**3. Experimental design**

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

#### **3.1 Model in experimental design**

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 *<sup>x</sup>* = (*x*1, *<sup>x</sup>*2,..., *xn*) <sup>∈</sup> *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*<sup>1</sup> *: new machine (level* 0*), old machine (level* 1*).*

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

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

*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*.*

**3.2 Orthogonal design**

**Definition 2.** *(Orthogonal design)*

*GF*(*q*) *for any given a*

*In this case,*

*Define v*(*a*) = {*i*|*ai* <sup>=</sup> 0, 1 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>n</sup>*}*. For A* <sup>⊆</sup> {0, 1}*n, let HA be the k* <sup>×</sup> *n matrix*

 

*. . . . . . ... . . .*

*h*<sup>11</sup> *h*<sup>12</sup> ... *h*1*<sup>n</sup> h*<sup>21</sup> *h*<sup>22</sup> ... *h*2*<sup>n</sup>*

A Description of Experimental Design on the Basis of an Orthonormal System 369

 

*<sup>C</sup>*<sup>⊥</sup> <sup>=</sup> {*x*|*<sup>x</sup>* <sup>=</sup> *<sup>r</sup>HA*, *<sup>r</sup>* <sup>∈</sup> *GF*(*q*)*k*}, (13)

. (12)

, (15)

*hk*<sup>1</sup> *hk*<sup>2</sup> ... *hkn*

*The components of this matrix, hij* ∈ *GF*(*q*) (1 ≤ *i* ≤ *k*, 1 ≤ *j* ≤ *n*)*, satisfy the following conditions. 1. The set* {*h*·*j*|*<sup>j</sup>* <sup>∈</sup> *<sup>v</sup>*(*a* <sup>+</sup> *<sup>a</sup>*)}1*, where <sup>h</sup>*·*<sup>j</sup> is the j-th column of HA, is linearly independent over*

*and* <sup>|</sup>*C*⊥| <sup>=</sup> *<sup>q</sup>k.*

*A* = {00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010}. (14)

10000 01011 00112

*2. The set* {*hi*·|<sup>1</sup> ≤ *<sup>i</sup>* ≤ *<sup>k</sup>*}*, where <sup>h</sup>i*· *is the i-th row of HA, is linearly independent over GF*(*q*)*.*

*An* orthogonal design *<sup>C</sup>*<sup>⊥</sup> *for main and interactive factors A* <sup>⊆</sup> {0, 1}*<sup>n</sup> is defined as*

*HA* =

 

*C*<sup>⊥</sup> = {00000, 00112, 00221, 01011, 01120, 01202, 02022, 02101, 02210, 10000, 10112, 10221, 11011, 11120, 11202, 12022, 12101, 12210, 20000, 20111, 20221, 21011,

*is an orthogonal design for A.* Many algorithms for constructing *HA* 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 *HA* 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.

<sup>1</sup> For *<sup>a</sup>*<sup>1</sup> = (*a*11, *<sup>a</sup>*12,..., *<sup>a</sup>*1*n*), *<sup>a</sup>*<sup>2</sup> = (*a*21, *<sup>a</sup>*22,..., *<sup>a</sup>*2*n*) <sup>∈</sup> {0, 1}*n*, the addition of vectors *<sup>a</sup>*<sup>1</sup> and *<sup>a</sup>*<sup>2</sup> is defined

as *a*<sup>1</sup> + *a*<sup>2</sup> = (*a*<sup>11</sup> ⊕ *a*21, *a*<sup>12</sup> ⊕ *a*22,..., *a*1*<sup>n</sup>* ⊕ *a*2*n*), where ⊕ is the *exclusive OR* operator.

*HA* =

, *a* ∈ *A.*

**Example 6.** *We consider the case q* = 3, *n* = 5 *and*

*satisfies the conditions in Definition 2. Therefore,*

21120, 21202, 22022, 22101, 22210},

It is usually assumed that the set *A* satisfies the following monotonicity condition (Okuno & Haga, 1969).

**Definition 1.** *Monotonicity*

$$a \in A \to b \in A \qquad \forall b \ (b \sqsubseteq a), \tag{7}$$

*where* (*b*1, *b*2,..., *bn*) (*a*1, *a*2,..., *an*) *indicates that if ai* = 0 *then bi* = 0, *i* = 1, 2, . . . , *n.*

**Example 4.** *Consider A* <sup>=</sup> {00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010}*. 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

$$y(\mathbf{x}) = \mu + \sum\_{l \in MF} a\_l(\mathbf{x}\_l) + \sum\_{\{l, m\} \in IF} \beta\_{l, m}(\mathbf{x}\_l, \mathbf{x}\_m) + \varepsilon\_{\mathbf{X}\prime} \tag{8}$$

where *µ* is the general mean, *αl*(*xl*) is the effect of the *xl*-th level of Factor *Fl*, *βl*,*m*(*xl*, *xm*) is the effect of the interaction of the *xl*-th level of Factor *Fl* and the *xm*-th level of Factor *Fm* and *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

$$\sum\_{\substack{\varrho=0\\\varrho=0}}^{q-1} \alpha\_l(\varrho) = 0,\tag{9}$$

$$\sum\_{\varphi=0}^{q-1} \beta\_{l,m}(\varphi, \psi) = 0,\tag{10}$$

$$\sum\_{\psi=0}^{q-1} \beta\_{l,m}(\varphi, \psi) = 0,\tag{11}$$

are assumed, the model contains redundant parameters.

**Example 5.** *Consider q* <sup>=</sup> 3, *<sup>n</sup>* <sup>=</sup> <sup>5</sup> *and A* <sup>=</sup> {00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010}*. Then,*

*µ*, *α*1(0), *α*1(1), *α*1(2), *α*2(0), *α*2(1), *α*2(2), *α*3(0), *α*3(1), *α*3(2), *α*4(0), *α*4(1), *α*4(2), *α*5(0), *α*5(1), *α*5(2), *β*1,2(0, 0), *β*1,2(0, 1), *β*1,2(0, 2), *β*1,2(1, 0), *β*1,2(1, 1), *β*1,2(1, 2), *β*1,2(2, 0), *β*1,2(2, 1), *β*1,2(2, 2), , *β*1,3(0, 0), *β*1,3(0, 1), *β*1,3(0, 2), *β*1,3(1, 0), *β*1,3(1, 1), *β*1,3(1, 2), *β*1,3(2, 0), *β*1,3(2, 1), *β*1,3(2, 2), *β*1,4(0, 0), *β*1,4(0, 1), *β*1,4(0, 2), *β*1,4(1, 0), *β*1,4(1, 1), *β*1,4(1, 2), *β*1,4(2, 0), *β*1,4(2, 1), *β*1,4(2, 2)

*are parameters. The number of parameters is* 43*, but the number of the independent parameters is* 23 *by the constraints.*

In experimental design, we are presented with a model of an experiment, which consists of a set *<sup>A</sup>* <sup>⊆</sup> {0, 1}*n*. First, we determine a set of level combinations *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*, *<sup>X</sup>* <sup>⊆</sup> *GF*(*q*)*n*. The set *<sup>X</sup>* 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).

#### **3.2 Orthogonal design**

4 Will-be-set-by-IN-TECH

It is usually assumed that the set *A* satisfies the following monotonicity condition (Okuno &

*where* (*b*1, *b*2,..., *bn*) (*a*1, *a*2,..., *an*) *indicates that if ai* = 0 *then bi* = 0, *i* = 1, 2, . . . , *n.* **Example 4.** *Consider A* <sup>=</sup> {00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010}*. 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

*αl*(*xl*) + ∑

{*l*,*m*}∈*IF*

where *µ* is the general mean, *αl*(*xl*) is the effect of the *xl*-th level of Factor *Fl*, *βl*,*m*(*xl*, *xm*) is the effect of the interaction of the *xl*-th level of Factor *Fl* and the *xm*-th level of Factor *Fm* and

Since the model is expressed through the effect of each factor, it is easy to understand how

*y*(*x*) = *µ* + ∑

are assumed, the model contains redundant parameters.

*l*∈*MF*

each factor affects the response variable. However, because the constraints

*q*−1 ∑ *ϕ*=0

*q*−1 ∑ *ϕ*=0

*q*−1 ∑ *ψ*=0

*X* and estimate the effects from the obtained results {(*x*, *y*(*x*))|*x* ∈ *X*}.

**Example 5.** *Consider q* <sup>=</sup> 3, *<sup>n</sup>* <sup>=</sup> <sup>5</sup> *and A* <sup>=</sup> {00000, 10000, 01000, 00100, 00010, 00001, 11000,

*µ*, *α*1(0), *α*1(1), *α*1(2), *α*2(0), *α*2(1), *α*2(2), *α*3(0), *α*3(1), *α*3(2), *α*4(0), *α*4(1), *α*4(2), *α*5(0), *α*5(1), *α*5(2), *β*1,2(0, 0), *β*1,2(0, 1), *β*1,2(0, 2), *β*1,2(1, 0), *β*1,2(1, 1), *β*1,2(1, 2), *β*1,2(2, 0), *β*1,2(2, 1), *β*1,2(2, 2), , *β*1,3(0, 0), *β*1,3(0, 1), *β*1,3(0, 2), *β*1,3(1, 0), *β*1,3(1, 1), *β*1,3(1, 2), *β*1,3(2, 0), *β*1,3(2, 1), *β*1,3(2, 2), *β*1,4(0, 0), *β*1,4(0, 1), *β*1,4(0, 2), *β*1,4(1, 0), *β*1,4(1, 1), *β*1,4(1, 2), *β*1,4(2, 0), *β*1,4(2, 1),

*are parameters. The number of parameters is* 43*, but the number of the independent parameters is* 23 *by the constraints.* In experimental design, we are presented with a model of an experiment, which consists of a set *<sup>A</sup>* <sup>⊆</sup> {0, 1}*n*. First, we determine a set of level combinations *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*, *<sup>X</sup>* <sup>⊆</sup> *GF*(*q*)*n*. The set *<sup>X</sup>* is referred to as a design. Then, we perform a set of experiments in accordance to the design

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 &

*x* is a random error with a zero mean and a constant variance *σ*2.

