Image Processing Reference
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18
A Description of Experimental Design on the
Basis of an Orthonormal System
Yoshifumi Ukita 1 and Toshiyasu Matsushima 2
1 Yokohama College of Commerce
2 Waseda University
Japan
1. Introduction
The Fourier series representation of a function is a classic representation which is widely
used to approximate real functions (Stein & Shakarchi, 2003). In digital signal processing
(Oppenheim & Schafer, 1975), the sampling theorem states that any real valued function f can
be reconstructed from a sequence of values of f that are discretely sampled with a frequency
at least twice as high as the maximum frequency of the spectrum of f . This theorem can also
be applied to functions over finite domains (Stankovic & Astola, 2007; Takimoto & Maruoka,
1997). Then, the range of frequencies of f can be expressed in more detail by using a bounded
set instead of the maximum frequency. A function whose range of frequencies is confined to
a bounded set I is referred to as “bandlimited to I ”. Ukita et al. obtained a sampling theorem
for bandlimited functions over Boolean (Ukita et al., 2003) and GF
n domains (Ukita et al.,
(
)
q
(
)
2010a), where q is a prime power and GF
is Galois field of order q . The sampling theorem
can be applied in various fields as well as in digital signal processing, and one of the fields is
the experimental design.
In most areas of scientific research, experimentation is a major tool for acquiring new
knowledge or a better understanding of the target phenomenon. Experiments usually aim
to study how changes in various factors affect the response variable of interest (Cochran
& Cox, 1992; Toutenburg & Shalabh, 2009). Since the model used most often at present in
experimental design is expressed through the effect of each factor, it is easy to understand
how each factor affects the response variable. However, since the model contains redundant
parameters and is not expressed in terms of an orthonormal system, a considerable amount of
time is often necessary to implement the procedure for estimating the effects.
In this chapter, we propose that the model of experimental design be expressed as an
orthonormal system, and show that the model contains no redundant parameters. Then, the
model is expressed by using Fourier coefficients instead of the effect of each factor. As there
is an abundance of software for calculating the Fourier transform, such a system allows for a
straightforward implementation of the procedures for estimating the Fourier coefficients by
using Fourier transform. In addition, the effect of each factor can be easily obtained from
the Fourier coefficients (Ukita & Matsushima, 2011). Therefore, it is possible to implement
easily the estimation procedures as well as to understand how each factor affects the response
variable in a model based on an orthonormal system. Moreover, the analysis of variance can
also be performed in a model based on an orthonormal system (Ukita et al., 2010b). Hence,
q
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