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

4

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

6

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

7

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

8

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

9

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

12

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

14

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

18

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

19

END

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