*a* ∈ *A* → *b* ∈ *A* ∀*b* (*b a*), (7)

*βl*,*m*(*xl*, *xm*) + *x*, (8)

*αl*(*ϕ*) = 0, (9)

*βl*,*m*(*ϕ*, *ψ*) = 0, (10)

*βl*,*m*(*ϕ*, *ψ*) = 0, (11)

Haga, 1969).

**Definition 1.** *Monotonicity*

10100, 10010}*. Then,*

Matsushima, 2011).

*β*1,4(2, 2)

**Definition 2.** *(Orthogonal design)*

*Define v*(*a*) = {*i*|*ai* <sup>=</sup> 0, 1 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>n</sup>*}*. For A* <sup>⊆</sup> {0, 1}*n, let HA be the k* <sup>×</sup> *n matrix*

$$H\_A = \begin{bmatrix} h\_{11} \ h\_{12} \ \dots \ h\_{1n} \\ h\_{21} \ h\_{22} \ \dots \ h\_{2n} \\ \vdots \ \vdots \ \ddots \ \vdots \\ h\_{k1} \ h\_{k2} \ \dots \ h\_{kn} \end{bmatrix} \tag{12}$$

*The components of this matrix, hij* ∈ *GF*(*q*) (1 ≤ *i* ≤ *k*, 1 ≤ *j* ≤ *n*)*, satisfy the following conditions.*


*An* orthogonal design *<sup>C</sup>*<sup>⊥</sup> *for main and interactive factors A* <sup>⊆</sup> {0, 1}*<sup>n</sup> is defined as*

$$\mathbb{C}^{\perp} = \{ \mathfrak{x} | \mathfrak{x} = \mathfrak{r} H\_{A\prime} \ \mathfrak{r} \in GF(q)^{k} \}, \tag{13}$$

*and* <sup>|</sup>*C*⊥| <sup>=</sup> *<sup>q</sup>k.*

**Example 6.** *We consider the case q* = 3, *n* = 5 *and*

$$A = \{00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010\}. \tag{14}$$

*In this case,*

$$H\_A = \begin{bmatrix} 1 \ 0 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 1 \ 1 \\ 0 \ 0 \ 1 \ 1 \ 2 \end{bmatrix} \tag{15}$$

*satisfies the conditions in Definition 2. Therefore,*

$$\begin{aligned} \mathbb{C}^{\perp} &= \{00000, 00112, 00221, 01011, 01120, 01202, 02022, 02101, 02210, 10000, 10112, 10121, 111210, 11210, 12210, 12210, 20000, 20111, 20221, 21011, \\ &10221, 11011, 11120, 11202, 12022, 12101, 12210, 20111, 20221, 21011, \end{aligned}$$

*is an orthogonal design for A.*

Many algorithms for constructing *HA* 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 *HA* 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.

<sup>1</sup> For *<sup>a</sup>*<sup>1</sup> = (*a*11, *<sup>a</sup>*12,..., *<sup>a</sup>*1*n*), *<sup>a</sup>*<sup>2</sup> = (*a*21, *<sup>a</sup>*22,..., *<sup>a</sup>*2*n*) <sup>∈</sup> {0, 1}*n*, the addition of vectors *<sup>a</sup>*<sup>1</sup> and *<sup>a</sup>*<sup>2</sup> is defined as *a*<sup>1</sup> + *a*<sup>2</sup> = (*a*<sup>11</sup> ⊕ *a*21, *a*<sup>12</sup> ⊕ *a*22,..., *a*1*<sup>n</sup>* ⊕ *a*2*n*), where ⊕ is the *exclusive OR* operator.

*Next, the following values are obtained.*

*Last, by using (19)–(21),*

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

amount of time.

**3.4 Analysis of variance**

1,2(0, 0) = <sup>−</sup>0.52, *<sup>β</sup>*<sup>ˆ</sup>

1,2(1, 1) = 1.15, *β*ˆ

1,2(2, 2) = <sup>−</sup>0.07, *<sup>β</sup>*<sup>ˆ</sup>

1,3(1, 0) = <sup>−</sup>1.41, *<sup>β</sup>*<sup>ˆ</sup>

1,3(2, 1) = <sup>−</sup>1.52, *<sup>β</sup>*<sup>ˆ</sup>

1,4(0, 2) = 0.04, *β*ˆ

1,4(2, 0) = 2.26, *β*ˆ

due to error. These can be computed as follows.

*y*¯ = 96.15, *y*¯1(0) = 95.89, *y*¯1(1) = 106.00, *y*¯1(2) = 86.56, *y*¯2(0) = 96.78, *y*¯2(1) = 97.00, *y*¯2(2) = 94.67, *y*¯3(0) = 93.56, *y*¯3(1) = 96.44, *y*¯3(2) = 98.44, *y*¯4(0) = 97.00, *y*¯4(1) = 94.22, *y*¯4(2) = 97.22, *y*¯5(0) = 94.11, *y*¯5(1) = 96.00, *y*¯5(2) = 98.33, *y*¯1,2(0, 0) = 96.00, *y*¯1,2(0, 1) = 96.00, *y*¯1,2(0, 2) = 95.67, *y*¯1,2(1, 0) = 106.67, *y*¯1,2(1, 1) = 108.00, *y*¯1,2(1, 2) = 103.33, *y*¯1,2(2, 0) = 87.67, *y*¯1,2(2, 1) = 87.00, *y*¯1,2(2, 2) = 85.00, *y*¯1,3(0, 0) = 93.33, *y*¯1,3(0, 1) = 96.00, *y*¯1,3(0, 2) = 98.33, *y*¯1,3(1, 0) = 102.00, *y*¯1,3(1, 1) = 108.00, *y*¯1,3(1, 2) = 108.00, *y*¯1,3(2, 0) = 85.33, *y*¯1,3(2, 1) = 85.33, *y*¯1,3(2, 2) = 89.00, *y*¯1,4(0, 0) = 96.67, *y*¯1,4(0, 1) = 94.00, *y*¯1,4(0, 2) = 97.00, *y*¯1,4(1, 0) = 104.67, *y*¯1,4(1, 1) = 104.00, *y*¯1,4(1, 2) = 109.33,

A Description of Experimental Design on the Basis of an Orthonormal System 371

*µ*ˆ = 96.15, *α*ˆ <sup>1</sup>(0) = −0.26, *α*ˆ <sup>1</sup>(1) = 9.85, *α*ˆ <sup>1</sup>(2) = −9.59, *α*ˆ <sup>2</sup>(0) = 0.63, *α*ˆ <sup>2</sup>(1) = 0.85, *α*ˆ <sup>2</sup>(2) = −1.48, *α*ˆ <sup>3</sup>(0) = −2.59, *α*ˆ <sup>3</sup>(1) = 0.30, *α*ˆ <sup>3</sup>(2) = 2.30, *α*ˆ <sup>4</sup>(0) = 0.85, *α*ˆ <sup>4</sup>(1) = −1.93, *α*ˆ <sup>4</sup>(2) = 1.07, *α*ˆ <sup>5</sup>(0) = −2.04, *α*ˆ <sup>5</sup>(1) = −0.15, *α*ˆ <sup>5</sup>(2) = 2.19,

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

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

The statistics needed in analysis of variance are the following. *SSMean* is the correction term (the sum of squares due to the mean), *SSFl* is the sum of squares due to the effect of *Fl*, *SSFl*×*Fm* is the sum of squares due to the interaction effect of *Fl* × *Fm*, and *SSError* is the sum of squares

*SSMean* <sup>=</sup> <sup>1</sup>

*q*−1 ∑ *ϕ*=0 *Y*2

1,2(0, 2) = 1.26, *β*ˆ

1,2(2, 0) = 0.48, *β*ˆ

1,3(0, 1) = <sup>−</sup>0.19, *<sup>β</sup>*<sup>ˆ</sup>

1,3(1, 2) = <sup>−</sup>0.30, *<sup>β</sup>*<sup>ˆ</sup>

1,4(0, 0) = <sup>−</sup>0.07, *<sup>β</sup>*<sup>ˆ</sup>

1,4(1, 1) = <sup>−</sup>0.07, *<sup>β</sup>*<sup>ˆ</sup>

1,4(2, 2) = −2.30.

1,2(1, 0) = 0.04,

1,3(0, 2) = 0.15,

1,3(2, 0) = 1.37,

1,4(0, 1) = 0.04,

1,4(1, 2) = 2.26,

*<sup>q</sup><sup>k</sup> <sup>Y</sup>*2, (22)

*<sup>l</sup>* (*ϕ*) − *SSMean*, (23)

1,2(2, 1) = −0.41,

*y*¯1,4(2, 0) = 89.67, *y*¯1,4(2, 1) = 84.67, *y*¯1,4(2, 2) = 85.33.

1,2(0, 1) = <sup>−</sup>0.74, *<sup>β</sup>*<sup>ˆ</sup>

1,2(1, 2) = <sup>−</sup>1.19, *<sup>β</sup>*<sup>ˆ</sup>

1,3(0, 0) = 0.04, *β*ˆ

1,3(1, 1) = 1.70, *β*ˆ

1,3(2, 2) = 0.15, *β*ˆ

1,4(1, 0) = <sup>−</sup>2.19, *<sup>β</sup>*<sup>ˆ</sup>

1,4(2, 1) = 0.04, *β*ˆ

explanation of analysis of variance, refer to (Toutenburg & Shalabh, 2009).

