Metodolosˇki zvezki, Vol. 4, No. 1, 2007, 1-7 A-Optimal Chemical Balance Weighing Design under Certain Condition Bronisław Ceranka and Małgorzata Graczyk1 Abstract The paper is studying the estimation problem of individual weights of p objects using the design matrix X of the A-optimal chemical balance weighing design under the restriction p1 + p2 = q < p, where p1 and p2 represent the number of objects placed on the left pan and on the right pan, respectively, in each of the measurement / __1 operations. The lower bound of tr(X X) is attained and the necessary and sufficient conditions for this lower bound to be obtained are given. There are given new construction methods of the A-optimal chemical balance weighing designs based on incidence matrices of the balanced bipartite weighing designs and the ternary balanced block designs. 1 Introduction Suppose specifically that there are p objects of true unknown weights w1, w2, ..., wp, respectively, and we wish to estimate them employing n measuring operations using a chemical balance. Let y1, y2, ..., yn denote the recorded observations in these n opera-tions, respectively. It is assumed that the observations follow the standard linear model y = Xw + e, E(e) = 0n, E(ee ) = ? In, (1.1) where X is of order n x p and is called the weighing design matrix. The elements of X are xij, i = 1,2,.., n, j = 1,2, ...,p, and a typical element xij is —1 if the jth object is placed on the left pan during the ith weighing operation, +1 if the jth object is placed on the right pan during the ith weighing operation and 0 if the jth object is not utilized ' in either pan during the ith weighing operation. Hence w = (w1,w2, .., wp) is the vector of true unknown weights (of parameters). The vector e is the so-called vector of error components satisfying the usual homoscedasticy conditions. The inference problem centres around estimation of true individual weights of all objects. The optimality problem is concerned with efficient estimation in some sense by a proper choice of the design matrix X among designs at our disposal. The model (1.1) is the 1Department of Mathematical and Statistical Methods, Agriculture University of Poznan´, Wojska Polskiego 28, 60-637 Poznan´, Poland; bronicer@au.poznan.pl, magra@au.poznan.pl 2 Bronisław Ceranka and Małgorzata Graczyk standard Gauss - Markov model and the following results are well known. The parameter vector w is estimable if and only if r(X) = p in which case w = (X X) X y, V(w^) = ?(XX) , (1.2) where w is the blue and V is the dispersion matrix. In the literature some optimality criterions, i.e. A-, D-, E-optimality, are considered. Some construction methods of optimal designs are known. They are formulated for design matrices with elements equal to —1 and 1, only and they are based on incidence matrices of block designs. Some problems connected with the optimality of chemical balance weighing designs were considered in the books of Raghavarao (1971), Banerjee (1975), Shah and Sinha (1989). Among all possible designs an A-optimal design are considered. There are designs in which the sum of variances of esimators, or equivalently tr (V(w)) = ?2tr(X X)-1, is minimal. Wong and Masaro (1984) gave the lower bound for tr(X X) and some construction methods of the A-optimal chemical balance weighing designs. In the paper we consider A-optimal criterion for designs for which in each measurement operation not all objects are included. In other words, in each column of the design matrix exist elements equal to —1, 1 and 0. In the next section we give a lower bound for tr(X X) and the necessary and sufficient conditions to this lower bound to be attained under given restriction on the number of objects included in the particular measurement operation. We present new construction method of the A-optimal design based on balanced bipartite weighing designs and ternary balanced block designs. 2 Some results on A-optimality Let X be an n x p design matrix of a chemical balance weighing design. The following results from the paper Wong and Masaro (1984) give the lower bound for tr(X X) Lemma 1 For ann x p design matrix X of rank p we have the inequality x [ (X X) 2 '-1 p tr (X X) > , , tr(X X) the equality being attained if and only if XX is equal to the p x p identity matrix / I Ip multiplied by scalar, i.e. X X = zp. f r'--1 ~i f f f The lower bound o tr [(X X) J is attained i and only i the elements o the design matrix X are equal to — 1 or 1, only. It implies that in each measurement operation all objects must be included in different combinations. Some times it is not possible. Therefore in the present paper we consider the situation the elements of the design matrix X are equal to 0, either. It other words, in each weighing not all objects are included. Thus we give new lower bound of tr [(X X) J . We have A-Optimal Chemical Balance Weighing Design... 