Some Remarks about Optimum Chemical Balance Weighing Design for p = v + 1 Objects
Bronislaw Ceranka and Malgorzata Graczyk1
Abstract
The problem of the estimation of unknown weights of p = v + 1 objects in the model of the chemical balance weighing design under the assumption that the measurement errors are uncorrelated and they have different variances is considered. The existence conditions determining the optimum design are presented.
1 Introduction
We consider the linear model
y = Xw + e,	(1.1)
which describe how to determine unknown measurements of p objects using n weighing operations according to the design matrix X = (xj), xij = — 1, 0, 1, i = 1, 2,..., n, j = 1, 2,..., p. For each i, the result of experiment yi is linear combination of unknown measurements of Wj with factors equal to xij. Each object is weighed at most m times. In the model (1.1) y is a n x 1 random vector of the observed weights, and w is a p x 1 vector representing unknown weights of objects. If we have at our disposal two measurements installations then we assume that there are no systematic errors and the errors are uncorrelated and have different variances, i.e., for the n x 1 random vector of errors e we have E(e) = 0n and E(ee ) = a2 G, where 0n is the n x 1 column vector of zeros, G is an n x n positive definite diagonal matrix of known elements
G
a ibi 0&!0
0b„ 0^ I bo
(1.2)
a > 0 is known scalar, n = b1 + b2. For the estimation of unknown weights of objects, we use the weighed least squares method and we get
W = (x' G-1X)-1 X' G-1y and the dispersion matrix of W is
1
Var (W) = a2 (XG-1 X provided X is full column rank, i.e., r(X) = p.
1 Department of Mathematical and Statistical Methods, Poznan University of Life Sciences, Wojska Polskiego 28, 60-637 Poznali, Poland; bronicer@up.poznan.pl, magra@up.poznan.pl
2 The optimality criterion
The concept of optimality comes from statistical theory of weighing designs.The optimality criterions which deal with the weighing designs were presented in Wong and Masaro (1984), Shah and Sinha (1989), Pukelsheim (1993). For G = In the optimality criterions for chemical designs were presented in Raghavarao (1971), and Banerjee (1975). For the case that the errors are correlated with equal variances, the conditions determining the existence of the optimum chemical balance weighing design were considered in Ceranka and Graczyk (2003b). They gave the lower bound of variance of each of the estimators and the construction methods of the optimal design.
Let us consider the design matrix of the chemical balance weighing design for p = v + 1 objects as
X
Xi 0b X2 1b
(2.1)
Definition 1 Nonsingular chemical balance weighing design with the design matrix X and with the dispertion matrix of errors a2G, where G is given in (1.2), is optimal if the variance of each of the estimators attains the lower bound.
Now, we can formulate the conditions determining the optimality criterion. From Ceranka and Graczyk (2003a), we have
Theorem 1 Any chemical balance weighing design with the design matrix X in the form (2.1) and with the dispertion matrix of errors a2G, where G is given in (1.2), is optimal if and only if
(i)	aXiXi + X'2X2 = (am 1 + m,2)Ip
(ii)	X2lb2 = 0p and
(iii)	am1 + m2 = b2.
We note that in the optimum chemical balance weighing design with the design matrix given by (2.1), for each of p = v + 1 objects, we have
Var (wj ) =
a2	a2
am1 + m2 b2
In the next sections we will consider the methods of construction of the optimum chemical balance weighing design based on the incidence matrices of the balanced incomplete block designs and the ternary balanced block designs.
3 Balanced designs
Now, we recall the definitions of a balanced incomplete block design given in Raghavarao (1971) and of a ternary balanced block design given in Billington (1984).
In a balanced incomplete block design we replace v treatments in b blocks, each of size k, in such a way, that each treatment occurs at most once in each block, occurs in exactly r blocks and every pair of treatments occurs together in exactly A blocks. The integers v, b, r, k, A are called the parameters of the balanced incomplete block design. The incidence relation between the treatments and blocks is denoted by the matrix N = (n j ), known as the incidence matrix, where nj denotes the number of times the ith treatment occurs in the j th block, N1b = r, N1v = k. It is straightforward to verify that vr = bk,
A(v - 1) = r(k - 1), NN' = (r - A)Iv + A1v 1V, where 1v is the v x 1 vector of units.
