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11
2006 International Conference on Design of
Experiments and Its Applications
July 9-13, 2006, Tianjin, P.R. China
Sung Hyun Park, Hyuk Joo Kim and Jae-Il Cho
Optimal central composite designsfor fitting second order response surface
regression models
2
Introduction
Contents
11
Orthogonality, rotatability and slope rotatability 22
The alphabetic design optimality 33
Optimal CCDs when the true model is of third order 44
Concluding remarks 55
3
1. Introduction
The central composite design (CCD) is a design
widely used for estimating second order response
surfaces. It is perhaps the most popular class of
second order designs.
Let denote the explanatory variables
being considered. Much of the motivation of the
CCD evolves from its use in sequential
experimentation. It involves the use of a two-level
factorial or fraction (resolution Ⅴ) combined with
the following 2k axial points:
1 2, , , kx x x
4
1. Introduction
As a result, the design involves, say, F=2k factorial
points (or F=2k-p fractional factorial points), 2k axial
points, and n0 center points. The CCDs were first
introduced by Box and Wilson(1951).
-α 0 … 0
α 0 … 0
0 -α … 0
0 α … 0
…
0 0 … -α
0 0 … α
1x 2x kx
5
2. Orthogonality, rotatability and slope rotatability
Let us consider the model represented by
where xiu is the value of the variable xi at the uth
experimental point, and εu's are uncorrelated
random errors with mean zero and variance σ2. This
is the second order response surface model.
20
1 1
, 1, 2, , 1k k k
u i iu ii iu ij iu ju ui i i j
y x x x x u N
6
2. Orthogonality, rotatability and slope rotatability
2.1. Orthogonality
In this subsection, we consider the model with the
pure quadratic terms corrected for their means, that
is,
where and . In regard to
orthogonality, this model is often used for the sake
of simplicity in calculation.
' 2 20
1 1
, 1, 2, , 2k k k
u i iu ii iu i ij iu ju ui i i j
y x x x x x u N
' 20 0
1
k
ii ii
x
2 2
1
/N
i ii
x x N
7
2. Orthogonality, rotatability and slope rotatability
Let denote the least squares estimators of
respectively. In the CCD, all the covariances
between the estimated regression coefficients except
are zero. But if the matrix is a diagonal
matrix, then also becomes zero. This property is
called orthogonality.
It is well-known (See Myers (1976, p.134) and Khuri
and Cornell (1996, p.122).) that the condition for a
CCD to be an orthogonal design is that
' , , , o i ii ijb b b b
' , , , o i ii ij
,ii jjCov b b 1'X X
1/ 2
02
2
F F k n F
8
2. Orthogonality, rotatability and slope rotatability
9
2. Orthogonality, rotatability and slope rotatability
2.2. Rotatability
It is important for a second order design to possess
a reasonably stable distribution of throu-
ghout the experimental design region. Here is
the estimated response at the point .
A rotatable design is one for which has
the same value at any two locations that have the
same distance from the design center. In other
words, is constant on spheres. The rotatability
property was first introduced by Box and
Hunter(1957).
2x /NVar y
xy
'1 2x , , , kx x x
2x /NVar y
10
2. Orthogonality, rotatability and slope rotatability
It is well-known that the condition for a CCD to be rotatable is that
This means that the value of α for a rotatable CCD does not depend on the number of center points.
1/ 4F
11
2. Orthogonality, rotatability and slope rotatability
Table 2.2 gives the values of α for rotatable CCDs
for various k. Note that for k=5 and 6, a CCD is also
suggested in which a fractional factorial is used
instead of a complete factorial. Also tabulated are F
and T, where T=2k+1.
The designs considered in the table contain a
single center point. This by no means implies that
one would always use only one center point.
