1、 First edition 2008 ISO/IEC 2008 GUIDE 98-3/Suppl.1 Uncertainty of measurement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) Supplement 1: Propagation of distributions using a Monte Carlo method ISO/IEC GUIDE 98-3/Suppl.1:2008(E) PDF disclaimer This PDF file may contain em
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6、e requester. 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 2008 All rights reservedISO/IEC GUIDE 98-3/Suppl.1:2008(E) ISO/IEC 2008 All rights reserved iiiContents Page Fo
7、reword .v Introductionvi 1 Scope1 2 Normative references2 3 Terms and definitions .2 4 Conventions and notation 6 5 Basic principles .8 5.1 Main stages of uncertainty evaluation 8 5.2 Propagation of distributions 9 5.3 Obtaining summary information9 5.4 Implementations of the propagation of distribu
8、tions10 5.5 Reporting the results 11 5.6 GUM uncertainty framework 12 5.7 Conditions for valid application of the GUM uncertainty framework for linear models 13 5.8 Conditions for valid application of the GUM uncertainty framework for non-linear models 14 5.9 Monte Carlo approach to the propagation
9、and summarizing stages 15 5.10 Conditions for the valid application of the described Monte Carlo method .16 5.11 Comparison of the GUM uncertainty framework and the described Monte Carlo method .17 6 Probability density functions for the input quantities.18 6.1 General .18 6.2 Bayes theorem19 6.3 Pr
10、inciple of maximum entropy.19 6.4 Probability density function assignment for some common circumstances .20 6.4.1 General .20 6.4.2 Rectangular distributions.20 6.4.3 Rectangular distributions with inexactly prescribed limits 20 6.4.4 Trapezoidal distributions22 6.4.5 Triangular distributions 23 6.4
11、.6 Arc sine (U-shaped) distributions24 6.4.7 Gaussian distributions25 6.4.8 Multivariate Gaussian distributions 25 6.4.9 t-distributions.26 6.4.10 Exponential distributions .28 6.4.11 Gamma distributions.28 6.5 Probability distributions from previous uncertainty calculations .29 7 Implementation of
12、a Monte Carlo method.29 7.1 General .29 7.2 Number of Monte Carlo trials .29 7.3 Sampling from probability distributions.29 7.4 Evaluation of the model30 7.5 Discrete representation of the distribution function for the output quantity30 7.6 Estimate of the output quantity and the associated standard
13、 uncertainty.31 7.7 Coverage interval for the output quantity.31 7.8 Computation time32 7.9 Adaptive Monte Carlo procedure.32 7.9.1 General .32 7.9.2 Numerical tolerance associated with a numerical value.32 7.9.3 Objective of adaptive procedure33 ISO/IEC GUIDE 98-3/Suppl.1:2008(E) iv ISO/IEC 2008 Al
14、l rights reserved7.9.4 Adaptive procedure .33 8 Validation of results 35 8.1 Validation of the GUM uncertainty framework using a Monte Carlo method35 8.2 Obtaining results from a Monte Carlo method for validation purposes 35 9 Examples 36 9.1 Illustrations of aspects of this Supplement 36 9.2 Additi
15、ve model .37 9.2.1 Formulation 37 9.2.2 Normally distributed input quantities37 9.2.3 Rectangularly distributed input quantities with the same width39 9.2.4 Rectangularly distributed input quantities with different widths .41 9.3 Mass calibration.42 9.3.1 Formulation 42 9.3.2 Propagation and summari
16、zing .43 9.4 Comparison loss in microwave power meter calibration45 9.4.1 Formulation 45 9.4.2 Propagation and summarizing: zero covariance46 9.4.3 Propagation and summarizing: non-zero covariance51 9.5 Gauge block calibration 53 9.5.1 Formulation: model .53 9.5.2 Formulation: assignment of PDFs .55
17、 9.5.3 Propagation and summarizing .58 9.5.4 Results 59 Annex A Historical perspective.61 Annex B Sensitivity coefficients and uncertainty budgets 62 Annex C Sampling from probability distributions.63 C.1 General63 C.2 General distributions.63 C.3 Rectangular distribution .64 C.4 Gaussian distributi
18、on65 C.5 Multivariate Gaussian distribution.66 C.6 t-distribution.67 Annex D Continuous approximation to the distribution function for the output quantity69 Annex E Coverage interval for the four-fold convolution of a rectangular distribution .72 Annex F Comparison loss problem 74 F.1 Expectation an