3 Theorem 1 For any nonsingular chemical balance weighing design with the design matrix X = (xij) we have / 1 p2 tr(X X) > ------, (2.1) q ¦ n where q = max(q1, q2,..., qn), qi = YTj=1x2ij, i=1,2,..,n. Proof. Because the design matrix X is of full column rank we have n p n tr(X X) = y y xij = y qi < q ¦ n. (2.2) Thus, from Lemma 1 we get (2.2). Hence the result. In the case q = p we get the inequality given in Wong and Masaro (1984). Definition 1 Any nonsingular chemical balance weighing design with the design matrix X = (xij) is said to be A-optimal if z1 p2 tr(X X) =-----. q ¦ n Theorem 2 Any nonsingular chemical balance weighing design with the design matrix X = (xij) is A-optimal if and only if / q • n X X =-----Ip. Proof. To prove the necessity part we observe that from Lemma 1 we have X X = zIp and the equality in (2.1) is satisfi The sufficiency part is obvious. f f (, q'n and the equality in (2.1) is satisfied i and only i tr X X) = q ¦ n. It implies that z = —. In the present paper we will construct an A-optimal chemical balance weighing design under the restriction p1 + p2 = q ? p, where p1 and p2 represent the number of objects placed on the left and on the right pan, respectively, in each of the measurement opera-tions. They are based on the incidence matrices of the balanced bipartite weighing designs and the ternary balanced block designs. 3 Balanced designs In this section we recall the definitions of the balanced bipartite weighing design given in Huang (1976) and of the ternary balanced block design given in Billington (1984). A balanced bipartite weighing design is a design which describes how to replace v treat-ments in b blocks such that each block containing k distinct treatments is divided into 2 subblocks containing k1 and k2 treatments, respectively, where k = k1 + k2. Each 4 Bronisław Ceranka and Małgorzata Graczyk treatment appears in r blocks. Every pair of the treatments from different subblocks appears together in ?i blocks and every pair of treatments from the same subblock appears together in ?2 blocks. The integers v, b, r, k\, k2, ?i, ?2 are all parameters of the balanced bipartite weighing design. The parameters are not independent. They are related by the following identities vr = bk, b ?1v(v - l) = 2k 1 k 2 , ?2 = 1- 2k k 2 2- ?1k(v-l) r = 2k k . Let N? be the incidence matrix of such a design with elements equal to 0 or 1, then N?N? = (r — ?i — ?2) Iv + (?i + ?2) 1v1v. 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 appears at least ones). Each of the distinct pairs appears ? times. Any ternary balanced block design is regular, that is, each treatment occurs alone in ?\ blocks and is repeated two times in ?2 blocks, where ?\ and ?2 are constant for the design. Let N be the incidence matrix of the ternary balanced block design. It is straightforward to verify that vr = bk, r = ?i + 2?2, ?(v — 1) = ?\(k — 1) + 2?2(k — 2) = r(k — 1) — 2?2, NN' = (?i + 4?2 — ?)Iv + ?1v1^v = (r + 2?2 — ?)Iv + ?1v1^v. 4 Construction of the design matrix Let N? be the incidence matrix of the balanced bipartite weighing design with the parameters v, b, r, k\, k2, ?i, ?2. From the matrix N? we form the matrix N by replacing k\ elements equal to +1 of each column which correspond to the elements belonging to the first subblock by —1. Thus each column of the matrix N will contain k\ elements equal to —1 and k2 elements equal to +1. From the matrix N we construct the design matrix X I ' of the chemical balance weighing design in the form X = N . In this design p = v and n = b. The following result is from Ceranka and Graczyk (2002) ' Lemma 2 Any chemical balance weighing design with the design matrix X = N is non-singular if and only if k\ = k2. Theorem 3 Any nonsingular chemical balance weighing design with the design matrix X = N is A-optimal if and only if ?2 = ?i (4.1) and q = k. (4.2) A-Optimal Chemical Balance Weighing Design... 