*SSFl* <sup>=</sup> <sup>1</sup>

*qk*−<sup>1</sup>

#### **3.3 Estimation of effects in experimental design**

First, we adopt the following definitions.

$$Y = \sum\_{\mathbf{x} \in \mathbb{C}^\perp} y(\mathbf{x}),\tag{16}$$

where <sup>|</sup>*C*⊥| <sup>=</sup> *<sup>q</sup>k*.

$$Y\_l(\boldsymbol{\varphi}) = \sum\_{\substack{\mathbf{x} \in \mathbb{C}\_l^\perp(\boldsymbol{\varphi})}} y(\mathbf{x})\_\prime \tag{17}$$

where *C*⊥ *<sup>l</sup>* (*ϕ*) = {*x*|*xl* = *ϕ*, *x* ∈ *C*⊥} and |*C*<sup>⊥</sup> *<sup>l</sup>* (*ϕ*)<sup>|</sup> <sup>=</sup> *<sup>q</sup>k*−1.

$$Y\_{l,m}(\boldsymbol{\varrho},\boldsymbol{\psi}) = \sum\_{\substack{\mathfrak{X} \in \mathbb{C}\_{l,m}^{\perp}(\boldsymbol{\varrho},\boldsymbol{\psi})}} y(\boldsymbol{\mathfrak{x}}),\tag{18}$$

where *C*⊥ *<sup>l</sup>*,*m*(*ϕ*, *ψ*) = {*x*|*xl* = *ϕ*, *xm* = *ψ*, *x* ∈ *C*⊥} and |*C*<sup>⊥</sup> *<sup>l</sup>*,*m*(*ϕ*, *<sup>ψ</sup>*)<sup>|</sup> <sup>=</sup> *<sup>q</sup>k*−2. Let *y*¯ = <sup>1</sup> *qk <sup>Y</sup>*, *<sup>y</sup>*¯*l*(*ϕ*) = <sup>1</sup> *qk*−<sup>1</sup> *Yl*(*ϕ*), *<sup>y</sup>*¯*l*,*m*(*ϕ*, *<sup>ψ</sup>*) = <sup>1</sup> *qk*−<sup>2</sup> *Yl*,*m*(*ϕ*, *ψ*). Then, the unbiased estimators of the parameters in (8) are given as

$$
\mathfrak{A} = \mathfrak{F}\_{\prime} \tag{19}
$$

$$
\hat{u}\_l(\varphi) = \bar{y}\_l(\varphi) - \hat{\mu}\_l \tag{20}
$$

$$
\hat{\mathfrak{H}}\_{l,m}(\varphi,\psi) = \overline{y}\_{l,m}(\varphi,\psi) - \mathfrak{A}\_l(\varphi) - \mathfrak{A}\_m(\psi) - \hat{\mathfrak{A}}.\tag{21}
$$

**Example 7.** *Consider the case that a set A is given by (14) and the result of experiments is given by Table 1.*


Table 1. Result of experiments

*First, using (16)–(18),*

*Y* = 2596, *Y*1(0) = 863, *Y*1(1) = 954, *Y*1(2) = 779, *Y*2(0) = 871, *Y*2(1) = 873, *Y*2(2) = 852, *Y*3(0) = 842, *Y*3(1) = 868, *Y*3(2) = 886, *Y*4(0) = 873, *Y*4(1) = 848, *Y*4(2) = 875, *Y*5(0) = 847, *Y*5(1) = 864, *Y*5(2) = 885, *Y*1,2(0, 0) = 288, *Y*1,2(0, 1) = 288, *Y*1,2(0, 2) = 287, *Y*1,2(1, 0) = 320, *Y*1,2(1, 1) = 324, *Y*1,2(1, 2) = 310, *Y*1,2(2, 0) = 263, *Y*1,2(2, 1) = 261, *Y*1,2(2, 2) = 255, *Y*1,3(0, 0) = 280, *Y*1,3(0, 1) = 288, *Y*1,3(0, 2) = 295, *Y*1,3(1, 0) = 306, *Y*1,3(1, 1) = 324, *Y*1,3(1, 2) = 324, *Y*1,3(2, 0) = 256, *Y*1,3(2, 1) = 256, *Y*1,3(2, 2) = 267, *Y*1,4(0, 0) = 290, *Y*1,4(0, 1) = 282, *Y*1,4(0, 2) = 291, *Y*1,4(1, 0) = 314, *Y*1,4(1, 1) = 312, *Y*1,4(1, 2) = 328, *Y*1,4(2, 0) = 269, *Y*1,4(2, 1) = 254, *Y*1,4(2, 2) = 256.

*Next, the following values are obtained.*

6 Will-be-set-by-IN-TECH

*y*(*x*), (16)

*y*(*x*), (17)

*y*(*x*), (18)

*<sup>l</sup>*,*m*(*ϕ*, *<sup>ψ</sup>*)<sup>|</sup> <sup>=</sup> *<sup>q</sup>k*−2.

*µ*ˆ = *y*¯, (19) *α*ˆ*l*(*ϕ*) = *y*¯*l*(*ϕ*) − *µ*ˆ, (20)

*<sup>l</sup>*,*m*(*ϕ*, *ψ*) = *y*¯*l*,*m*(*ϕ*, *ψ*) − *α*ˆ*l*(*ϕ*) − *α*ˆ *<sup>m</sup>*(*ψ*) − *µ*ˆ. (21)

*qk*−<sup>2</sup> *Yl*,*m*(*ϕ*, *ψ*). Then, the unbiased estimators of

*Y* = ∑ *x*∈*C*<sup>⊥</sup>

*Yl*(*ϕ*) = ∑

*Yl*,*m*(*ϕ*, *ψ*) = ∑

*x*∈*C*<sup>⊥</sup> *<sup>l</sup>* (*ϕ*)

> *x*∈*C*<sup>⊥</sup> *<sup>l</sup>*,*m*(*ϕ*,*ψ*)

**Example 7.** *Consider the case that a set A is given by (14) and the result of experiments is given by*

*x y*(*x*) *x y*(*x*) *x y*(*x*) 93 10000 99 20000 87 97 10112 109 20111 86 98 10221 112 20221 90 90 11011 102 21011 85 96 11120 111 21120 82 102 11202 111 21202 94 97 12022 105 22022 84 95 12101 104 22101 88 95 12210 101 22210 83

*Y* = 2596, *Y*1(0) = 863, *Y*1(1) = 954, *Y*1(2) = 779, *Y*2(0) = 871, *Y*2(1) = 873, *Y*2(2) = 852, *Y*3(0) = 842, *Y*3(1) = 868, *Y*3(2) = 886, *Y*4(0) = 873, *Y*4(1) = 848, *Y*4(2) = 875, *Y*5(0) = 847, *Y*5(1) = 864, *Y*5(2) = 885, *Y*1,2(0, 0) = 288, *Y*1,2(0, 1) = 288, *Y*1,2(0, 2) = 287, *Y*1,2(1, 0) = 320, *Y*1,2(1, 1) = 324, *Y*1,2(1, 2) = 310, *Y*1,2(2, 0) = 263, *Y*1,2(2, 1) = 261, *Y*1,2(2, 2) = 255, *Y*1,3(0, 0) = 280, *Y*1,3(0, 1) = 288, *Y*1,3(0, 2) = 295, *Y*1,3(1, 0) = 306, *Y*1,3(1, 1) = 324, *Y*1,3(1, 2) = 324, *Y*1,3(2, 0) = 256, *Y*1,3(2, 1) = 256, *Y*1,3(2, 2) = 267, *Y*1,4(0, 0) = 290, *Y*1,4(0, 1) = 282, *Y*1,4(0, 2) = 291, *Y*1,4(1, 0) = 314, *Y*1,4(1, 1) = 312, *Y*1,4(1, 2) = 328,

*Y*1,4(2, 0) = 269, *Y*1,4(2, 1) = 254, *Y*1,4(2, 2) = 256.

*<sup>l</sup>* (*ϕ*)<sup>|</sup> <sup>=</sup> *<sup>q</sup>k*−1.

**3.3 Estimation of effects in experimental design** First, we adopt the following definitions.

*<sup>l</sup>* (*ϕ*) = {*x*|*xl* = *ϕ*, *x* ∈ *C*⊥} and |*C*<sup>⊥</sup>

*β*ˆ

*<sup>l</sup>*,*m*(*ϕ*, *ψ*) = {*x*|*xl* = *ϕ*, *xm* = *ψ*, *x* ∈ *C*⊥} and |*C*<sup>⊥</sup>

*qk*−<sup>1</sup> *Yl*(*ϕ*), *<sup>y</sup>*¯*l*,*m*(*ϕ*, *<sup>ψ</sup>*) = <sup>1</sup>

where <sup>|</sup>*C*⊥| <sup>=</sup> *<sup>q</sup>k*.

