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Research 2: J Clare
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Equivalence of published GLS solutions to comparison analysis
John Clare and Annette KooMeasurement Standards Laboratory
of New Zealand, IRL
2
Synopsis
• Measurement standards, comparisons• Comparison data, model, analysis• Least squares• Published approaches to comparison analysis
using GLS• Result 1: equality of estimates from these
approaches• Result 2: equality of variances/covariances of
estimates
3
Measurement standards
• Measurement standards: realization• Comparisons: purpose, nature, complexity• Aim: biases of participant laboratories
(degree of equivalence), uncertainties• Participants report: measurements,
uncertainties, correlations• Complexity: multiple artefacts, star, linked
loops
Pilot1
2
3
4 5 6
7
8
9
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CCPR.K6-2010
MSL
NMIJNPL
NIST
A*STAR
KRISS
VNIIOFI
LNE-INM
NMISA
MKEH
NRC
PTB
5
CCT-K3
7 artefacts, 15 laboratories, two sub pilots, cascaded loops, mixed numbers of artefacts, different numbers of repeats difficult to audit, difficult to confirm minimum uncertainty
NIST
VSL
NRC
IMGC
NML
KRISS
NIMNRLM
MSL
PTB
BIPM
NPL
VNIIM
BNM SMU
3, 1
1
3, 1, 1
1
3
3, 1
3, 1, 1
1
24
2
3, 1
1, 1
3, 1
3, 1
1, 1
1, 1, 1, 1, 1
1, 1, 1, 1
33
3
4
3
2
3
2
2
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Data and analysis
• ( participant, j artefact, r repeat)
• model • weights based on • one artefact
– = weighted average– differences from
• multiple artefacts– step-by-step, or– least-squares fit — minimize
rjy ,,
2ˆ ii yy
rjjrj ey ,,,,
rje ,,w
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Model for data
• unknowns • values taken • random variables • design matrix• fully linked column rank • no unique solution • constraint required
• key comparison, reference value, KCRV• add constraint new full column rank
,,, jβ
eβy X
E βY Xnm
rjjrj ey ,,,,
bwL
1
0b
X~
X
LJn
m
1nX
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Model for data
• Inclusion of constraint– (1) use it to eliminate one or
– (2) form a matrix such that• is orthonormal,
• vector of weights,
• has full column rank,
•
•
S
Sv
vXS
E βY
XS
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Least-squares regression
• covariance matrix of measurements • OLS --- no weighting
• WLS --- weights on diagonal of • GLS --- full covariance matrix
• covariance estimates
• uncertainty estimates
V
yyβ RVXXVX 111 )(ˆ
yβ XXX 1)(ˆ
V
RRV )ˆcov(β
JJRRVu ,2 ˆ
V
10
Comparison run by MSL
• our need, simulations• GLS: auditable, complexity, correlations, sound• 3 differing implementations• role of systematic-error estimates
• Sutton:
• Woolliams:
• White:
• Find: estimates and uncertainties same in each case
yβ RSuttonˆ RRV Sutton)ˆcov(β
yβ 0Woolliamsˆ R 00Woolliams)ˆcov( RVR β
yβ 0Whiteˆ R
L
JJ duwduwRVRu
,1
2222,000
2 )()()1()()ˆ(
yy 10
1100 )( VXXVXR
11
Errors
• Uncertainties encompass errors:– random– “round-dependent”– intra-participant– inter-participant
Γ
Φ
ΦΓ E
LLL ,,,,,,,,,,,, 222111 Φ
VVyV 0ˆcov ΦΦΓΓ EEE EE
random errors
systematic errors
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Measurement covariance matrix
96444
69444
44855
44585
44558
44444
44444
44444
44444
44444
22
22
111
111
111
3
3
3
3
3
random dependentround systematic
192
115
511
213
16
96444
69444
44855
44585
44558
V
13
Proof (1) Estimators equal
• Postulate: • Define • There exists non-singular such that • R = (X'V -1X)-1X'V -1
= (X'WW -1V -1X)-1G-1GX'WW -1V -1
= (GX'WW -1V -1X)-1GX'WW -1V -1
= (GX'WW -1V -1X)-1GX'WW -1V -1
= (X' V0-1X)-1X' V0
-1
= R0
0RR VVW 1
01
G XWXG
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Proof (2) Estimators equal
• condensed
• model
• Rao (1967, Corollary to Lemma 5a)
if
then
LLL ,,,,,,,,,,,, 222111 Φ
L,,,,0,,0 21 Φ LJ
ΓΦββY XXX
ΓβΓΦβ XXXAXXX covcov
yy 0RR
15
Uncertainties equal• variances, covariances of
• Sutton, Woolliams:
• White:
• Proof: column space within column space of
– projection operator projectson to self
– symmetries of
,ˆ,ˆ,ˆ j
RRVRVRRR 000ˆcov
UURVRRRVRRV 00
V X
XXXXH
HVHVVH
11 XXXVXXXRVRU
UVXUX
V
16
Symmetry of
*****
*****
*****
*****
*****
yyy
xxx
www
vvv
uuu
fedcbaaa
fedcbaaa
fedcbaaa
U
U
17
Uncertainties equal• variances, covariances of
• Sutton, Woolliams:
• White:
• Proof: column space within column space of
– projection operator projectson to self
– deduce symmetries of
,ˆ,ˆ,ˆ j
RRVRVRRR 000ˆcov
UURVRRRVRRV 00
V X
XXXXH
HVHVVH
L
JJ duwduwRVRu
,1
2222,000
2 )()()1()()ˆ(
U
L
k
L
k
L
k
kkkkkkkk awwawwawUu
1 1,1
2,4
2 )1(2)1(ˆ
11 XXXVXXXRVRU
VXUX
V
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Degrees of equivalence
• unilateral degree of equivalence = bias
• bilateral degree of equivalence =
• GLS result can be written
which matches ‘step-by-step’ formalism
JJRRVu ,2 ˆ
ˆˆBD
JJJJJJ RRVRRVRRVDu ,,,B2 2
cuwcucuu 2KCRV
222total 2ˆ
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END
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