ISO IEC GUIDE 98-3 SUPP 2-2011 Uncertainty of measurement - Part 3 Guide to the expression of uncertainty in measurement (GUM 1995) - Supplement 2 Extension to .pdf

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

2、 typefaces. In accordance with Adobes licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this file, parties accept therein the responsibility of not

3、infringing Adobes licensing policy. The ISO Central Secretariat accepts no liability in this area. Adobe is a trademark of Adobe Systems Incorporated. Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters were

4、 optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below. COPYRIGHT PROTECTED DOCUMENT ISO/IEC 2011 All rights re

5、served. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISOs member body in the country of the requ

6、ester. ISO copyright office Case postale 56 CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyrightiso.org Web www.iso.org Published in Switzerland ii ISO/IEC 2011 All rights reserved ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E) ISO/IEC 2011 All rights reserved iiiISO/IEC Foreword

7、 ISO (the International Organization for Standardization) and IEC (the International Electrotechnical Commission) form the specialized system for worldwide standardization. National bodies that are members of ISO or IEC participate in the development of International Standards through technical comm

8、ittees established by the respective organization to deal with particular fields of technical activity. ISO and IEC technical committees collaborate in fields of mutual interest. Other international organizations, governmental and non-governmental, in liaison with ISO and IEC, also take part in the

9、work. Draft Guides adopted by the responsible Committee or Group are circulated to the member bodies for voting. Publication as a Guide requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this document may be the

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

16、member organizations (BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML). Copyright Even if electronic versions are available free of charge on the website of one or more of the JCGM member organizations, economic and moral copyrights related to all JCGM publications are internationally protected. T

17、he JCGM does not, without its written authorisation, permit third parties to rewrite or re-brand issues, to sell copies to the public, or to broadcast or use on-line its publications. Equally, the JCGM also objects to distortion, augmentation or mutilation of its publications, including its titles,

18、slogans and logos, and those of its member organizations. Ocial versions and translations The only ocial versions of documents are those published by the JCGM, in their original languages. The JCGMs publications may be translated into languages other than those in which the documents were originally

19、 published by the JCGM. Permission must be obtained from the JCGM before a translation can be made. All translations should respect the original and ocial format of the formul and units (without any conversion to other formul or units), and contain the following statement (to be translated into the

20、chosen language): All JCGMs products are internationally protected by copyright. This translation of the original JCGM document has been produced with the permission of the JCGM. The JCGM retains full internationally protected copyright on the design and content of this document and on the JCGMs tit

21、les, slogan and logos. The member organizations of the JCGM also retain full internationally protected right on their titles, slogans and logos included in the JCGMs publications. The only ocial version is the document published by the JCGM, in the original languages. The JCGM does not accept any li

22、ability for the relevance, accuracy, completeness or quality of the information and materials oered in any translation. A copy of the translation shall be provided to the JCGM at the time of publication. Reproduction The JCGMs publications may be reproduced, provided written permission has been gran

23、ted by the JCGM. A sample of any reproduced document shall be provided to the JCGM at the time of reproduction and contain the following statement: This document is reproduced with the permission of the JCGM, which retains full internationally protected copyright on the design and content of this do

24、cument and on the JCGMs titles, slogans and logos. The member organizations of the JCGM also retain full internationally protected right on their titles, slogans and logos included in the JCGMs publications. The only ocial versions are the original versions of the documents published by the JCGM. Di

25、sclaimer The JCGM and its member organizations have published this document to enhance access to information about metrology. They endeavor to update it on a regular basis, but cannot guarantee the accuracy at all times and shall not be responsible for any direct or indirect damage that may result f

26、rom its use. Any reference to commercial products of any kind (including but not restricted to any software, data or hardware) or links to websites, over which the JCGM and its member organizations have no control and for which they assume no responsibility, does not imply any approval, endorsement

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 . . . . . . . . . . . . . . . . . . . . .