ON THE CONSTRUCTION OF REGULAR A-OPTIMAL SPRING BALANCE WEIGHING DESIGNS

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1 Colloquium Biometricum 43 3, 3 9 ON THE CONSTRUCTION OF REGUAR A-OPTIMA SPRING BAANCE WEIGHING DESIGNS Bronisław Ceranka, Małgorzata Graczyk Department of Matematical and Statistical Metods Poznań Uniersity of ife Sciences Wojska Polskiego 8, Poznań, Poland s: bronicer@up.poznan.pl, magra@up.poznan.pl Summary In tis paper we study te problem of te estimation of indiidual measurements of objects in spring balance weiging design under te assumption tat errors are uncorrelated and ae different ariances. Te incidence matrices of te balanced incomplete block designs are used for new construction of te regular A-optimal spring balance weiging design. Teoretical researc is illustrated by an example. Keywords and prases: A-optimal spring balance weiging design, balanced incomplete block design Classification AMS : 6K5, 6K. Introduction Te optimality of designs plays a main role in te teory of te experimental designs. In many papers concerning te optimality, te weiging designs are considered. In a spring balance, tere is only one pan and any number of objects can be placed on te pan. Ten te pointer proides a reading wic represents te total weigt of te objects on te pan.

2 4 BRONISŁAW CERANKA, MAŁGORZATA GRACZYK Nowadays, te spring balance weiging design is te name for te experimental design connected not only wit a spring balance, but wit any experiment in tat te results we can describe as te linear combination of unknown measurements of objects wit coefficients of tis combination equal to or. In fact, te weiging designs are applicable to a great ariety of problems of measurements, not only for weigts, but of lengt, oltages, resistance and concentrations of cemical in mixture, analyzing te lines of legume. Te main idea is to determine unknown measurements of p objects in n weiging operations. We sall make two standing assumptions on te maps under consideration. Recorded obserations are independent and tere are not systematic errors. Te second basic assumption is tat te errors ae different ariances. Of course, te experimenter wants to coose a weiging design tat is optimal wit respect to some criterion. In literature, seeral criteria are often expressed in terms of te information matrix. One of tem is A-optimality, in tat we study te trace of te inerse of information matrix.. Te linear model Suppose, tere are p objects of unknown measurements w,...,, w wp, respectiely, and we wis to estimate tem employing n measurement operations using a spring balance. et y, y,..., yn denote recorded obserations in tese n operations. It is assumed tat te obserations follow te model were ( ) y Xw + e (.) y y, y,..., y is an n n random ector of te obserations. Te X, usually called weiging matrix, belongs to te class design matrix ( x ij ) {, } Ψ -wic denotes te class of n p matrices of known elements x n p or as in te i t weiging operation te j t object is not placed on te pan or is placed. A p ector ( ) w w contains unknown weigts of, w,..., wp objects and e is an n random ector of errors. We assume tat in te model (.) te errors are uncorrelated and ae different ariances, i.e. E ( ee ) σ G, and moreoer E ( e ), were n is an n null ector, n G is te known n n diagonal positie definite matrix. Accordingly, any spring balance weiging design is nonsingular if and only if X is of full column rank, i.e. r ( X) p. In ij

3 ON THE CONSTRUCTION OF REGUAR A-OPTIMA SPRING BAANCE 5 suc a design for te estimation of unknown weigts of p objects, we use te general weigted least squares metod and we get Te coariance matrix of ŵ is equal to ( X G X) X G y wˆ. (.) ( wˆ ) σ ( X G X) Var. (.3) Te matrix X G X is called te information matrix. In te literature, some study on optimality criterions are presented. For a deeper discussion we refer to te paper Jacroux and Notz (983). In many papers concerning te weiging designs, te A-optimal design is considered. For te gien coariance matrix of errors σ G, te design X is A-optimal if te sum of ariances of estimators for unknown parameters is minimal, i.e. ( X G X) tr is minimal in te class Ψ {,} n p. Moreoer, te design for wic te sum of ariances of estimators of parameters attains te lowest bound in Ψ, is called te regular A-optimal design. et us note, in te set n p{ } of design matrices Ψ {,} n p, te regular A-optimal design may not exist, wereas A-optimal design always exist. Te concept of te A-optimality was sown in Pukelseim (983), Sa and Sina (989), Ceranka and Graczyk (4), Ceranka, Graczyk, Katulska (6, 7), Masaro and Wong (8), Graczyk (, ). In tis paper we consider te experimental situation were we determine unknown measurements of p objects in n n s measurement operations under model (.). It is assumed tat n s measurements are taken in different conditions or at different installations, s,,...,. So, te coariance matrix of errors σ G is gien by te matrix G g I n G n n n n n n I n g n n g n n I n n n, (.4)