5 ' Proof. F or the design matrix X = N we have and X = (r — ?2 + ?1)Iv + (?2 — ?1)1v1v X = — Iv. v Comparing these two equalities we get ?2 = ?1 and r — ?2 + ?1 = —. If (4.1) is satisfied then we get (4.2) from the last equation. Hence we get the thesis of the Theorem. Now, we consider the chemical balance weighing design with the design matrix X = N/ ' — 1b1v, where N is the incidence matrix of the ternary balanced block design with the parameters v, b, r, k, ?, ?1, ?2. In this design p = v and n = b. From Ceranka, Katulska and Mizera (1998) we have ' ' Lemma 3 Any chemical balance weighing design with the design matrix X = N — 1b1 v is nonsingular if and only if v = k. Thus we get Theorem 4 Any nonsingular chemical balance weighing design with the design matrix X = N — 1b1 v is A-optimal if and only if b + ? — 2r = 0 (4.3) and q ¦ b b — ?1 =------. (4.4) v Proof. For the design matrix X = N — 1b1v we have — 1b1 X = (r + 2?2 — ?)Iv and XI ' X = (r + 2?2 — ?)Iv + (b + ? — 2r)1v1v X = — Iv. Comparing these two equalities we get b + ? — 2r = 0 and r + 2?2 — ? = —.If (4.3) is satisfied we get (4.4) from the last equation. Hence the claim of the Theorem. 5 Balanced bipartite weighing designs leading to the A-optimal designs We have seen in the Theorem 3 that if the parameters of the balanced bipartite weighing design satisfy the condition (4.1) then the chemical balance weighing design with the de- ' sign matrix X = N is A-optimal. Under this condition we have the following Theorem given in Ceranka and Graczyk (2005) 6 Bronisław Ceranka and Małgorzata Graczyk Theorem 5 The existence of the balanced bipartite weighing design with the parameters 2sv(v — 1) 2s(v—1) k c(c— 1) k c(c+1) ? ? v, b = 221) , r = c21 , 1 = 2 , 2 = 2 , 1 = s, 2 = s, c = 2, 3,..., s = 1,2,... implies the existence of the A-optimal chemical balance weighing design, v > c2, q = c2. 6 Ternary balanced block designs leading to the A-optimal designs We have seen in the Theorem 4 that if the parameters of the ternary balanced block design satisfy the condition (4.3) then a chemical balance weighing design with the design ma- ' ' trix X = N — 1b1v is A-optimal. Under this condition we have the following Theorem Theorem 6 The existence of the ternary balanced block design with the parameters (i) v = s, b = us, r = u(s — 2), k = s — 2, ? = u(s — 4), ?1 = u(s — 4), ?2 = u, u = 1, 2,..., s = 5, 6,..., except the case u = 1 and s = 5, (ii) v = s, b = us, r = u(s — 3), k = s — 3, ? = u(s — 6), ?1 = u(s — 9), ?2 = 3u, u = 1,2,..., s = 10,11,..., (iii) v = s, b = us, r = u(s — 4), k = s — 4, ? = u(s — 8), ?1 = u(s — 16), ?2 = 6u, u = 1, 2,..., s = 17,18,..., implies the existence of the A-optimal chemical balance weighing design. Proof. It is easy to see that the parameters (i)-(iii) satisfy the conditions (4.3) and (4.4). Example We determine unknown measurements of p = 6 objects by weighing them n = 6 times and each object is weighed at most q = 4 times. For construction the design we consider the ternary balanced block design with the parameters v = b = 6, r = k = 4, ? = ?1 = 2, ?2 = 1 (see Theorem 6(i)) and with the incidence matrix N = . 2 0 10 0 1 12 0 10 0 0 12 0 10 0 0 12 0 1 10 0 12 0 0 10 0 12 Based on the matrix N we form the design matrix X of the A-optimal chemical balance weighing design in the form X = N — 1b1v A-Optimal Chemical Balance Weighing Design... 7 X = 1 0 -1 -1 0 -1 1 1 0-1-1 0 0 -1 1 0 -1 -1 1 0-1 1 0-1 1-1 0-1 1 0 0 -1 -1 0 -1 1 1 -1 -1 We have X X = 4I6 and V(w^,) = ^4 for i = 1,2,..., 6. In the paper were presented conditions determined A-optimal chemical balance weighing design and some new methods of construction of designs under assumption no all objects are included in each measurement operation. These methods are based on the incidence matrices of balanced bipartite weighing designs and ternary balanced block design. References [1] Banerjee, K.S. (1975): Weighing Designs for Chemistry, Medicine, Economics, Op-erations Research, Statistics. New York: Marcel Dekker Inc. [2] Billington, E.J. (1984): Balanced n-ary designs: A combinatorial survey and some new results. Ars Combinatoria, 17A, 37-72. [3] Ceranka, B. and Graczyk, M. (2002): Optimum chemical balance weighing design based on balanced bipartite block designs. Listy Biometryczne-Biometrical Letters, 39, 71-84. [4] Ceranka, B. and Graczyk, M. (2005): About relations between the parameters of the balanced bipartite weighing designs. In S.M. Ermakov, V.B. Melas, and A.N. Pepelyshev (Eds): Proceedings of the 5th St. Petersburg Workshop on Simulation, 197-203. [5] Ceranka, B., Katulska, K., and Mizera, D. (1998): The application of the ternary balanced block designs. Discussiones Mathematicae-Algebra and Stochastics Meth-ods, 18, 179-185. [6] Huang, Ch. (1976): Balanced bipartite block designs. Journal of Combinatorial The-ory, 21, 20-34. [7] Shah, K.R. and Sinha, B.K. (1989): Theory of Optimal Designs. Berlin, Heidelberg: Springer-Verlag. [8] Raghavarao, D. (1971): Constructions and Combinatorial Problems in Design of Experiments. New Yok: John Wiley Inc. [9] Wong, C.S. and Masaro, J.C. (1984): A-optimal design matrices X = (xij)N×n with xij = -1, 0, 1. Linear and Multilinear Algebra, 15, 23-46. .