where *C*⊥

where *C*⊥

Let *y*¯ = <sup>1</sup>

*Table 1.*

*qk <sup>Y</sup>*, *<sup>y</sup>*¯*l*(*ϕ*) = <sup>1</sup>

the parameters in (8) are given as

Table 1. Result of experiments

*First, using (16)–(18),*


*Last, by using (19)–(21),*

*µ*ˆ = 96.15, *α*ˆ <sup>1</sup>(0) = −0.26, *α*ˆ <sup>1</sup>(1) = 9.85, *α*ˆ <sup>1</sup>(2) = −9.59, *α*ˆ <sup>2</sup>(0) = 0.63, *α*ˆ <sup>2</sup>(1) = 0.85, *α*ˆ <sup>2</sup>(2) = −1.48, *α*ˆ <sup>3</sup>(0) = −2.59, *α*ˆ <sup>3</sup>(1) = 0.30, *α*ˆ <sup>3</sup>(2) = 2.30, *α*ˆ <sup>4</sup>(0) = 0.85, *α*ˆ <sup>4</sup>(1) = −1.93, *α*ˆ <sup>4</sup>(2) = 1.07, *α*ˆ <sup>5</sup>(0) = −2.04, *α*ˆ <sup>5</sup>(1) = −0.15, *α*ˆ <sup>5</sup>(2) = 2.19, *β*ˆ 1,2(0, 0) = <sup>−</sup>0.52, *<sup>β</sup>*<sup>ˆ</sup> 1,2(0, 1) = <sup>−</sup>0.74, *<sup>β</sup>*<sup>ˆ</sup> 1,2(0, 2) = 1.26, *β*ˆ 1,2(1, 0) = 0.04, *β*ˆ 1,2(1, 1) = 1.15, *β*ˆ 1,2(1, 2) = <sup>−</sup>1.19, *<sup>β</sup>*<sup>ˆ</sup> 1,2(2, 0) = 0.48, *β*ˆ 1,2(2, 1) = −0.41, *β*ˆ 1,2(2, 2) = <sup>−</sup>0.07, *<sup>β</sup>*<sup>ˆ</sup> 1,3(0, 0) = 0.04, *β*ˆ 1,3(0, 1) = <sup>−</sup>0.19, *<sup>β</sup>*<sup>ˆ</sup> 1,3(0, 2) = 0.15, *β*ˆ 1,3(1, 0) = <sup>−</sup>1.41, *<sup>β</sup>*<sup>ˆ</sup> 1,3(1, 1) = 1.70, *β*ˆ 1,3(1, 2) = <sup>−</sup>0.30, *<sup>β</sup>*<sup>ˆ</sup> 1,3(2, 0) = 1.37, *β*ˆ 1,3(2, 1) = <sup>−</sup>1.52, *<sup>β</sup>*<sup>ˆ</sup> 1,3(2, 2) = 0.15, *β*ˆ 1,4(0, 0) = <sup>−</sup>0.07, *<sup>β</sup>*<sup>ˆ</sup> 1,4(0, 1) = 0.04, *β*ˆ 1,4(0, 2) = 0.04, *β*ˆ 1,4(1, 0) = <sup>−</sup>2.19, *<sup>β</sup>*<sup>ˆ</sup> 1,4(1, 1) = <sup>−</sup>0.07, *<sup>β</sup>*<sup>ˆ</sup> 1,4(1, 2) = 2.26, *β*ˆ 1,4(2, 0) = 2.26, *β*ˆ 1,4(2, 1) = 0.04, *β*ˆ 1,4(2, 2) = −2.30.

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.

#### **3.4 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. *SSMean* is the correction term (the sum of squares due to the mean), *SSFl* is the sum of squares due to the effect of *Fl*, *SSFl*×*Fm* is the sum of squares due to the interaction effect of *Fl* × *Fm*, and *SSError* is the sum of squares due to error. These can be computed as follows.

$$SS\_{Mean} = \frac{1}{q^k} Y^2 \,\, ^\prime\tag{22}$$

$$SS\_{F\_l} = \frac{1}{q^{k-1}} \sum\_{\varphi=0}^{q-1} Y\_l^2(\varphi) - SS\_{Mean} \tag{23}$$

$$SS\_{F\_l \times F\_m} = \frac{1}{q^{k-2}} \sum\_{q=0}^{q-1} \sum\_{\psi=0}^{q-1} Y\_{l,m}^2(\rho, \psi) - SS\_{F\_l} - SS\_{F\_m} - SS\_{Mean} \tag{24}$$

$$SS\_{Error} = \sum\_{\mathbf{x} \in \mathbb{C}^\perp} y^2(\mathbf{x}) - SS\_{Mean} - \sum\_{l \in MF} SS\_{\overline{l}l} - \sum\_{\{l, m\} \in IF} SS\_{\overline{l}l \times F\_m}.\tag{25}$$

**Example 8.** *Consider the case that a set A is given by (14) and the result of experiments is given by Table 1. Then, using (22)–(25),*

$$\begin{array}{ccccccccc} \text{SS}\_{\text{Mean}} & = & 249600.6 \text{, } \text{SS}\_{\text{F}\_{\text{1}}} & = & 1702.3 \text{, } \text{SS}\_{\text{F}\_{\text{2}}} & = & 29.9 \text{, } \text{SS}\_{\text{F}\_{\text{3}}} & = & 108.7 \text{, } \text{SS}\_{\text{F}\_{\text{1}}} \\\\ \text{SS}\_{\text{F}\_{4}} & = & 50.3 \text{, } \text{SS}\_{\text{F}\_{\text{5}}} & = & 80.5 \text{, } \text{SS}\_{\text{F}\_{\text{1}} \times \text{F}\_{\text{2}}} = 16.6 \text{, } \text{SS}\_{\text{F}\_{\text{1}} \times \text{F}\_{\text{3}}} & = & 27.7 \\\\ \text{SS}\_{\text{F}\_{\text{1}} \times \text{F}\_{4}} & = & 60.8 \text{, } \text{SS}\_{\text{Error}} & = & 16.6. \end{array}$$

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

*Assume that A* <sup>⊆</sup> {0, 1}*<sup>n</sup> is monotonic and*

*as follows:*

any monotonic set *A*.

*Table 1. Then,*

**Theorem 1.** *Sampling Theorem for Bandlimited Functions over a GF*(*q*)*<sup>n</sup> Domain*

*<sup>f</sup><sup>a</sup>* <sup>=</sup> <sup>1</sup>

ˆ *<sup>f</sup><sup>a</sup>* <sup>=</sup> <sup>1</sup>

X ∗ *a*(*x*) = *e*

*f*<sup>00000</sup> = 2596/27,

*Using (29), (30) and e*2*πik* = 1 *for any integer k,*

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

*f*(*x*) = ∑

*a*∈*IA*

*<sup>q</sup><sup>k</sup>* ∑ *x*∈*C*<sup>⊥</sup>

*<sup>q</sup><sup>k</sup>* ∑ *x*∈*C*<sup>⊥</sup>

*where IA* = {(*b*1*a*1,..., *bnan*)|*a* ∈ *A*, *bi* ∈ *GF*(*q*)}*. Then, the Fourier coefficients can be computed*

A Description of Experimental Design on the Basis of an Orthonormal System 373

*where C*<sup>⊥</sup> *is an orthogonal design for A (*|*C*⊥| <sup>=</sup> *<sup>q</sup>k).* When an experiment is conducted in accordance to the orthogonal design *C*⊥, unbiased estimators of *fa* in (26) can be obtained by using Theorem 1 and assuming that *E*(*x*) = 0:

Then, the Fourier coefficients can be easily estimated by using Fourier transform. There are a number of software packages for Fourier transform, which can be used to calculate (29) for

**Example 10.** *Consider the case that a set A is given by (14) and the result of experiments is given by*

*f*<sup>10000</sup> = (863 + 954*e*−2*πi*/3 + 779*e*−4*πi*/3)/27,

*f*<sup>20000</sup> = (863 + 779*e*−2*πi*/3 + 954*e*−4*πi*/3)/27,

*f*<sup>01000</sup> = (871 + 873*e*−2*πi*/3 + 852*e*−4*πi*/3)/27,

*f*<sup>02000</sup> = (871 + 852*e*−2*πi*/3 + 873*e*−4*πi*/3)/27,

*f*<sup>00100</sup> = (842 + 868*e*−2*πi*/3 + 886*e*−4*πi*/3)/27,

*f*<sup>00200</sup> = (842 + 886*e*−2*πi*/3 + 868*e*−4*πi*/3)/27,

*f*<sup>00010</sup> = (873 + 848*e*−2*πi*/3 + 875*e*−4*πi*/3)/27,

*f*<sup>00020</sup> = (873 + 875*e*−2*πi*/3 + 848*e*−4*πi*/3)/27,

*f*<sup>00001</sup> = (847 + 864*e*−2*πi*/3 + 885*e*−4*πi*/3)/27,

*f*<sup>00002</sup> = (847 + 885*e*−2*πi*/3 + 864*e*−4*πi*/3)/27,

*f*<sup>11000</sup> = (859 + 863*e*−2*πi*/3 + 874*e*−4*πi*/3)/27,

*f*<sup>12000</sup> = (867 + 868*e*−2*πi*/3 + 861*e*−4*πi*/3)/27,

*f*<sup>21000</sup> = (867 + 861*e*−2*πi*/3 + 868*e*−4*πi*/3)/27,

*f*<sup>22000</sup> = (859 + 874*e*−2*πi*/3 + 863*e*−4*πi*/3)/27,

*f*<sup>10100</sup> = (860 + 861*e*−2*πi*/3 + 875*e*−4*πi*/3)/27,

*f*<sup>10200</sup> = (871 + 857*e*−2*πi*/3 + 868*e*−4*πi*/3)/27,

*f*(*x*)X <sup>∗</sup>

*y*(*x*)X <sup>∗</sup>

*fa*X*a*(*x*), (27)

*a*(*x*), (28)

*a*(*x*). (29)

<sup>−</sup>2*πi*(*a*<sup>1</sup> *<sup>x</sup>*1+*a*<sup>2</sup> *<sup>x</sup>*2+*a*<sup>3</sup> *<sup>x</sup>*3+*a*<sup>4</sup> *<sup>x</sup>*4+*a*<sup>5</sup> *<sup>x</sup>*<sup>5</sup> )/3. (30)

#### **4. 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.

#### **4.1 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:

$$y(\mathbf{x}) = \sum\_{\mathbf{a} \in I\_A} f\_{\mathbf{a}} \mathcal{X}\_{\mathbf{a}}(\mathbf{x}) + \epsilon\_{\mathbf{X}\prime} \tag{26}$$

where *IA* = {(*b*1*a*1,..., *bnan*)|*a* ∈ *A*, *bi* ∈ *GF*(*q*)} and *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 { *fa*|*a* ∈ *IA*}. 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.