12
2. Orthogonality, rotatability and slope rotatability
2.3. Slope rotatability
Suppose that estimation of the first derivative of η
is of interest (η is the expected value of the response
variable y). For the second order model,
The variance of this derivative is a function of the
point x at which the derivative is estimated and also
a function of the design.
x2i ii i ij j
j ii
yb b x b x
x
13
2. Orthogonality, rotatability and slope rotatability
Hader and Park (1978) proposed an analog of the
Box-Hunter rotatability criterion, which requires that
the variance of be constant on circles (k=2),
spheres (k=3), or hyperspheres (k≥4) centered at
the design origin.
Estimates of the derivative over axial directions
would then be equally reliable for all points
equidistant from the design origin. They referred to
this property as slope rotatability, and showed that
the condition for a CCD to be a slope-rotatable
design is as follows:
x / iy x
x
14
2. Orthogonality, rotatability and slope rotatability
8 6 40
2 2 2
2 4 4 8 1
8 1 2 1 0
F n kF F N k kF k
k F F k N F
15
2. Orthogonality, rotatability and slope rotatability
16
2. Orthogonality, rotatability and slope rotatability
Table 2.3 gives slope-rotatable values of α for 2≤k≤6. For k=5 and 6, CCDs involving fractional factorials are also considered.
17
3. The alphabetic design optimality
3.1. D-optimality
The best known and most often used criterion is D-
optimality. D-optimality is based on the notion that
the experimental design should be chosen so as to
achieve certain properties in the matrix . Here is the
following matrix:
2 211 1 11 1 11 12 1,1 1
2 212 2 12 2 12 22 1,2 2
2 21 1 1 2 1,
1
1
1
k k k k
k k k k
N kN N kN N N k N kN
x x x x x x x x
x x x x x x x xX
x x x x x x x x
18
3. The alphabetic design optimality
Suppose the maximum, arithmetic mean, and
geometric mean of the eigenvalues of
are indicated by and . It turns out that an
important norm on the moment matrix is the
determinant; that is,
where p is the number of parameters in the model.
' 1
1
pp
ii
D X X
1 2, , , p 1'X X
max ,
19
3. The alphabetic design optimality
Under the assumption of independent normal
errors with constant variance, the determinant of
is inversely proportional to the square of the volume
of the confidence region for the regression
coefficients. The volume of the confidence region is
relevant because it reflects how well the set of
coefficients are estimated. A D-optimal design is one
in which is maximized; that is,
where maxξ implies that the maximum is taken over
all design ξ’s.
'max X X
'X X
'X X
20
3. The alphabetic design optimality
3.3. E-optimality
The criterion E, evaluation of the smallest
eigenvalue, also gains in understanding by a passage
to variances. It is the same as minimizing the largest
eigenvalue of the dispersion matrix; that is,
where i=1,2,…,p
In terms of variance, it is a minimax approach.
Thus the E-optimal design is defined as
max i iE
min max i i
21
3. The alphabetic design optimality
3.4. Application to the CCD
For fitting the two factor second order model, we
can consider the following CCD. It consists of (i) a 22
factorial, at levels ±1, (ii) a one-factor-at-a-time
array and (iii) n0 center points. That is, the matrix X
is given by
22
3. The alphabetic design optimality
Then the matrix is given by
where N is the number of experimental points, F is
the number of factorial points, a=F+2α2 and b=F+2α4
For the two factor CCD, for example, the value of D
is
where n0 is the number of center points.
'X X0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0
0 0 0
0 0 0 0 0
N a a
a
a
a b F
a F b
F
2 2 2 20 2D n Fa N b F a b F
23
3. The alphabetic design optimality
Figure 3.1 shows a plot of D versus for α the
indicated values of n0 for a CCD in k=2 factors.
24
3. The alphabetic design optimality
In CCDs, the determinant of moment matrix has a
tendency of increase as α increases. That is, a larger
value of α is recommendable for D-optimal sense.
But in a practical experiment, the region of interest
is usually restricted and the conditions of
experiment cannot be set for a large α. So it is
necessary for the experimenter to choose as large as
possible within the controllable region of interest.