A ternary balanced block design is defined as the design consisting of b blocks, each of size k, chosen from a set of objects of size v, in such a way that each of the v treatments occurs r times altogether and 0, 1 or 2 times in each block, (2 treatments appear together at least once). Each of the distinct pairs of objects appears A times. Any ternary balanced block design is regular, that is, each treatment occurs alone in p1 blocks and is repeated two times in p2 blocks, where p1 and p2 are constant for the design. Let N be the incidence matrix. It is straightforward to verify that vr = bk, r = pi + 2p2,
A(v - 1) = pi(k - 1) + 2p2(k - 2) = r(k - 1) - 2p2, NN' = (pi + 4p2 - A)Iv + A1v 1v = (r + 2p2 - A)Iv + A1vK.
4 The optimality designs
Let N1 be the incidence matrix of the balanced incomplete block design with the parameters v, b1, r1, k1, A1, and, let N2 be the incidence matrix of the ternary balanced block design with the parameters v, b2, r2, k2, A2, p12, p22. From the matrices N1 and N2, we construct the design matrix X of the chemical balance weighing design in the form (2.1) for X1 = 2N1 - 16l 1v and X2 = N; - 1&2 ïv,
X
2N1 - 1bi 1v 0b
N2 - 1b2 1v 1b
(4.1)
Lemma 1 Any chemical balance weighing design with the design matrix X given in (4.1) is nonsingular.
Theorem 2 Any chemical balance weighing design with the design matrix X in the form (4.1) and with the dispersion matrix a2G, where G is given in (1.2), is optimal if and only if the conditions
(i)	r2 = &2
(ii)	a = fi
(iii)	a [bi - 4(ri - Ai)] + 62 + A2 - 2r2 = 0 are simultaneously fulfilled.
Proof. For the design matrix X in (4.1) and G in (1.2) we have
XG-iX
T ^ (r2 - 62) 1v
(r2 - 62) l'v	62
where T =
[4a (ri - Ai) + r2 - A2 + 2P22] Iv + [a (bi - 4(ri - Ai)) + 62 - 2r2 + A2] lvlV. Using the optimality conditions given in the Theorem 2 our result is proved.
Based on Raghavarao (1971), Billington and Robinson (1983), Ceranka and Graczyk (2004), we can formulate
Theorem 3 Let a = 2. If the parameters of the balanced incomplete block design and the parameters of the ternary balanced block design are equal to one of
(i)	v = 7, 6i = 21, ri = 6, ki = 2, Ai = 1 and v = k2 = 7, 62 = r2 = 54, A2 = 52, pi2 = 42, P22 = 6;
(ii)	v = 12, 6i = 33, ri = 11, ki = 4, Ai = 3 and v = k2 = 12, 62 = r2 = 88, A2 = 86, pi2 = 66, P22 = 11; or
(iii)	v = 6i = 13, ri = ki = 4, Ai = 1 and v = k2 = 13, 62 = r2 = 50, A2 = 48, pi2 = 26, P22 = 12,
then X in the form (4.1) is the design matrix of the optimum chemical balance weighing design with the dispersion matrix of errors a2G, where G is in (1.2).
Proof. It is easy to check that the parameters of the balanced bipartite weighing design and the ternary balanced block design satisfy the conditions (i) - (iii) of Theorem 2.
Theorem 4 Let a = 2. If the parameters of the balanced incomplete block design are equal to v = 15, 6i = 42, ri = 14, ki = 5, Ai = 4 and the parameters of the ternary balanced block design are equal to v = k2 = 15, 62 = r2 = 35, A2 = 34, pi2 =
21, p22 = 7, then X in the form (4.1) is the design matrix of the optimum chemical balance weighing design with the dispersion matrix of errors a2G where G is in (1.2).
Let us consider the design matrix X of the chemical balance weighing design in the form (2.1) for Xi = 2N1 - 1bl l'v and X2 = N'., - 1b2 l'v. We have
X
2Ni - lbi lv lb
N2 - lb2 lV 0b
(4.2)
Theorem 5 Any chemical balance weighing design with the design matrix X in the form (4.2) and with the dispersion matrix a2G, where G is given in (1.2), does not exist.
Proof. For the design matrix X in (4.1) and G in (1.2), we have
X' G-iX =\ T / (2ari - bi)lv I (r2 - b2)l'v abi
where T =
[4a (ri - Ai) + r2 - A2 + 2p^] Iv + [a (bi - 4(ri - Ai)) + b2 - 2r2 + A2] lv lv. Using the optimality conditions given in the Theorem 1 and comparing diagonal elements of X'G-iX, we obtain b2 = pi2. Therefore, a ternary balanced block design with parameters v, b2, r2, k2, A2, pi2 = b2, p22 does not exist.