**Example 9.** *Consider q* <sup>=</sup> 3, *<sup>n</sup>* <sup>=</sup> <sup>5</sup> *and A* <sup>=</sup> {00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010}*. Then, IA is given by*

*IA* = {00000, 10000, 20000, 01000, 02000, 00100, 00200, 00010, 00020, 00001, 00002, 11000,

12000, 21000, 22000, 10100, 10200, 20100, 20200, 10010, 10020, 20010, 20020},

*and Fourier coefficients*

*f*00000, *f*10000, *f*20000, *f*01000, *f*02000, *f*00100, *f*00200, *f*00010, *f*00020, *f*00001, *f*00002, *f*11000, *f*12000, *f*21000, *f*22000, *f*10100, *f*10200, *f*20100, *f*20200, *f*10010, *f*10020, *f*20010, *f*<sup>20020</sup> *are parameters. The number of parameters is* 23*, and these parameters are independent.*

#### **4.2 Estimation of Fourier coefficients in experimental design**

First, we present the following theorem (Ukita et al., 2010a).

8 Will-be-set-by-IN-TECH

*l*∈*MF*

**Example 8.** *Consider the case that a set A is given by (14) and the result of experiments is given by*

*SSMean* = 249600.6, *SSF*<sup>1</sup> = 1702.3, *SSF*<sup>2</sup> = 29.9, *SSF*<sup>3</sup> = 108.7, *SSF*<sup>4</sup> = 50.3, *SSF*<sup>5</sup> = 80.5, *SSF*1×*F*<sup>2</sup> = 16.6, *SSF*1×*F*<sup>3</sup> = 27.7,

**4. Description of experimental design on the basis of an orthonormal system**

**4.1 Model on the basis of an orthonormal system in experimental design**

*y*(*x*) = ∑

*a*∈*IA*

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

We use *y*(*x*) to denote the response of an experiment with a level combination *x*, and assume

where *IA* = {(*b*1*a*1,..., *bnan*)|*a* ∈ *A*, *bi* ∈ *GF*(*q*)} and *x* is a random error with a zero mean

Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. The effects are represented by the parameters { *fa*|*a* ∈ *IA*}. In addition, there are no constraints between the parameters, and the parameters are independent. Hence, it is clear

**Example 9.** *Consider q* <sup>=</sup> 3, *<sup>n</sup>* <sup>=</sup> <sup>5</sup> *and A* <sup>=</sup> {00000, 10000, 01000, 00100, 00010, 00001, 11000,

*IA* = {00000, 10000, 20000, 01000, 02000, 00100, 00200, 00010, 00020, 00001, 00002, 11000,

12000, 21000, 22000, 10100, 10200, 20100, 20200, 10010, 10020, 20010, 20020},

*f*00000, *f*10000, *f*20000, *f*01000, *f*02000, *f*00100, *f*00200, *f*00010, *f*00020, *f*00001, *f*00002, *f*11000, *f*12000, *f*21000,

*are parameters. The number of parameters is* 23*, and these parameters are independent.*

*SSFl* − ∑

{*l*,*m*}∈*IF*

*<sup>l</sup>*,*m*(*ϕ*, *ψ*) − *SSFl* − *SSFm* − *SSMean*, (24)

*fa*X*a*(*x*) + *x*, (26)

*SSFl*×*Fm* . (25)

*SSFl*×*Fm* <sup>=</sup> <sup>1</sup>

*SSError* = ∑

*Table 1. Then, using (22)–(25),*

system.

the following model:

and a constant variance.

*and Fourier coefficients*

10100, 10010}*. Then, IA is given by*

that the model contains no redundant parameters.

*f*22000, *f*10100, *f*10200, *f*20100, *f*20200, *f*10010, *f*10020, *f*20010, *f*<sup>20020</sup>

*qk*−<sup>2</sup>

*SSF*1×*F*<sup>4</sup> = 60.8, *SSError* = 16.6.

*x*∈*C*<sup>⊥</sup>

*q*−1 ∑ *ϕ*=0

*q*−1 ∑ *ψ*=0 *Y*2

*<sup>y</sup>*2(*x*) <sup>−</sup> *SSMean* <sup>−</sup> ∑

**Theorem 1.** *Sampling Theorem for Bandlimited Functions over a GF*(*q*)*<sup>n</sup> Domain Assume that A* <sup>⊆</sup> {0, 1}*<sup>n</sup> is monotonic and*

$$f(\mathbf{x}) = \sum\_{\mathbf{a} \in I\_A} f\_{\mathbf{a}} \mathcal{X}\_{\mathbf{a}}(\mathbf{x}),\tag{27}$$

*where IA* = {(*b*1*a*1,..., *bnan*)|*a* ∈ *A*, *bi* ∈ *GF*(*q*)}*. Then, the Fourier coefficients can be computed as follows:*

$$f\_{\mathbf{d}} = \frac{1}{q^k} \sum\_{\mathbf{x} \in \mathbb{C}^\perp} f(\mathbf{x}) \mathcal{X}\_{\mathbf{d}}^\*(\mathbf{x}),\tag{28}$$

*where C*<sup>⊥</sup> *is an orthogonal design for A (*|*C*⊥| <sup>=</sup> *<sup>q</sup>k).*

When an experiment is conducted in accordance to the orthogonal design *C*⊥, unbiased estimators of *fa* in (26) can be obtained by using Theorem 1 and assuming that *E*(*x*) = 0:

$$\hat{f}\_{\mathfrak{d}} = \frac{1}{q^{\hat{k}}} \sum\_{\mathfrak{X} \in \mathbb{C}^{\perp}} y(\mathfrak{x}) \mathcal{X}\_{\mathfrak{d}}^{\*}(\mathfrak{x}). \tag{29}$$

Then, the Fourier coefficients can be easily estimated by using Fourier transform. There are a number of software packages for Fourier transform, which can be used to calculate (29) for any monotonic set *A*.

**Example 10.** *Consider the case that a set A is given by (14) and the result of experiments is given by Table 1. Then,*

$$\mathcal{X}\_{\mathbf{f}}^{\*}(\mathbf{x}) = e^{-2\pi i \left(a\_1 \mathbf{x}\_1 + a\_2 \mathbf{x}\_2 + a\_3 \mathbf{x}\_3 + a\_4 \mathbf{x}\_4 + a\_5 \mathbf{x}\_5\right)/3}.\tag{30}$$

*Using (29), (30) and e*2*πik* = 1 *for any integer k,*

ˆ *f*<sup>00000</sup> = 2596/27, ˆ *f*<sup>10000</sup> = (863 + 954*e*−2*πi*/3 + 779*e*−4*πi*/3)/27, ˆ *f*<sup>20000</sup> = (863 + 779*e*−2*πi*/3 + 954*e*−4*πi*/3)/27, ˆ *f*<sup>01000</sup> = (871 + 873*e*−2*πi*/3 + 852*e*−4*πi*/3)/27, ˆ *f*<sup>02000</sup> = (871 + 852*e*−2*πi*/3 + 873*e*−4*πi*/3)/27, ˆ *f*<sup>00100</sup> = (842 + 868*e*−2*πi*/3 + 886*e*−4*πi*/3)/27, ˆ *f*<sup>00200</sup> = (842 + 886*e*−2*πi*/3 + 868*e*−4*πi*/3)/27, ˆ *f*<sup>00010</sup> = (873 + 848*e*−2*πi*/3 + 875*e*−4*πi*/3)/27, ˆ *f*<sup>00020</sup> = (873 + 875*e*−2*πi*/3 + 848*e*−4*πi*/3)/27, ˆ *f*<sup>00001</sup> = (847 + 864*e*−2*πi*/3 + 885*e*−4*πi*/3)/27, ˆ *f*<sup>00002</sup> = (847 + 885*e*−2*πi*/3 + 864*e*−4*πi*/3)/27, ˆ *f*<sup>11000</sup> = (859 + 863*e*−2*πi*/3 + 874*e*−4*πi*/3)/27, ˆ *f*<sup>12000</sup> = (867 + 868*e*−2*πi*/3 + 861*e*−4*πi*/3)/27, ˆ *f*<sup>21000</sup> = (867 + 861*e*−2*πi*/3 + 868*e*−4*πi*/3)/27, ˆ *f*<sup>22000</sup> = (859 + 874*e*−2*πi*/3 + 863*e*−4*πi*/3)/27, ˆ *f*<sup>10100</sup> = (860 + 861*e*−2*πi*/3 + 875*e*−4*πi*/3)/27, ˆ *f*<sup>10200</sup> = (871 + 857*e*−2*πi*/3 + 868*e*−4*πi*/3)/27,

$$\begin{array}{l} f\_{20100} = (871 + 868e^{-2\pi i/3} + 857e^{-4\pi i/3})/27, \\ f\_{20200} = (860 + 875e^{-2\pi i/3} + 861e^{-4\pi i/3})/27, \\ f\_{10010} = (872 + 852e^{-2\pi i/3} + 872e^{-4\pi i/3})/27, \\ f\_{10020} = (858 + 859e^{-2\pi i/3} + 879e^{-4\pi i/3})/27, \\ f\_{20010} = (858 + 879e^{-2\pi i/3} + 859e^{-4\pi i/3})/27, \\ f\_{20020} = (872 + 872e^{-2\pi i/3} + 852e^{-4\pi i/3})/27. \end{array}$$

In particular, when *<sup>q</sup>* <sup>=</sup> <sup>2</sup>*m*, where *<sup>m</sup>* is an integer and *<sup>m</sup>* <sup>≥</sup> 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* ∈ *IA*. The complexity of the vector-radix FFT is *O*(*q<sup>k</sup>* log *qk*). 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).

#### **4.3 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.

**Theorem 2.** *Let <sup>µ</sup>*<sup>ˆ</sup> *be the unbiased estimator of the general mean <sup>µ</sup> in the model of Sect.3.1, and let* <sup>ˆ</sup> *f*0...0 *be that of the Fourier coefficient f*0...0 *in the model of Sect.4.1. Then, the following equation holds:*

$$
\hat{\mu} = \hat{f}\_{0\dots0}.\tag{31}
$$

**Theorem 4.** *Let β*ˆ

Fourier coefficients.

*First, using (31), µ*ˆ = ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

*β*ˆ

orthonormal system.

