14.20 o2 j clare
- 1. Equivalence of published GLS solutions to comparison analysis John Clare and Annette Koo Measurement 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 Pilot 1 2 3 4 5 6 7 8 9
- 4. CCPR.K6-2010 MSL NMIJ NPL 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
- 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
- 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
- 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,
- 9. Least-squares regression covariance matrix of measurements OLS --- no weighting WLS --- weights on diagonal of GLS --- full covariance matrix covariance estimates uncertainty estimates
- 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
- 11. Errors Uncertainties encompass errors: random “ round-dependent” intra-participant inter-participant random errors systematic errors
- 13. Proof (1) Estimators equal Postulate: Define There exists non-singular such that R = ( X ' V -1 X ) -1 X ' V - 1 = ( X ' WW -1 V -1 X ) -1 G -1 G X ' WW -1 V - 1 = ( G X 'WW -1 V -1 X ) -1 G X 'WW -1 V - 1 = ( GX 'W W -1 V -1 X ) -1 G X 'W W -1 V - 1 = ( X ' V 0 -1 X ) -1 X ' V 0 -1 = R 0
- 14. Proof (2) Estimators equal condensed model Rao (1967, Corollary to Lemma 5a) if then
- 15. Uncertainties equal variances, covariances of Sutton, Woolliams: White: Proof: column space within column space of projection operator projects on to self symmetries of
- 16. Symmetry of
- 17. Uncertainties equal variances, covariances of Sutton, Woolliams: White: Proof: column space within column space of projection operator projects on to self deduce symmetries of
- 18. Degrees of equivalence unilateral degree of equivalence = bias bilateral degree of equivalence = GLS result can be written which matches ‘step-by-step’ formalism
- 19. END Pilot 1 2 3 4 5 6 7 8 9