![David C. Wyld et al. (Eds) : NETCOM, NCS, WiMoNe, CSEIT, SPM - 2015
pp. 237–247, 2015. © CS & IT-CSCP 2015 DOI : 10.5121/csit.2015.51620
A GENERALIZED SAMPLING THEOREM
OVER GALOIS FIELD DOMAINS FOR
EXPERIMENTAL DESIGN
Yoshifumi Ukita
Department of Management and Information,
Yokohama College of Commerce, Yokohama, Japan
ukita@shodai.ac.jp
ABSTRACT
In this paper, the sampling theorem for bandlimited functions over ܨܩሺݍሻ
domains is
generalized to one over ∏ୀଵ
ܨܩሺݍሻ domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEYWORDS
Digital Signal Processing, Sampling Theorem, Experimental Design, Orthogonal Arrays,
Fourier Analysis
1. INTRODUCTION
In digital signal processing [3], the sampling theorem states that any real valued function ݂ can be
reconstructed from a sequence of values of ݂ that are discretely sampled with a frequency at least
twice as high as the maximum frequency of the spectrum of ݂. This theorem can also be applied
to functions over finite domain [4] [8]. For example, Ukita et al. obtained a sampling theorem
over ܨܩሺݍሻ
domains [8], which is applicable to the experimental design model in which all
factors have the same number of levels. However, this sampling theorem is not applicable to the
model in which each factor has a different number of levels, even though they often do [2], [7].
Moreover, a sampling theorem for such a model has not been provided so far. In this paper, the
sampling theorem for bandlimited functions over ܨܩሺݍሻ
domains is generalized to one over
∏ୀଵ
ܨܩሺݍሻ domains. The generalized theorem is applicable to the experimental design model in
which each factor has a different number of levels and enables us to estimate the parameters in
the model using Fourier transforms. In addition, recently, the volume of the data has grown up
rapidly in the field of Big Data and Cloud Computing [11] [12], and the generalized theorem can
also be used to estimate the parameters for Big Data efficiently. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays [1] is provided.](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-1-320.jpg)
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2. PRELIMINARIES
2.1 Fourier Analysis on Finite Abelian Groups
Here, a brief explanation of Fourier analysis on finite Abelian groups is provided. Characters are
important in the context of finite Fourier series.
2.1.1 Characters [5]
Let ܩ be a finite Abelian group (with the additive notation), and let ܵଵ
be the unit circle in the
complex plane. A character on ܩ is a complex-valued function ߯: ܩ → ܵଵ
that satisfies the
condition
߯൫࢞ + ࢞′
൯ = ߯ሺ࢞ሻ߯൫࢞′
൯ ∀࢞, ࢞′
∈ .ܩ (1)
In other words, a character is a homomorphism from ܩ to the circle group.
2.1.2 Fourier Transform [4]
Let ܩ, ݅ = 1,2, ⋯ , ݊, be Abelian groups of respective orders |ܩ| = ݃, ݅ = 1,2, ⋯ ݊, ݃ଵ ≤ ݃ଶ ≤
⋯ ≤ ݃, and
ܩ =×ୀଵ
ܩ ܽ݊݀ ݃ = ෑ ݃
ୀଵ
. ሺ2ሻ
Since the character group of ܩ is isomorphic to ,ܩ we can index the characters by the elements of
,ܩ that is, ሼ ߯࢝ሺ࢞ሻ|࢝ ∈ }ܩ are the characters of .ܩ Note that ߯ሺ࢞ሻ is the principal character, and
it is identically equal to 1. The charactersሼ ߯࢝ሺ࢞ሻ|࢝ ∈ }ܩ form an orthonormal system:
1
݃
߯࢝ሺ࢞ሻ߯ࢠ
∗ሺ࢞ሻ
࢞∈ீ
= ቄ
1, ࢝ = ࢠ,
0, ࢝ ≠ ࢠ,
ሺ3ሻ
where ߯ࢠ
∗ሺ࢞ሻ is the complex conjugate of ߯ࢠሺ࢞ሻ.
Any function ݂: ܩ → ℂ, where ℂ is the field of complex numbers, can be uniquely expressed as a
linear combination of the following characters:
݂ሺ࢞ሻ = ݂࢝
࢝ ∈ீ
߯࢝ሺ࢞ሻ, ሺ4ሻ
where the complex number
݂࢝ =
1
݃
݂ሺ࢞ሻ
࢞ ∈ீ
߯࢝
∗ ሺ࢞ሻ, ሺ5ሻ
is the ࢝-th Fourier coefficient of ݂.](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-2-320.jpg)
![Computer Science & Information Technology (CS & IT) 239
2.2 Fourier Analysis on ∏ୀ
ࡳࡲሺሻ
Assume that ݍ, ݅ = 1,2, ⋯ ݊, are prime powers. Let ܨܩሺݍሻ, ݅ = 1,2, ⋯ ݊, be a Galois fields of
respective orders ݍ, ݅ = 1,2, ⋯ ݊, which contain finite numbers of elements. We also use
∏ୀଵ
ܨܩሺݍሻ to denote the set of all ݊-tuples with entries from ܨܩሺݍሻ, ݅ = 1,2, ⋯ ݊. The
elements of ∏ୀଵ
ܨܩሺݍሻ are expressed as vectors.
Example 1: Consider ܨܩሺ2ሻ = ሼ0,1} and ܨܩሺ3ሻ = ሼ0,1,2}. Then, if ݊ = 3 and ݍଵ = 2, ݍଶ =
2, ݍଷ = 3,
ෑ ܨܩሺݍሻ
ଷ
ୀଵ
= ሼ000,001,002,010,011,012,100,101,102,110,111,112}.
Specifying the group ܩ in Sect. 2.1.2 to be the group of ∏ୀଵ
ܨܩሺݍሻ and ݃ = ∏ୀଵ
ݍ, the
relations (3),(4) and (5) also hold over the ∏ୀଵ
ܨܩሺݍሻ domain.