*of Sect.4.1.*

*model of Sect.3.1, and let* ˆ

*Then, the following equation holds:*

*β*ˆ

*effect of the interaction of F*<sup>1</sup> *and F*2*. Then,*

*computed Fourier coefficients (*2 *parameters). Last, using (33) and (34), the following equations*

1,2(0, 0) = ˆ

1,2(0, 1) = *e*2*πi*/3 ˆ

1,2(0, 2) = *e*4*πi*/3 ˆ

1,2(1, 0) = *e*2*πi*/3 ˆ

1,2(1, 1) = *e*4*πi*/3 ˆ

1,2(1, 2) = ˆ

1,2(2, 0) = *e*4*πi*/3 ˆ

1,2(2, 1) = ˆ

1,2(2, 2) = *e*2*πi*/3 ˆ

*<sup>l</sup>*,*m*(*ϕ*, *ψ*) = ∑

*f*<sup>00000</sup> *holds. Next, using (32) and (34), the following equations*

*al*∈*GF*(*q*) *al*=0

∑*am*∈*GF*(*q*) *am*=0

From these theorems, the effect of each factor can be easily obtained from the computed

A Description of Experimental Design on the Basis of an Orthonormal System 375

**Example 11.** *Let q* = 3 *and n* = 5*. Consider the general mean, the effect of main factor F*1*, and the*

X*l*(*k*) = *e*

*α*ˆ <sup>1</sup>(0) = ˆ

*α*ˆ <sup>1</sup>(1) = *e*2*πi*/3 ˆ

*α*ˆ <sup>1</sup>(2) = *e*4*πi*/3 ˆ

*f*<sup>11000</sup> + ˆ

*f*<sup>11000</sup> + *e*4*πi*/3 ˆ

*f*<sup>11000</sup> + *e*2*πi*/3 ˆ

*f*<sup>11000</sup> + *e*2*πi*/3 ˆ

*f*<sup>11000</sup> + ˆ

*f*<sup>11000</sup> + *e*4*πi*/3 ˆ

*f*<sup>11000</sup> + *e*4*πi*/3 ˆ

*f*<sup>11000</sup> + *e*2*πi*/3 ˆ

*f*<sup>11000</sup> + ˆ

*<sup>l</sup>*,*m*(*ϕ*, *ψ*) *be the unbiased estimator of the effect of the interaction βl*,*m*(*ϕ*, *ψ*) *in the*

<sup>X</sup>*al*(*ϕ*)X*am* (*ψ*) <sup>ˆ</sup>

*f*<sup>10000</sup> + ˆ

*f*<sup>10000</sup> + *e*4*πi*/3 ˆ

*f*<sup>10000</sup> + *e*2*πi*/3 ˆ

*f*<sup>12000</sup> + ˆ

*f*<sup>12000</sup> + *e*2*πi*/3 ˆ

*f*<sup>12000</sup> + *e*4*πi*/3 ˆ

*f*<sup>12000</sup> + *e*4*πi*/3 ˆ

*f*<sup>12000</sup> + ˆ

*f*<sup>12000</sup> + *e*2*πi*/3 ˆ

*f*<sup>12000</sup> + *e*2*πi*/3 ˆ

*f*<sup>12000</sup> + *e*4*πi*/3 ˆ

*f*<sup>12000</sup> + ˆ

*hold. Hence, it is clear that the effects of the interaction of F*<sup>1</sup> *and F*<sup>2</sup> *(*9 *parameters) can be obtained from the computed Fourier coefficients (*4 *parameters).* 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

*hold. Hence, it is clear that the effects of main factor F*<sup>1</sup> *(*3 *parameters) can be obtained from the*

*f*20000,

*f*20000,

*f*20000,

*f*0...0*al*0...0*am*0...0 *be that of the Fourier coefficient f*0...0*al*0...0*am*0...0 *in the model*

*f*0...0*al*0...0*am*0...0. (33)

<sup>2</sup>*πilk*/3. (34)

*f*<sup>21000</sup> + ˆ

*f*<sup>21000</sup> + *e*4*πi*/3 ˆ

*f*<sup>21000</sup> + *e*2*πi*/3 ˆ

*f*<sup>21000</sup> + *e*4*πi*/3 ˆ

*f*<sup>21000</sup> + *e*2*πi*/3 ˆ

*f*<sup>21000</sup> + ˆ

*f*<sup>21000</sup> + *e*2*πi*/3 ˆ

*f*<sup>21000</sup> + ˆ

*f*<sup>21000</sup> + *e*4*πi*/3 ˆ

*f*22000,

*f*22000,

*f*22000,

*f*22000,

*f*22000,

*f*22000,

*f*22000,

*f*22000,

*f*22000,

Next, we present a theorem of the relation between the Fourier coefficients and the effect of the main factor.

**Theorem 3.** *Let α*ˆ*l*(*ϕ*) *be the unbiased estimator of the effect of the main factor αl*(*ϕ*) *in the model of Sect.3.1, and let* ˆ *f*0...0*al*0...0 *be that of the Fourier coefficient f*0...0*al*0...0 *in the model of Sect.4.1. Then, the following equation holds:*

$$\mathfrak{A}\_{l}(\varphi) = \sum\_{\substack{a\_l \in GF(q) \\ a\_l \neq 0}} \mathcal{X}\_{a\_l}(\varphi) \widehat{f}\_{0\dots 0a\_l 0 \dots 0}. \tag{32}$$

Last, we present a theorem of the relation between the Fourier coefficients and the effect of the interaction.

**Theorem 4.** *Let β*ˆ *<sup>l</sup>*,*m*(*ϕ*, *ψ*) *be the unbiased estimator of the effect of the interaction βl*,*m*(*ϕ*, *ψ*) *in the model of Sect.3.1, and let* ˆ *f*0...0*al*0...0*am*0...0 *be that of the Fourier coefficient f*0...0*al*0...0*am*0...0 *in the model of Sect.4.1.*

*Then, the following equation holds:*

10 Will-be-set-by-IN-TECH

*f*<sup>20100</sup> = (871 + 868*e*−2*πi*/3 + 857*e*−4*πi*/3)/27,

*f*<sup>20200</sup> = (860 + 875*e*−2*πi*/3 + 861*e*−4*πi*/3)/27,

*f*<sup>10010</sup> = (872 + 852*e*−2*πi*/3 + 872*e*−4*πi*/3)/27,

*f*<sup>10020</sup> = (858 + 859*e*−2*πi*/3 + 879*e*−4*πi*/3)/27,

*f*<sup>20010</sup> = (858 + 879*e*−2*πi*/3 + 859*e*−4*πi*/3)/27,

*f*<sup>20020</sup> = (872 + 872*e*−2*πi*/3 + 852*e*−4*πi*/3)/27.

In particular, when *<sup>q</sup>* <sup>=</sup> <sup>2</sup>*m*, where *<sup>m</sup>* is an integer and *<sup>m</sup>* <sup>≥</sup> 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* ∈ *IA*. The complexity of the vector-radix FFT is *O*(*q<sup>k</sup>* log *qk*). 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

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

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

First, we present a theorem of the relation between the Fourier coefficient and the general

**Theorem 2.** *Let <sup>µ</sup>*<sup>ˆ</sup> *be the unbiased estimator of the general mean <sup>µ</sup> in the model of Sect.3.1, and let* <sup>ˆ</sup>

Next, we present a theorem of the relation between the Fourier coefficients and the effect of

**Theorem 3.** *Let α*ˆ*l*(*ϕ*) *be the unbiased estimator of the effect of the main factor αl*(*ϕ*) *in the model of*

Last, we present a theorem of the relation between the Fourier coefficients and the effect of the

*α*ˆ*l*(*ϕ*) = ∑

*al*∈*GF*(*q*) *al*=0

*f*0...0*al*0...0 *be that of the Fourier coefficient f*0...0*al*0...0 *in the model of Sect.4.1.*

<sup>X</sup>*al*(*ϕ*) <sup>ˆ</sup>

*f*0...0. (31)

*f*0...0*al*0...0. (32)

*µ*ˆ = ˆ

**4.3 The relation between the Fourier coefficients and the effect of each factor**

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

direct representation of the effect of each factor.

*Then, the following equation holds:*

*Then, the following equation holds:*

the effect of each factor (Ukita & Matsushima, 2011).

*f*0...0 *be that of the Fourier coefficient f*0...0 *in the model of Sect.4.1.*

of (29).

mean.

the main factor.

*Sect.3.1, and let* ˆ

interaction.

$$\hat{\beta}\_{l,m}(\boldsymbol{\varphi},\boldsymbol{\psi}) = \sum\_{\substack{\boldsymbol{a}\_l \in \mathrm{GF}(q) \\ \boldsymbol{a}\_l \neq 0 \\ \boldsymbol{a}\_l \neq 0}} \sum\_{\substack{\boldsymbol{a}\_l \in \mathrm{GF}(q) \\ \boldsymbol{a}\_m \neq 0}} \mathcal{X}\_{\boldsymbol{a}\_l}(\boldsymbol{\varphi}) \mathcal{X}\_{\boldsymbol{a}\_m}(\boldsymbol{\psi}) \boldsymbol{f}\_{0\dots 0 \boldsymbol{a}\_l 0\dots 0 \boldsymbol{a}\_m 0\dots 0} \,. \tag{33}$$

From these theorems, the effect of each factor can be easily obtained from the computed Fourier coefficients.

