ISO/IEC Guide 98-3:2008/Suppl 2:2011
Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) — Supplement 2: Extension to any number of output quantities
ISO/IEC Guide 98-3:2008/Suppl.2:2011 is concerned with measurement models having any number of input quantities and any number of output quantities. The quantities involved might be real or complex. Two approaches are considered for treating such models. The first approach is a generalization of the GUM uncertainty framework. The second is a Monte Carlo method as an implementation of the propagation of distributions. Appropriate use of the Monte Carol method would be expected to provide valid results when the applicability of the GUM uncertainty framework is questionable.
For a prescribed coverage probability, ISO/IEC Guide 98-3:2008/Suppl.2:2011 can be used to provide a coverage region for the output quantities of a multivariate model, the counterpart of a coverage interval for a single scalar output quantiy. The provision of coverage regions includes those taking the form of a hyper-ellipsoid or a hyper-rectangle. These coverage regions are produced from the results of the two approaches described here. A procedure for providing an approximation to the smallest coverage region, obtained from results provided by the Monte Carol method, is also given. Detailed examples to illustrate the guidance are provided.
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Foreword
Introduction
1 Scope
2 Normative references
3 Terms and definitions
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
4 Conventions and notation
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
5 Basic principles
5.1 General
5.1.1
5.1.2
5.1.3
5.1.4
5.2 Main stages of uncertainty evaluation
5.2.1
5.2.2
5.3 Probability density functions for the input quantities
5.3.1 General
5.3.2 Multivariate t-distribution
5.3.3 Construction of multivariate probability density functions
5.4 Propagation of distributions
5.4.1
5.4.2
5.4.3
5.4.4
5.5 Obtaining summary information
5.5.1
5.5.2
5.5.3
5.6 Implementations of the propagation of distributions
5.6.1
5.6.2
5.6.3
5.6.4
6 GUM uncertainty framework
6.1 General
6.1.1
6.1.2
6.1.3
6.1.4
6.2 Propagation of uncertainty for explicit multivariate measurement models
6.2.1 General
6.2.2 Examples
6.3 Propagation of uncertainty for implicit multivariate measurement models
6.3.1 General
6.3.2 Examples
6.4 Propagation of uncertainty for models involving complex quantities
6.5 Coverage region for a vector output quantity
6.5.1 General
6.5.2 Bivariate case
6.5.3 Multivariate case
6.5.4 Coverage region for the expectation of a multivariate Gaussian distribution
7 Monte Carlo method
7.1 General
7.1.1
7.1.2
7.1.3
7.1.4
7.1.5
7.1.6
7.1.7
7.1.8
7.2 Number of Monte Carlo trials
7.2.1
7.2.2
7.3 Making draws from probability distributions
7.3.1
7.3.2
7.3.3
7.4 Evaluation of the vector output quantity
7.4.1
7.4.2
7.4.3
7.5 Discrete representation of the distribution function for the output quantity
7.6 Estimate of the output quantity and the associated covariance matrix
7.7 Coverage region for a vector output quantity
7.7.1 General
7.7.2 Hyper-ellipsoidal coverage region
7.7.3 Hyper-rectangular coverage region
7.7.4 Smallest coverage region
7.8 Adaptive Monte Carlo procedure
7.8.1 General
7.8.2 Numerical tolerance associated with a numerical value
7.8.3 Adaptive procedure
8 Validation of the GUM uncertainty framework using a Monte Carlo method
8.1
8.2
8.3
9 Examples
9.1 Illustrations of aspects of this Supplement
9.1.1
9.1.2
9.1.3
9.1.4
9.1.5
9.1.6
9.2 Additive measurement model
9.2.1 Formulation
9.2.2 Propagation and summarizing: case 1
9.2.3 Propagation and summarizing: case 2
9.2.4 Propagation and summarizing: case 3
9.3 Co-ordinate system transformation
9.3.1 Formulation
9.3.2 Propagation and summarizing: zero covariance
9.3.3 Propagation and summarizing: non-zero covariance
9.3.4 Discussion
9.4 Simultaneous measurement of resistance and reactance
9.4.1 Formulation
9.4.2 Propagation and summarizing
9.5 Measurement of Celsius temperature using a resistance thermometer
9.5.1 General
9.5.2 Measurement of a single Celsius temperature
9.5.3 Measurement of several Celsius temperatures
