A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
B r on i sł aw C e r a n k a * , M a ł g o r z a t a G r a c z y k * *
C O N ST R U C T IO N S OF O PTIM UM CH EM ICAL BALANCE W EIGHING D E SIG N S BASED ON BALANCED BLOCK D E SIG N S
Abstract
T h e p a p e r is stu d y in g the p ro b lem o f estim atio n o f the individual u n k n o w n m easurem ents (w eights) o f p o b jects w hen we have at o u r disposal n m ea su rem en t o p e ratio n s (weighings). In this p ro b lem we use the linear m odel called chem ical balance w eighing design u n d e r the re stric tio n on the n u m b er tim es in w hich each o bject is m easu red . A low er b o u n d for the v arian ce o f each o f the estim ated m easurem ents and a necessary an d sufficient co n d itio n s for this low er b o u n d to be a tta in e d are given. T h e incidence m atrices o f balan ced incom plete block designs an d tern a ry b alanced block designs arc used to c o n stru c t the design m atrix X o f o p tim u m chem ical b alan ce weighing design.
Key words: balanced in com plete block design, chem ical b alan ce w eighing design, tern a ry b alanced block design.
I. IN T R O D U C T IO N
The results o f n m easurem ent (weighing) operations arc used to determ ine the individual weights o f p objects w ith a balance with tw o pans th a t is corrected fo r bias. T h e linear m odel for this experim ent is given in the form
y = Xw + e, (1)
w here у is n x 1 ra n d o m colum n vector o f the observed m easurem ents (weights), X = (Xij), i = 1 , 2 , n, j = 1 , 2 , p, is the n x p m atrix o f know n
* Professor, D e p a rtm e n t o f M ath em atical and S tatistical M eth o d s, A gricultural U niversity in P oznań.
elem ents with x tJ = — 1, 1 or 0 if the у-th object is k ep t on the right p an, left pan or is n o t included in the p articu larly m easu rem en t o p e ra tion (weighing), respectively, w is the p x 1 colum n vector representing the unknow n m easurem ents (weights) o f objects and e is the n x 1 co lum n vector o f ran d o m errors between observed and expected readings such th a t E(e) = 0„, and E(ee') = o 2l n. By the o th er w ords, we assum e th a t ran d o m erro rs are uncorrelated and they have the sam e variances. 0„ is the n x 1 vector with elem ents equal to 0 everyw here and I„ is the n x n identity m atrix , E stands for expectation and c' is used for tra n s pose o f c.
T h e n o rm al equ atio n s estim ating w are o f the form
X'Xw = X'y, (2)
w here w is th e vector o f the unkno w n m easurem ents (weights) estim ated by the least squares m ethod.
A chem ical balance weighing design is said to be singular o r nonsingular, depending on w hether the m atrix X'X is singular o r n on singular, respectively. It is obvious th a t the m atrix X'X is n onsing ular if and only if th e design m atrix X is o f full colum n ra n k ( = p). N ow , if X'X is n on sin gu lar, the least squares estim ato r o f \v is given by
w = (X 'X )- 1X'y (3)
and the variance - covariance m atrix o f w is given by
Var(w) = ct2(X 'X )- 1 . (4)
V arious aspects o f chem ical balance weighing designs have been studied by R ag h av a rao (1971) and Banerjee (1975). H otelling (1944) have showed th a t the m inim um attain ab le variance for each o f the estim ated weights for a chem ical balance weighing design is a 2/n and proved the theorem th at each o f th e variances o f the estim ated weights atta in s the m inim um if and only if X'X = n Ip. T his design is said to be optim um chem ical balance weighing design. In the o th er w ords, m atrix X o f an op tim um chemical balance weighing design has as elem ents only -1 and 1. T h a t m eans, in each m easurem ent o p eratio n all objects are included. In this case several m eth o d s o f co nstructing optim um chem ical balance w eighing designs are available in the literature.
Som e m eth o d s o f co nstructin g chem ical balance weighing designs in which the estim ated weights are uncorrelated in the case w hen the design
m atrix X has elem ents -1 , 1 and 0 was given by Swam y (1982), C era n k a et al. (1998) and C era n k a and K a tu lsk a (1999).
In the present p ap e r we study an o th er m ethod o f co n stru ctin g the design m atrix X o f an optim um chem ical balance weighing design, which has elem ents equal to -1 , 1 and 0, under the restriction on the n u m ber times in which each object is weighted. T his m ethod is based on the incidence m atrices o f balanced incom plete block designs and tern ary balanced block designs.
