1、 Reference number ISO/TR 13587:2012(E) ISO 2012TECHNICAL REPORT ISO/TR 13587 First edition 2012-07-15 Three statistical approaches for the assessment and interpretation of measurement uncertainty Trois approches statistiques pour lvaluation et linterprtation de lincertitude de mesure ISO/TR 13587:20
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4、 reserved iiiContents Page Foreword . v Introduction vi 1 Scope 1 2 Normative references 1 3 Terms and definitions . 1 4 Symbols (and abbreviated terms) 2 5 The problem addressed 3 6 Statistical approaches 4 6.1 Frequentist approach 4 6.2 Bayesian approach 5 6.3 Fiducial approach 5 6.4 Discussion .
5、6 7 Examples 6 7.1 General . 6 7.2 Example 1a . 6 7.3 Example 1b . 7 7.4 Example 1c . 7 8 Frequentist approach to uncertainty evaluation 7 8.1 Basic method . 7 8.2 Bootstrap uncertainty intervals . 10 8.3 Example 1 . 13 8.3.1 General . 13 8.3.2 Example 1a . 14 8.3.3 Example 1b . 15 8.3.4 Example 1c
6、. 15 9 Bayesian approach for uncertainty evaluation 16 9.1 Basic method . 16 9.2 Example 1 . 18 9.2.1 General . 18 9.2.2 Example 1a . 18 9.2.3 Example 1b . 20 9.2.4 Example 1c . 21 9.2.5 Summary of example 21 10 Fiducial inference for uncertainty evaluation . 21 10.1 Basic method . 21 10.2 Example 1
7、 . 23 10.2.1 Example 1a . 23 10.2.2 Example 1b . 25 10.2.3 Example 1c . 26 11 Example 2: calibration of a gauge block . 26 11.1 General . 26 11.2 Frequentist approach 28 11.3 Bayesian approach 30 11.4 Fiducial approach 33 12 Discussion . 35 12.1 Comparison of uncertainty evaluations using the three
8、statistical approaches 35 ISO/TR 13587:2012(E) iv ISO 2012 All rights reserved12.2 Relation between the methods proposed in GUM Supplement 1 (GUMS1) and the three statistical approaches .38 13 Summary .40 Bibliography 42 ISO/TR 13587:2012(E) ISO 2012 All rights reserved vForeword ISO (the Internatio
9、nal Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been
10、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 collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standar
11、dization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for v
12、oting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. In exceptional circumstances, when a technical committee has collected data of a different kind from that which is normally published as an International Standard (“state of the ar
13、t”, for example), it may decide by a simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely informative in nature and does not have to be reviewed until the data it provides are considered to be no longer valid or useful. Attention is drawn to
14、 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 or all such patent rights. ISO/TR 13587:2012 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods, Subcommittee SC 6, M
15、easurement methods and results. This Technical Report is primarily based on Reference 10. ISO/TR 13587:2012(E) vi ISO 2012 All rights reservedIntroduction The adoption of ISO/IEC Guide 98-3 (GUM) 1has led to an increasing recognition of the need to include uncertainty statements in measurement resul
16、ts. Laboratory accreditation based on International Standards like ISO 170252 has accelerated this process. Recognizing that uncertainty statements are required for effective decision-making, metrologists in laboratories of all types, from National Metrology Institutes to commercial calibration labo
17、ratories, are exerting considerable effort on the development of appropriate uncertainty evaluations for different types of measurement using methods given in the GUM. Some of the strengths of the procedures outlined and popularized in the GUM are its standardized approach to uncertainty evaluation,
18、 its accommodation of sources of uncertainty that are evaluated either statistically (Type A) or non-statistically (Type B), and its emphasis on reporting all sources of uncertainty considered. The main approach to uncertainty propagation in the GUM, based on linear approximation of the measurement
19、function, is generally simple to carry out and in many practical situations gives results that are similar to those obtained more formally. In short, since its adoption, the GUM has sparked a revolution in uncertainty evaluation. Of course, there will always be more work needed to improve the evalua
20、tion of uncertainty in particular applications and to extend it to cover additional areas. Among such other work, the Joint Committee for Guides in Metrology (JCGM), responsible for the GUM since the year 2000, has completed Supplement 1 to the GUM, namely, “Propagation of distributions using a Mont
21、e Carlo method” (referred to as GUMS1)3 . The JCGM is developing other supplements to the GUM on topics such as modelling and models with any number of output quantities. Because it should apply to the widest possible set of measurement problems, the definition of measurement uncertainty in ISO/IEC
22、Guide 99:20074 as a “non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used” cannot reasonably be given at more than a relatively conceptual level. As a result, defining and understanding the appropriate roles of dif
23、ferent statistical quantities in uncertainty evaluation, even for relatively well-understood measurement applications, is a topic of particular interest to both statisticians and metrologists. Earlier investigations have approached these topics from a metrological point of view, some authors focusin
24、g on characterizing statistical properties of the procedures given in the GUM. Reference 5 shows that these procedures are not strictly consistent with either a Bayesian or frequentist interpretation. Reference 6 proposes some minor modifications to the GUM procedures that bring the results into clo
25、ser agreement with a Bayesian interpretation in some situations. Reference 7 discusses the relationship between procedures for uncertainty evaluation proposed in GUMS1 and the results of a Bayesian analysis for a particular class of models. Reference 8 also discusses different possible probabilistic
26、 interpretations of coverage intervals and recommends approximating the posterior distributions for this class of Bayesian analyses by probability distributions from the Pearson family of distributions. Reference 9 compares frequentist (“conventional”) and Bayesian approaches to uncertainty evaluati
27、on. However, the study is limited to measurement systems for which all sources of uncertainty can be evaluated using Type A methods. In contrast, measurement systems with sources of uncertainty evaluated using both Type A and Type B methods are treated in this Technical Report and are illustrated us