5 Example
We consider two measurements installations and let a = | be the factor determining the relation between the variances of both of them. We present the construction of the
0b 0'
design matrix X given in Theorem 3 (ii). Let G
2 Ibl 0b 0'
Ib
, and let Ni
be the incidence matrix of the balanced incomplete block design with the parameters v = 12, bi = 33, ri = 11, ki = 4, Ai = 3 and let N2 be the incidence matrix of the ternary balanced block design with the parameters v = k2 = 12, b2 = r2 = 88, A2 =
86, pi2 = 66, p22 = 11, where
Ni
1	1	1	1	11111	1	1	00000000		00000000000000
1	1	1	0	00000	0	0	111	11111	00000000000000
0	0	0	1	110 0 0	0	0	111	00000	11111000000000
0	1	0	0	00100	0	0	000	10000	11100111100000
0	0	0	1	00110	0	0	110	01000	00000110011100
1	0	1	0	00001	0	0	000	01000	11010000011011
0	0	0	0	00100	1	1	101	0 0 10 0	00010101000011
0	0	0	0	1 0 0 0 0	1	1	000	01111	10100001010100
0	1	0	0	01001	0	0	000	0 0 0 1 1	00011110000110
0	0	0	0	01001	1	0	0 1 1	10100	00001001111000
0	0	0	1	10010	0	0	000	10 110	01000000100111
0	0	1	0	00010	0	1	0 0 1	00001	00101010101001
and N2
N*
112 1
12166
where
N*
2	2	2	2	2	2	2	2	2	2	2	0	0	0	0	0	0	0	0	0	0	0
2	2	2	2	2	0	0	0	0	0	0	2	2	2	2	2	2	0	0	0	0	0
2	2	0	0	0	2	2	2	0	0	0	2	2	2	0	0	0	2	2	2	0	0
0	0	2	2	0	2	2	0	2	0	0	2	2	0	2	0	0	2	0	0	2	2
2	0	2	0	0	2	0	0	0	2	2	0	0	2	2	2	0	2	2	0	2	0
0	0	2	0	2	0	2	2	0	2	0	2	0	2	0	0	2	0	2	0	2	2
0	2	0	2	0	0	2	0	0	2	2	0	0	2	2	0	2	2	0	2	0	2
2	0	0	0	2	0	2	0	2	0	2	0	2	0	2	0	2	0	2	2	2	0
0	2	0	0	2	2	0	0	2	2	0	2	0	0	2	2	0	0	2	2	0	2
0	2	2	0	0	0	0	2	2	0	2	0	2	0	0	2	2	2	2	0	0	2
0	0	0	2	2	2	0	2	0	0	2	0	2	2	0	2	0	0	0	2	2	2
2	0	0	2	0	0	0	2	2	2	0	2	0	0	0	2	2	2	0	2	2	0
From the matrices N1 and N2, we construct the design matrix X of the chemical balance weighing design in the form (4) as
X
T1	033
T2	122
T3	166
where T1 = 2N1 - 133112,
To
(N2) - I22I
12 '
T3
-1661
66112.
In this design, we estimate measurements of 12 objects with Var (Wj ) = fg for j
1, 2.....12.
References
[1]	Banerjee, K.S. (1975): Weighing Designs for Chemistry, Medicine, Economics, Operations research, Statistics. New York: Marcel Dekker Inc..
[2]	Billington, E.J. (1984): Balanced n-array designs: a combinatorial survey and some new results. Ars Combinatoria, 17, 37-72.
[3]	Billington, E.J. and Robinson, P.J. (1983): A list of balanced ternary designs with R < 15 and some necessary existence conditions. Ars Combinatoria, 16, 235-258.
[4]	Ceranka, B. and Graczyk, M. (2003a): Optimum chemical balance weighing designs. Tatra Mountains Math. Publ., 26, 49-57.
[5]	Ceranka, B. and Graczyk, M. (2003b): On the estimation of parameters in the chemical balance weighing designs under the covariance matrix of errors a2G. 18th Inter-ational Workshop on Statistical Modelling, G. Verbeke, G. Molenberghs, M. Aerts, S. Fieuws, Eds., Leuven, 69-74.
[6]	Ceranka, B. and Graczyk, M. (2004): Balanced ternary block under the certain condition. Colloquium Biometryczne, 34, 63-75.
[7]	Pukelsheim, F. (1993): Optimal Design of Experiment. New York: John Willey and Sons.
[8]	Raghavarao, D. (1971): Constructions and Combinatorial Problems in designs of Experiments. New York: John Willey and Sons.
[9]	Shah, K.R., Sinha, B.K. (1989): Theory of Optimal Designs. Berlin, Heidelberg: Springer-Verlag.
[10] Wong, C.S. and Masaro, J.C. (1984): A-optimal design matrices X = (xj)Nxn with x»j = — 1,0,1. Linear and Multilinear Algebra, 15, 23-46.