Research Journal of Mathematics and Statistics 4(1): 1-5, 2012
ISSN: 2040-7505
© Maxwell Scientific Organization, 2012
Submitted: October 12, 2011
Accepted: November 18, 2011
Published: February 25, 2012
On D-Optimality Criterion of Non-Overlapping and Overlapping Segmentation of
the Response Surfaces
1
1
T.A. Ugbe and 2P.E Chigbu
Department of Mathematics/Statistics and Computer Science, University of Calabar, Calabar,
Cross River State, Nigeria
2
Departments of Statistics, University of Nigeria, Nsukka, Enugu State, Nigeria
Abstract: The aim of this study was to find out the Segment which generates a design that is D-optimal(has
minimum variance). This was achieved by partitioning the Response Surface into two equal segments for Nonoverlapping and Overlapping Segments and then select support points which formed the design matrices for
the segments. Further more, the determinant of the information Matrices of the designs were compared for both
Non-overlapping and Overlapping Segments using the Unbiased response function and the biased response
function respectively. It was found that design 1 is D-optimal and also G-optimal (by equivalence theorem
given in theorem 1),that is the Non-Overlapping Segmentation of the Response Surface forms a design which
is D- and G-optimal for the first order unbiased response function and second order biased response function,
respectively. In other words, the Overlapping Segmentation of the Response Surfaces form a design that was
not D-optimal.
Key words: Design matrix, determinant, information matrix, optimality, segmentation, support points
‘‘Prediction Criterion’’.An experimental design
which is optimal with respect to an estimation
Criterion like the D-optimality is the one that
maximizes Parameter information by minimizing
Variabilities of the parameters. A design which is
optimal with respect to a prediction Criterion like Qoptimality maximizes information about a response
Surface by focusing on the Prediction qualities of the
fitted model. D-optimality is the most important and
popular design Criterion in the life applications,
which is introduced by Wald (1943), put the
emphasis on the quality of the parameter estimates.
D-optimality Criterion is also known as determinant
Criterion and is defined as;
INTRODUCTION
Optimal designs are experimental designs that are
generated based on a particular Optimality Criterion and
are generally Optimal only for a Specific Statistical
Model. The proper Meaning of ‘Optimal’ depends on the
situation, and can include: Most effective, Minimum
Variance, Minimum bias, etc.
Kiefer and Wolfowitz (1959) developed a theoretical
frame work for optimal design Criteria by expressing a
design as a probability Measure representing the
allocation of observations at any point in the design space:
Their theoretical approach to design optimality and their
introduction of the D- and E Optimality Criteria for the
linear regression model laid the ground work for other
design Criteria A-, F-, G- and Q- Optimality Criteria etc.
Each design optimality Criterion addresses a Specific goal
in the experiment to be performed or achieves a Specific
property in the final fitted regression model.
Many of these design Criteria were originally
developed for the homogeneous Variance Linear Models,
but most have been adopted for use in a non-linear and
non-homogeneous Variance Situation as well. The basic
idea underlying design optimality theory is that statistical
inference about quantities of interest can be improved by
Max det (X/X) = Min det (X/X)G1
xi, i = 1, …, n
xi, i = 1, …, n
Which means maximizing the determinant of the
information matrix or equivalently, minimizing the
determinant of the inverse of the information matrix.
The aim of D-optimality is essentially a parameter
estimation Criterion. This was called lately, D-optimality
by Kiefer and Wolfowitz (1959). It is the most well
studied problem which is Seen in the Literature by Kiefer
(1959), Fedorov (1972), Silvey (1980), Pazman (1986),
Atkinson and Donev (1992), Pukelsheim (1993), Mandal
(2000) and Bamanga and Asiribo (2006) etc. The
objective of this paper is to partition the Response Surface
into 2 Segments, obtain design matrices from each
‘‘Optimally’’ selecting levels of the control
Variables. In general, a design Optimality Criterion
can be characterized as an ‘‘Estimation Criterion’’ or
Corresponding Author: T.A. Ugbe, Department of Mathematics/Statistics and Computer Science, University of Calabar, Calabar,
Cross River State, Nigeria, Tel.: 08160980759
1
Res. J. Math Stat., 4(1): 1-5, 2012
segment, find out which segment is D-optimal using Nonoverlapping and overlapping Segments for both unbiased
and biased response functions respectively.
