1、 GUIDE 98-3/Suppl.2 Uncertainty of measurement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) Supplement 2: Extension to any number of output quantities First edition 2011 ISO/IEC 2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E) PDF disclaimer This PDF file may contain embedded
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10、subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO/IEC Guide 98-3/Suppl.2 was prepared by Working Group 1 of the Joint Committee for Guides in Metrology (as JCGM 102:2011), and was adopted by the national bodies of ISO and IEC. ISO/IEC Guid
11、e 98 consists of the following parts, under the general title Uncertainty of measurement: Part 1: Introduction to the expression of uncertainty in measurement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) Part 4: Role of measurement uncertainty in conformity assessment The
12、 following parts are planned: Part 2: Concepts and basic principles Part 5: Applications of the least-squares method ISO/IEC Guide 98-3 has three supplements: Supplement 1: Propagation of distributions using a Monte Carlo method Supplement 2: Models with any number of output quantities Supplement 3:
13、 Modelling Given that ISO/IEC Guide 98-3:2008/Suppl.2:2011 is identical in content to JCGM 102:2011, the decimal symbol is a point on the line in the English version. Annex ZZ has been appended to provide a list of corresponding ISO/IEC Guides and JCGM guidance documents for which equivalents are no
14、t given in the text. Joint Committee for Guides in Metrology JCGM 102 2011 Evaluation of measurement data | Supplement 2 to the Guide to the expression of uncertainty in measurement“ | Extension to any number of output quantities Evaluation des donn ees de mesure | Suppl ement 2 du Guide pour lexpre
15、ssion de lincertitude de mesure“ | Extension a un nombre quelconque de grandeurs de sortie c JCGM 2011| All rights reserved JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E) ISO/IEC JCGM 2011 All rights reserved ic JCGM 2011 Copyright of this JCGM guidance document is shared jointly by the JCGM
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27、or recommendation by the JCGM and its member organizations. ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E) JCGM 102:2011 ii ISO/IEC JCGM 2011 All rights reservedContents Page Foreword: v Introduction: vi 1 Scope : 1 2 Normative references: 2 3 Terms and denitions : 2 4 Conventions and notation : 8 5 Basic
28、principles: 10 5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Main stages of uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 P
29、robability density functions for the input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3.2 Multivariate t-distribution . .
30、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3.3 Construction of multivariate probability density functions . . . . . . . . . . . . . . . . . . . . . . . . 12 5.4 Propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31、. . . . . . . . . . . . . . . . . . . . . . 12 5.5 Obtaining summary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.6 Implementations of the propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32、. . 13 6 GUM uncertainty framework : 14 6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.2 Propagation of uncertainty for explicit multivariate measurement models . . . . . . . . . . . . . . . . . . 1
33、5 6.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.3 Propagation of unce
34、rtainty for implicit multivariate measurement models . . . . . . . . . . . . . . . . . . 17 6.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
35、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.4 Propagation of uncertainty for models involving complex quantities . . . . . . . . . . . . . . . . . . . . . . 19 6.5 Coverage region for a vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36、. . . . . . 19 6.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.5.2 Bivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.5.3 Mult
37、ivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.5.4 Coverage region for the expectation of a multivariate Gaussian distribution . . . . . . . . . . . 22 7 Monte Carlo method : 23 7.1 General . . . . . . . . . . . . . . .
38、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 Number of Monte Carlo trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.3 Making draws from probability distributions . . . . . .
39、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.4 Evaluation of the vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.5 Discrete representation of the distribution function for the output quantity . . . . . . . . .
40、 . . . . . . . 27 7.6 Estimate of the output quantity and the associated covariance matrix . . . . . . . . . . . . . . . . . . . . 27 7.7 Coverage region for a vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.7.1 General . . . . . . . . . . .
41、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.7.2 Hyper-ellipsoidal coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.7.3 Hyper-rectangular coverage region . . . . . . . . . . . . . . .
42、. . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.7.4 Smallest coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.8 Adaptive Monte Carlo procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43、. . . . . . . . 31 7.8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.8.2 Numerical tolerance associated with a numerical value . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.8.3 Adaptive procedure .
44、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8 Validation of the GUM uncertainty framework using a Monte Carlo method: 34 JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E) ISO/IEC JCGM 2011 All rights reserved iii9 Examples: 35 9.1 Illu
45、strations of aspects of this Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.2 Additive measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.2.1 Formulation . . . . . . . . . . . .
46、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.2.2 Propagation and summarizing: case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.2.3 Propagation and summarizing: case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9.2.4 Propagation and summarizing: case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.3 Co-ordinate system transformation . . . . . . . . . . . . . . . . . . . . .
