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Chapter 4
Design of Experiments
4.1 Introduction
In Chapter 3 we have considered the location of the data points fixed and studied how to
pass a good response surface through the given data. However, the choice of points
where experiments (whether numerical or physical) are performed has very large effect
on the accuracy of the response surface, and in this chapter we will explore methods for
selecting a good set of points for carrying out experiments. The selection of these points
is known as
design of experiments.
Design of experiments is inherently a multiobjective optimization problem. We
would like to select points so that we maximize the accuracy of the information that we
get from the experiments. We usually also would like to minimize the number of
experiments, because these are expensive. In some cases the objective of the
experiments is to estimate some physical characteristics, and in these cases, we would
like to maximize the accuracy of these characteristics. However, in the design
applications, which are of primary interest to us, we would like to construct a response
surface that could be used to predict the performance of other designs. In this case, our
primary goal is to choose the points for the experiments so as to maximize the predictive
capability of the model.
A lot of work has been done on experimental designs in regular design domains.
Such domains occur when each design variable is bounded by simple lower and upper
limits, so that the design domain is box like. Occasionally, spherical domains are also
considered. Sometimes each design variable can take only two or three values, often
called levels. These levels are termed low, nominal and high. In other cases, the design
space is approximately box like, but it is possible to carry experiments with the design
variables taking values outside the box for the purpose of improving the properties of the
response surface. In the next section we will summarize briefly some of the properties of
experimental design in boxlike domains, and present some of the more popular
experimental designs in such domains.
For design optimization, however, it is common for us to try and create response
surfaces in irregularly shaped domains. In that case, we have to create our own
experimental design. Section 4.3 will discuss several techniques available for finding
good designs in an irregular shaped domain.
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4.2 Design of Experiments in Boxlike Domains
In this case the region of interest is defined by simple lower and upper limits on each of
the design variables
,
,....
,
1
,
'
n
i
x
x
x
iu
i
il
(4.2.1)
where
il
x
, and
iu
x
are the lower and upper limits, respectively, on the design variable
'
i
x
.
The prime indicates that the design variable has not been normalized. For convenience
we normalize the design variable as
,
2
'
il
iu
in
il
i
i
x
x
x
x
x
x
(
4
.
2
.
2
)
The normalized variables are then all bound in the cube
,
1
1
i
x
(
4
.
2
.
3
)
4.2.1 Interpolation, extrapolation and prediction variance
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 Spring '08
 PETERIFJU

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