X2
(0,1)
(1,1)
MATERIALS AND METHODS
This study was carried out in the University of
Nigeria,Nsukka,Enugu State,Nigeria.The essence of this
work was to explore the Response Surface and segment it
into Non-overlapping and Overlapping segments for
unbiased and biased models, and then find out the
segment that is D-optimal.
Segmentation is the partitioning of the Response
~
Surface [(Experimental Space), X ] , into Subspaces
called Segments. Support points are selected from each
Segment which give rise to design matrices of the Kth
segments: Onukogu and Chigbu (2002). The Segments
used here are Non-overlapping and Overlapping and are
Partitioned into two equal Segments. Illustrative
description of Segmentation of the Response Surfaces are
shown in the Fig. 1 and 2.
(0,1/2)
S1
(1,1/2)
S2
X1
(0,0)
(1/2,0)
Fig. 1: Segmentation{(Non-overlapping);describing the
partitioning of the response surface into two segments,
S1 and S2}
X2
(0,2)
Equivalence of D- and G-optimality criteria: In design
of experiment, the main tool for checking the optimality
of a candidate design are equivalence theorems, Kiefer
(1974), Pukelsheim and Torsney (1991) show that the Dand G-optimality Criteria are equivalent if Fe-2 equal to a
constant. In other words, a D-optimal design is also
minimax and on the other hand a minimax design is Doptimal: Fedorov (1972). The equivalence theorems have
been stated and proved by many authors, Kiefer and
Wolfowitz (1960), Kiefer (1974), Pukelsheim (1980) and
Pazman (1986). The statement of the theorem as given:
(0,1)
(0,0)
Theorem1: Given M(.*) to be non-singular and Fe = 1,
a design is D-optimal if it is G-optimal, that is:
(1.2)
(1/2,2)
(2.2)
(3/2,2)
(3/2,1)
(1/2,1)
S1
S12
S21
(1,0)
(1/2,0)
S2
(2,0)
(3/2,0)
X1
-2
Max
x ∈s ( x )
det M (.*) =
Fig. 2 : Segmentation{(overlapping);describing the partitioning
of the response surface into two segments,S1 and S2 such
that S1 overlaps into S2,and is written as S21 and S2
overlaps into S1,and is witten as S12}
– det M(.)
Max
= x ∈s ( x ) X/ M -1(.*)X
= Min{
X2
2
7/4
3/2
X/ M G1(.*) X}, S(x)< X~ ,
M (.),M (.*) m M,n×m
Max
x ∈s ( x )
5/4
Max X/M-1(.*)X = n, n is the number of linearly
independent parameters in the model, that is the rank of
M(.*) .
1*
3/4
1/2
RESULTS AND DISCUSSION
*
*
*
*
*
S1
*
*
S2
*
*
1/4
To consider D-optimality criterion of Nonoverlapping and overlapping segmentation of responses
surfaces, we first of all consider the response functions
(a) First order unbiased response function (Linear) is
