# ISO GUM, Uncertainty Quantification, and Philosophy of Statistics

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1 ISO GUM, Uncertainty Quantification, and Philosophy of Statistics Gunnar Taraldsen NTNU and SINTEF January 21th

2 2

3 Abstract and history In 1978, recognizing the lack of international consensus on the expression of uncertainty in measurement, the world's highest authority in metrology, the Comité International des Poids et Mesures (CIPM), requested the Bureau International des Poids et Mesures (BIPM) to address the problem in conjunction with the national standards laboratories and to make a recommendation. As a result the International Organization for Standardization (ISO) published the first version of the Guide to the expression of Uncertainty of Measurements (GUM) in

4 This session will discuss uncertainty quantification in relation to the ISO GUM, which states: "Just as the nearly universal use of the International System of Units (SI) has brought coherence to all scientific and technological measurements, a worldwide consensus on the evaluation and expression of uncertainty in measurement would permit the significance of a vast spectrum of measurement results in science, engineering, commerce, industry, and regulation to be readily understood and properly interpreted. In this era of the global marketplace, it is imperative that the method for evaluating and expressing uncertainty be uniform throughout the world so that measurements performed in different countries can be easily compared." Additionally, we will discuss the relation between uncertainty quantification and the philosophical foundations of statistics. 4

5 What is a measurement? Measurement of temperature outside of the hotell. Measurements of height to obtain a terrain model... If X is the result of a measurement, then Y=f(X) is a measurement. Any function applied on a measurement is itself a measurement. The function can be simple as exemplified by Y=1/X, or it can be more complicated. The mapping from the initial data of a system of conservation equations into the solution of the equations is a more complicated example. 5

6 6 The ISO GUMs main message. Always report:

7 ISO and GUM [1, British foreword] The International Organization for Standardization (ISO) requires that the 1993 edition of the Guide to the expression of uncertainty in measurement (GUM) be referenced when writing standards concerning the expression of uncertainty in measurement. The purpose of such guidance is: (1) to promote full information on how uncertainty statements are arrived at; (2) to provide a basis for the international comparison of measurement results. 7

8 ISO GUM introduction [1] Just as the nearly universal use of the International System of Units (SI) has brought coherence to all scientific and technological measurements, a worldwide consensus on the evaluation and expression of uncertainty in measurement would permit the significance of a vast spectrum of measurement results in science, engineering, commerce, industry, and regulation to be readily understood and properly interpreted. In this era of the global marketplace, it is imperative that the method for evaluating and expressing uncertainty be uniform throughout the world so that measurements performed in different countries can be easily compared. 8

9 ISO GUM introduction [1] The ideal method for evaluating and expressing the uncertainty of the result of a measurement should be: universal: the method should be applicable to all kinds of measurements and to all types of input data used in measurements; The actual quantity used to express uncertainty should be: internally consistent: it should be directly derivable from the components that contribute to it, as well as independent of how these components are grouped and of the decomposition of the components into subcomponents; transferable: it should be possible to use directly the uncertainty evaluated for one result as a component in evaluating the uncertainty of another measurement in which the first result is used. 9

10 The 8 step GUM procedure [1,8.1-8] 1. Define the measurand Y = f(x1, X2,, XN) 2. Determine input quantity (xi) 3. Evaluate the standard uncertainty u(xi) 4. Evaluate the covariances 5. Calculate y = f(x1, x2,, xn) 6. Evaluate the combined standard uncertainty uc(y) 7. Evaluate the expanded uncertainty U 8. Report y, uc(y), U(y) with level I would prefer u(y). The xi s and the y are conceptually similar. 10

11 Interpretation of the ISO GUM [1] standard uncertainty uncertainty of the result of a measurement expressed as a standard deviation type A evaluation (of uncertainty) method of evaluation of uncertainty by the statistical analysis of series of observations type B evaluation (of uncertainty) method of evaluation of uncertainty by means other than the statistical analysis of series of observations 11

12 Taraldsen: On the blackboard Noise mapping example: Terrain, sources, meteorology, MC evaluation of uncertainty (GUM supplement) Elementary evaluation of uncertainty Bayes, Frequentist, Fiducial statistical philosophies Improper priors Mapping of probability distributions. Uncertainty on spheres, circles, manifolds: Variance??? Confidence distributions: What is uncertainty? Linear system and error: Not always least squares solution! Savage book on decision theory: Estimation of error is meaningless! Mishra: Example of a different kind of mapping of a probability measure into a new probability measure. Foundation? MC evaluation of statistics versus drawing from distribution. Experimental design and MC error reduction and MLMC. 12

13 The end But you may glance through the rest if you like 13

14 Gravimetry: the pendulum 14

15 The simple pendulum The acceleration g of gravity is given by where T is the period and l is the length of the pendulum. Measurement of g can hence be done by measurement of T and l. There are several possible estimation strategies, and some of them will be investigated. 15

16 Measurement of l The length was measured with manually with a measuring tape, and the result was where the number in parentheses is the numerical value of the expanded uncertainty U(l) referred to the corresponding last digits of the quoted result. This defines an interval estimated to have a level of confidence of 95 percent. The standard uncertainty is estimated to be u(l) = 0.5cm The uncertainty has been determined by a Type B evaluation. 16

17 Measurement of T The period was measured manually 10 times, and the result for an average of 10 periods was T = 3.8s + Assuming a normal distribution, and application of the Student t distribution gives The standard error is u (T) = 3.1 m s corresponding to a coverage factor k = 2.26 and 9 degrees of freedom. 17