19、d standard deviation obtained analytically .74 F.2 Analytic solution for zero estimate of the voltage reflection coefficient having associated zero covariance75 F.3 GUM uncertainty framework applied to the comparison loss problem .76 Annex G Glossary of principal symbols.78 Bibliography 83 Alphabeti
20、cal index 86 ISO/IEC GUIDE 98-3/Suppl.1:2008(E) ISO/IEC 2008 All rights reserved vForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out throug
21、h ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborat
22、es closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. Draft Guides adopted by the responsible Committee or Group are circulated
23、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 subject of patent rights. ISO shall not be held responsible for identifying any
24、or all such patent rights. This first edition of Supplement 1 to ISO/IEC Guide 98-3 has been prepared by Working Group 1 of the JCGM, and has benefited from detailed reviews undertaken by member organizations of the JCGM and National Metrology Institutes. For further information, see the Introductio
25、n (0.2). ISO/IEC Guide 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) The following parts are planned: Part 2: Concep
26、ts and basic principles Part 4: Role of measurement uncertainty in conformity assessment Part 5: Applications of the least-squares method ISO/IEC Guide 98-3 has one supplement. Supplement 1: Propagation of distributions using a Monte Carlo method The following supplements to ISO/IEC Guide 98-3 are p
27、lanned: Supplement 2: Models with any number of output quantities Supplement 3: Modelling Note that in this document, GUM is used to refer to the industry-recognized publication, adopted as ISO/IEC Guide 98-3:2008. When a specific clause or subclause number is cited, the reference is to ISO/IEC Guid
28、e 98-3:2008. ISO/IEC GUIDE 98-3/Suppl.1:2008(E) vi ISO/IEC 2008 All rights reservedIntroduction 0.1 General This Supplement to the Guide to the expression of uncertainty in measurement (GUM) is concerned with the propagation of probability distributions through a mathematical model of measurement IS
29、O/IEC Guide 98-3:2008, 3.1.6 as a basis for the evaluation of uncertainty of measurement, and its implementation by a Monte Carlo method. The treatment applies to a model having any number of input quantities, and a single output quantity. The described Monte Carlo method is a practical alternative
30、to the GUM uncertainty framework ISO/IEC Guide 98-3:2008, 3.4.8. It has value when a) linearization of the model provides an inadequate representation or b) the probability density function (PDF) for the output quantity departs appreciably from a Gaussian distribution or a scaled and shifted t-distr
31、ibution, e.g. due to marked asymmetry. In case a), the estimate of the output quantity and the associated standard uncertainty provided by the GUM uncertainty framework might be unreliable. In case b), unrealistic coverage intervals (a generalization of “expanded uncertainty” in the GUM uncertainty
32、framework) might be the outcome. The GUM ISO/IEC Guide 98-3:2008, 3.4.8 “provides a framework for assessing uncertainty ”, based on the law of propagation of uncertainty ISO/IEC Guide 98-3:2008, Clause 5 and the characterization of the output quantity by a Gaussian distribution or a scaled and shift
33、ed t-distribution ISO/IEC Guide 98-3:2008, G.6.2, G.6.4. Within that framework, the law of propagation of uncertainty provides a means for propagating uncertainties through the model. Specifically, it evaluates the standard uncertainty associated with an estimate of the output quantity, given 1) bes
34、t estimates of the input quantities, 2) the standard uncertainties associated with these estimates, and, where appropriate, 3) degrees of freedom associated with these standard uncertainties, and 4) any non-zero covariances associated with pairs of these estimates. Also within the framework, the PDF
35、 taken to characterize the output quantity is used to provide a coverage interval, for a stipulated coverage probability, for that quantity. The best estimates, standard uncertainties, covariances and degrees of freedom summarize the information available concerning the input quantities. With the ap
36、proach considered here, the available information is encoded in terms of PDFs for the input quantities. The approach operates with these PDFs in order to determine the PDF for the output quantity. Whereas there are some limitations to the GUM uncertainty framework, the propagation of distributions w