4 6 BRONISŁAW CERANKA, MAŁGORZATA GRACZYK were g s > denotes te factor of precision, s,,...,. Consequently, X, as according to te partition of G we write te design matrix { } X X X, (.5) X were X s is te n s p design matrix of any spring balance weiging design. Graczyk () gae te following teorems and definition. Teorem.. et p be odd. In any nonsingular spring balance weiging design {,} X in (.5) wit te coariance matrix of errors σ G, were G is of (.4), tr ( X G X) 4 p ( p + ) tr( G ) Definition.. et p be odd. Any {,} G 3. (.6) X in (.5) wit te coariance matrix of errors σ, were G is gien by (.4), is said to be te regular A-optimal spring balance weiging design if tr ( X G X) 4 3 p. (.7) ( p + ) tr( G ) Teorem.. et p be odd. Any {,} X in (.5) wit te coariance Ψ n p matrix of errors σ G, were G is gien by (.4), is te regular A-optimal spring balance weiging design if and only if ( G )( I ) p + tr. (.8) p p 4 p X G X p +

5 ON THE CONSTRUCTION OF REGUAR A-OPTIMA SPRING BAANCE 7 3. Construction of te regular A-optimal designs In Graczyk () some construction metods of te regular A-optimal spring balance weiging design are gien. In tis paper we present some new experimental plans of suc designs. Te construction of a regular A-optimal spring balance weiging design proceeds as follow. It is wort pointing out te incidence matrices of te block designs may be used for te construction of te design matrix {,} X, ten we take n b s and p. Now, we present some series of te balanced incomplete block designs. Based on teir X, of te spring incidence matrices we form te design matrix { } balance weiging design wit σ G, were G is of (.4), tat is te regular A-optimal design. Summarizing, we can formulate our main result. Teorem 3.. et be odd and let N be te incidence matrix of balanced incomplete block design wit te parameters (i) b 4 t + 3, r k ( t +), λ t +, were 4 t + 3 is a prime or a prime power (te complementary design to te design wic is described in Ragaarao (97, Teorem 5.7.4) or Ragaarao and Padgett (5, Corollary 4.5.), (ii) 4 t +, b ( 4t + ), r ( t + ), k t +, λ t +, were 4 t + is a prime or a prime power (te complementary design to te design wic is described in Ragaarao (97, Teorem 5.75), * * (iii) b 4( k ), r k ( k ) *, ( ) λ k, (if tere exists a balanced incomplete block design wit λ * and r * k * +, Ragaarao (97, Teorem 5.9.) or Ragaarao and Padgett (5, Teorem 4.3), (i), b, r, k t, λ of an irreducible balance t t t incomplete block design, Ragaarao (97, p. 9), or Ragaarao and Padgett (5, p. 86). Any {,} X in te form Ψ b X N is te regular A-optimal spring balance weiging design wit te coariance matrix of errors is gien by (.4). σ G, were G

6 8 BRONISŁAW CERANKA, MAŁGORZATA GRACZYK Proof. For te design matrix X N and G in (.4), we ae X G X NN g s. N is te incidence matrix of balanced incomplete block design terefore NN ( r λ) I + λ. We ae NN λ( I + ) because r λ. On te account of te aboe remark, we get ( I ) X G X λ s + g. So, it is eident tat te condition (.8) olds. 4. Example As an example let us consider te experiment in tat we determine unknown measurements of p 5 objects using n measuring operations. Te g coariance matrix of errors σ I G is gien by te matrix G, g I were g, g >. To construct te design matrix X Ψ 5 {,} we can use te balanced incomplete block design wit te parameters 5, b, r 6, k 3, λ 3 gien by te incidence matrix N. N In tis case we ae X G X 3 g + g I From (.8) N 6 we obtain X G X ( g + g )( I ). Hence 4 5 ( X G X) I5 55, 3( g + g ) 6 X and ( )( )

7 ON THE CONSTRUCTION OF REGUAR A-OPTIMA SPRING BAANCE 9 5 X. Te same conclusion can be drawn from 8( g + g ) (.7). In tis case we obtain tr ( X G X). 6 terefore tr ( X G ) ( g + g ) 8( g + g ) References Ceranka B., Graczyk M. (4). A-optimal cemical balance weiging design. Folia Facultatis Scientiarum Naturalium Uniersitatis Masarykianae Brunensis. Matematica 5, Ceranka B., Graczyk M., Katulska K. (6). A-optimal cemical balance weiging design wit nonomogeneity of ariances of errors. Statistics and Probability etters 76, Ceranka B., Graczyk M., Katulska K. (7). On certain A-optimal cemical balance weiging designs. Computational Statistics and Data Analysis 5, Graczyk M. (). A-optimal biased spring balance design. Kybernetika 47, Graczyk M. (). Regular A-optimal spring balance weiging designs. Restat, Jacroux M., Notz W. (983). On te optimality of spring balance weiging designs. Te Annals of Statistics, Masaro J., Wong C.S. (8). Robustness of optimal designs for correlated random ariables. inear Algebra and its Applications 49, Pukelseim F. (993). Optimal design of experiment. Jon Wiley and Sons, New York. Ragaarao D. (97). Constructions and Combinatorial Problems in designs of Experiments. Jon Wiley Inc., New York. Ragaarao, D., Padgett,.V. (5). Block Designs, Analysis, Combinatorics and Applications. Series of Applied Matematics, Word Scientific Publising Co. Pte. td. Sa K.R., Sina B.K. (989). Teory of optimal designs. Springer-Verlag, Berlin, Heidelberg.