Then, the characters ሼ ߯࢝ሺ࢞ሻ|࢝ ∈ ∏ୀଵ
ܨܩሺݍሻ} form an orthonormal system:
1
∏ୀଵ
ݍ
߯࢝ሺ࢞ሻ߯ࢠ
∗ሺ࢞ሻ
࢞∈∏సభ
ீிሺሻ
= ቄ
1, ࢝ = ࢠ,
0, ࢝ ≠ ࢠ,
ሺ6ሻ
Any function ݂: ∏ୀଵ
ܨܩሺݍሻ → ℂ, can be uniquely expressed as a linear combination of the
following characters:
݂ሺ࢞ሻ = ݂࢝
࢝ ∈∏సభ
ீிሺሻ
߯࢝ሺ࢞ሻ, ሺ7ሻ
where the complex number
݂࢝ =
1
∏ୀଵ
ݍ
݂ሺ࢞ሻ
࢞ ∈∏సభ
ீிሺሻ
߯࢝
∗ ሺ࢞ሻ, ሺ8ሻ
is the ࢝-th Fourier coefficient of ݂.
3. EXPERIMENTAL DESIGN
In this section, a short introduction to experimental design [2], [7] is provided.
3.1 Experimental Design Model
Let ܨଵ, ܨଶ, ⋯ , ܨ denote the ݊ factors to be included in an experiment. The levels of factor ܨ can
be represented by ܨܩሺݍሻ, and the level combinations can be represented by the ݊-tuples
࢞ = ሺݔଵ, ݔଶ, ⋯ , ݔሻ ∈ ∏ୀଵ
ܨܩሺݍሻ.
Example 2:
Let Machine (ܨଵ) and Worker (ܨଶ) be factors that might influence the quantity of a product.
ܨଵ : new machine (level 0), old machine (level 1),](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-3-320.jpg)
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ܨଶ: highly skilled worker (level 0), average skilled worker (level 1), unskilled worker (level 2).
For example, ࢞ = 01 represents a combination of new machine and average skilled worker.
Then, the effect of the machine, averaged over all workers, is referred to as the effect of main
factor ܨଵ. Similarly, the effect of the worker, averaged over both machines, is referred to as the
effect of main factor ܨଶ. The contrast between the effect of the machine for a highly skilled
worker, the effect of the machine for an average skilled worker, and the effect of the machine for
an unskilled worker is referred to as the effect of the interaction of ܨଵ and ܨଶ.
Next, an explanation of the model in the context of experimental design is given. In previous
works [8], [9], [10], all factors were restricted to have the same number of levels. In this paper, I
give the definition of the generalized model in which each factor has a different number of levels
as follows.
Definition 1: Generalized Model
y(x) is used to denote the response of the experiment with level combination x and assume the
model
ݕሺ࢞ሻ = ݂࢝߯࢝ሺ࢞ሻ
࢝∈ூಲ
+ ߳࢞, ሺ9ሻ
where
ܫ = ሼ ሺbଵaଵ, bଶaଶ, … , b୬a୬ሻ|ࢇ ∈ ,ܣ b୧ ∈ ܨܩሺݍሻ}. ሺ10ሻ
The set ܣ ⊆ ሼ 0,1 }୬
represents the general mean, main factors, and interactive factors included
in the model.
(For example, consider ܣ ⊆ ሼ 000,100,010,001,110 }.Then, 000,100,010,001,110 indicate the
general mean, main factor of ܨଵ, main factor of ܨଶ, main factor of ܨଷ, and interactive factor of ܨଵ
and ܨଶ, respectively.) The model includes a random error
߳࢞ satisfying the expected value ܧሺ߳࢞ሻ = 0 and constant variance ߪଶ
.
In addition, it is usually assumed that the set ܣ satisfies the following monotonicity condition [2].
Definition 2: Monotonicity
ࢇ ∈ ܣ → ࢇ′ ∈ ܣ ∀ ࢇ′ ሺࢇ′ ⊑ ࢇሻ, ሺ11ሻ
where ࢇ = ሺܽଵ, ܽଶ, ⋯ , ܽሻ, ࢇ′ = ሺܽ′ଵ, ܽ′ଶ, ⋯ , ܽ′ሻ and ࢇ′ ⊑ ࢇ means that if ܽ = 0 then
ܽ′ = 0, ݅ = 1,2, ⋯ , ݊.
Example 3:
Consider ܣ = ሼ 00000,10000,01000,00100,00010,00001,11000,10100,10010 }.
Since the set ܣ satisfies (11), ܣ is monotonic.](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-4-320.jpg)
![Computer Science & Information Technology (CS & IT) 241
Next, let ࢝ = ሺݓଵ, ݓଶ, ⋯ , ݓሻ. The main effect of ܨ is represented by ሼ ݂࢝| ݓ ≠ 0 ܽ݊݀ ݓ =
0 ݂ݎ ݇ ≠ ݅}. The interaction of ܨ and ܨ is represented by ሼ ݂࢝| ݓ ≠ 0 ܽ݊݀ ݓ ≠ 0 ܽ݊݀ ݓ =
0 ݂ݎ ݇ ≠ ݅, ݆}
Example 4:
Consider ܣ given in Example 3 and ݍଵ = 2, ݍ = 3, ݅ = 2, … , 5. Then, ܫ is given by
ܫ = ሼ00000,10000,01000,02000,00100,00200,00010,00020,
00001,00002,11000,12000}.
For example, the main effect of ܨଵ is represented by ݂ଵ, and the interaction of ܨଵ and ܨଶ is
represented by ݂ଵଵ and ݂ଵଶ.
In experimental design, we are given a model of the experiment. In other words, we are given a
set ܣ ⊆ ሼ 0,1 }
. Then, we determine a set of level combinations ݔ ∈ ܺ, ܺ ⊆ ∏ୀଵ
ܨܩሺݍሻ. The
set ܺ is called a design. Next, we perform a set of experiments according to the design ܺ and
estimate the effects from the result, ሼ ሺ࢞, ݕሺ࢞ሻሻ|࢞ ∈ ܺ }.
An important standard for evaluating designs is the maximum of the variances of the unbiased
estimators of effects calculated from the result of the experiments. It is known that, for a given
number of experiments, this criterion is minimized in an orthogonal design [6].
3.2 Orthogonal Designs
In this subsection, a definition of Orthogonal Designs for the generalized model is provided.
Definition 3: Orthogonal Designs
At first, define ݒሺࢇሻ = ሼ݅ |ܽ ≠ 0, 1 ≤ ݅ ≤ ݊ }.
