1.5 Evaluation of regional key comparison data
1.5.3 Different principles for transforming the results of regional
Expression (1.7) underlies many algorithms for the evaluation of the results of regional comparisons by transforming them to the level of CIPM key comparisons [109, 121, 135, 260, 261]. In fact, (1.7) can be rewritten in the form
dODyC XL
1
wi.xiyi/xrefDytransfxref, ytransfDyC.
Some algorithms of the RMO KC results transformation are inferred from the con- dition of minimization of uncertainty associated with correction, i.e., the uncertainty of the difference of measurands in the CIPM and RMO key comparisons. At the heart of an algorithm used in the COOMET Recommendation GM/RU/14:2006 [121], the condition of uncertainty minimization underlies:u.PL
1 wi.xiyi// D u./. This condition is provided at a choice of weights, inversely proportional to the squares of the uncertainty of linking results differences:
i D
1 .1i/u2i
PL
1 1 .1i/u2i
.
The comparison with the optimal weights given by expression (1.8) shows that the weights are close to the optimal ones in those cases where it is possible to neglect the uncertainty of KCRV.
Section 1.5 Evaluation of regional key comparison data 23 Recommendation COOMET GM/RU/14:2006 [121] contains two data processing procedures which are conventionally called procedures C and D. These procedures correspond to the two methods of RMO KC data transformation. The transformation can be realized by introducing an additive or multiplicative correction (multiplication by a factor).
Procedure C calls for the introduction of an additive correction for the measurement results obtained in RMO KC, whereas procedure D requires a multiplicative correc- tion. The multiplicative correction itself is calculated in a way similar to that of the additive one, where, instead of the differences of measurement results of the linking institutes, their ratios are taken.
An undoubted advantage of [121, 261] is the fact that it considers not only the sit- uation where the RMO KC results are independent of each other and independent of CIPM KC results except the results of linking measurement standards, but also the situation where some of the RMO KC results are correlated among themselves or with the CIPM KC results. The main reason for such a correlation is the traceability of the laboratories participating in RMO KC to the laboratory participants of the CIPM KC, which is rather frequently met in practice. Then procedure D [261] is considered by comparing it with other multiplicative corrections.
The algorithms for transforming the RMO KC results by introducing a multiplica- tive correction are treated in [157, 261]. In [157] a correction is suggested in the form of the ratio of a KCRV to a weighted mean of the linking institutes results obtained in RMO KC:
r1D Pn 1
1u2.xi/ Xn
1
xi u2.xi/ PL 1
1 u2.yi/ XL
1
yi u2.yi/
. (1.10)
In [261] the multiplicative correction is introduced as a weighted mean of linking the institute ratios obtained in CIPM and RMO key comparisons:
r2D XL
1
1
u2rel.xi/ .1i/xi yi XL
1
1 u2rel.xi/ .1i/
. (1.11)
Since a criterion of correction selection is the minimum of uncertainty associated with data transformation, it has been suggested to consider the multiplicative correc- tion of the general type [112]:
r D Xn
1!ixi XL
1 jyj
. (1.12)
In addition, the condition of normalization Pn
1!i D1, PL
1 i D1 has been in- troduced. Thus, the weighted mean of all the results of the CIPM key comparisons is located in the numerator and the weighted mean of the results which the linking institutes have obtained in RMO comparisons are in the denominator.
The weights are calculated reasoning from the condition of minimization of the relative uncertaintyu2rel.r/:
u2rel.r/D Xn
1
!i2u2rel.xi/C XL
1
i2u2rel.xi/2 XL
1
!iiiu2rel.xi/.
The solution of the problem gives the expression for the weights!i, ithrough the constants, :
!i D i C
u2rel.xi/.1i2/, i D1,: : :,L,
!i D
u2.xi/, i DLC1,: : :,n, i D Ci
u2.xi/
12i, iD1,: : :,L.
These constants are determined from the condition of normalization Pn
1!i D1, PL
1 i D1 by solving the linear system of equations:
D ./
, D ./
, D XL
1
1
u2.xi/.1Ci/XL
1
1 u2.xi/ .1i/
XL 1
1 u2.xi/
12i Xn LC1
1 u2.xi/, ./D
XL 1
1
u2.xi/ .1Ci/, ./DXL
1
1
u2.xi/ .1Ci/CXn
LC1
1 u2.xi/.