**Example 11.** *Let q* = 3 *and n* = 5*. Consider the general mean, the effect of main factor F*1*, and the effect of the interaction of F*<sup>1</sup> *and F*2*. Then,*

$$\mathcal{X}\_l(k) = e^{2\pi ilk/3}. \tag{34}$$

*First, using (31), µ*ˆ = ˆ *f*<sup>00000</sup> *holds. Next, using (32) and (34), the following equations*

$$\begin{array}{ll} \pounds\_1(0) = \bigwedge\_{10000} + \bigwedge\_{20000} \\ \pounds\_1(1) = e^{2\pi i/3} f\_{10000} + e^{4\pi i/3} f\_{20000} \\ \pounds\_1(2) = e^{4\pi i/3} f\_{10000} + e^{2\pi i/3} f\_{20000} \end{array}$$

*hold. Hence, it is clear that the effects of main factor F*<sup>1</sup> *(*3 *parameters) can be obtained from the computed Fourier coefficients (*2 *parameters).*

*Last, using (33) and (34), the following equations*

*β*ˆ 1,2(0, 0) = ˆ *f*<sup>11000</sup> + ˆ *f*<sup>12000</sup> + ˆ *f*<sup>21000</sup> + ˆ *f*22000, *β*ˆ 1,2(0, 1) = *e*2*πi*/3 ˆ *f*<sup>11000</sup> + *e*4*πi*/3 ˆ *f*<sup>12000</sup> + *e*2*πi*/3 ˆ *f*<sup>21000</sup> + *e*4*πi*/3 ˆ *f*22000, *β*ˆ 1,2(0, 2) = *e*4*πi*/3 ˆ *f*<sup>11000</sup> + *e*2*πi*/3 ˆ *f*<sup>12000</sup> + *e*4*πi*/3 ˆ *f*<sup>21000</sup> + *e*2*πi*/3 ˆ *f*22000, *β*ˆ 1,2(1, 0) = *e*2*πi*/3 ˆ *f*<sup>11000</sup> + *e*2*πi*/3 ˆ *f*<sup>12000</sup> + *e*4*πi*/3 ˆ *f*<sup>21000</sup> + *e*4*πi*/3 ˆ *f*22000, *β*ˆ 1,2(1, 1) = *e*4*πi*/3 ˆ *f*<sup>11000</sup> + ˆ *f*<sup>12000</sup> + ˆ *f*<sup>21000</sup> + *e*2*πi*/3 ˆ *f*22000, *β*ˆ 1,2(1, 2) = ˆ *f*<sup>11000</sup> + *e*4*πi*/3 ˆ *f*<sup>12000</sup> + *e*2*πi*/3 ˆ *f*<sup>21000</sup> + ˆ *f*22000, *β*ˆ 1,2(2, 0) = *e*4*πi*/3 ˆ *f*<sup>11000</sup> + *e*4*πi*/3 ˆ *f*<sup>12000</sup> + *e*2*πi*/3 ˆ *f*<sup>21000</sup> + *e*2*πi*/3 ˆ *f*22000, *β*ˆ 1,2(2, 1) = ˆ *f*<sup>11000</sup> + *e*2*πi*/3 ˆ *f*<sup>12000</sup> + *e*4*πi*/3 ˆ *f*<sup>21000</sup> + ˆ *f*22000, *β*ˆ 1,2(2, 2) = *e*2*πi*/3 ˆ *f*<sup>11000</sup> + ˆ *f*<sup>12000</sup> + ˆ *f*<sup>21000</sup> + *e*4*πi*/3 ˆ *f*22000,

*hold. Hence, it is clear that the effects of the interaction of F*<sup>1</sup> *and F*<sup>2</sup> *(*9 *parameters) can be obtained from the computed Fourier coefficients (*4 *parameters).*

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.

#### **4.4 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.

**Theorem 5.** *Let SSMean be the sum of squares due to the mean in Sect.3.4, and let* ˆ *f*0...0 *be the unbiased estimator of the Fourier coefficient f*0...0 *in the model of Sect.4.1. Then,*

$$q^k |\hat{f}\_{0\dots0}|^2 = \mathcal{SS}\_{\text{Mean}\prime} \tag{35}$$

*where*

$$f\_{0\ldots0} = \frac{1}{q^k} \sum\_{\mathbf{x} \in \mathbb{C}^\perp} y(\mathbf{x}) \mathcal{X}\_{0\ldots0}^\*(\mathbf{x}).\tag{36}$$

**Example 12.** *Consider the case that a set A is given by (14) and the result of experiments is given by*

A Description of Experimental Design on the Basis of an Orthonormal System 377

= (863<sup>2</sup> <sup>+</sup> 9542 <sup>+</sup> <sup>779</sup><sup>2</sup> <sup>−</sup> <sup>863</sup> · <sup>954</sup> <sup>−</sup> <sup>863</sup> · <sup>779</sup> <sup>−</sup> <sup>954</sup> · <sup>779</sup>)/272

*<sup>k</sup>*=<sup>0</sup> *<sup>e</sup>*2*πik*/3 <sup>=</sup> <sup>0</sup> *and e*2*π<sup>i</sup>* <sup>=</sup> <sup>1</sup>*,*

<sup>−</sup>4*πi*/3)(863 + 954*e*

= 31.52401. (43)

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

*f*00200|

*f*00002|

*f*21000|

*f*20100|

*f*20010|

*f*00100|

*f*00001|

*f*12000|

*f*10200|

*f*10020|

<sup>2</sup>) = 1702.3,

<sup>2</sup>) = 29.9,

<sup>2</sup>) = 108.7,

<sup>2</sup>) = 50.3,

<sup>2</sup>) = 80.5,

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

*f*21000|

*f*20100|

*f*20010|

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

*f*22000|

*f*20200|

*f*20020|

<sup>00000</sup> = (2596/27)(2596/27) = 9244.466, (42)

<sup>2</sup>*πi*/3 + 779*e*

<sup>2</sup> = 2.013717,

<sup>2</sup> = 1.491084,

<sup>2</sup> = 0.058985,

<sup>2</sup> = 0.223594,

<sup>2</sup> = 0.577503.

<sup>2</sup>) = 16.6,

<sup>2</sup>) = 27.7,

<sup>2</sup>) = 60.8,

<sup>4</sup>*πi*/3)/272

*Table 1. Then, using the result of Example10,* ∑<sup>2</sup>

*f*<sup>10000</sup> ˆ *f* ∗ 10000

= (863 + 954*e*

<sup>2</sup> = ˆ

<sup>2</sup> = 31.52401,

*f*02000|

*f*00020|

*f*22000|

*f*20200|

*f*20020|

*f*00000|

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

*f*11000|

*f*10100|

*f*10010|

*f*10000|

*f*01000|

*f*00100|

*f*00010|

*f*00001|

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> <sup>|</sup> <sup>ˆ</sup>

*SSMean* <sup>=</sup> <sup>27</sup><sup>|</sup> <sup>ˆ</sup>

*SSF*<sup>1</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*<sup>2</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*<sup>3</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*<sup>4</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*<sup>5</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*1×*F*<sup>2</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*1×*F*<sup>3</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSF*1×*F*<sup>4</sup> <sup>=</sup> <sup>27</sup>(<sup>|</sup> <sup>ˆ</sup>

*SSError* = 16.6.

design on the basis of an orthonormal system.

experimental design.

**5. Conclusion**

*f*<sup>00000</sup> ˆ *f* ∗

<sup>−</sup>2*πi*/3 + 779*e*

<sup>2</sup> <sup>=</sup> 0.552812, <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> 0.931413, <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> 0.248285, <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> 0.289438, <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> 0.548697, <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> = 249600.6,

*f*20000|

*f*02000|

*f*00200|

*f*00020|

*f*00002|

*f*12000|

*f*10200|

*f*10020|

Therefore, the analysis of variance can be executed in the proposed description of

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

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

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup> <sup>ˆ</sup>


<sup>2</sup> = ˆ







*Hence, using Theorem 5–7 and (41),*


*Similarly,*

**Theorem 6.** *Let SSFl be the sum of squares due to the effect of Fl in Sect.3.4, and let* <sup>ˆ</sup> *f*0...0*al*0...0 *be the unbiased estimator of the Fourier coefficient f*0...0*al*0...0 *in the model of Sect.4.1. Then,*

$$\sum\_{\substack{a\_l \in GF(q) \\ a\_l \neq 0}} q^k |f\_{0\dots 0a\_l 0\dots 0}|^2 = \mathcal{SS}\_{F\_l \prime} \quad l = 1, 2, \dots, n,\tag{37}$$

*where*

$$f\_{0\ldots0a\_l0\ldots0} = \frac{1}{q^k} \sum\_{\mathbf{x} \in \mathbb{C}^\perp} \mathcal{Y}(\mathbf{x}) \mathcal{X}\_{0\ldots0a\_l0\ldots0}^\*(\mathbf{x}).\tag{38}$$

**Theorem 7.** *Let SSFl*×*Fm be the sum of squares due to the interaction effect of Fl* <sup>×</sup> *Fm in Sect.3.4, and let* ˆ *f*0...0*al*0...0*am*0...0 *be the unbiased estimator of the Fourier coefficient f*0...0*al*0...0*am*0...0 *in the model of Sect.4.1. Then,*

$$\sum\_{a\_l \neq 0} \sum\_{a\_m \neq 0} q^k |f\_{0\dots 0a\_l 0\dots 0a\_m 0\dots 0}|^2 = \mathcal{SS}\_{F\_l \times F\_m \nu}$$

$$l \,\, m = 1 \, 2 \, \cdots \, \, \_ \nu \,\, \_ \nu \,\, (l < m) \,\, \, \tag{39}$$

*where the sums are taken over al*, *am* ∈ *GF*(*q*) *and*

$$f\_{0\ldots0a/0\ldots0a\_{m}0\ldots0} = \frac{1}{q^k} \sum\_{\mathbf{x}\in\mathbb{C}^\perp} y(\mathbf{x}) \mathcal{X}\_{0\ldots0a/0\ldots0a\_{m}0\ldots0}^\*(\mathbf{x}).\tag{40}$$

By these theorems, *SSMean*, *SSFl* and *SSFl*×*Fm* can be obtained in the proposed description of experimental design. In addition, using the Parseval-Plancherel formula and these theorems, *SSError* can be computed as follows.

$$SS\_{Error} = \sum\_{\mathbf{x} \in \mathbb{C}^\perp} y^2(\mathbf{x}) - SS\_{Mean} - \sum\_{l \in MF} SS\_{\overline{l}l} - \sum\_{\{l, m\} \in IF} SS\_{\overline{l} \times F\_m}.\tag{41}$$

**Example 12.** *Consider the case that a set A is given by (14) and the result of experiments is given by Table 1. Then, using the result of Example10,* ∑<sup>2</sup> *<sup>k</sup>*=<sup>0</sup> *<sup>e</sup>*2*πik*/3 <sup>=</sup> <sup>0</sup> *and e*2*π<sup>i</sup>* <sup>=</sup> <sup>1</sup>*,*

$$|f\_{00000}|^2 = f\_{00000} f\_{00000}^\* = (2596/27)(2596/27) = 9244.466,\tag{42}$$

$$\begin{split} |f\_{10000}|^2 &= \hat{f}\_{10000} \hat{f}\_{10000}^\* \\ &= (863 + 954e^{-2\pi i/3} + 779e^{-4\pi i/3})(863 + 954e^{2\pi i/3} + 779e^{4\pi i/3})/27^2 \\ &= (863^2 + 954^2 + 779^2 - 863 \cdot 954 - 863 \cdot 779 - 954 \cdot 779)/27^2 \\ &= 31.52401. \end{split} \tag{43}$$

*Similarly,*

12 Will-be-set-by-IN-TECH

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.