Annex A (informative) Derivatives of complex multivariate measurement functions
A.1
A.2
A.3
A.4
A.5
Annex B (informative) Evaluation of sensitivity coefficients and covariance matrix for multivariate measurement models
B.1
B.2
Annex C (informative) Co-ordinate system transformation
C.1 General
C.2 Analytical solution for a special case
C.2.1
C.2.2
C.2.3
C.2.4
C.2.5
C.2.6
C.3 Application of the GUM uncertainty framework
C.3.1
C.3.2
C.3.3
C.3.4
C.3.5
C.3.6
Annex D (informative) Glossary of principal symbols
Bibliography
Alphabetical index
Annex ZZ (informative) Corresponding ISO/IEC Guides for the JCGM guidance documents for which their equivalents are not given in the text
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Foreword
Introduction
1 Scope
2 Normative references
3 Terms and definitions
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
4 Conventions and notation
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
5 Basic principles
5.1 General
5.1.1
5.1.2
5.1.3
5.1.4
5.2 Main stages of uncertainty evaluation
5.2.1
5.2.2
5.3 Probability density functions for the input quantities
5.3.1 General
5.3.2 Multivariate t-distribution
5.3.3 Construction of multivariate probability density functions
5.4 Propagation of distributions
5.4.1
5.4.2
5.4.3
5.4.4
5.5 Obtaining summary information
5.5.1
5.5.2
5.5.3
5.6 Implementations of the propagation of distributions
5.6.1
5.6.2
5.6.3
5.6.4
6 GUM uncertainty framework
6.1 General
6.1.1
6.1.2
6.1.3
6.1.4
6.2 Propagation of uncertainty for explicit multivariate measurement models
6.2.1 General
6.2.2 Examples
6.3 Propagation of uncertainty for implicit multivariate measurement models
6.3.1 General
6.3.2 Examples
6.4 Propagation of uncertainty for models involving complex quantities
6.5 Coverage region for a vector output quantity
6.5.1 General
6.5.2 Bivariate case
6.5.3 Multivariate case
6.5.4 Coverage region for the expectation of a multivariate Gaussian distribution
7 Monte Carlo method
7.1 General
7.1.1
7.1.2
7.1.3
7.1.4
7.1.5
7.1.6
7.1.7
7.1.8
7.2 Number of Monte Carlo trials
7.2.1
7.2.2
7.3 Making draws from probability distributions
7.3.1
7.3.2
7.3.3
7.4 Evaluation of the vector output quantity
7.4.1
7.4.2
7.4.3
7.5 Discrete representation of the distribution function for the output quantity
7.6 Estimate of the output quantity and the associated covariance matrix
7.7 Coverage region for a vector output quantity
7.7.1 General
7.7.2 Hyper-ellipsoidal coverage region
7.7.3 Hyper-rectangular coverage region
7.7.4 Smallest coverage region
7.8 Adaptive Monte Carlo procedure
7.8.1 General
7.8.2 Numerical tolerance associated with a numerical value
7.8.3 Adaptive procedure
8 Validation of the GUM uncertainty framework using a Monte Carlo method
8.1
8.2
8.3
9 Examples
9.1 Illustrations of aspects of this Supplement
9.1.1
9.1.2
9.1.3
9.1.4
9.1.5
9.1.6
9.2 Additive measurement model
9.2.1 Formulation
9.2.2 Propagation and summarizing: case 1
9.2.3 Propagation and summarizing: case 2
9.2.4 Propagation and summarizing: case 3
9.3 Co-ordinate system transformation
9.3.1 Formulation
9.3.2 Propagation and summarizing: zero covariance
9.3.3 Propagation and summarizing: non-zero covariance
9.3.4 Discussion
9.4 Simultaneous measurement of resistance and reactance
9.4.1 Formulation
9.4.2 Propagation and summarizing
9.5 Measurement of Celsius temperature using a resistance thermometer
9.5.1 General
9.5.2 Measurement of a single Celsius temperature
9.5.3 Measurement of several Celsius temperatures
Annex A (informative) Derivatives of complex multivariate measurement functions
A.1
A.2
A.3
A.4
A.5
Annex B (informative) Evaluation of sensitivity coefficients and covariance matrix for multivariate measurement models
B.1
B.2
Annex C (informative) Co-ordinate system transformation
C.1 General
C.2 Analytical solution for a special case
C.2.1
C.2.2
C.2.3
C.2.4
C.2.5
C.2.6
C.3 Application of the GUM uncertainty framework
C.3.1
C.3.2
C.3.3
C.3.4
C.3.5
C.3.6
Annex D (informative) Glossary of principal symbols
Bibliography
Alphabetical index
Annex ZZ (informative) Corresponding ISO/IEC Guides for the JCGM guidance documents for which their equivalents are not given in the text