It should be noted that the weights of the linking institutes exceed the weights of the remaining ones; this difference increases with the growth of the correlation coefficient between the results of the linking institutes. To make the results of the comparative analysis more obvious, let us assume thati , i L. This assumption allows the
Section 1.5 Evaluation of regional key comparison data 25 expression used for the weights to be simplified:
i D u2.xi/ PL
1 u2.xi/, iD1,: : :,L,
!i Du2.xi/ PL
1 u2.xi/C 1 Pn
1urel2.xi/
!
i D1,: : :,L,
!i D
u2rel.xi/D.1/ urel2.xi/ Pn
1urel2.xi/
, i > L.
In Table 1.1 the uncertainties of the multiplicative corrections considered and their comparison are illustrated. Two cases are treated: with several linking laboratories and with one linking laboratory.
Table 1.1.Uncertainties of the multiplicative corrections and their comparison.
General case LD1
u2rel.r /D .1/2 Pn
1u2rel .xi/C 12 PL
1 u2rel.xi/ u2rel.r /D.12/ u2rel.x1/C .1/2 Pn
1u2rel.xi/ u2rel.r1/D 12
Pn
1urel2.xi/C 1 PL
1 urel2.xi/ u2rel.r1/Du2rel.x1/C 12 Pn
1urel2.xi/ u2rel.r2/D2.1/L 1
P 1u2rel.xi/
u2rel.r2/D2.1/ u2rel.x1/
u2rel.r1/u2rel.r /D
2 1
PL
1 u2
rel.i
1 Pn
1urel2.xi/
! u2rel.r1/u2rel.r /D 2
u2rel.x1/ 1 Pn
1u2rel.xi/
u2rel.r2/u2rel.r /D
.1/2 1
PL
1 urel2.xi/ 1 Pn
1u2rel.xi/
!
u2rel.r2/u2rel.r /D .1/2
u2rel.x1/ 1 Pn
1u2rel .xi/
The multiplicative correctionr can be considered to be the generalization of cor- rectionsr1 andr2, since at D 0 it coincides withr1. Moreover, if the corrections r1,r2 are compared with each other, then at < 0.5 the correctionr1 has a lesser uncertainty, and, vice versa,r2is more preferable at >0.5.
Differences in uncertainties associated with the corrections considered depend first of all on the coefficient of correlation among the results of linking institutes. It can be assumed that the CIPM KC participants have approximately similar uncertainties
u .xi/u, i D1,: : :,n, and then the difference between the generalized correctionr and the correctionsr1,r2has a clear expression and is determined by the number of linking institutes and a coefficient of the correlation of their results:
u2rel.r1/u2rel.r/D2u2relnL nL , u2rel.r2/u2rel.r/D.1/2u2relnL
nL .
Further on, a significant correlation among linking laboratories results is assumed;
therefore the correction r2 is considered for linking comparisons [261]. The trans- formed results of the regional comparisonzk D r2yk, k > L can be added to the results of the corresponding CIPM comparison with the following uncertainties:
u2rel.zk/Du2rel.yk/Cu2rel.r2/, u2rel.r2/D 2 PL
kD1 1 u2rel.yk/.1i/
.
The degree of equivalence for the RMO KC (k > L) participants are given in the form
dk Dzkxref
with the corresponding uncertainty
u2.dk/Du2.zk/Cu2.xref/2u.zk,xref/.
Let us note that the correlation with the results of the institutes, which are the CIPM KC participants leads to a decrease in the uncertainty of degrees of equivalence. The following expressions for the uncertainties of degrees of equivalence have been ob- tained:
u2.di/Dr22u2.xi/Cu2.xref/C2 XL
1
1 u2.xk/.1k/
!1
1u2.xref/ XL
1
1 u2.xk/
!
(1.13) if there is no correlation between the regional comparison result and the results of the CIPM KC participants, and
u2.di/Šr22u2.xi/u2.xref/C2 XL
1
1 u2.xk/.1k/
!1
1u2.xref/ XL
1
1 u2.xi/
!
Section 1.6 Bayesian approach to the evaluation of systematic biases 27 if there is a correlation between the regional comparison results and the results of the linking institute (for example, the result of the regional comparison is traceable to the measurement standard of a CIPM KC participant).
Expression (1.13) for the uncertainty of the equivalence degrees of transformed data is the sum of three components: initial uncertainties, uncertainty of the reference value, and uncertainty caused by transformation.