*f*0...0 *be the unbiased*

*f*0...0*al*0...0 *be the*

<sup>2</sup> = *SSMean*, (35)

0...0(*x*). (36)

, *l* = 1, 2, ··· , *n*, (37)

0...0*al*0...0(*x*). (38)

0...0*al*0...0*am*0...0(*x*). (40)

*SSFl*×*Fm* . (41)

**Theorem 5.** *Let SSMean be the sum of squares due to the mean in Sect.3.4, and let* ˆ

*<sup>q</sup>k*<sup>|</sup> <sup>ˆ</sup> *f*0...0|

> *<sup>q</sup><sup>k</sup>* ∑ *x*∈*C*<sup>⊥</sup>

> > <sup>2</sup> <sup>=</sup> *SSFl*

*<sup>q</sup><sup>k</sup>* ∑ *x*∈*C*<sup>⊥</sup>

**Theorem 7.** *Let SSFl*×*Fm be the sum of squares due to the interaction effect of Fl* <sup>×</sup> *Fm in Sect.3.4, and*

*f*0...0*al*0...0*am*0...0|

By these theorems, *SSMean*, *SSFl* and *SSFl*×*Fm* can be obtained in the proposed description of experimental design. In addition, using the Parseval-Plancherel formula and these theorems,

*<sup>q</sup><sup>k</sup>* ∑ *x*∈*C*<sup>⊥</sup>

*<sup>y</sup>*2(*x*) <sup>−</sup> *SSMean* <sup>−</sup> ∑

*f*0...0*al*0...0*am*0...0 *be the unbiased estimator of the Fourier coefficient f*0...0*al*0...0*am*0...0 *in the model of*

*y*(*x*)X <sup>∗</sup>

*l*∈*MF*

**Theorem 6.** *Let SSFl be the sum of squares due to the effect of Fl in Sect.3.4, and let* <sup>ˆ</sup>

*unbiased estimator of the Fourier coefficient f*0...0*al*0...0 *in the model of Sect.4.1. Then,*

*f*0...0*al*0...0|

*y*(*x*)X <sup>∗</sup>

*y*(*x*)X <sup>∗</sup>

<sup>2</sup> <sup>=</sup> *SSFl*×*Fm* ,

*SSFl* − ∑

{*l*,*m*}∈*IF*

*l*, *m* = 1, 2, ··· , *n*,(*l* < *m*), (39)

*estimator of the Fourier coefficient f*0...0 *in the model of Sect.4.1. Then,*

ˆ *<sup>f</sup>*0...0 <sup>=</sup> <sup>1</sup>

∑*al*∈*GF*(*q*) *al*=0

ˆ

∑ *al*=0

*where the sums are taken over al*, *am* ∈ *GF*(*q*) *and*

ˆ

*SSError* can be computed as follows.

*SSError* = ∑

*x*∈*C*<sup>⊥</sup>

∑ *am*=0

*<sup>f</sup>*0...0*al*0...0*am*0...0 <sup>=</sup> <sup>1</sup>

*<sup>q</sup>k*<sup>|</sup> <sup>ˆ</sup>

*<sup>q</sup>k*<sup>|</sup> <sup>ˆ</sup>

*<sup>f</sup>*0...0*al*0...0 <sup>=</sup> <sup>1</sup>

**4.4 Analysis of variance in experimental design**

*where*

*where*

*let* ˆ

*Sect.4.1. Then,*

$$\begin{array}{l} \left|\hat{f}\_{2000}\right|^2 = 31.52401, \\ \left|\hat{f}\_{0100}\right|^2 = \left|\hat{f}\_{0200}\right|^2 = 0.552812, \left|\hat{f}\_{00100}\right|^2 = \left|\hat{f}\_{0200}\right|^2 = 2.013717, \\ \left|\hat{f}\_{0010}\right|^2 = \left|\hat{f}\_{0020}\right|^2 = 0.931413, \left|\hat{f}\_{0000}\right|^2 = \left|\hat{f}\_{0000}\right|^2 = 1.491084, \\ \left|\hat{f}\_{1000}\right|^2 = \left|\hat{f}\_{2200}\right|^2 = 0.248285, \left|\hat{f}\_{1200}\right|^2 = \left|\hat{f}\_{2100}\right|^2 = 0.058985, \\ \left|\hat{f}\_{10100}\right|^2 = \left|\hat{f}\_{2020}\right|^2 = 0.289438, \left|\hat{f}\_{1020}\right|^2 = \left|\hat{f}\_{2010}\right|^2 = 0.223594, \\ \left|\hat{f}\_{1010}\right|^2 = \left|\hat{f}\_{2020}\right|^2 = 0.548697, \left|\hat{f}\_{10020}\right|^2 = \left|\hat{f}\_{2010}\right|^2 = 0.577503. \end{array}$$

*Hence, using Theorem 5–7 and (41),*

$$\begin{array}{l} \begin{array}{l} \left| SS\_{Mean} = 27 \right| |\hat{f}\_{00000}|^{2} = 249600.6, \\ SS\_{F\_1} = 27 | (\hat{f}\_{10000})^{2} + |\hat{f}\_{20000}|^{2}) = 1702.3, \\ SS\_{F\_2} = 27 (| \hat{f}\_{01000}|^{2} + |\hat{f}\_{02000}|^{2}) = 29.9, \\ SS\_{F\_3} = 27 (| \hat{f}\_{00100}|^{2} + |\hat{f}\_{00200}|^{2}) = 108.7, \\ SS\_{F\_4} = 27 (| \hat{f}\_{00010}|^{2} + |\hat{f}\_{00020}|^{2}) = 50.3, \\ SS\_{F\_5} = 27 (| \hat{f}\_{10000}|^{2} + |\hat{f}\_{00002}|^{2}) = 80.5, \\ SS\_{F\_1 \times F\_2} = 27 (| \hat{f}\_{10100}|^{2} + |\hat{f}\_{12000}|^{2} + |\hat{f}\_{21000}|^{2} + |\hat{f}\_{22000}|^{2}) = 16.6, \\ SS\_{F\_1 \times F\_3} = 27 (| \hat{f}\_{10100}|^{2} + |\hat{f}\_{10200}|^{2} + |\hat{f}\_{20100}|^{2} + |\hat{f}\_{20200}|^{2}) = 27.7, \\ SS\_{F\_1 \times F\_4} = 27 (| \hat{f}\_{10010}|^{2} + |\hat{f}\_{10020}|^{2} + |\hat{f}\_{2010}|^{2} + |\hat{f}\_{2020}|^{2}) = 60.8, \\ SS\_{F\_$$

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.

#### **5. 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

**0**

**19**

Zbigniew Szadkowski

*Department of Physics and Applied Informatics, Faculty of High Energy Astrophysics, Łód´z*

*University of Łód´z*

*Poland*

**An Optimization of 16-Point Discrete Cosine**

**in Extensive Air Shower Experiments**

**Transform Implemented into a FPGA as a Design**

**for a Spectral First Level Surface Detector Trigger**

The Pierre Auger Observatory is a ground based detector located in Malargue (Argentina) (Auger South) at 1400 m above the sea level and dedicated to the detection of ultra high-energy cosmic rays with energies above 10<sup>18</sup> eV with unprecedented statistical and systematical accuracy. The main goal of cosmic rays investigation in this energy range is to determine the origin and nature of particles produced at these enormous energies as well as their energy spectrum. These cosmic particles carry information complementary to neutrinos and photons and even gravitational waves. They also provide an extremely energetic stream for the study of particle interactions at energies orders of magnitude above energies reached

The flux of cosmic rays above 10<sup>19</sup> eV is extraordinarily low: on the order of one event per square-kilometer per century. Only detectors of exceptional size, thousands of square-kilometers, may acquire a significant number of events. The nature of the primary particles must be inferred from properties of the associated extensive air showers (EAS). The Pierre Auger Observatory consists of a surface detectors (SD) array spread over 3000 km2 for measuring the charged particles of EAS and their lateral density profile of muon and electromagnetic components in the shower front at ground, and of 24 wide-angle Schmidt telescopes installed at 4 locations at the boundary of the ground array measuring the fluorescence light associated with the evolution of air showers: the growth and subsequent deterioration during a development. Such a "hybrid" measurements allow cross-calibrations between different experimental techniques, controlling and reducing the

Very inclined showers are different from the ordinary vertical ones. At large zenith angles the slant atmospheric depth to ground level is enough to absorb the part of the shower that follows from the standard cascading interactions, both of electromagnetic and hadronic type. Only penetrating particles such as muons and neutrinos can traverse the atmosphere at large zenith angles to reach the ground or to induce secondary showers deep in the atmosphere and

**1. Introduction**

systematic uncertainties.

close to an air shower detector.

at terrestrial accelerators (Abraham J. et al., 2004).

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.

#### **6. References**