II. V A R IA N C E L IM IT O F E S T IM A T E D W E IG H T S
Let X be an n x p m atrix o f ran k p o f a chem ical balance weighing design an d let nij be the num ber o f tim es in which y'-th object is weighted, j = 1 ,2 ,..., p. C era n k a and G raczyk (2001) proved the follow ing theorem s:
Theorem 1. F o r any nonsingular chem ical balance w eighing design given by m atrix X the variance o f w} for a p artic u la r j such th a t 1 ś j ś p cann ot be less th an (r2/m, where m = m ax (m,).
J = 1,2,...p
Theorem 2. F o r any n x p m atrix X o f a no n sin g u lar chem ical balance weighing design, in which m axim um n um ber o f elem ents eq ual to -1 and 1 in colum ns is equal to m, each o f the variances o f the estim ated weights attain s the m inim um if and only if
X'X = m l p. (5)
Definition 1. A non singular chem ical balance weighing design is said to be optim al for the estim ation individual weights o f objects if the variances o f th eir estim ato rs attain the lower bound given by T h eorem 1., i.e., if
a 2
V ar(wj) = — ’ j = 1 ,2 , . ..,p. (6)
In the other words, the optim um design is given by the m atrix X satisfying co n d itio n (5).
In the next sections we will present constru ctio n o f the design m atrix X o f optim um chem ical balance weighing design based o n incidence m atrices o f balanced incom plete block designs an d tern ary balanced block designs.
Ш . B A L A N C E D B L O C K D E S IG N
A balanced incom plete block design there is an arran g e m e n t o f v tre a t m ents in to h blocks, each o f size k, in such a way, th a t each trea tm e n t occurs at m o st ones in each block, occurs in exactly r blocks and every p air o f trea tm e n ts occurs together in exactly X blocks. T h e integers v, h, r, к, X are called the param eters o f the balanced incom plete block design. L et N be th e incidence m atrix o f balanced incom plete block design. It is straig h tfo rw ard to verify th at
vr = bk, X(v— l ) = r(k — 1), N N ' = ( r — X)lv + Al„l
where 1„ is v x 1 vector with elem ents equal to 1 everyw here.
A tern a ry balanced block design is defined as the design consisting o f h blocks, each o f size k, chosen from a set o f size v in such a way th at each o f the v elem ents occurs r times altog eth er and 0, 1 or 2 times in each block, (2 appears at least ones) and each o f th e distinct pairs ap p ears X times. A ny tern ary balanced block design is regular, th a t is, each elem ent occurs alone in p l blocks and is repeated tw o times in p 2 blocks, w here p t and p 2 are c o n sta n t for the design. Let N be th e incidence m atrix o f the ternary balanced block design. T he param eters are n o t all independent and they are related by the follow ing identities
vr = bk, r = Pi + 2 p2,
A(v - 1) = P l (k - 1) + 2p z{k - 2) = r(k - 1) - 2p 2, N N ' = (P l + 4p 2 X)lv + X I X = (r + 2P2 Q K + M X
-IV. C O N S T R U C T IO N O F T H E D E S IG N M A T R IX
L et N j be the incidence m atrix o f balanced incom plete block design w ith the param eters v, b v, r 1, k lt and N 2 be th e incidence m atrix of tern a ry balanced block design w ith the p aram eters v, b2, r2, k 2, X2, p 12,
p22- T h e m atrix X o f a chem ical balance weighing design we define in the follow ing form
Thus each colum n o f X contains hl — r i + b2 — p l2 — P22 elements equal -1, r t + Pzz elem ents equal 1 and p l2 elem ents equal 0. In this design each o f the p = v objects is m easured m = + h 2 — p i2 times in n = b x + b 2 weighing operations.
Lemma 1. T h e chem ical balance weighing design w ith th e design m atrix X given in the form (7) is nonsingular if and only if
2/cj Ф k 2 (8)
or
2/cj. = k 2 Фv. (9)
P ro o f. F o r the design m atrix X given by (7) we have
X'X = [4(r j — Ax) + r 2 + 2p 22 — Я 2] ^ + [fej — 4 (rj — Aj) + b2 + X2 — 2r2\ \ v\'v ( 10) It is easy to calculate th a t the determ in an t (10) is equal
det(X 'X ) = [4(rx - A,) + r 2 + 2p 22 - A J " - 1 ■
• T- j - [v2(rx^ 2 + r2k i) - 2v k l k 2(2rl + r 2) + k 1k 2(4r1k 1 + r 2k 2)]. 1 2
O D Evidently 4( r t —
Aj)
+ r 2 + 2 p22 —Az
is positive and hence dct(X 'X ) is positive if and only if 2 к у Ф к 2 or 2 k t = k 2 ф v . So, the lem m a is proved.Theorem 3. T he nonsingular chemical balance weighing design with m atrix X given by (7) is optim al if and only if
fei-4(r1-A1) + (fe2 + A2 —2r2) = 0.
