XP ISO/TS 28037
Détermination and use of straight-line calibration functions
ISO/TS 28037:2010 is concerned with linear, that is, straight-line, calibration functions that describe the relationship between two variables X and Y, namely, functions of the form Y = A + BX. Although many of the principles apply to more general types of calibration function, the approaches described exploit the simple form of the straight-line calibration function wherever possible. Values of the parameters A and B are determined on the basis of measured data points (xi, yi), i = 1, ... , m. Various cases are considered relating to the nature of the uncertainties associated with these data. No assumption is made that the errors relating to the yi are homoscedastic (having equal variance), and similarly for the xi when the errors are not negligible. Estimates of the parameters A and B are determined using least squares methods. The emphasis is on choosing the least squares method most appropriate for the type of measurement data, in particular methods that reflect the associated uncertainties. The most general type of covariance matrix associated with the measurement data is treated, but important special cases that lead to simpler calculations are described in detail. For all cases considered, methods for validating the use of the straight-line calibration functions and for evaluating the uncertainties and covariance associated with the parameter estimates are given. ISO/TS 28037:2010 also describes the use of the calibration function parameter estimates and their associated uncertainties and covariance to predict a value of X and its associated standard uncertainty given a measured value of Y and its associated standard uncertainty.
ISO/TS 28037:2010 is concerned with linear, that is, straight-line, calibration functions that describe the relationship between two variables X and Y, namely, functions of the form Y = A + BX. Although many of the principles apply to more general types of calibration function, the approaches described exploit the simple form of the straight-line calibration function wherever possible.
Values of the parameters A and B are determined on the basis of measured data points (xi, yi), i = 1, ... , m. Various cases are considered relating to the nature of the uncertainties associated with these data. No assumption is made that the errors relating to the yi are homoscedastic (having equal variance), and similarly for the xi when the errors are not negligible.
Estimates of the parameters A and B are determined using least squares methods. The emphasis is on choosing the least squares method most appropriate for the type of measurement data, in particular methods that reflect the associated uncertainties. The most general type of covariance matrix associated with the measurement data is treated, but important special cases that lead to simpler calculations are described in detail.
For all cases considered, methods for validating the use of the straight-line calibration functions and for evaluating the uncertainties and covariance associated with the parameter estimates are given.
ISO/TS 28037:2010 also describes the use of the calibration function parameter estimates and their associated uncertainties and covariance to predict a value of X and its associated standard uncertainty given a measured value of Y and its associated standard uncertainty.
- Avant-proposv
- Introductionvi
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1 Domaine d'application1
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2 Références normatives1
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3 Termes et définitions1
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4 Conventions et notation4
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5 Principes de l'étalonnage linéaire5
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5.1 Généralités5
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5.2 Éléments d'entrée pour la détermination de la fonction d'étalonnage5
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5.3 Détermination de la fonction d'étalonnage6
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5.4 Traitement numérique7
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5.5 Incertitudes et covariances associées aux paramètres de la fonction d'étalonnage7
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5.6 Validation du modèle8
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5.7 Utilisation de la fonction d'étalonnage8
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5.8 Détermination de la droite de meilleur ajustement des données par la méthode des moindres carrés ordinaires9
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6 Modèle applicable aux incertitudes associées à yi9
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6.1 Généralités9
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6.2 Estimations des paramètres d'étalonnage, des incertitudes-types et de la covariance associées10
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6.3 Validation du modèle11
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6.4 Organisation des calculs12
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7 Modèle applicable aux incertitudes associées à xi et yi16
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7.1 Généralités16
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7.2 Estimations des paramètres d'étalonnage, des incertitudes-types et de la covariance associées18
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7.3 Validation du modèle19
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7.4 Organisation des calculs20
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8 Modèle applicable aux incertitudes associées à xi et yi et aux covariances associées aux paires (xi, yi)24
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8.1 Généralités24
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8.2 Estimations des paramètres d'étalonnage et incertitudes-types et covariance associées.25
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9 Modèle applicable aux incertitudes et aux covariances associées à yi25
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9.1 Généralités25
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9.2 Estimations des paramètres d'étalonnage, incertitudes-types et covariance associées26
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9.3 Validation du modèle28
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9.4 Organisation des calculs28
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10 Modèle applicable aux incertitudes et aux covariances associées à xi et yi32
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10.1 Généralités32
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10.2 Estimations des paramètres d'étalonnage, des incertitudes-types et covariances associées33
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10.3 Validation du modèle35
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11 Utilisation de la fonction d'étalonnage39
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11.1 Prédiction directe39
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11.2 Prédiction inverse40
- Annexe A (informative) Opérations matricielles42
- Annexe B (informative) Application de l'algorithme de Gauss-Newton à la régression selon le critère de distance généralisée48
- Annexe C (informative) Approche de factorisation orthogonale pour résoudre le problème de Gauss-Markov généralisé50
- Annexe D (informative) Disposition relative aux incertitudes et covariances associées aux valeurs mesurées x et y56
- Annexe E (informative) Incertitudes connues avec un facteur d'échelle donné61
- Annexe F (informative) Application logicielle des algorithmes décrits66
- Annexe G (informative) Glossaire des principaux symboles68
- Bibliographie70
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