For ࢇ = ሺܽଵଵ, ܽଵଶ, … , ܽଵሻ, ࢇ = ሺܽଶଵ, ܽଶଶ, … , ܽଶሻ ∈ ሼ0,1 }
, the addition of vectors ࢇ and
ࢇ is defined by ࢇ + ࢇ = ሺܽଵଵ ⊕ ܽଶଵ, ܽଵଶ ⊕ ܽଶଶ, … , ܽଵ ⊕ ܽଶሻ, where ⊕ is the exclusive
or operation.
An orthogonal design ܥୄ
for ܣ ⊆ ሼ0,1 }
is satisfies the condition that for any ࢇ, ࢇ′ ∈ ,ܣ
หܥభ,…,
ୄ
ሺ߮ଵ, … , ߮ሻห =
|ܥୄ|
ݍభ
ݍమ
… ݍ
,
߮ଵ ∈ ܨܩ൫ݍభ
൯, … , ߮ ∈ ܨܩ൫ݍ
൯ (12)
where ݅ଵ, … , ݅ are defined by ݒሺࢇ + ࢇᇱሻ = ሼ݅ଵ, … , ݅}, and ܥభ,…,
ୄ
ሺ߮ଵ, … , ߮ሻ = ሼ࢞|ݔభ
=
߮ଵ, … , ݔ
= ߮, ࢞ ∈ ܥୄ
}.
Example 5:
Consider ܣ given in Example 3 and ݍଵ = 2, ݍ = 3, ݅ = 2, … , 5.
Then, an orthogonal design ܥୄ
for ܣ is given as follows.](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-5-320.jpg)
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Table 1. Example of orthogonal design ܥୄ
.
ݔଵ ݔଶ ݔଷ ݔସ ݔହ
1 0 0 0 0 0
2 0 0 1 1 1
3 0 0 2 2 2
4 0 1 0 0 1
5 0 1 1 1 2
6 0 1 2 2 0
7 0 2 0 1 0
8 0 2 1 2 1
9 0 2 2 0 2
10 1 0 0 2 2
11 1 0 1 0 0
12 1 0 2 1 1
13 1 1 0 1 2
14 1 1 1 2 0
15 1 1 2 0 1
16 1 2 0 2 1
17 1 2 1 0 2
18 1 2 2 1 0
The Hamming weight ݓܪሺࢇሻ of a vector ࢇ = ሺܽଵ, ܽଶ, ⋯ , ܽሻ is defined as the number of nonzero
components. As a special case, if ܣ = ሼࢇ|ݓܪሺࢇሻ ≤ ,ݐ ࢇ ∈ ሼ0,1 }
}. ܥୄ
corresponds to the set of
rows of a subarray in a mixed level orthogonal array of strength 2ݐ [1]. Hence, ܥୄ
can be easily
obtained by using the results of orthogonal arrays.
However, because it is generally not easy to construct an orthogonal design for ,ܣ it is important
to consider efficiency in making the algorithm to produce the design. However, because the main
purpose of this paper is not to construct the orthogonal design, the algorithm is not included in
this paper.
4. SAMPLING THEOREM FOR FUNCTIONS OVER GALOIS FIELD
DOMAINS FOR EXPERIMENTAL DESIGN
In this section, I provide a sampling theorem for bandlimited functions over Galois field domains,
which is applicable to the experimental design model in which each factor has a different number
of levels.
4.1 Bandlimited Functions
The range of frequencies of ݂ is defined by a bounded set ܫ ⊂ ∏ୀଵ
ܨܩሺݍሻ. Then, ݂࢝ = 0 for all
࢝ ∈ ∏ୀଵ
ܨܩሺݍሻ ∖ .ܫ Any function whose range of frequencies is confined to a bounded set ܫ is
referred to as bandlimited to .ܫ](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-6-320.jpg)
![Computer Science & Information Technology (CS & IT) 243
4.2 A Sampling Theorem for Bandlimited Functions over ∏ୀ
ࡳࡲሺሻ Domains
Theorem 1:
Suppose a set ܣ is monotonic and ݂ሺ࢞ሻ is expressed as
݂ሺ࢞ሻ = ݂࢝
࢝ ∈ ூಲ
߯࢝ሺ࢞ሻ, ሺ13ሻ
where ܫ = ሼ ሺbଵaଵ, bଶaଶ, … , b୬a୬ሻ|ࢇ ∈ ,ܣ b୧ ∈ ܨܩሺݍሻ}. Then, the Fourier coefficients can be
computed by
݂࢝ =
1
|ܥୄ|
݂ሺ࢞ሻ
࢞ ∈఼
߯࢝
∗ ሺ࢞ሻ, ሺ14ሻ
where ܥୄ
is an orthogonal design for .ܣ
The proof of Theorem 1 requires the following three lemmas.
Lemma 1:
For any non principal character ߯ of ,ܪ
߯ሺࢎሻ
ࢎ ∈ு
= 0, ሺ15ሻ
Proof: This follows immediately [5, Lemma 2.4].
Lemma 2:
Suppose a set ܣ is monotonic, and ܥୄ
is an orthogonal design for .ܣ
Then, for ࢝, ࢠ ∈ ܫ,
หܥభ,…,
ୄ
ሺ߮ଵ, … , ߮ሻห =
|ܥୄ|
ݍభ
ݍమ
… ݍ
,
߮ଵ ∈ ܨܩ൫ݍభ
൯, … , ߮ ∈ ܨܩ൫ݍ
൯
(16)
where ݅ଵ, … , ݅are defined by ݒሺࢠ − ݓሻ = ሼ݅ଵ, … , ݅}.
Proof: Let ܵ = ሼ ݒሺࢇ + ࢇᇱሻ|ࢇ, ࢇᇱ
∈ ܣ }. Because a set ܣ is monotonic, ݒሺࢠ − ݓሻ ∈ ܵ holds for
࢝, ࢠ ∈ ܫ. Hence, by the definition of ܥୄ
, equation (16) holds.
Lemma 3:
Suppose a set ܣ is monotonic, and ܥୄ
is an orthogonal design for .ܣ Then,](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-7-320.jpg)
![Computer Science & Information Technology (CS & IT) 245
Hence, the parameters can be estimated by using Fourier transforms.
5. RELATIONSHIP BETWEEN THE SAMPLING THEOREM AND
ORTHOGONAL ARRAYS
Experiments are frequently conducted according to an orthogonal array. Here, the relationship
between the proposed sampling theorem and orthogonal arrays will be provided.
At first, mixed level orthogonal arrays of strength ݐ are defined as follows.