given by:
(0,0) 1/4
1/2
3/4
1
5/4
3/2
7/4
2 X1
Fig. 3: Non-overlapping segmentation of the response surfaces
{showing different support points, marked with asterisk
which are picked to form design design matrices from S1
and S2, respectively}
f (x1,x2) = a00 +a10x1 +a22x2
2
Res. J. Math Stat., 4(1): 1-5, 2012
X2
2
7/4
3/2
X2
2
7/4
3/2
*
*
**
5/4
S1
3/2
**
3/4
S21
1/2
1/4
1/2
*
3/4
1
5/4
*
S2
*
*
1/4
*
0
*
1/2
*
1/4
3/2
7/4
*
3/4
*
S12
*
*
S1
1
S2
*
*
5/4
*
1
*
2
*
*
(0)
X1
Fig. 4: Overlapping segmentation of the response surface,
{showing different support points marked with asterisk
which are picked to form design matrices from S12 and
S21, respectively}
1/4
1/2
3/4
5/4
1
3/2
7/4
2
X1
Fig. 5: Non-overlapping segmentation of the response surface
{same as Fig. 3}
SA = {x1, x2; 3/4 #x1 #1, 0 #x2 #2}
SB = {x1, x2; 9/8 #x1 #3/2,1/4 #x2 #2}
We define the segments (from Fig. 3) by:
where SA = S12 (overlapping portion of S2 into S1 )
SB = S21 (overlapping portion of S1 into S2 )
S1 = {x1, x2; 0 #x1 #2, 0 # x2 #2}
S2 = {x1, x2; 0 #x1 #2, 1/2 #x2 # 2}
The design matrices are:
The design matrices are:
⎛
⎜1
⎜
⎜1
⎜
⎜1
X A = ⎜⎜
1
⎜
⎜1
⎜
⎜
⎜1
⎝
⎛ 1 0 0⎞
⎛ 1 2 0⎞
⎜
⎟
⎟
⎜
1
7
1
1
2
⎜
⎟
⎜1 2 4⎟
⎜1 1 1⎟
⎟
⎜
3
2
2
⎟ , X = ⎜ 1 2 1⎟
X1 = ⎜
2
⎜ 1 21 2⎟
⎜ 1 2 1⎟
⎜
⎟
⎜
7
1⎟
⎜ 1 0 1⎟
⎜1 4 2⎟
⎟
⎜
⎜
⎟
⎝ 1 2 2⎠
⎝ 1 21 23 ⎠
)
.
12708
0.9583⎞
⎛1
⎟
⎜
1
.
.
.
X B/ X B = ⎜ 12708
16328
12083
⎟
N
⎟
⎜ 0.9583 12083
.
.
12396
⎠
⎝
(
15417
13750
.
.
⎛1
⎞
⎜
⎟
1
/
0.8854 19792
.
.
X 2 X 2 = ⎜15417
⎟
N
⎜13750
⎟
19792
2
2188
.
.
.
⎝
⎠
(
7
8
3
4
⎛1
⎞
.
0.8750 11662
⎜
⎟
1
/
.
X A X A ) = ⎜ 0.8750 0.7760 10521
(
⎟
N
⎜
⎟
.
.
.
10521
17500
⎝ 11667
⎠
0.3353 1
⎛1
⎞
⎜
⎟
1
/
X 1 X 1 = ⎜ 0.3333 01667
.
0.4167⎟
N
⎜1
⎟
.
0.4167 14167
⎝
⎠
(
9 1⎞
⎛
⎜1
⎟
⎞
8 2⎟
⎜
0⎟
5
⎜
⎟
⎟
⎜1 4 1 ⎟
1⎟
⎜
⎟
3⎟
⎜ 1 9 3⎟
⎟
⎜
8 4⎟
2⎟
, XB = ⎜
⎟
5
2⎟
⎜1
2⎟
⎟
4
⎜
⎟
1 ⎟⎟
⎜ 11 5 ⎟
⎜1
⎟
⎟
8 4⎟
3⎟
⎜
⎜ 2 1⎟
2⎠
⎜1
⎟
⎝ 2 4⎠
3
4
1
7
8
1
)
M (ς N 2 ) =
⎛ 2.0000 18750
.
2.3750 ⎞
⎜
⎟
1
1
M (ζ N 1 ) = ( X 1/ X 1 ) +
X 2/ X 2 ) = ⎜ 18750
.
3.0521 2.3959⎟
(
N
N
⎜
⎟
⎝ 2.3750 2.3959 3.6355⎠
)
2.1458 2.1250 ⎞
⎛2
⎜
⎟
1
1
X A/ X A +
X B/ X B = ⎜ 2.1458 2.4089 2.2604⎟
N
N
⎜ 2.1250 2.2604 2.9896⎟
⎝
⎠
(
)
(
)
where M(.N2) is the information matrix of overlapping
segmentation.
where M(.N1) is the information matrix of the Nonoverlapping segmentation
det M(.N2) = 0.1552
|det M(.N1)>det M(.N2)
det M (.N1) = 2.053
Thus by comparison det M (.N1) is maximized, this
implies that .N1 is D-optimal. By Equivalence theorem
stated in theorem 1, the design .N1 is also G-optimal.