(12)
P roof. F ro m the conditions (5) and (10) it follows th a t a chemical balance w eighing design is optim al if and only if the co n d itio n (12) holds. H ence th e theorem .I f the chem ical balance weighing design with th e design m atrix X given by (7) is optim al th en
f f 2
Var(vvj) = j ~ r r --- > j — 1 . 2 , p. (13) bi + b2 - p l2
In particular the equality (12) is true when by = 4(rl — Xy) and b2 = 2r2 — X2. Corollary 1. If the conditions
bi = 4 ( r l - X 1) (14)
and
b2 — 2r2 — X2 (15)
are tru e th en a n onsingular chem ical balance w eighing design w ith the design m atrix X given by (7) is optim al.
T h e balanced incom plete block designs for w hich the co nd itio n (14) holds belong to the family A (R ag h av arao , 1971: 69). T he series o f ternary balanced block designs for which the con ditio n (15) is tru e was given by B illington and R o b in so n (1983).
Corollary 2. A chemical balance weighing design w ith th e design m a t rix X given by (7) based o n balan ced in co m p lete b lo ck desig n s for which the co n d itio n (14) holds and tern ary balanced block designs for w hich the condition (15) holds for the sam e n u m b er o f treatm en ts is optim al.
We have seen in the Theorem 3 th at if param eters o f balanced incom plete block designs satisfied the cond ition h l — 4(rl — Xl ) = a and param eters of tern ary balanced block designs satisfied the co nd ition b2 — 2r2 + X2 = — a, a Ф 0, th en a chem ical balance weighing design with the design m atrix X given by (7) is o ptim al. F o r a = - 2 , -1 , 1, 2 we have
Corollary 3. A chemical balance weighing design with the design m atrix X given by (7) based on the incidence m atrices o f balanced incom plete block designs and tern a ry balanced block designs w ith p aram eters
(i) v = 5, = 10, r x = 4, ky = 2, Xy = I an d v = 5, b 2 = 5(s + 4), r2 = 3(s + 4), k 2 - 3, X2 = s+ 6, p i2 = s + 12, p 22 = s, s = 1 ,2, (ii) v = 7, b t = 42, r L = 12, k i = 2, = 2 and v = 7, b 2 = s + 13, r2 = s + 13, k 2 = 7, A2 = s + l l , p 12 = s + l , p 22 = 6, s = 1 ,2 ,..., (iii) v = 11, b [ = 11, Гу - 5, ký = 5, Xi = 2 and v = 11, b 2 = 11,
r2
= 7, k 2 = 7, X2 = 4, p 12 = 5, p 22 = 1, (iv) v = 12, by = 3 3 , Гу = 11, ky = 4 , Xy = 3 and v = 12, b 2 = 18, r 2 = 15, k 2 = 10, A2 = 1 1 , p y 2 = 1, p 22 = 7,(v) v = 15, = 15, r L = 7, k t = 7, A1 = 3 and v = 15, b2 = 3(s + 4), r2 = 2(s + 4), k 2 = 10, A2 = s + 5, p l2 = 6 - 2 s, p 22 = 2s + 1, s = 1 ,2
is optim al.
In a special case, when r t = A, th en con dition (12) is o f th e form
Corollary 4. T h e existence o f tern ary balanced block designs with the param eters v, b 2, r2, k 2, A2, p i2, p 22 for w hich b 2 < 2 r 2 — A2 im plies the existence o f optim u m chem ical balance weighing design w ith
where b x — 2 r2 — A2 — h2.