Definition 4: Orthogonal Arrays of strength ݐ [1]
An Orthogonal Array of strength ݐ is ܰ × ݊ matrix whose ݅-th column contains ݍ different
factor-levels in such a way that, for any $t$ columns, every $t$-tuple of levels appears equally
often in the matrix.
The ܰ rows specify the different experiments to be performed.
Next, the definition of orthogonal arrays of strength ݐ can be generalized by using a bounded set
ܣ instead of the strength .ݐ The definition of the generalized mixed level orthogonal arrays is
provided as follows.
Definition 5: Orthogonal Arrays for ܣ
An orthogonal array for ܣ is an ܰ × ݊ matrix whose ݅-th column contains ݍ different factor-
levels in such a way that, for any ࢇ, ࢇ′ ∈ ,ܣ and for any ݉ columns which are ݅ଵ-th column, …,
݅-th column, where ݅ଵ, … , ݅ are defined by ݒሺࢇ + ࢇᇱሻ = ሼ݅ଵ, … , ݅}, every ݉-tuple of levels
appears equally often in the matrix.
If ܣ = ሼࢇ|ݓܪሺࢇሻ ≤ ,ݐ ࢇ ∈ ሼ0,1}
}, an orthogonal array for ܣ is identical to a mixed level
orthogonal array of strength 2.ݐ In other words, Definition 4 is a special case of Definition 5.
Moreover, by Definition 3 and Definition 5, it is clear that the set of rows of an orthogonal array
for ܣ is an orthogonal design ܥୄ
for .ܣ Hence the following Corollary is obtained from Theorem
1 immediately.
Corollary 1:
Suppose a set ܣ is monotonic and ݂ሺ࢞ሻ is expressed as
݂ሺ࢞ሻ = ݂࢝
࢝ ∈ ூಲ
߯࢝ሺ࢞ሻ, ሺ24ሻ
where ܫ = ሼ ሺbଵaଵ, bଶaଶ, … , b୬a୬ሻ|ࢇ ∈ ,ܣ b୧ ∈ ܨܩሺݍሻ}. Then, the Fourier coefficients can be
computed by
݂࢝ =
1
|ܥୄ|
݂ሺ࢞ሻ
࢞ ∈఼
߯࢝
∗ ሺ࢞ሻ, ሺ25ሻ](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-9-320.jpg)
![246 Computer Science & Information Technology (CS & IT)
where ܥୄ
is the set of rows of an orthogonal array for ܣ defined in Definition 5 and |ܥୄ
| = ܰ.
This corollary shows the relationship between the proposed sampling theorem and orthogonal
arrays.
6. CONCLUSIONS
In this paper, I have generalized the sampling theorem for bandlimited functions over ܨܩሺݍሻ
domains to one over ∏ୀଵ
ܨܩሺݍሻ domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. I have also provided the
relationship between the proposed sampling theorem and orthogonal arrays.
ACKNOWLEDGEMENTS
This paper was partially supported by the Academic Committee of Yokohama College of
Commerce.
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[4] R.S. Stankovic and J. Astola, (2007) “Reading the Sampling Theorem in Multiple-Valued Logic: A
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[5] E.M. Stein and R. Shakarchi, (2003) Fourier Analysis: An Introduction, Princeton University Press.
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[8] Y. Ukita, T. Saito, T. Matsushima and S. Hirasawa, (2010) “A Note on a Sampling Theorem for
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[9] Y. Ukita and T. Matsushima, (2011) “A Note on Relation between the Fourier Coefficients and the
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AUTHORS
Yoshifumi Ukita has been a professor of the Department of Management Information at Yokohama
College of Commerce, Kanagawa, Japan since 2011. His research interests are artificial intelligence, signal
processing and experimental designs. He is a member of the Information Processing Society of Japan, the
Japan Society for Artificial Intelligence and IEEE.](https://image.slidesharecdn.com/20-180206122505/85/A-GENERALIZED-SAMPLING-THEOREM-OVER-GALOIS-FIELD-DOMAINS-FOR-EXPERIMENTAL-DESIGN-11-320.jpg)
A GENERALIZED SAMPLING THEOREM OVER GALOIS FIELD DOMAINS FOR EXPERIMENTAL DESIGN
- 1. David C. Wyld et al. (Eds) : NETCOM, NCS, WiMoNe, CSEIT, SPM - 2015 pp. 237–247, 2015. © CS & IT-CSCP 2015 DOI : 10.5121/csit.2015.51620 A GENERALIZED SAMPLING THEOREM OVER GALOIS FIELD DOMAINS FOR EXPERIMENTAL DESIGN Yoshifumi Ukita Department of Management and Information, Yokohama College of Commerce, Yokohama, Japan ukita@shodai.ac.jp ABSTRACT In this paper, the sampling theorem for bandlimited functions over ܨܩሺݍሻ domains is generalized to one over ∏ୀଵ ܨܩሺݍሻ domains. The generalized theorem is applicable to the experimental design model in which each factor has a different number of levels and enables us to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship between the proposed sampling theorem and orthogonal arrays is also provided. KEYWORDS Digital Signal Processing, Sampling Theorem, Experimental Design, Orthogonal Arrays, Fourier Analysis 1. INTRODUCTION In digital signal processing [3], the sampling theorem states that any real valued function ݂ can be reconstructed from a sequence of values of ݂ that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of ݂. This theorem can also be applied to functions over finite domain [4] [8]. For example, Ukita et al. obtained a sampling theorem over ܨܩሺݍሻ domains [8], which is applicable to the experimental design model in which all factors have the same number of levels. However, this sampling theorem is not applicable to the model in which each factor has a different number of levels, even though they often do [2], [7]. Moreover, a sampling theorem for such a model has not been provided so far. In this paper, the sampling theorem for bandlimited functions over ܨܩሺݍሻ domains is generalized to one over ∏ୀଵ ܨܩሺݍሻ domains. The generalized theorem is applicable to the experimental design model in which each factor has a different number of levels and enables us to estimate the parameters in the model using Fourier transforms. In addition, recently, the volume of the data has grown up rapidly in the field of Big Data and Cloud Computing [11] [12], and the generalized theorem can also be used to estimate the parameters for Big Data efficiently. Moreover, the relationship between the proposed sampling theorem and orthogonal arrays [1] is provided.
