1.5 Evaluation of regional key comparison data
1.5.2 Equation of linking RMO and CIPM KC. Optimization of the
It is natural to begin solving the problem of evaluating the regional comparison data by formulating a measurement model. In the case given an output quantity is the degree of equivalence of the results obtained by a laboratory which is a RMO KC participant, or more precisely, the vector of the values of the degrees of equivalence, since in the general case in the regional comparisons several laboratories participate and have to be “linked”.
The measurands determined in the process of the CIPM and RMO key comparisons and which influence the estimate of output quantities, i.e., the estimate of the degrees of equivalence of national measurement standards, are considered as input quantities.
In accordance with the MRA the pair-wise degree of equivalence of two measure- ment standards is expressed as the difference of their degrees of equivalence and its corresponding uncertainty:
Dij DDiDj. (1.3)
Section 1.5 Evaluation of regional key comparison data 19 Expression (1.3) is general and also valid in the case where the degrees of equiva- lencei andj are evaluated in different key comparisons [156]:
di Dxixref1, u .di/, dj Dxjxref2, u
dj , dij Ddidj, u2
dij
Du2.di/Cu2 dj
2cov di,dj
.
Provided that the degrees of equivalence are established in one comparison and for the same measurand, then the pair-wise degree of equivalence can be established without using the reference value:
dij Dxi xj,
u2.dij/Du2.x/Cu2.xj/2cov.xi,xj/
If we take the second term from the right part of equation (1.3) and place it in the left part, we will obtain
Dj DDj i CDi. (1.4)
Expression (1.4) is the equation of linking, where in the left-hand side is the quantity looked for, i.e., the degree of equivalenceDj of the RMO KC laboratory participant, and in the right-hand side there are quantities through which are defined
the degree of equivalence of the measurement standard of the linking laboratoryDi, the estimate of which has been obtained in CIPM KC -ạdi,u .di/ºL1;
the pair-wise degree of equivalence of measurement standardsDj i, the estimate of which has been obtained in RMO KC:ạyjyi,u.yjyi/Dquj2Cu2iº.
From equation (1.4) the role of linking laboratories in the procedure of ensuring the traceability to the KCRV is evident. This traceability can be established only in com- parisons with a linking laboratory, i.e., under the condition of establishing the pair- wise equivalence with the laboratory, the degree of equivalence of which has already been established.
Since the linking can be realized through any linking laboratory 1iLand for any RMO KC participant, apart from linking laboratoriesLC1j m, then one obtainsL.mL/equations
DLC1DDLC1,1CD1,
DLC1DDLC1,LCDL,
DmDDm1CD1,
DmDDmLCDL.
Application of the LSM gives a vector of estimates the degrees of equivalence DET D.DLC1,: : :,Dm/with the corresponding covariance matrix of these estimates U.D/E [109]:
DOE D.ˆTU1ˆ/1ˆTU1ZE U.D/E D.ˆTU1ˆ/1, (1.5) where
ZET D.yLC1y1Cd1,: : :,yLCŠyLCdL,: : :,ymy1Cd1,: : :,ymyLCdL/ is the vector of estimates of the input quantities of a dimensionL.mL/, formed of pair-wise differences of measurement results and estimates of the degrees of equiva- lence;
U is the covariance matrix of the input quantities estimates of dimensionL.mL/
L .mL/;
ˆis the matrix of dimensionL.mL/.mL/, having a block structure and con- sisting of vector columnsj .L/E of the dimensionL:
ˆD 0 BB
@
j.L/000...0000 0j .L/0000 ...
00000 j.L/
1 CC
A, j.L/D 0
@ 1
1 1 A.
Despite the fact that the matrixˆhas a block structure, in the general case the system of equations does not fall to independent subsystems relative to elements of the vector DET D .DLC1,: : :,Dm/, since covariance of the estimates of input data differ from zero.
Since in practice it is preferable to have an explicit solution, let us consider a certain simplification of the general problem. For this purpose some additional conditions will be introduced: (1) linking of the results of each laboratory is realized independently of the results of other laboratories; (2) the reference value is estimated as the weighted mean of the results of the laboratories participating in CIPM KC; (3) the degree of equivalence of linking institutes is estimated asdi Dxixref.
Conditions (2) and (3) correspond to the case most common in practice and condi- tion (1) means a transition from the dimensionL.mL/to.mL/systems of the dimensionL. Expression (1.5) for the degree of equivalence of the laboratory that has obtained the resultyin RMO KC is rewritten in the form
dO D.jTU1j /1jTU1ZE D.jTU1j /1jTU1
.yxref/jT C EX EY DyxrefC.jTU1j /1jTU1.XE EY /, (1.6) XET D.x1,: : :,xL/, YET D.y1,: : :,yL/.
In the additional assumptions mentioned above and related to the stability of ac- curacy characteristics of the linking laboratories results, namely i D u.xcovi.x/u.yi,yii//,
Section 1.5 Evaluation of regional key comparison data 21 u.xi/Du.yi/, the elements of the covarianceUij are equal to
cov di,dj
D
´u2.y/.1ij/ u2.xref/ i ¤j
u2.y/.12i/ u2.xref/C2.1i/ u2.xi/ iDj. Expression (1.6) means that the estimate of the degree of equivalence can be pre- sented in the form
dO DyxrefCXL
1
wi.xiyi/ (1.7)
In other words, in order to evaluate a degree of equivalence a correction has to be made to the result of RMO KC, which is equal to a weighted sum of the difference of linking laboratories results, and then the KCRV value is subtracted. Thus, the quote al- gorithm based on the LSM confirms the rule of transformation of the RMO KC results, which is natural at first sight.
Let the optimal weightswi be determined, reasoning from the condition of an un- certainty minimumu.d /O and assuming the independence of results of the laboratory- participant of RMO KC as well as that of the results of the laboratory participants of CIPM KC:
u2.d /O Du2.y/Cu2.xref/C2 XL
1
wi2.1i/ u2i 2 XL
1
wi.1i/ u2ref. In order to find the weights it is necessary to solve the following optimization prob- lem under the condition where their sum is equal to 1:
minwi
XL 1
wi2.1i/ u2i XL
1
wi.1i/ u2refC XL
1
wi1
. Hence the following expressions are obtained for the optimal weights:
wi D 1u2refPL
1 1 2u2i
2.1i/ u2i PL
1 1 2.1i/u2i
Cu2ref 2u2i
D
1 .1i/u2i
PL
1 1 .1i/u2i
Cu2ref 0 B@ 1
2u2i XL
1
1 2u2i
1 2.1i/u2i
PL
1 1 2.1i/u2i
1
CA, (1.8)
wherei D cov.xi,yi/
u.xi/u.yi/ andu.xi/Du.yi/Dui,u.xref/Duref.
The corresponding standard uncertainty of the degree of equivalence is given by the expression
u2.d /O Du2.y/Cu2ref XL
1
wiiC 2 PL
1 1
.1i/u2i
1u2ref
XL 1
1 2u2i
. (1.9)
Provided that in (1.9) the uncertainty of the reference value is neglected regarding a first approximation, then it is obvious that at the transformation of RMO KC results their uncertainty increases by a quantity
PL 2
1 1 .1i/u2i
.
This uncertainty may be called the uncertainty of the linking algorithm. It is the smaller at the greater number of linking laboratories and higher coefficients of correlation between the results of these laboratories take place.
In conclusion it should be noted that equation (1.4) contains only the degrees of equivalence and does not directly include KCRV directly.