37、ill always provide a PDF for the output quantity that is consistent with the PDFs for the input quantities. This PDF for the output quantity describes the knowledge of that quantity, based on the knowledge of the input quantities, as described by the PDFs assigned to them. Once the PDF for the outpu
38、t quantity is available, that quantity can be summarized by its expectation, taken as an estimate of the quantity, and its standard deviation, taken as the standard uncertainty associated with the estimate. Further, the PDF can be used to obtain a coverage interval, corresponding to a stipulated cov
39、erage probability, for the output quantity. ISO/IEC GUIDE 98-3/Suppl.1:2008(E) ISO/IEC 2008 All rights reserved viiThe use of PDFs as described in this Supplement is generally consistent with the concepts underlying the GUM. The PDF for a quantity expresses the state of knowledge about the quantity,
40、 i.e. it quantifies the degree of belief about the values that can be assigned to the quantity based on the available information. The information usually consists of raw statistical data, results of measurement, or other relevant scientific statements, as well as professional judgement. In order to
41、 construct a PDF for a quantity, on the basis of a series of indications, Bayes theorem can be applied 27, 33. When appropriate information is available concerning systematic effects, the principle of maximum entropy can be used to assign a suitable PDF 51, 56. The propagation of distributions has w
42、ider application than the GUM uncertainty framework. It works with richer information than that conveyed by best estimates and the associated standard uncertainties (and degrees of freedom and covariances when appropriate). Decimal sign: The decimal sign in the English text is the point on the line,
43、 and the comma on the line is the decimal sign in the French text. (See 4.12) An historical perspective is given in Annex A. NOTE 1 The GUM provides an approach when linearization is inadequate ISO/IEC Guide 98-3:2008, 5.1.2 Note. The approach has limitations: only the leading non-linear terms in th
44、e Taylor series expansion of the model are used, and the PDFs for the input quantities are regarded as Gaussian. NOTE 2 Strictly, the GUM characterizes the variable (Y y)/u(y) by a t-distribution, where Y is the output quantity, y an estimate of Y, and u(y) the standard uncertainty associated with y
45、 ISO/IEC Guide 98-3:2008, G.3.1. This characterization is also used in this Supplement. The GUM in fact refers to the variable (y Y)/u(y). NOTE 3 A PDF for a quantity is not to be understood as a frequency density. NOTE 4 “The evaluation of uncertainty is neither a routine task nor a purely mathemat
46、ical one; it depends on detailed knowledge of the nature of the measurand and of the measurement method and procedure used. The quality and utility of the uncertainty quoted for the result of a measurement therefore ultimately depends on the understanding, critical analysis, and integrity of those w
47、ho contribute to the assignment of its value.” 17. 0.2 JCGM background information In 1997, the Joint Committee for Guides in Metrology (JCGM), chaired by the Director of the Bureau International des Poids et Mesures (BIPM), was created by the seven international organizations that had originally in
48、 1993 prepared the Guide to the expression of uncertainty in measurement (GUM) and the International vocabulary of basic and general terms in metrology (VIM). The JCGM assumed responsibility for these two documents from the ISO Technical Advisory Group 4 (TAG4). The Joint Committee is formed by the
49、BIPM with the International Electrotechnical Commission (IEC), the International Federation of Clinical Chemistry and Laboratory Medicine (IFCC), the International Organization for Standardization (ISO), the International Union of Pure and Applied Chemistry (IUPAC), the International Union of Pure and Applied Physics (IUPAP), and the International Organization of Legal Metrology (OIML). A further organization joined these seven international organizations, namely, the International Laboratory Accreditation Cooperat
