormal system in experimental design
We use y(x) to denote the response of an experiment with a level combination x, and assume the following model: where I A = {(b 1 a 1 ,...,b n a n )|a ∈ A, b i ∈ GF(q)} and ǫ x is a random error with a zero mean and a constant variance. Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. The effects are represented by the parameters { f a |a ∈ I A }. In addition, there are no constraints between the parameters, and the parameters are independent. Hence, it is clear that the model contains no redundant parameters.

Estimation of Fourier coefficients in experimental design
First, we present the following theorem (Ukita et al., 2010a).

Theorem 1. Sampling Theorem for Bandlimited Functions over a GF(q) n Domain
Assume that A ⊆{0, 1} n is monotonic and where I A = {(b 1 a 1 ,...,b n a n )|a ∈ A, b i ∈ GF(q)}. Then, the Fourier coefficients can be computed as follows: where C ⊥ is an orthogonal design for A (|C ⊥ | = q k ).
When an experiment is conducted in accordance to the orthogonal design C ⊥ , unbiased estimators of f a in (26) can be obtained by using Theorem 1 and assuming that E(ǫ x )=0: In particular, when q = 2 m , where m is an integer and m ≥ 1, it is possible to use the vector-radix fast Fourier transform (FFT), which is a multidimensional implementation of the FFT algorithm, for calculating (29) for all a ∈ I A . The complexity of the vector-radix FFT is O(q k log q k ). In addition, it can be shown that the Yates' Method (Yates, 1937) for efficient calculation of (19)-(21) in the case of q = 2 is equivalent to the vector-radix FFT for calculation of (29).

The relation between the Fourier coefficients and the effect of each factor
In a description of experimental design on the basis of an orthonormal system, the model is expressed by using Fourier coefficients. Fourier coefficients themselves do not provide a direct representation of the effect of each factor. On the other hand, since the previous model in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable.
In this section, we present three theorems of the relation between the Fourier coefficients and the effect of each factor (Ukita & Matsushima, 2011). First, we present a theorem of the relation between the Fourier coefficient and the general mean.
Next, we present a theorem of the relation between the Fourier coefficients and the effect of the main factor.
Last, we present a theorem of the relation between the Fourier coefficients and the effect of the interaction.
From these theorems, the effect of each factor can be easily obtained from the computed Fourier coefficients.
From these theorems, the effect of each factor can be easily obtained from the Fourier coefficients. Therefore, it is possible to implement easily the estimation procedures as well as to understand how each factor affects the response variable in a model based on an orthonormal system.

Analysis of variance in experimental design
On the other hand, it is already shown that the analysis of variance can also be performed in the model of experimental design on the basis of an orthonormal system (Ukita et al., 2010b). We present three theorems with respect to the sum of squares needed in analysis of variance.
where the sums are taken over a l , a m ∈ GF(q) and By these theorems, SS Mean , SS F l and SS F l ×F m can be obtained in the proposed description of experimental design. In addition, using the Parseval-Plancherel formula and these theorems, SS Error can be computed as follows.
Therefore, the analysis of variance can be executed in the proposed description of experimental design. Hence, it is clear that two main procedures in the experimental design, that is, the estimation of the effects and the analysis of variance can be executed in a description of experimental design on the basis of an orthonormal system.

Conclusion
In this chapter, we have proposed that the model of experimental design be expressed as an orthonormal system, and shown that the model contains no redundant parameters. Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a system allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. In addition, the effect of each factor can be easily obtained from the Fourier coefficients. Therefore, it is possible to implement easily the estimation procedures as well as to understand how each factor affects the response variable in a model based on an orthonormal system. Moreover, it is already shown that the analysis of variance can also be performed in a model based on an orthonormal system. Hence, it is clear that two main procedures in the experimental design, that is, the estimation of the effects and the analysis of variance can be executed in a description of experimental design on the basis of an orthonormal system. In this book the reader will find a collection of chapters authored/co-authored by a large number of experts around the world, covering the broad field of digital signal processing. This book intends to provide highlights of the current research in the digital signal processing area, showing the recent advances in this field. This work is mainly destined to researchers in the digital signal processing and related areas but it is also accessible to anyone with a scientific background desiring to have an up-to-date overview of this domain. Each chapter is self-contained and can be read independently of the others. These nineteenth chapters present methodological advances and recent applications of digital signal processing in various domains as communications, filtering, medicine, astronomy, and image processing.

How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: Yoshifumi Ukita and Toshiyasu Matsushima (2011). A Description of Experimental Design on the Basis of an Orthonormal System, Applications of Digital Signal Processing, Dr. Christian Cuadrado-Laborde (Ed.), ISBN: 978-953-307-406-1, InTech, Available from: http://www.intechopen.com/books/applications-of-digital-signalprocessing/a-description-of-experimental-design-on-the-basis-of-an-orthonormal-system