Corollary 5. T h e chem ical balance weighing design w ith the design m atrix X given by (17) based on tern ary balanced block designs with param eters
(i) v = 2 s + l , />2 = 4s + u + 1, r 2 = 4s + u + l , k2 = 2 s + l , A2 = 4s + u —1, P12 = u + 1, p 22 = 2s, s = 2, 3 ,..., u = 1,2,
(ii) v = 2s, b 2 = 4s + u — 2, r 2 = 4s + и — 2, k 2 = 2s, A2 = 4s + и — 4, P12 = w, P2 2 = 2 s - 1, s = 2 ,3 ,..., и = 1 ,2 ,...
is optim al, w here b x = 2.
Corollary 6. A chemical balance weighing design with th e design m a t rix X given by (17) based on tern ary balanced block designs with p a ra m eters
p22 — 2, s — 1 ,2 ,...,
(ii) V = 12, b2 = 18, r 2 = 15, k 2 = 10, A2 = 11, p l2 = 1, p 22 = 1,
(iii) v = s, b 2 = s + u — 1, r 2 = s + u — 1, k 2 = s, A2 = s + u — 2, P i 2 = u,
is optim al, w here = 1.
In a p artic u la r case, w hen A2 = 2 r2 then the co n ditio n (12) is o f the form b x h2 -\- A.2 — 2 r2 = 0. (16)
(17)
(i) v = 5, b 2 = 5(s 4- 1), r2 — 4 (s + 1), k 2 = 4, A2 = 3s + 2, p 12 = 4s,
P22 = ^ 2 ^ ’ s = 5, 9, 11, 15, u = 1 ,2 ,...
fci — 4 (rL — Ax) + b2 = 0. (18)
Corollary 7. T h e existence o f balanced incom plete block designs with the p aram eters v, b lt r t , /c1; Ax for which b { < 4 (r, — Ax) implies the existence o f o p tim u m chem ical balance weighing design w ith
w here b2 = 4 (rj — Xy) — b v
C orollary 8. A chem ical balance weighing design with the design m atrix X given by (19) based on balanced incom plete block designs with param eters
(i) v =
4s
+ 1, by =2(4s
+ 1), Гу =4s,
ky =2s,
Ax
=2s -
1,s
=1,2, ...,(4s+
1) is prim er or prim er pow er,(ii) v = 4(s + 1), b!=2(4s + 3), r,=4s + 3,
k y = 2 ( s + \ ) ,A1 = 2s + 1,
s = 1,2,...
is o p tim al, w here b2 = 2.
C orollary 9. A chem ical balance weighing design with the design m atrix X given by (19) based on balanced incom plete block designs with param eters
( i ) v =
4s2—
1,
b 1 =4s2- l ,
rj =2s2
—1,
ky = 2s2 —1, Ax
=s2
—1,
s = 1,2,...,
( i i ) v =
4s + 3,
fcj=4s + 3,
r1 = 2s+l,
k y = 2 s + l , Xy =s,
s= 1,2,..., (4s
+3)
is prim er o r prim er pow er,(iii) v = 8s + 7, bj = 8s4-7, rx = 4s + 3,
k y = 4 s + 3,A1 = 2s+1,
s = 1,2,...,
is optim al, w here b2 = 1.
REFERENCES
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B r o n is la w C e r a n k a , M a łg o r z a ta G r a c z y k
K O N S T R U K C J E M A C IE R Z Y O P T Y M A L N Y C H C H E M IC Z N Y C H U K Ł A D Ó W W A G O W Y C H N A P O D S T A W IE M A C IE R Z Y IN C Y D E N C JI
Z R Ó W N O W A Ż O N Y C H IJK L A D Ó W B L O K Ó W Streszczenie
W p ra c y o m ó w io n a z o sta ła tem a ty k a estym acji n iezn an y ch m ia r (w ag) p o b iek tó w w sytuacji, gdy dysp o n u jem y n operacjam i pom iarow ym i. Z asto so w an y m odel określany jest m ianem chem icznego u k ład u w agow ego, p rzy czym o g ran iczo n a je s t liczba p o m ia ró w p o szczególnych o biektów . Z o stało p o d a n e d o ln e ograniczenie w ariancji każdej składow ej es ty m a to ra w ek to ra n ieznanych m ia r o b iek tó w o raz w aru n k i konieczne i do statecz n e, przy spełnieniu k tó ry ch w ariancje esty m ato ró w osiąg n ą to d o ln e ograniczenie. D o konstrukcji m acierzy o p ty m aln eg o chem icznego u k ład u w agow ego zostały w y k o rzy stan e m acierze incyden- cji u k ład ó w zrów now ażonych o blokach n iekom pletnych o ra z tró jk o w y ch zrów now ażonych u k ład ó w bloków .
