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

Yoshifumi Ukita1 and Toshiyasu Matsushima2

<sup>1</sup>*Yokohama College of Commerce* <sup>2</sup>*Waseda University Japan*

### **1. Introduction**

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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*)*<sup>n</sup>* 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,

**2.2 Fourier analysis on** *GF*(*q*)*<sup>n</sup>*

*Moreover, consider n* = 5*.*

**3. Experimental design**

**3.1 Model in experimental design**

*product. Assume each factor has two levels. F*<sup>1</sup> *: new machine (level* 0*), old machine (level* 1*). F*<sup>2</sup> *: skilled worker (level* 0*), unskilled worker (level* 1*).*

1, *am* = 1, *a* ∈ *A*2}, where *A*<sup>2</sup> = {*a*|*w*(*a*) = 2, *a* ∈ *A*}.

{1, 2, 3}*, A*<sup>2</sup> = {110} *and IF* = {{1, 2}}*.*

*<sup>x</sup>* = (*x*1, *<sup>x</sup>*2,..., *xn*) <sup>∈</sup> *GF*(*q*)*n*.

*GF*(*q*). The elements of *GF*(*q*)*<sup>n</sup>* are referred to as vectors.

relations (3), (4) and (5) also hold over the *GF*(*q*)*<sup>n</sup>* domain.

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

*For example, x* = 01 *represents a combination of new machine and unskilled worker.*

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*)*<sup>n</sup>* to denote the set of all *n*-tuples with entries from

A Description of Experimental Design on the Basis of an Orthonormal System 367

*and* <sup>|</sup>*GF*(3)5<sup>|</sup> <sup>=</sup> <sup>243</sup>*.* Specifying the group *G* in Section 2.1.2 to be the support group of *GF*(*q*)*<sup>n</sup>* and *g* = *qn*, the

Let *F*1, *F*2,..., *Fn* 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

**Example 2.** *Let Machine (F*1*) and Worker (F*2*) be factors that might influence the total amount of the*

*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*<sup>1</sup> *and F*2*.* Let the set *<sup>A</sup>* <sup>⊆</sup> {0, 1}*<sup>n</sup>* represent all factors that might influence the response of an experiment. The *Hamming weight w*(*a*) of a vector *a* = (*a*1, *a*2,..., *an*) ∈ *A* is defined as the number of nonzero components. The main factors are represented by *MF* = {*l*|*al* = 1, *a* ∈ *A*1}, where *A*<sup>1</sup> = {*a*|*w*(*a*) = 1, *a* ∈ *A*}. The interactive factors are represented by *IF* = {{*l*, *m*}|*al* =

**Example 3.** *Consider A* <sup>=</sup> {000, 100, 010, 001, 110}*. Then, A*<sup>1</sup> <sup>=</sup> {100, 010, 001} *and MF* <sup>=</sup>

*For example,* 1 ∈ *MF indicates the main factor F*1*, and* {1, 2} ∈ *IF indicates the interactive factors F*<sup>1</sup> *and F*2*.*

*GF*(3)<sup>5</sup> <sup>=</sup> {00000, 10000, ··· , 22222}, (6)

**Example 1.** *Consider GF*(3) = {0, 1, 2}*. Addition and multiplication are defined as follows:*

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.

#### **2. Preliminaries**

#### **2.1 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.

#### **2.1.1 Characters**

Let *G* be a finite Abelian group (with additive notation), and let *S*<sup>1</sup> be the unit circle in the complex plane. A character on *<sup>G</sup>* is a complex-valued function <sup>X</sup> : *<sup>G</sup>* <sup>→</sup> *<sup>S</sup>*<sup>1</sup> that satisfies the condition

$$\mathcal{X}(\mathbf{x} + \mathbf{x}') = \mathcal{X}(\mathbf{x})\mathcal{X}(\mathbf{x}') \quad \forall \mathbf{x}, \mathbf{x}' \in \mathcal{G}. \tag{1}$$

In other words, a character is a homomorphism from *G* to the circle group.

#### **2.1.2 Fourier transform**

Let *Gi*, *i* = 1, 2, . . . , *n*, be Abelian groups of respective orders |*Gi*| = *gi*, *i* = 1, 2, . . . , *n*, *g*<sup>1</sup> ≤ *g*<sup>2</sup> ≤ ··· ≤ *gn*, and let

$$\mathbf{G} = \times\_{i=1}^{n} \mathbf{G}\_{i} \quad \text{and} \quad \mathbf{g} = \prod\_{i=1}^{n} \mathbf{g}\_{i} . \tag{2}$$

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:

$$\frac{1}{g} \sum\_{\mathbf{x} \in \mathcal{G}} \mathcal{X}\_{\mathbf{d}}(\mathbf{x}) \mathcal{X}\_{\mathbf{b}}^{\*}(\mathbf{x}) = \begin{cases} 1, \ a = b, \\ 0, \ a \neq b, \end{cases} \tag{3}$$

where X <sup>∗</sup> *<sup>b</sup>*(*x*) is the complex conjugate of <sup>X</sup>*b*(*x*).

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:

$$f(\mathbf{x}) = \sum\_{\mathbf{a} \in G} f\_{\mathbf{a}} \mathcal{X}\_{\mathbf{a}}(\mathbf{x}),\tag{4}$$

where the complex number

$$f\_{\mathbf{d}} = \frac{1}{\mathcal{S}} \sum\_{\mathbf{x} \in G} f(\mathbf{x}) \mathcal{X}\_{\mathbf{d}}^{\*}(\mathbf{x}) \tag{5}$$

is the *a*-th *Fourier coefficient* of *f* .