We define the segments (from Fig.4) by:
3
Res. J. Math Stat., 4(1): 1-5, 2012
Therefore the Non-overlapping segmentation of the
response surface generates or forms a design which is Dand-G-optimal for a first order unbiased model.
(b) Second-order biased response function is given by:
X2
2
15/8
7/4
13/8
3/2
11/8
5/4
9/8
1
7/8
3/4
5/8
1/2
3/8
1/4
1/8
f(x1,x2) = a00+a10x1+a20x2+a12x1x2+a11x12+a22x22
We define the segments (from Fig. 5) by:
S1 = {x1,x2;0 # x1 #7/8, 0 # x2 # 2}
S2 = {x1 , x2;9/8 #x1 #2, 1/2 #x2 #2}
The design matrices are:
*
*
1
2
0
3
4
1
4
7
8
3
4
1
0
3
2
1
2
2
1
4
⎛1 3
2
⎜
⎜1 2
⎜
⎜ 1 74
X2 = ⎜
9
⎜1 8
⎜
5
⎜1 4
⎜
⎝ 1 13
8
1
2
0
9
8
1
8
14
8
3
16
1
4
0
9
16
1
16
49
64
9
16
1
2
⎤
1⎥
0⎥
9⎥
⎥
4⎥
1⎥
4⎥
⎥
4⎥
1⎥
⎥
16 ⎦
3
2
S12
*
*
S1
1/4 1/2
S21
3/4
S2
5/4 3/2 7/4
1
2
X1
Fig. 6: Overlapping segmentation of the response Surfaces
{same as Fig. (4)}
9
4
1
2
4
4
3
2
21
8
7
4
63
32
49
16
81
64
15
8
75
32
25
16
13
16
*
*
0
⎡
⎢1
⎢1
⎢
⎢1
⎢
X1 = ⎢
1
⎢
⎢
⎢1
⎢
⎢1
⎣
*
169
64
⎛2
⎜
⎜ 2.0625
⎜ 2.3125
M (ς N 1 ) = ⎜
⎜ 2.8307
⎜ 2.8307
⎜
⎝ 3.6068
1⎞
⎟
4 ⎟
⎟
9
4 ⎟
⎟
49
16 ⎟
225 ⎟
64 ⎟
1 ⎟
4 ⎠
2.0625 2.3125 2.8229 2.8307 3.6068 ⎞
⎟
2.8307 2.8229 4.0176 4.3428 4.5801⎟
⎟
2.8229 3.6068 4.5801 4.0175 61611
.
⎟
4.0176 4.5801 6.6331 6.3207 7.9921 ⎟
4.3428 4.0175 6.3207 7.1238 6.6331 ⎟
⎟
4.5801 61611
7.9921 6.6331 10.9987⎠
.
det M (.N1) = 0.0021
We define the segments (from Fig. 6)by:
SA = {1/2 #x1 #7/8, 1/4 #x2 #3/2}
SA = {1#x1# 3/2,1#x2#2}
where: SA = S12 (overlapping portion of S2 into S1)
SB=S21 (overlapping portion of S1 into S2)
The design matrices are:
0.5208 0.8750
⎛1
⎜
0
5208
0.3672 0.6146
.
⎜
⎜ 0.8750 0.6146 12604
.
1
( X / X )= ⎜
N 1 1 ⎜ 0.6146 0.4661 0.9661
⎜ 0.3672 0.2757 0.4661
⎜
0.9661 2.0859
.
⎝ 12604
0.6146 0.3672 12604
.
⎞
⎟
0.4661 0.2757 0.9661 ⎟
0.9661 0.4661 2.0859 ⎟
⎟
0.7715 0.3685 16790
.
⎟
0.3685 0.2142 0.7715⎟
⎟
16790
0.7715 3.6882 ⎠
.
⎡1
⎢
⎢1
⎢1
XA = ⎢
⎢1
⎢1
⎢
⎢⎣1
.
.
15417
14375
2.2083 2.4635 2.3464 ⎞
⎛1
⎜
⎟
.
.
2.4635 2.2083 35514
4.0671 3.6139 ⎟
⎜ 15417
⎜ 14375
.
.
2.2083 2.3464 3.6139 35514
4.0752⎟
1
⎟
( X / X )= ⎜
.