28、ing several examples, including one of the examples from Annex H of the GUM. Statisticians have historically placed strong emphasis on using methods for uncertainty evaluation that have probabilistic justification or interpretation. Through their work, often outside metrology, several different appr
29、oaches for statistical inference relevant to uncertainty evaluation have been developed. This Technical Report presents some of those approaches to uncertainty evaluation from a statistical point of view and relates them to the methods that are currently being used in metrology or are being develope
30、d within the metrology community. The particular statistical approaches under which different methods for uncertainty evaluation will be described are the frequentist, Bayesian, and fiducial approaches, which are discussed further after outlining the notational conventions needed to distinguish diff
31、erent types of quantities. TECHNICAL REPORT ISO/TR 13587:2012(E) ISO 2012 All rights reserved 1Three statistical approaches for the assessment and interpretation of measurement uncertainty 1 Scope This Technical Report is concerned with three basic statistical approaches for the evaluation and inter
32、pretation of measurement uncertainty: the frequentist approach including bootstrap uncertainty intervals, the Bayesian approach, and fiducial inference. The common feature of these approaches is a clearly delineated probabilistic interpretation or justification for the resulting uncertainty interval
33、s. For each approach, the basic method is described and the fundamental underlying assumptions and the probabilistic interpretation of the resulting uncertainty are discussed. Each of the approaches is illustrated using two examples, including an example from ISO/IEC Guide 98-3 (Uncertainty of measu
34、rement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995). In addition, this document also includes a discussion of the relationship between the methods proposed in the GUM Supplement 1 and these three statistical approaches. 2 Normative references The following referenced docu
35、ments are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 3534-1:2006, Statistics Vocabulary and symbols Part 1: General statistical
36、 terms and terms used in probability ISO 3534-2:2006, Statistics Vocabulary and symbols Part 2: Applied statistics ISO/IEC Guide 98-3:2008, Uncertainty of measurement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) ISO/IEC Guide 98-3:2008/Suppl 1:2008, Uncertainty of measure
37、ment Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) Supplement 1: Propagation of distributions using a Monte Carlo method 3 Terms and definitions For the purposes of this document, the terms and definitions in ISO 3534-1, ISO 3534-2 and the following apply. 3.1 empirical di
38、stribution function empirical cumulative distribution function distribution function that assigns probability 1 n to each of the items in a random sample, i.e., the empirical distribution function is a step function defined by n () i n x x Fx n , where 1 ,., n x x is the sample and A is the number o
39、f elements in the set A . ISO/TR 13587:2012(E) 2 ISO 2012 All rights reserved3.2 Bayesian sensitivity analysis study of the effect of the choices of prior distributions for the parameters of the statistical model on the posterior distribution of the measurand 3.3 sufficient statistic function of a r
40、andom sample 1 ,., n X X from a probability density function with parameter for which the conditional distribution of 1 ,., n X X given this function does not depend on NOTE A sufficient statistic contains as much information about as 1 ,., n X X . 3.4 observation model mathematical relation between
41、 a set of measurements (indications), the measurand, and the associated random measurement errors 3.5 structural equation statistical model relating the observable random variable to the unknown parameters and an unobservable random variable whose distribution is known and free of unknown parameters
42、 3.6 non-central chi-squared distribution probability distribution that generalizes the typical (or central) chi-squared distribution NOTE 1 For independent, normally distributed random variables k i X with mean i and variance 2 i , the random variable 2 1 k XX () ii i is non-central chi-squared dis
43、tributed. The non-central chi-squared distribution has two parameters: , the degrees of freedom (i.e., the number of k i X ), and , which is related to the means of the random variables i X by 2 1 ( k ii i ) and called the non-centrality parameter. NOTE 2 The corresponding probability density functi
44、on is expressed as a mixture of central 2 probability density functions as given by 2 2 0 () 1 22 2 0 2 (2 ) () () ! 2! 2 2 ki i XY i k i i k i i e gg i e k ii , where is distributed as chi-squared with degrees of freedom. q Y q 4 Symbols (and abbreviated terms) In 4.1.1 of the GUM, it is stated tha
45、t Latin letters are used to represent both physical quantities to be determined by measurement (i.e., measurands in GUM terminology) as well as random variables that may take different observed values of a physical quantity. This use of the same symbols, whose different meanings are only indicated b
46、y context, can be difficult to interpret and sometimes leads to unnecessary ambiguities or misunderstandings. To mitigate this potential source of confusion, the more traditional notation often used in the statistical literature is employed in this Technical Report. In this notation, Greek letters a
47、re used to represent parameters in a statistical model (e.g., measurands), which can be either random variables or ISO/TR 13587:2012(E) ISO 2012 All rights reserved 3constants depending on the statistical approach being used and nature of the model. Upper-case Latin letters are used to represent ran
48、dom variables that can take different values of an observable quantity (e.g., potential measured values), and lower-case Latin letters to represent specific observed values of a quantity (e.g., specific measured values). Since additional notation may be required to denote other physical, mathematica
49、l, or statistical concepts, there will still always be some possibility for ambiguity 1) . In those cases the context clarifies the appropriate interpretation. 5 The problem addressed 5.1 The concern in this Technical Report is with a measurement model in which 1 ,., p are input quantities and is the output quantity: 1 ., , p f (1) where f is known as the measurement function. The function f is specified math
