1 2006 International Conference on Design of Experiments and Its Applications July 9-13, 2006,...

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


2.1. 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


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


9


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


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


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


2.3. 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


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


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


16


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


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


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


3.3. 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


3.4. 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


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


Figure 3.1 shows a plot of D versus for α the

indicated values of n0 for a CCD in k=2 factors.

24


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


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


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


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


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


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


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


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


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


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


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


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

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


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


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


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


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


Table 5.2 shows values of α recommended for the

CCD considering the order d2.

4444

Thank you

1 2006 International Conference on Design of Experiments and Its Applications July 9-13, 2006,...

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