.
3.6139 58617
5.9522 6.3131 ⎟
N 2 2 ⎜ 2.2083 35514
⎜ 2.4635 4.0671 35514
⎟
.
.
5.9522 6.9096 58617
⎜
⎟
.
.
.
.
.
.
2
3464
3
6139
4
0752
6
3131
58617
7
3106
⎝
⎠
⎛1
⎜
⎜1
⎜
⎜1
XB = ⎜
1
⎜
⎜1
⎜
⎝1
1
1
M (ς N 1 ) = ( X 1/ X 1 ) +
( X 2/ X 2 )
N
N
4
1
2
3
2
3
4
1
4
3
4
3
4
9
16
9
16
7
8
7
8
49
64
49
64
5
8
1
4
5
32
25
64
3
4
1
3
4
9
16
1
2
5
4
5
8
1
4
1
1
1
9
8
9
8
81
64
5
4
3
2
25
6
11
8
2
121
64
3
2
7
4
9
4
5
4
13
8
25
16
⎤
⎥
⎥
49 ⎥
64
1 ⎥
16 ⎥
1⎥
⎥
25
16 ⎥
⎦
9
4
9
4
1⎞
⎟
⎟
9 ⎟
4 ⎟
4 ⎟
⎟
49
16 ⎟
⎟
169
64 ⎠
81
64
Res. J. Math Stat., 4(1): 1-5, 2012
⎛1
⎜
⎜ 0.6667
⎜ 0.9375
1
( X A/ X A ) = ⎜ 0.6016
N
⎜
⎜ 0.4635
⎜
.
⎝ 10339
⎛ 1
⎜
.
⎜ 125
⎜ 15
.
1
( X / X )= ⎜ .
N B B ⎜ 15807
⎜ 15885
.
⎜
⎝ 2.3698
M (ς N 2 ) =
(
.
0.6667 0.9375 0.6016 0.4635 10339
⎞
⎟
0.4635 0.6016 0.4066 0.3346 0.6312⎟
⎟
.
.
0.6016 10339
0.6312 0.4066 12393
⎟
0.4066 0.6312 0.4071 0.2882 0.7211⎟
0.3346 0.4066 0.2882 0.2494 0.4071⎟
⎟
.
.
0.6312 12393
0.7211 0.4071 15684
⎠
125
.
15885
.
15
.
15807
.
19245
2.3698 2.4176 2.5042
.
2.0316 2.4176 2.9344 2.6519
2.0508 2.5042 2.6519 2.6869
31051
3.9082 38289
.
.
)
(
1
1
X A/ X A +
X B/ X B
N
N
⎛ 2.0000
⎜
.
⎜ 19167
⎜ 2.4375
M (ς N 2 ) = ⎜
⎜ 2.1823
⎜ 2.0521
⎜
⎜ 3.4036
⎝
15885
.
19245
2.0316 2.0508
.
4.1161
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2.3698⎞
⎟
31051
.
⎟
⎟
31051
.
⎟
38289
.
⎟
4.1161⎟
⎟
6.6694⎠
)
19167
2.4375 2.1823 2.0521 3.4036 ⎞
.
⎟
2.0521 2.5260 2.4382 2.3854 3.7367 ⎟
⎟
2.5260 3.4036 3.0488 2.9108 51475
.
⎟
2.4382 3.6488 3.3416 2.9401 4.5500 ⎟
2.3854 2.9108 2.9401 2.9364 4.5232 ⎟⎟
3.7363 51475
4.5500 4.5232 8.2378⎟⎠
.
det M(.N12) = 0.00001406
|det M(.N1)> det M(.N2)
In the same manner .N1 is D-optimal since det M(.N1)
is maximized, again by equivalence theorem it is also Goptimal. Therefore, the Non-overlapping segmentation of
the response surface generate a design which is D- and Goptimal for a second order biased model.
CONCLUSION
From the foregoing, we conclude that .N1 is Doptimal and also G-optimal (by equivalence theorem
given in theorem 1). That is, the Non-overlapping
segmentation of the response surface forms a design
which is D- and G-optimal for a first order unbiased
response function and second order biased response
function, respectively.
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