14.20 o2 j clare

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Research 2: J Clare

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14.20 o2 j clare

  1. 1. Equivalence of published GLS solutions to comparison analysis John Clare and Annette Koo Measurement Standards Laboratory of New Zealand, IRL
  2. 2. Synopsis <ul><li>Measurement standards, comparisons </li></ul><ul><li>Comparison data, model, analysis </li></ul><ul><li>Least squares </li></ul><ul><li>Published approaches to comparison analysis using GLS </li></ul><ul><li>Result 1: equality of estimates from these approaches </li></ul><ul><li>Result 2: equality of variances/covariances of estimates </li></ul>
  3. 3. Measurement standards <ul><li>Measurement standards: realization </li></ul><ul><li>Comparisons: purpose, nature, complexity </li></ul><ul><li>Aim: biases of participant laboratories (degree of equivalence), uncertainties </li></ul><ul><li>Participants report: measurements, uncertainties, correlations </li></ul><ul><li>Complexity: multiple artefacts, star, linked loops </li></ul>Pilot 1 2 3 4 5 6 7 8 9
  4. 4. CCPR.K6-2010 MSL NMIJ NPL NIST A*STAR KRISS VNIIOFI LNE- INM NMISA MKEH NRC PTB
  5. 5. CCT-K3 <ul><li>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 </li></ul>
  6. 6. Data and analysis <ul><li>( participant, j artefact, r repeat) </li></ul><ul><li>model </li></ul><ul><li>weights based on </li></ul><ul><li>one artefact </li></ul><ul><ul><li>= weighted average </li></ul></ul><ul><ul><li>differences from </li></ul></ul><ul><li>multiple artefacts </li></ul><ul><ul><li>step-by-step, or </li></ul></ul><ul><ul><li>least-squares fit — minimize </li></ul></ul>
  7. 7. Model for data <ul><li>unknowns </li></ul><ul><li>values taken </li></ul><ul><li>random variables </li></ul><ul><li>design matrix </li></ul><ul><li>fully linked column rank </li></ul><ul><li>no unique solution </li></ul><ul><li>constraint required </li></ul><ul><li>key comparison, reference value, KCRV </li></ul><ul><li>add constraint new full column rank </li></ul>
  8. 8. Model for data <ul><li>Inclusion of constraint </li></ul><ul><ul><li>(1) use it to eliminate one or </li></ul></ul><ul><ul><li>(2) form a matrix such that </li></ul></ul><ul><ul><ul><li>is orthonormal, </li></ul></ul></ul><ul><ul><ul><li>vector of weights, </li></ul></ul></ul><ul><ul><ul><li>has full column rank, </li></ul></ul></ul>
  9. 9. Least-squares regression <ul><li>covariance matrix of measurements </li></ul><ul><li>OLS --- no weighting </li></ul><ul><li>WLS --- weights on diagonal of </li></ul><ul><li>GLS --- full covariance matrix </li></ul><ul><li>covariance estimates </li></ul><ul><li>uncertainty estimates </li></ul>
  10. 10. Comparison run by MSL <ul><li>our need, simulations </li></ul><ul><li>GLS: auditable, complexity, correlations, sound </li></ul><ul><li>3 differing implementations </li></ul><ul><li>role of systematic-error estimates </li></ul><ul><li>Sutton: </li></ul><ul><li>Woolliams: </li></ul><ul><li>White: </li></ul><ul><li>Find: estimates and uncertainties same in each case </li></ul>
  11. 11. Errors <ul><li>Uncertainties encompass errors: </li></ul><ul><ul><li>random </li></ul></ul><ul><ul><li>“ round-dependent” </li></ul></ul><ul><ul><li>intra-participant </li></ul></ul><ul><ul><li>inter-participant </li></ul></ul>random errors systematic errors
  12. 12. Measurement covariance matrix
  13. 13. Proof (1) Estimators equal <ul><li>Postulate: </li></ul><ul><li>Define </li></ul><ul><li>There exists non-singular such that </li></ul><ul><li>R = ( X ' V -1 X ) -1 X ' V - 1 </li></ul><ul><li>= ( X ' WW -1 V -1 X ) -1 G -1 G X ' WW -1 V - 1 </li></ul><ul><li>= ( G X 'WW -1 V -1 X ) -1 G X 'WW -1 V - 1 </li></ul><ul><li>= ( GX 'W W -1 V -1 X ) -1 G X 'W W -1 V - 1 </li></ul><ul><li>= ( X ' V 0 -1 X ) -1 X ' V 0 -1 </li></ul><ul><li>= R 0 </li></ul>
  14. 14. Proof (2) Estimators equal <ul><li>condensed </li></ul><ul><li>model </li></ul><ul><li>Rao (1967, Corollary to Lemma 5a) if then </li></ul>
  15. 15. Uncertainties equal <ul><li>variances, covariances of </li></ul><ul><li>Sutton, Woolliams: </li></ul><ul><li>White: </li></ul><ul><li>Proof: column space within column space of </li></ul><ul><ul><li>projection operator projects on to self </li></ul></ul><ul><ul><li>symmetries of </li></ul></ul>
  16. 16. Symmetry of
  17. 17. Uncertainties equal <ul><li>variances, covariances of </li></ul><ul><li>Sutton, Woolliams: </li></ul><ul><li>White: </li></ul><ul><li>Proof: column space within column space of </li></ul><ul><ul><li>projection operator projects on to self </li></ul></ul><ul><ul><li>deduce symmetries of </li></ul></ul>
  18. 18. Degrees of equivalence <ul><li>unilateral degree of equivalence = bias </li></ul><ul><li>bilateral degree of equivalence = </li></ul><ul><li>GLS result can be written which matches ‘step-by-step’ formalism </li></ul>
  19. 19. END Pilot 1 2 3 4 5 6 7 8 9

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