On the other hand, for the two factor CCD, the value of A
is
2
2 1 1
2i
N b FA
a F b F N b F a
25
3. The alphabetic design optimality
Figure 3.2 shows plots of A versus for the
indicated values of n0 for CCDs in k=2 factors. Table
3.1 shows the results of optimal α values for two
factor CCDs.
26
3. The alphabetic design optimality
27
4. Optimal CCDs when the true model is of third order
Suppose that we fit the second order response
surface model, but the true model is of third order.
For this case, what value of α should be used in the
CCD?
We can generally formulate the problem by
supposing that the experimenter fits a model
of order d1 in a region R of the explanatory variables.
However, the true model is a polynomial
of order d2 , where d2>d1 . Then, a reasonable design
criterion is the minimization of
2
2x x x/ x 3
R R
NM E y g d d
1 2ˆ , , , ky x x x
1 2, , , kg x x x
28
4. Optimal CCDs when the true model is of third order
The multiple integral in Eq. (3) actually represents the average of the expected squared deviations of the true response from the estimated response over the region R.
Writing the integral
2
2
2
2
22
2
x x x
x x x
x x x x . 4
R
R
R R
NKM E y g d
NKE y E y E y g d
NKE y E y d E y g d
x 1/ ,Rd k
29
4. Optimal CCDs when the true model is of third order
The first quantity in Eq. (4) is the variance of , integrated or, rather averaged over the region R, whereas the second quantity is the square of the bias, similarly averaged. Thus M is naturally divided as follows:
M=V+B
where V is the average variance of , and B is the average squared bias of .
In this section, as a reasonable choice of design we will consider the design which minimizes B. Such a design is called the all-bias design.
y
y
y
30
4. Optimal CCDs when the true model is of third order
It is assumed here that the experimenter desires to
fit a quadratic response surface in a cuboidal region
R but that the true function is best described by a
cubic polynomial. The actual measured variables
have been transformed to which are scaled
so that the region of interest R is a unit cube. Also
the assumption on the design is made that its center
of gravity is at the origin (0,0,…,0) of the cube.
1 2, , , kx x x
31
4. Optimal CCDs when the true model is of third order
The equation of the fitted model is
where
The true relationship is written as
where
contains the cubic contribution to the actual model.
The vector contains the coefficients corresponding
to terms in ; terms such as are included.
'1 1x ,y
' 2 21 1 1 1 2 1
1 0 1 11 12 1,
x 1, , ; , , ; , ,
, , ; , , ; , , .
k k k k
k kk k k
x x x x x x x x
b b b b b b b
' '1 1 2 2x xE y
' 3 2 2 3 2 2 3 2 22 1 1 2 1 2 2 1 2 1 1
1 2 3 1 2 4 2 1
x [ , , ; , , , ; ; , , , ;
, , , ]k k k k k k
k k k
x x x x x x x x x x x x x x x
x x x x x x x x x
'2
'2x 111 122, ,
32
4. Optimal CCDs when the true model is of third order
The matrix X1 is given by
In this case the matrix X2 is
2 211 1 11 1 11 21 1,1 ,1
2 212 2 12 2 12 22 1,2 ,2
1
2 21 1 1 2 1, ,
1
1
1
k k k k
k k k k
N kN N kN N N k N k N
x x x x x x x x
x x x x x x x xX
x x x x x x x x
3 2 2 3 2 211 11 21 11 1 21 21 11 21 13 2 2 3 2 212 12 22 12 2 22 22 12 22 2
2
3 2 2 3 2 21 1 2 1 2 2 1 2
3 2 21 1 11 1 1,1 11 21 31 2,1 1,1 1
32 2
k k
k k
N N N N kN N N N N kN
k k k k k k k
k k
x x x x x x x x x x
x x x x x x x x x xX
x x x x x x x x x x
x x x x x x x x x x x
x x x
2 212 2 1,2 12 22 32 2,2 1,2 2
3 2 21 1, 1 2 3 2, 1,
k k k k k
kN kN N kN k N N N N k N k N kN
x x x x x x x x
x x x x x x x x x x x
33
4. Optimal CCDs when the true model is of third order
Let us now write
where . One can write the bias term as
where the a2 vector is merely .