- 2. 238 Computer Science & Information Technology (CS & IT) 2. PRELIMINARIES 2.1 Fourier Analysis on Finite Abelian Groups Here, a brief explanation of Fourier analysis on finite Abelian groups is provided. Characters are important in the context of finite Fourier series. 2.1.1 Characters [5] Let ܩ be a finite Abelian group (with the additive notation), and let ܵଵ be the unit circle in the complex plane. A character on ܩ is a complex-valued function ߯: ܩ → ܵଵ that satisfies the condition ߯൫࢞ + ࢞′ ൯ = ߯ሺ࢞ሻ߯൫࢞′ ൯ ∀࢞, ࢞′ ∈ .ܩ (1) In other words, a character is a homomorphism from ܩ to the circle group. 2.1.2 Fourier Transform [4] Let ܩ, ݅ = 1,2, ⋯ , ݊, be Abelian groups of respective orders |ܩ| = ݃, ݅ = 1,2, ⋯ ݊, ݃ଵ ≤ ݃ଶ ≤ ⋯ ≤ ݃, and ܩ =×ୀଵ ܩ ܽ݊݀ ݃ = ෑ ݃ ୀଵ . ሺ2ሻ Since the character group of ܩ is isomorphic to ,ܩ we can index the characters by the elements of ,ܩ that is, ሼ ߯࢝ሺ࢞ሻ|࢝ ∈ }ܩ are the characters of .ܩ Note that ߯ሺ࢞ሻ is the principal character, and it is identically equal to 1. The charactersሼ ߯࢝ሺ࢞ሻ|࢝ ∈ }ܩ form an orthonormal system: 1 ݃ ߯࢝ሺ࢞ሻ߯ࢠ ∗ሺ࢞ሻ ࢞∈ீ = ቄ 1, ࢝ = ࢠ, 0, ࢝ ≠ ࢠ, ሺ3ሻ where ߯ࢠ ∗ሺ࢞ሻ is the complex conjugate of ߯ࢠሺ࢞ሻ. Any function ݂: ܩ → ℂ, where ℂ is the field of complex numbers, can be uniquely expressed as a linear combination of the following characters: ݂ሺ࢞ሻ = ݂࢝ ࢝ ∈ீ ߯࢝ሺ࢞ሻ, ሺ4ሻ where the complex number ݂࢝ = 1 ݃ ݂ሺ࢞ሻ ࢞ ∈ீ ߯࢝ ∗ ሺ࢞ሻ, ሺ5ሻ is the ࢝-th Fourier coefficient of ݂.
- 3. Computer Science & Information Technology (CS & IT) 239 2.2 Fourier Analysis on ∏ୀ ࡳࡲሺሻ Assume that ݍ, ݅ = 1,2, ⋯ ݊, are prime powers. Let ܨܩሺݍሻ, ݅ = 1,2, ⋯ ݊, be a Galois fields of respective orders ݍ, ݅ = 1,2, ⋯ ݊, which contain finite numbers of elements. We also use ∏ୀଵ ܨܩሺݍሻ to denote the set of all ݊-tuples with entries from ܨܩሺݍሻ, ݅ = 1,2, ⋯ ݊. The elements of ∏ୀଵ ܨܩሺݍሻ are expressed as vectors. Example 1: Consider ܨܩሺ2ሻ = ሼ0,1} and ܨܩሺ3ሻ = ሼ0,1,2}. Then, if ݊ = 3 and ݍଵ = 2, ݍଶ = 2, ݍଷ = 3, ෑ ܨܩሺݍሻ ଷ ୀଵ = ሼ000,001,002,010,011,012,100,101,102,110,111,112}. Specifying the group ܩ in Sect. 2.1.2 to be the group of ∏ୀଵ ܨܩሺݍሻ and ݃ = ∏ୀଵ ݍ, the relations (3),(4) and (5) also hold over the ∏ୀଵ ܨܩሺݍሻ domain. Then, the characters ሼ ߯࢝ሺ࢞ሻ|࢝ ∈ ∏ୀଵ ܨܩሺݍሻ} form an orthonormal system: 1 ∏ୀଵ ݍ ߯࢝ሺ࢞ሻ߯ࢠ ∗ሺ࢞ሻ ࢞∈∏సభ ீிሺሻ = ቄ 1, ࢝ = ࢠ, 0, ࢝ ≠ ࢠ, ሺ6ሻ Any function ݂: ∏ୀଵ ܨܩሺݍሻ → ℂ, can be uniquely expressed as a linear combination of the following characters: ݂ሺ࢞ሻ = ݂࢝ ࢝ ∈∏సభ ீிሺሻ ߯࢝ሺ࢞ሻ, ሺ7ሻ where the complex number ݂࢝ = 1 ∏ୀଵ ݍ ݂ሺ࢞ሻ ࢞ ∈∏సభ ீிሺሻ ߯࢝ ∗ ሺ࢞ሻ, ሺ8ሻ is the ࢝-th Fourier coefficient of ݂. 3. EXPERIMENTAL DESIGN In this section, a short introduction to experimental design [2], [7] is provided. 3.1 Experimental Design Model Let ܨଵ, ܨଶ, ⋯ , ܨ denote the ݊ factors to be included in an experiment. The levels of factor ܨ can be represented by ܨܩሺݍሻ, and the level combinations can be represented by the ݊-tuples ࢞ = ሺݔଵ, ݔଶ, ⋯ , ݔሻ ∈ ∏ୀଵ ܨܩሺݍሻ. Example 2: Let Machine (ܨଵ) and Worker (ܨଶ) be factors that might influence the quantity of a product. ܨଵ : new machine (level 0), old machine (level 1),