18 Measurement of g The acceleration of gravity is 18

19 The ISO GUMs main message 19

20 References It may be beneficial to consider (download your own private version from British standard [A] ISO, Guide to the expression of uncertainty in measurement (GUM) [B] IS0 3534, 1985, Statistics - Vocabulary and symbols [C] IS0 5725, 1994, Accuracy (trueness and precision) of measurement methods and results [D] INCE (2005). Managing uncertainties in noise measurements and predictions : a new challenge for acousticians. Uncertainty Noise Symposium, LeMans, INCE. 20

21 References [1] J.O. Berger. Statistical Decision Theory and Bayesian Analysis. Springer, [2] W. Bich, M.G. Cox, and P.M. Harris. Evolution of the Guide to the Expression of Uncertainty in Measurement. Metrologia, 43: , [3] B. Blanquart. Basic of uncertainties for acousticians. Symposium Le Mans, June [4] M. Born. Physics in my generation. Springer, [5] K. Brinkmann, R. Higginson, and L. Nielsen. Treatment of measurement uncertainties in international and european standards on acoustics. In Managing Uncertainty in Noise Measurement and Prediction - Symposium Le Mans (France). INCE EUROPE, [6] G. Casella and R.L. Berger. Statistical inference. Duxbury,

22 References [7] D.R. Cox and D.V. Hinkley. Theoretical statistics. Chapman- Hall (2000), [8] S.V. Crowder and S.L. Kupferman. Use of Welch-Satterthwaite Approximation in Calibration of Voltage Standards. Journal of Quality Technology, 36:38 52, [9] R.J. Douglas, A.G. Steele, B.M. Wood, and K.D. Hill. A useful reflection. Metrologia, 42:35 39, [10] M. Evans and T. Swartz. Approximating integrals via Monte Carlo and deterministic methods. Oxford, [11] L.J. Gleser. Assessing Uncertainty in Measurement. Statistical Science, 13: , [12] ISO. Guide to the expression of uncertainty of measurements. International Organisation for Standardisation,1995. [13] ISO Statistics - vocabulary and symbols

23 References [14] ISO Accuracy (trueness and precision) of measurement methods and results [15] H.K. Iyer, C.M.J. Wang, and T. Mathew. Models and Confidence Intervals for True Values in Interlaboratory Trials. Journal of the American Statistical Association, 99: , [16] R. Kacker and A. Jones. On use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistent. Metrologia, 40: , [17] R.N. Kacker. Bayesian alternative to the ISO-GUMs use of the Welch-Satterthwaite formula. Metrologia, 43:1 11, [18] E.L. Lehmann. Testing statistical hypotheses. Springer (1997), [19] E.L. Lehmann and G. Casella. Theory of point estimation. Springer (1998),

24 References [20] I. Lira. Evaluating the Measurement Uncertainty: Fundamentals and practical guidance. Institute of Physics Publishing, Bristol, [21] I. Lira. Resolution revisited. Metrologia, 43:14 17, [22] B. Ripley. Stochastic simulation. Wiley, [23] M.J. Schervish. Theory of Statistics. Springer, [24] R. Willink. A procedure for the evaluation of measurement uncertainty based on moments. Metrologia, 42: , [25] R. Willink. Principles of probability and statistics for metrology. Metrologia, 43: , [26] R. Willink and I. Lira. A united interpretation of different uncertainty intervals. Measurement, 38:61 66,

25 A simple example [4,Blanquart, B. 2005a] 25

26 A simple example [4,Blanquart, B. 2005a] 26

27 A simple example [4,Blanquart, B. 2005a] 27

28 A simple example [4,Blanquart, B. 2005a] 28

29 A simple example [Blanquart, B. 2005a]: Interpretation The standard uncertainty u(y) is the ISO GUM notation for the squareroot of the estimated variance of the estimator corresponding to y. This is an estimate of the standard deviation. The quantity y +- k u(y) defines an interval estimated to have a given level of confidence - typically 95%. The coverage factor k is usually in the range 2 to 3. The quantity U(y) = k u(y) is the expanded uncertainty. The following concepts are hence essential: Variance and standard deviation. Estimator. Estimate. Level of confidence. 29

30 Abstract of resolution paper A procedure is presented to evaluate the expanded uncertainty of a quantity about which discretized measurement data are available. The method is based on conventional statistics and depends on the value of the experimental variance and the resolution. The suggested procedure is compared with a recently suggested procedure based on Bayesian statistics. The ISO Guide to the expression of Uncertainty of Measurements (GUM) is discussed briefly. It is argued that both conventional and Bayesian statistics give a consistent interpretation of the GUM procedure, and the two approaches supplement each other. Conventional statistics estimates the uncertainty of the measurement procedure, while Bayesian statistics gives the uncertainty of the measurand. 30

31 31 Micrometer example Let 7.489, 7.503, 7.433, 7.549, 7.526, 7.396, 7.543,7.509, 7.504, be the result in mm of the measurement of some length μ with a micrometer. [Lira, I: 2006] Model where are independent random variables. The idealization is that the measurand μ is a property of the object, and σ is a property of the experiment as a whole.

32 Micrometer example 32

33 Micrometer example 33

34 Calliper example 34

35 Calliper example 35

36 Calliper and micrometer examples 36

37 Calliper and micrometer examples 37

38 Calliper and micrometer examples 38

39 Calliper and micrometer examples 39

40 Calliper and micrometer examples 40

41 Calliper example 41

42 Micrometer example 42

43 The ISO GUMs main message 43

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