(See Myers (1976, p.213))
1 ' 1 '11 1 1 12 1 2
' '11 1 1 12 1 2
, ,
x x x, x x x,R R
M N X X M N X X
K d K d
1 x
RK d
'' ' 1 1 1 1 12 22 12 11 12 11 12 11 12 11 11 12 11 12 2a a , 5B M M M M
2 /n
34
4. Optimal CCDs when the true model is of third order
The first term in the square brackets in Eq. (5)
contains only the region moment matrices and thus
is independent of the design. The bias term can be
no smaller than the positive semidefinite quadratic
form
. So the experimenter has to use
designs which minimize the positive semidefinite
quadratic form
' ' 12 22 12 11 12 2a a
'' 1 1 1 12 11 12 11 12 11 11 12 11 12 2a aM M M M
35
4. Optimal CCDs when the true model is of third order
Now we will find out the value of which makes the
optimal design in the CCDs. But, let's assume that a2
is a vector of ones. That is, a2 is (1,1,1,1) for the two
factor CCDs when d1=2 and d2=3. And, if we assume
that the region of interest -1≤xi≤1 is where i=1,2,…,k
then we can obtain region moment matrices(μ11 and
μ12).
36
4. Optimal CCDs when the true model is of third order
For example, let's consider the second order CCD
which minimizes the squared bias from the third
order terms for k=2. The design consists of four
factorial points, four axial points at a distance α from
the origin, and two center points. Then we obtain the
following design moment matrices and region
moment matrices.2 2
2
2
11 2 2
2 2
10 0 0 4 2 4 2 0
0 4 2 0 0 0 0
0 0 4 2 0 0 01
10 4 2 0 0 4 2 4 0
4 2 0 0 4 4 2 0
0 0 0 0 0 4
M
37
4. Optimal CCDs when the true model is of third order
4
2 2
4
12 2 2
0 0 0 0
4 2 20 0
4 2 4 2
1 2 4 20 0
10 4 2 4 20 0 0 0
0 0 0 0
0 0 0 0
M
11
4 44 0 0 0
3 34
0 0 0 0 03
40 0 0 0 0
1 34 4 44
0 0 03 5 94 4 4
0 0 03 9 5
40 0 0 0 0
9
12
0 0 0 0
12 40 0
5 34 121 0 03 54
0 0 0 0
0 0 0 0
0 0 0 0
38
4. Optimal CCDs when the true model is of third order
So is obtained as
The value of which minimizes is found to be
A very interesting fact is that f(α) has nothing to
do with the number of center points. Table 4.1 gives
the appropriate values of α for second order CCD
which minimize the squared bias from the third order
terms for k factors.
1/ 2
2 2 1 0.91018
'' 1 1 1 12 11 12 11 12 11 11 12 11 12 2a aM M M M
22 4
22
2 32 14 15.
675 2f
39
4. Optimal CCDs when the true model is of third order
40
5. Concluding remarks
In this paper, we found out values of α which
optimize CCDs for fitting second order response
surface models under several criteria. Table 5.1
gives the value of α in Tables 2.1, 2.2 and 4.1.
41
5. Concluding remarks
From Table 5.1, we can find that the values of tend
to increase in the following order :
Minimum bias<Orthogonality<Rotatability
<Slope rotatability<Alphabetic
optimality
42
5. Concluding remarks
Note that the optimal value of α under the minimum
bias and rotatability criteria does not depend on the
number of center points. Also, an interesting fact is
that the optimal value of α under the minimum bias
criterion is very similar to that under the orthogonality
criterion with one center point.
In conclusion, we will consider reasonable choice of
CCD for fitting the second order model according to
the following cases:
1. when the true model is of second order (d2=2)
2. when the true model is of third order (d2=3)
43
5. Concluding remarks
Table 5.2 shows values of α recommended for the
CCD considering the order d2.
4444
Thank you