- 4. 240 Computer Science & Information Technology (CS & IT) ܨଶ: highly skilled worker (level 0), average skilled worker (level 1), unskilled worker (level 2). For example, ࢞ = 01 represents a combination of new machine and average skilled worker. Then, the effect of the machine, averaged over all workers, is referred to as the effect of main factor ܨଵ. Similarly, the effect of the worker, averaged over both machines, is referred to as the effect of main factor ܨଶ. The contrast between the effect of the machine for a highly skilled worker, the effect of the machine for an average skilled worker, and the effect of the machine for an unskilled worker is referred to as the effect of the interaction of ܨଵ and ܨଶ. Next, an explanation of the model in the context of experimental design is given. In previous works [8], [9], [10], all factors were restricted to have the same number of levels. In this paper, I give the definition of the generalized model in which each factor has a different number of levels as follows. Definition 1: Generalized Model y(x) is used to denote the response of the experiment with level combination x and assume the model ݕሺ࢞ሻ = ݂࢝߯࢝ሺ࢞ሻ ࢝∈ூಲ + ߳࢞, ሺ9ሻ where ܫ = ሼ ሺbଵaଵ, bଶaଶ, … , b୬a୬ሻ|ࢇ ∈ ,ܣ b୧ ∈ ܨܩሺݍሻ}. ሺ10ሻ The set ܣ ⊆ ሼ 0,1 }୬ represents the general mean, main factors, and interactive factors included in the model. (For example, consider ܣ ⊆ ሼ 000,100,010,001,110 }.Then, 000,100,010,001,110 indicate the general mean, main factor of ܨଵ, main factor of ܨଶ, main factor of ܨଷ, and interactive factor of ܨଵ and ܨଶ, respectively.) The model includes a random error ߳࢞ satisfying the expected value ܧሺ߳࢞ሻ = 0 and constant variance ߪଶ . In addition, it is usually assumed that the set ܣ satisfies the following monotonicity condition [2]. Definition 2: Monotonicity ࢇ ∈ ܣ → ࢇ′ ∈ ܣ ∀ ࢇ′ ሺࢇ′ ⊑ ࢇሻ, ሺ11ሻ where ࢇ = ሺܽଵ, ܽଶ, ⋯ , ܽሻ, ࢇ′ = ሺܽ′ଵ, ܽ′ଶ, ⋯ , ܽ′ሻ and ࢇ′ ⊑ ࢇ means that if ܽ = 0 then ܽ′ = 0, ݅ = 1,2, ⋯ , ݊. Example 3: Consider ܣ = ሼ 00000,10000,01000,00100,00010,00001,11000,10100,10010 }. Since the set ܣ satisfies (11), ܣ is monotonic.
- 5. Computer Science & Information Technology (CS & IT) 241 Next, let ࢝ = ሺݓଵ, ݓଶ, ⋯ , ݓሻ. The main effect of ܨ is represented by ሼ ݂࢝| ݓ ≠ 0 ܽ݊݀ ݓ = 0 ݂ݎ ݇ ≠ ݅}. The interaction of ܨ and ܨ is represented by ሼ ݂࢝| ݓ ≠ 0 ܽ݊݀ ݓ ≠ 0 ܽ݊݀ ݓ = 0 ݂ݎ ݇ ≠ ݅, ݆} Example 4: Consider ܣ given in Example 3 and ݍଵ = 2, ݍ = 3, ݅ = 2, … , 5. Then, ܫ is given by ܫ = ሼ00000,10000,01000,02000,00100,00200,00010,00020, 00001,00002,11000,12000}. For example, the main effect of ܨଵ is represented by ݂ଵ, and the interaction of ܨଵ and ܨଶ is represented by ݂ଵଵ and ݂ଵଶ. In experimental design, we are given a model of the experiment. In other words, we are given a set ܣ ⊆ ሼ 0,1 } . Then, we determine a set of level combinations ݔ ∈ ܺ, ܺ ⊆ ∏ୀଵ ܨܩሺݍሻ. The set ܺ is called a design. Next, we perform a set of experiments according to the design ܺ and estimate the effects from the result, ሼ ሺ࢞, ݕሺ࢞ሻሻ|࢞ ∈ ܺ }. An important standard for evaluating designs is the maximum of the variances of the unbiased estimators of effects calculated from the result of the experiments. It is known that, for a given number of experiments, this criterion is minimized in an orthogonal design [6]. 3.2 Orthogonal Designs In this subsection, a definition of Orthogonal Designs for the generalized model is provided. Definition 3: Orthogonal Designs At first, define ݒሺࢇሻ = ሼ݅ |ܽ ≠ 0, 1 ≤ ݅ ≤ ݊ }. For ࢇ = ሺܽଵଵ, ܽଵଶ, … , ܽଵሻ, ࢇ = ሺܽଶଵ, ܽଶଶ, … , ܽଶሻ ∈ ሼ0,1 } , the addition of vectors ࢇ and ࢇ is defined by ࢇ + ࢇ = ሺܽଵଵ ⊕ ܽଶଵ, ܽଵଶ ⊕ ܽଶଶ, … , ܽଵ ⊕ ܽଶሻ, where ⊕ is the exclusive or operation. An orthogonal design ܥୄ for ܣ ⊆ ሼ0,1 } is satisfies the condition that for any ࢇ, ࢇ′ ∈ ,ܣ หܥభ,…, ୄ ሺ߮ଵ, … , ߮ሻห = |ܥୄ| ݍభ ݍమ … ݍ , ߮ଵ ∈ ܨܩ൫ݍభ ൯, … , ߮ ∈ ܨܩ൫ݍ ൯ (12) where ݅ଵ, … , ݅ are defined by ݒሺࢇ + ࢇᇱሻ = ሼ݅ଵ, … , ݅}, and ܥభ,…, ୄ ሺ߮ଵ, … , ߮ሻ = ሼ࢞|ݔభ = ߮ଵ, … , ݔ = ߮, ࢞ ∈ ܥୄ }. Example 5: Consider ܣ given in Example 3 and ݍଵ = 2, ݍ = 3, ݅ = 2, … , 5. Then, an orthogonal design ܥୄ for ܣ is given as follows.
- 6. 242 Computer Science & Information Technology (CS & IT) Table 1. Example of orthogonal design ܥୄ . ݔଵ ݔଶ ݔଷ ݔସ ݔହ 1 0 0 0 0 0 2 0 0 1 1 1 3 0 0 2 2 2 4 0 1 0 0 1 5 0 1 1 1 2 6 0 1 2 2 0 7 0 2 0 1 0 8 0 2 1 2 1 9 0 2 2 0 2 10 1 0 0 2 2 11 1 0 1 0 0 12 1 0 2 1 1 13 1 1 0 1 2 14 1 1 1 2 0 15 1 1 2 0 1 16 1 2 0 2 1 17 1 2 1 0 2 18 1 2 2 1 0 The Hamming weight ݓܪሺࢇሻ of a vector ࢇ = ሺܽଵ, ܽଶ, ⋯ , ܽሻ is defined as the number of nonzero components. As a special case, if ܣ = ሼࢇ|ݓܪሺࢇሻ ≤ ,ݐ ࢇ ∈ ሼ0,1 } }. ܥୄ corresponds to the set of rows of a subarray in a mixed level orthogonal array of strength 2ݐ [1]. Hence, ܥୄ can be easily obtained by using the results of orthogonal arrays. However, because it is generally not easy to construct an orthogonal design for ,ܣ it is important to consider efficiency in making the algorithm to produce the design. However, because the main purpose of this paper is not to construct the orthogonal design, the algorithm is not included in this paper. 4. SAMPLING THEOREM FOR FUNCTIONS OVER GALOIS FIELD DOMAINS FOR EXPERIMENTAL DESIGN In this section, I provide a sampling theorem for bandlimited functions over Galois field domains, which is applicable to the experimental design model in which each factor has a different number of levels. 4.1 Bandlimited Functions The range of frequencies of ݂ is defined by a bounded set ܫ ⊂ ∏ୀଵ ܨܩሺݍሻ. Then, ݂࢝ = 0 for all ࢝ ∈ ∏ୀଵ ܨܩሺݍሻ ∖ .ܫ Any function whose range of frequencies is confined to a bounded set ܫ is referred to as bandlimited to .ܫ
- 7. Computer Science & Information Technology (CS & IT) 243 4.2 A Sampling Theorem for Bandlimited Functions over ∏ୀ ࡳࡲሺሻ Domains Theorem 1: Suppose a set ܣ is monotonic and ݂ሺ࢞ሻ is expressed as ݂ሺ࢞ሻ = ݂࢝ ࢝ ∈ ூಲ ߯࢝ሺ࢞ሻ, ሺ13ሻ where ܫ = ሼ ሺbଵaଵ, bଶaଶ, … , b୬a୬ሻ|ࢇ ∈ ,ܣ b୧ ∈ ܨܩሺݍሻ}. Then, the Fourier coefficients can be computed by ݂࢝ = 1 |ܥୄ| ݂ሺ࢞ሻ ࢞ ∈఼ ߯࢝ ∗ ሺ࢞ሻ, ሺ14ሻ where ܥୄ is an orthogonal design for .ܣ The proof of Theorem 1 requires the following three lemmas. Lemma 1: For any non principal character ߯ of ,ܪ ߯ሺࢎሻ ࢎ ∈ு = 0, ሺ15ሻ Proof: This follows immediately [5, Lemma 2.4]. Lemma 2: Suppose a set ܣ is monotonic, and ܥୄ is an orthogonal design for .ܣ Then, for ࢝, ࢠ ∈ ܫ, หܥభ,…, ୄ ሺ߮ଵ, … , ߮ሻห = |ܥୄ| ݍభ ݍమ … ݍ , ߮ଵ ∈ ܨܩ൫ݍభ ൯, … , ߮ ∈ ܨܩ൫ݍ ൯ (16) where ݅ଵ, … , ݅are defined by ݒሺࢠ − ݓሻ = ሼ݅ଵ, … , ݅}. Proof: Let ܵ = ሼ ݒሺࢇ + ࢇᇱሻ|ࢇ, ࢇᇱ ∈ ܣ }. Because a set ܣ is monotonic, ݒሺࢠ − ݓሻ ∈ ܵ holds for ࢝, ࢠ ∈ ܫ. Hence, by the definition of ܥୄ , equation (16) holds. Lemma 3: Suppose a set ܣ is monotonic, and ܥୄ is an orthogonal design for .ܣ Then,
- 8. 244 Computer Science & Information Technology (CS & IT) ߯ࢠሺ࢞ሻ߯࢝ ∗ ሺ࢞ሻ ࢞∈఼ = ൜ |ܥୄ|, ࢝ = ࢠ; 0, ݐℎ݁,݁ݏ݅ݓݎ ሺ17ሻ for all ࢠ ∈ ܫ. Proof: If ࢝ = ࢠ, then ߯ࢠሺ࢞ሻ߯࢝ ∗ ሺ࢞ሻ = ߯ሺ࢞ሻ=1 for any ࢞. Hence, ∑ ߯ࢠሺ࢞ሻ߯࢝ ∗ ሺ࢞ሻ࢞∈఼ = |ܥୄ|. Next, consider the case that ࢝ ≠ ࢠ. Define ࢛ = ࢠ − ࢝ and let ݒሺ࢛ሻ = ሼ݅ଵ, … , ݅}. Then, ߯ࢠሺ࢞ሻ߯࢝ ∗ ሺ࢞ሻ ࢞∈఼ = ࢛߯ሺ࢞ሻ ࢞∈఼ ሺ18ሻ = ߯௨భ,…,௨ ൫ݔଵ , … , ݔ൯ ࢞∈఼ ሺ19ሻ = |ܥୄ| ݍభ ݍమ … ݍ ൮ ߯௨భ,…,௨ ሺࢎሻ ࢎ∈∏ೕసభ ீிቀೕቁ ൲ ሺ20ሻ where ߯௨ೕ ቀݔ ቁ = 1 for ݑ = 0, was used for the transformation from (18) to (19), and Lemma 2 was used for the transformation from (19) to (20). Then, by (20) and Lemma 1, ∑ ߯ࢠሺ࢞ሻ߯࢝ ∗ ሺ࢞ሻ࢞∈఼ = 0 is obtained. Proof of Theorem 1: The right hand side of Equation (14) is given by 1 |ܥୄ| ݂ሺ࢞ሻ ࢞ ∈఼ ߯࢝ ∗ ሺ࢞ሻ = 1 |ܥୄ| ൮ ݂ࢠ ࢠ ∈ ூಲ ߯ࢠሺ࢞ሻ൲ ࢞ ∈఼ ߯࢝ ∗ ሺ࢞ሻ = 1 |ܥୄ| ݂ࢠ ቌ ߯ࢠሺ࢞ሻ ࢞ ∈఼ ߯࢝ ∗ ሺ࢞ሻቍ ࢠ ∈ூಲ ሺ21ሻ = ݂࢝ ሺ22ሻ where Lemma 3 was used for the transformation from (21) to (22). Hence, Theorem 1 is obtained. Theorem 1 is applicable to the generalized model given in Definition 1. When we experiment according to an orthogonal design ܥୄ , we can obtain unbiased estimators of the ݂࢝ in (9) using Theorem 1 and the assumption that ϵ࢞ is a random error with zero mean, ݂መ࢝ = 1 |ܥୄ| ݂ሺ࢞ሻ ࢞ ∈఼ ߯࢝ ∗ ሺ࢞ሻ, ሺ23ሻ
- 9. Computer Science & Information Technology (CS & IT) 245 Hence, the parameters can be estimated by using Fourier transforms. 5. RELATIONSHIP BETWEEN THE SAMPLING THEOREM AND ORTHOGONAL ARRAYS Experiments are frequently conducted according to an orthogonal array. Here, the relationship between the proposed sampling theorem and orthogonal arrays will be provided. At first, mixed level orthogonal arrays of strength ݐ are defined as follows. Definition 4: Orthogonal Arrays of strength ݐ [1] An Orthogonal Array of strength ݐ is ܰ × ݊ matrix whose ݅-th column contains ݍ different factor-levels in such a way that, for any $t$ columns, every $t$-tuple of levels appears equally often in the matrix. The ܰ rows specify the different experiments to be performed. Next, the definition of orthogonal arrays of strength ݐ can be generalized by using a bounded set ܣ instead of the strength .ݐ The definition of the generalized mixed level orthogonal arrays is provided as follows. Definition 5: Orthogonal Arrays for ܣ An orthogonal array for ܣ is an ܰ × ݊ matrix whose ݅-th column contains ݍ different factor- levels in such a way that, for any ࢇ, ࢇ′ ∈ ,ܣ and for any ݉ columns which are ݅ଵ-th column, …, ݅-th column, where ݅ଵ, … , ݅ are defined by ݒሺࢇ + ࢇᇱሻ = ሼ݅ଵ, … , ݅}, every ݉-tuple of levels appears equally often in the matrix. If ܣ = ሼࢇ|ݓܪሺࢇሻ ≤ ,ݐ ࢇ ∈ ሼ0,1} }, an orthogonal array for ܣ is identical to a mixed level orthogonal array of strength 2.ݐ In other words, Definition 4 is a special case of Definition 5. Moreover, by Definition 3 and Definition 5, it is clear that the set of rows of an orthogonal array for ܣ is an orthogonal design ܥୄ for .ܣ Hence the following Corollary is obtained from Theorem 1 immediately. Corollary 1: Suppose a set ܣ is monotonic and ݂ሺ࢞ሻ is expressed as ݂ሺ࢞ሻ = ݂࢝ ࢝ ∈ ூಲ ߯࢝ሺ࢞ሻ, ሺ24ሻ where ܫ = ሼ ሺbଵaଵ, bଶaଶ, … , b୬a୬ሻ|ࢇ ∈ ,ܣ b୧ ∈ ܨܩሺݍሻ}. Then, the Fourier coefficients can be computed by ݂࢝ = 1 |ܥୄ| ݂ሺ࢞ሻ ࢞ ∈఼ ߯࢝ ∗ ሺ࢞ሻ, ሺ25ሻ
- 10. 246 Computer Science & Information Technology (CS & IT) where ܥୄ is the set of rows of an orthogonal array for ܣ defined in Definition 5 and |ܥୄ | = ܰ. This corollary shows the relationship between the proposed sampling theorem and orthogonal arrays. 6. CONCLUSIONS In this paper, I have generalized the sampling theorem for bandlimited functions over ܨܩሺݍሻ domains to one over ∏ୀଵ ܨܩሺݍሻ domains. The generalized theorem is applicable to the experimental design model in which each factor has a different number of levels and enables us to estimate the parameters in the model by using Fourier transforms. I have also provided the relationship between the proposed sampling theorem and orthogonal arrays. ACKNOWLEDGEMENTS This paper was partially supported by the Academic Committee of Yokohama College of Commerce. REFERENCES [1] A.S. Hedayat, N.J.A. Sloane and J. Stufken, (1999) Orthogonal Arrays: Theory and Applications, Springer. [2] T. Okuno and T. Haga, (1969) Experimental Designs, Baifukan, Tokyo. [3] A.V. Oppenheim and R.W. Schafer, (1975) Digital Signal Processing, Prentice-Hall. [4] R.S. Stankovic and J. Astola, (2007) “Reading the Sampling Theorem in Multiple-Valued Logic: A journey from the (Shannon) sampling theorem to the Shannon decomposition rule,” in Proc. 37th Int. Symp. on Multiple-Valued Logic, Oslo, Norway. [5] E.M. Stein and R. Shakarchi, (2003) Fourier Analysis: An Introduction, Princeton University Press. [6] I. Takahashi, (1979) Combinatorial Theory and its Application, Iwanami Syoten, Tokyo. [7] H. Toutenburg and Shalabh, (2009) Statistical Analysis of Designed Experiments (Third Edition), Springer. [8] Y. Ukita, T. Saito, T. Matsushima and S. Hirasawa, (2010) “A Note on a Sampling Theorem for Functions over GF(q)^n Domain,” IEICE Trans. Fundamentals, Vol.E93-A, no.6, pp.1024-1031. [9] Y. Ukita and T. Matsushima, (2011) “A Note on Relation between the Fourier Coefficients and the Effects in the Experimental Design,” in Proc. 8th Int. Conf. on Inf., Comm. and Signal Processing, pp.1-5. [10] Y. Ukita, T. Matsushima and S. Hirasawa, (2012) “A Note on Relation Between the Fourier Coefficients and the Interaction Effects in the Experimental Design,” in Proc. 4th Int. Conf. on Intelligent and Advanced Systems, Kuala Lumpur, Malaysia, pp.604-609.
- 11. Computer Science & Information Technology (CS & IT) 247 [11] Sharma, S., Tim, U. S., Wong, J., Gadia, S., and Sharma, S. (2014) “A Brief Review on Leading Big Data Models,” Data Science Journal, 13(0), pp.138-157. [12] Sharma, S., Shandilya, R., Patnaik, S., and Mahapatra, A. (2015) Leading NoSQL models for handling Big Data: a brief review, International Journal of Business Information Systems, Inderscience. AUTHORS Yoshifumi Ukita has been a professor of the Department of Management Information at Yokohama College of Commerce, Kanagawa, Japan since 2011. His research interests are artificial intelligence, signal processing and experimental designs. He is a member of the Information Processing Society of Japan, the Japan Society for Artificial Intelligence and IEEE.